3.4.45 \(\int \frac {(e+f x)^3 \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) [345]

Optimal. Leaf size=1432 \[ \frac {3 b f^3 x}{8 a^2 d^3}+\frac {3 \left (a^2-b^2\right ) f^3 x}{8 a^2 b d^3}-\frac {b (e+f x)^3}{4 a^2 d}-\frac {\left (a^2-b^2\right ) (e+f x)^3}{4 a^2 b d}+\frac {i b (e+f x)^4}{4 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^4}{4 a^2 b^3 f}-\frac {6 f (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^2}+\frac {6 f^3 \cos (c+d x)}{a d^4}+\frac {6 \left (a^2-b^2\right ) f^3 \cos (c+d x)}{a b^2 d^4}-\frac {3 f (e+f x)^2 \cos (c+d x)}{a d^2}-\frac {3 \left (a^2-b^2\right ) f (e+f x)^2 \cos (c+d x)}{a b^2 d^2}-\frac {(e+f x)^3 \csc (c+d x)}{a d}+\frac {\left (a^2-b^2\right )^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d}+\frac {\left (a^2-b^2\right )^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d}-\frac {b (e+f x)^3 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 i \left (a^2-b^2\right )^2 f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}-\frac {3 i \left (a^2-b^2\right )^2 f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (e^{2 i (c+d x)}\right )}{2 a^2 d^2}-\frac {6 f^3 \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {6 f^3 \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^4}+\frac {6 \left (a^2-b^2\right )^2 f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^3}+\frac {6 \left (a^2-b^2\right )^2 f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^3}-\frac {3 b f^2 (e+f x) \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a^2 d^3}+\frac {6 i \left (a^2-b^2\right )^2 f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^4}+\frac {6 i \left (a^2-b^2\right )^2 f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^4}-\frac {3 i b f^3 \text {Li}_4\left (e^{2 i (c+d x)}\right )}{4 a^2 d^4}+\frac {6 f^2 (e+f x) \sin (c+d x)}{a d^3}+\frac {6 \left (a^2-b^2\right ) f^2 (e+f x) \sin (c+d x)}{a b^2 d^3}-\frac {(e+f x)^3 \sin (c+d x)}{a d}-\frac {\left (a^2-b^2\right ) (e+f x)^3 \sin (c+d x)}{a b^2 d}-\frac {3 b f^3 \cos (c+d x) \sin (c+d x)}{8 a^2 d^4}-\frac {3 \left (a^2-b^2\right ) f^3 \cos (c+d x) \sin (c+d x)}{8 a^2 b d^4}+\frac {3 b f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 a^2 d^2}+\frac {3 \left (a^2-b^2\right ) f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 a^2 b d^2}-\frac {3 b f^2 (e+f x) \sin ^2(c+d x)}{4 a^2 d^3}-\frac {3 \left (a^2-b^2\right ) f^2 (e+f x) \sin ^2(c+d x)}{4 a^2 b d^3}+\frac {b (e+f x)^3 \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (a^2-b^2\right ) (e+f x)^3 \sin ^2(c+d x)}{2 a^2 b d} \]

[Out]

-b*(f*x+e)^3*ln(1-exp(2*I*(d*x+c)))/a^2/d-3*(a^2-b^2)*f*(f*x+e)^2*cos(d*x+c)/a/b^2/d^2+6*(a^2-b^2)*f^2*(f*x+e)
*sin(d*x+c)/a/b^2/d^3-3/8*(a^2-b^2)*f^3*cos(d*x+c)*sin(d*x+c)/a^2/b/d^4+3/4*b*f*(f*x+e)^2*cos(d*x+c)*sin(d*x+c
)/a^2/d^2-3/4*(a^2-b^2)*f^2*(f*x+e)*sin(d*x+c)^2/a^2/b/d^3+(a^2-b^2)^2*(f*x+e)^3*ln(1-I*b*exp(I*(d*x+c))/(a-(a
^2-b^2)^(1/2)))/a^2/b^3/d+(a^2-b^2)^2*(f*x+e)^3*ln(1-I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/b^3/d+6*(a^2-
b^2)^2*f^2*(f*x+e)*polylog(3,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a^2/b^3/d^3+6*(a^2-b^2)^2*f^2*(f*x+e)*pol
ylog(3,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/b^3/d^3+6*I*(a^2-b^2)^2*f^3*polylog(4,I*b*exp(I*(d*x+c))/(a
-(a^2-b^2)^(1/2)))/a^2/b^3/d^4+6*I*(a^2-b^2)^2*f^3*polylog(4,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/b^3/d
^4-3*f*(f*x+e)^2*cos(d*x+c)/a/d^2+6*f^2*(f*x+e)*sin(d*x+c)/a/d^3-(f*x+e)^3*csc(d*x+c)/a/d-6*f^3*polylog(3,-exp
(I*(d*x+c)))/a/d^4+6*f^3*polylog(3,exp(I*(d*x+c)))/a/d^4+1/4*I*b*(f*x+e)^4/a^2/f-3/2*b*f^2*(f*x+e)*polylog(3,e
xp(2*I*(d*x+c)))/a^2/d^3-6*I*f^2*(f*x+e)*polylog(2,exp(I*(d*x+c)))/a/d^3-3/4*I*b*f^3*polylog(4,exp(2*I*(d*x+c)
))/a^2/d^4+6*f^3*cos(d*x+c)/a/d^4+3/2*I*b*f*(f*x+e)^2*polylog(2,exp(2*I*(d*x+c)))/a^2/d^2-1/4*(a^2-b^2)*(f*x+e
)^3/a^2/b/d+1/2*b*(f*x+e)^3*sin(d*x+c)^2/a^2/d+3/8*b*f^3*x/a^2/d^3-6*f*(f*x+e)^2*arctanh(exp(I*(d*x+c)))/a/d^2
-1/4*b*(f*x+e)^3/a^2/d-(f*x+e)^3*sin(d*x+c)/a/d-3*I*(a^2-b^2)^2*f*(f*x+e)^2*polylog(2,I*b*exp(I*(d*x+c))/(a-(a
^2-b^2)^(1/2)))/a^2/b^3/d^2-3*I*(a^2-b^2)^2*f*(f*x+e)^2*polylog(2,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/
b^3/d^2+3/4*(a^2-b^2)*f*(f*x+e)^2*cos(d*x+c)*sin(d*x+c)/a^2/b/d^2+3/8*(a^2-b^2)*f^3*x/a^2/b/d^3+6*(a^2-b^2)*f^
3*cos(d*x+c)/a/b^2/d^4-(a^2-b^2)*(f*x+e)^3*sin(d*x+c)/a/b^2/d-3/8*b*f^3*cos(d*x+c)*sin(d*x+c)/a^2/d^4-3/4*b*f^
2*(f*x+e)*sin(d*x+c)^2/a^2/d^3+1/2*(a^2-b^2)*(f*x+e)^3*sin(d*x+c)^2/a^2/b/d-1/4*I*(a^2-b^2)^2*(f*x+e)^4/a^2/b^
3/f+6*I*f^2*(f*x+e)*polylog(2,-exp(I*(d*x+c)))/a/d^3

________________________________________________________________________________________

Rubi [A]
time = 1.90, antiderivative size = 1432, normalized size of antiderivative = 1.00, number of steps used = 85, number of rules used = 21, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {4639, 4493, 3392, 3377, 2718, 3391, 4495, 4268, 2611, 2320, 6724, 4490, 32, 2715, 8, 4489, 3798, 2221, 6744, 4621, 4615} \begin {gather*} -\frac {i \left (a^2-b^2\right )^2 (e+f x)^4}{4 a^2 b^3 f}+\frac {i b (e+f x)^4}{4 a^2 f}+\frac {b \sin ^2(c+d x) (e+f x)^3}{2 a^2 d}+\frac {\left (a^2-b^2\right ) \sin ^2(c+d x) (e+f x)^3}{2 a^2 b d}-\frac {\csc (c+d x) (e+f x)^3}{a d}+\frac {\left (a^2-b^2\right )^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) (e+f x)^3}{a^2 b^3 d}+\frac {\left (a^2-b^2\right )^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) (e+f x)^3}{a^2 b^3 d}-\frac {b \log \left (1-e^{2 i (c+d x)}\right ) (e+f x)^3}{a^2 d}-\frac {\left (a^2-b^2\right ) \sin (c+d x) (e+f x)^3}{a b^2 d}-\frac {\sin (c+d x) (e+f x)^3}{a d}-\frac {b (e+f x)^3}{4 a^2 d}-\frac {\left (a^2-b^2\right ) (e+f x)^3}{4 a^2 b d}-\frac {6 f \tanh ^{-1}\left (e^{i (c+d x)}\right ) (e+f x)^2}{a d^2}-\frac {3 \left (a^2-b^2\right ) f \cos (c+d x) (e+f x)^2}{a b^2 d^2}-\frac {3 f \cos (c+d x) (e+f x)^2}{a d^2}-\frac {3 i \left (a^2-b^2\right )^2 f \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) (e+f x)^2}{a^2 b^3 d^2}-\frac {3 i \left (a^2-b^2\right )^2 f \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) (e+f x)^2}{a^2 b^3 d^2}+\frac {3 i b f \text {PolyLog}\left (2,e^{2 i (c+d x)}\right ) (e+f x)^2}{2 a^2 d^2}+\frac {3 b f \cos (c+d x) \sin (c+d x) (e+f x)^2}{4 a^2 d^2}+\frac {3 \left (a^2-b^2\right ) f \cos (c+d x) \sin (c+d x) (e+f x)^2}{4 a^2 b d^2}-\frac {3 b f^2 \sin ^2(c+d x) (e+f x)}{4 a^2 d^3}-\frac {3 \left (a^2-b^2\right ) f^2 \sin ^2(c+d x) (e+f x)}{4 a^2 b d^3}+\frac {6 i f^2 \text {PolyLog}\left (2,-e^{i (c+d x)}\right ) (e+f x)}{a d^3}-\frac {6 i f^2 \text {PolyLog}\left (2,e^{i (c+d x)}\right ) (e+f x)}{a d^3}+\frac {6 \left (a^2-b^2\right )^2 f^2 \text {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) (e+f x)}{a^2 b^3 d^3}+\frac {6 \left (a^2-b^2\right )^2 f^2 \text {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) (e+f x)}{a^2 b^3 d^3}-\frac {3 b f^2 \text {PolyLog}\left (3,e^{2 i (c+d x)}\right ) (e+f x)}{2 a^2 d^3}+\frac {6 \left (a^2-b^2\right ) f^2 \sin (c+d x) (e+f x)}{a b^2 d^3}+\frac {6 f^2 \sin (c+d x) (e+f x)}{a d^3}+\frac {3 b f^3 x}{8 a^2 d^3}+\frac {3 \left (a^2-b^2\right ) f^3 x}{8 a^2 b d^3}+\frac {6 \left (a^2-b^2\right ) f^3 \cos (c+d x)}{a b^2 d^4}+\frac {6 f^3 \cos (c+d x)}{a d^4}-\frac {6 f^3 \text {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^4}+\frac {6 f^3 \text {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^4}+\frac {6 i \left (a^2-b^2\right )^2 f^3 \text {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^4}+\frac {6 i \left (a^2-b^2\right )^2 f^3 \text {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^4}-\frac {3 i b f^3 \text {PolyLog}\left (4,e^{2 i (c+d x)}\right )}{4 a^2 d^4}-\frac {3 b f^3 \cos (c+d x) \sin (c+d x)}{8 a^2 d^4}-\frac {3 \left (a^2-b^2\right ) f^3 \cos (c+d x) \sin (c+d x)}{8 a^2 b d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((e + f*x)^3*Cos[c + d*x]^3*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]

[Out]

(3*b*f^3*x)/(8*a^2*d^3) + (3*(a^2 - b^2)*f^3*x)/(8*a^2*b*d^3) - (b*(e + f*x)^3)/(4*a^2*d) - ((a^2 - b^2)*(e +
f*x)^3)/(4*a^2*b*d) + ((I/4)*b*(e + f*x)^4)/(a^2*f) - ((I/4)*(a^2 - b^2)^2*(e + f*x)^4)/(a^2*b^3*f) - (6*f*(e
+ f*x)^2*ArcTanh[E^(I*(c + d*x))])/(a*d^2) + (6*f^3*Cos[c + d*x])/(a*d^4) + (6*(a^2 - b^2)*f^3*Cos[c + d*x])/(
a*b^2*d^4) - (3*f*(e + f*x)^2*Cos[c + d*x])/(a*d^2) - (3*(a^2 - b^2)*f*(e + f*x)^2*Cos[c + d*x])/(a*b^2*d^2) -
((e + f*x)^3*Csc[c + d*x])/(a*d) + ((a^2 - b^2)^2*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b
^2])])/(a^2*b^3*d) + ((a^2 - b^2)^2*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b^3
*d) - (b*(e + f*x)^3*Log[1 - E^((2*I)*(c + d*x))])/(a^2*d) + ((6*I)*f^2*(e + f*x)*PolyLog[2, -E^(I*(c + d*x))]
)/(a*d^3) - ((6*I)*f^2*(e + f*x)*PolyLog[2, E^(I*(c + d*x))])/(a*d^3) - ((3*I)*(a^2 - b^2)^2*f*(e + f*x)^2*Pol
yLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^3*d^2) - ((3*I)*(a^2 - b^2)^2*f*(e + f*x)^2*PolyL
og[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b^3*d^2) + (((3*I)/2)*b*f*(e + f*x)^2*PolyLog[2, E^((
2*I)*(c + d*x))])/(a^2*d^2) - (6*f^3*PolyLog[3, -E^(I*(c + d*x))])/(a*d^4) + (6*f^3*PolyLog[3, E^(I*(c + d*x))
])/(a*d^4) + (6*(a^2 - b^2)^2*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^3*
d^3) + (6*(a^2 - b^2)^2*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b^3*d^3) -
(3*b*f^2*(e + f*x)*PolyLog[3, E^((2*I)*(c + d*x))])/(2*a^2*d^3) + ((6*I)*(a^2 - b^2)^2*f^3*PolyLog[4, (I*b*E^
(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^3*d^4) + ((6*I)*(a^2 - b^2)^2*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)
))/(a + Sqrt[a^2 - b^2])])/(a^2*b^3*d^4) - (((3*I)/4)*b*f^3*PolyLog[4, E^((2*I)*(c + d*x))])/(a^2*d^4) + (6*f^
2*(e + f*x)*Sin[c + d*x])/(a*d^3) + (6*(a^2 - b^2)*f^2*(e + f*x)*Sin[c + d*x])/(a*b^2*d^3) - ((e + f*x)^3*Sin[
c + d*x])/(a*d) - ((a^2 - b^2)*(e + f*x)^3*Sin[c + d*x])/(a*b^2*d) - (3*b*f^3*Cos[c + d*x]*Sin[c + d*x])/(8*a^
2*d^4) - (3*(a^2 - b^2)*f^3*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*b*d^4) + (3*b*f*(e + f*x)^2*Cos[c + d*x]*Sin[c +
d*x])/(4*a^2*d^2) + (3*(a^2 - b^2)*f*(e + f*x)^2*Cos[c + d*x]*Sin[c + d*x])/(4*a^2*b*d^2) - (3*b*f^2*(e + f*x
)*Sin[c + d*x]^2)/(4*a^2*d^3) - (3*(a^2 - b^2)*f^2*(e + f*x)*Sin[c + d*x]^2)/(4*a^2*b*d^3) + (b*(e + f*x)^3*Si
n[c + d*x]^2)/(2*a^2*d) + ((a^2 - b^2)*(e + f*x)^3*Sin[c + d*x]^2)/(2*a^2*b*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3391

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^
2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[b*(c + d*x)*Cos[e + f*x
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

Rule 3392

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*m*(c + d*x)^(m - 1)*((
b*Sin[e + f*x])^n/(f^2*n^2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[d^2*m*((m - 1)/(f^2*n^2)), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[b*(c + d*x)^m*Cos[e + f
*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(
m + 1))), x] - Dist[2*I, Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 4268

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*
x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[d*(m/f), Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 4489

Int[Cos[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[(c + d
*x)^m*(Sin[a + b*x]^(n + 1)/(b*(n + 1))), x] - Dist[d*(m/(b*(n + 1))), Int[(c + d*x)^(m - 1)*Sin[a + b*x]^(n +
1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]

Rule 4490

Int[Cos[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(-(c +
d*x)^m)*(Cos[a + b*x]^(n + 1)/(b*(n + 1))), x] + Dist[d*(m/(b*(n + 1))), Int[(c + d*x)^(m - 1)*Cos[a + b*x]^(
n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]

Rule 4493

Int[Cos[(a_.) + (b_.)*(x_)]^(n_.)*Cot[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Int[
(c + d*x)^m*Cos[a + b*x]^n*Cot[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cos[a + b*x]^(n - 2)*Cot[a + b*x]^p, x]
/; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 4495

Int[Cot[(a_.) + (b_.)*(x_)]^(p_.)*Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[
(-(c + d*x)^m)*(Csc[a + b*x]^n/(b*n)), x] + Dist[d*(m/(b*n)), Int[(c + d*x)^(m - 1)*Csc[a + b*x]^n, x], x] /;
FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 4615

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
Simp[(-I)*((e + f*x)^(m + 1)/(b*f*(m + 1))), x] + (Int[(e + f*x)^m*(E^(I*(c + d*x))/(a - Rt[a^2 - b^2, 2] - I*
b*E^(I*(c + d*x)))), x] + Int[(e + f*x)^m*(E^(I*(c + d*x))/(a + Rt[a^2 - b^2, 2] - I*b*E^(I*(c + d*x)))), x])
/; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]

Rule 4621

Int[(Cos[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol
] :> Dist[a/b^2, Int[(e + f*x)^m*Cos[c + d*x]^(n - 2), x], x] + (-Dist[1/b, Int[(e + f*x)^m*Cos[c + d*x]^(n -
2)*Sin[c + d*x], x], x] - Dist[(a^2 - b^2)/b^2, Int[(e + f*x)^m*(Cos[c + d*x]^(n - 2)/(a + b*Sin[c + d*x])), x
], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 4639

Int[(Cos[(c_.) + (d_.)*(x_)]^(p_.)*Cot[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin
[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Cos[c + d*x]^p*Cot[c + d*x]^n, x], x] - Dist[b/a
, Int[(e + f*x)^m*Cos[c + d*x]^(p + 1)*(Cot[c + d*x]^(n - 1)/(a + b*Sin[c + d*x])), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6744

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Dist[f*(m/(b*c*p*Log[F])), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rubi steps

\begin {align*} \int \frac {(e+f x)^3 \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\int (e+f x)^3 \cos ^3(c+d x) \cot ^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx}{a}\\ &=-\frac {\int (e+f x)^3 \cos ^3(c+d x) \, dx}{a}+\frac {\int (e+f x)^3 \cos (c+d x) \cot ^2(c+d x) \, dx}{a}-\frac {b \int (e+f x)^3 \cos ^4(c+d x) \cot (c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^3 \cos ^5(c+d x)}{a+b \sin (c+d x)} \, dx}{a^2}\\ &=-\frac {f (e+f x)^2 \cos ^3(c+d x)}{3 a d^2}-\frac {(e+f x)^3 \cos ^2(c+d x) \sin (c+d x)}{3 a d}-\frac {2 \int (e+f x)^3 \cos (c+d x) \, dx}{3 a}-\frac {\int (e+f x)^3 \cos (c+d x) \, dx}{a}+\frac {\int (e+f x)^3 \cos ^3(c+d x) \, dx}{a}+\frac {\int (e+f x)^3 \cot (c+d x) \csc (c+d x) \, dx}{a}-\frac {b \int (e+f x)^3 \cos ^2(c+d x) \cot (c+d x) \, dx}{a^2}-\left (1-\frac {b^2}{a^2}\right ) \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+b \sin (c+d x)} \, dx+\frac {\left (2 f^2\right ) \int (e+f x) \cos ^3(c+d x) \, dx}{3 a d^2}\\ &=\frac {2 f^3 \cos ^3(c+d x)}{27 a d^4}-\frac {(e+f x)^3 \csc (c+d x)}{a d}-\frac {5 (e+f x)^3 \sin (c+d x)}{3 a d}+\frac {2 f^2 (e+f x) \cos ^2(c+d x) \sin (c+d x)}{9 a d^3}+\frac {2 \int (e+f x)^3 \cos (c+d x) \, dx}{3 a}-\frac {b \int (e+f x)^3 \cot (c+d x) \, dx}{a^2}+\frac {b \int (e+f x)^3 \cos (c+d x) \sin (c+d x) \, dx}{a^2}-\frac {\left (a \left (1-\frac {b^2}{a^2}\right )\right ) \int (e+f x)^3 \cos (c+d x) \, dx}{b^2}-\frac {\left (-1+\frac {b^2}{a^2}\right ) \int (e+f x)^3 \cos (c+d x) \sin (c+d x) \, dx}{b}-\frac {\left (\left (a^2-b^2\right ) \left (-1+\frac {b^2}{a^2}\right )\right ) \int \frac {(e+f x)^3 \cos (c+d x)}{a+b \sin (c+d x)} \, dx}{b^2}+\frac {(2 f) \int (e+f x)^2 \sin (c+d x) \, dx}{a d}+\frac {(3 f) \int (e+f x)^2 \csc (c+d x) \, dx}{a d}+\frac {(3 f) \int (e+f x)^2 \sin (c+d x) \, dx}{a d}+\frac {\left (4 f^2\right ) \int (e+f x) \cos (c+d x) \, dx}{9 a d^2}-\frac {\left (2 f^2\right ) \int (e+f x) \cos ^3(c+d x) \, dx}{3 a d^2}\\ &=\frac {i b (e+f x)^4}{4 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^4}{4 a^2 b^3 f}-\frac {6 f (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^2}-\frac {5 f (e+f x)^2 \cos (c+d x)}{a d^2}-\frac {(e+f x)^3 \csc (c+d x)}{a d}+\frac {4 f^2 (e+f x) \sin (c+d x)}{9 a d^3}-\frac {(e+f x)^3 \sin (c+d x)}{a d}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \sin (c+d x)}{b^2 d}+\frac {b (e+f x)^3 \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \sin ^2(c+d x)}{2 b d}+\frac {(2 i b) \int \frac {e^{2 i (c+d x)} (e+f x)^3}{1-e^{2 i (c+d x)}} \, dx}{a^2}-\frac {\left (\left (a^2-b^2\right ) \left (-1+\frac {b^2}{a^2}\right )\right ) \int \frac {e^{i (c+d x)} (e+f x)^3}{a-\sqrt {a^2-b^2}-i b e^{i (c+d x)}} \, dx}{b^2}-\frac {\left (\left (a^2-b^2\right ) \left (-1+\frac {b^2}{a^2}\right )\right ) \int \frac {e^{i (c+d x)} (e+f x)^3}{a+\sqrt {a^2-b^2}-i b e^{i (c+d x)}} \, dx}{b^2}-\frac {(2 f) \int (e+f x)^2 \sin (c+d x) \, dx}{a d}-\frac {(3 b f) \int (e+f x)^2 \sin ^2(c+d x) \, dx}{2 a^2 d}+\frac {\left (3 a \left (1-\frac {b^2}{a^2}\right ) f\right ) \int (e+f x)^2 \sin (c+d x) \, dx}{b^2 d}-\frac {\left (3 \left (1-\frac {b^2}{a^2}\right ) f\right ) \int (e+f x)^2 \sin ^2(c+d x) \, dx}{2 b d}-\frac {\left (4 f^2\right ) \int (e+f x) \cos (c+d x) \, dx}{9 a d^2}+\frac {\left (4 f^2\right ) \int (e+f x) \cos (c+d x) \, dx}{a d^2}+\frac {\left (6 f^2\right ) \int (e+f x) \cos (c+d x) \, dx}{a d^2}-\frac {\left (6 f^2\right ) \int (e+f x) \log \left (1-e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (6 f^2\right ) \int (e+f x) \log \left (1+e^{i (c+d x)}\right ) \, dx}{a d^2}-\frac {\left (4 f^3\right ) \int \sin (c+d x) \, dx}{9 a d^3}\\ &=\frac {i b (e+f x)^4}{4 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^4}{4 a^2 b^3 f}-\frac {6 f (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^2}+\frac {4 f^3 \cos (c+d x)}{9 a d^4}-\frac {3 f (e+f x)^2 \cos (c+d x)}{a d^2}-\frac {3 a \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \cos (c+d x)}{b^2 d^2}-\frac {(e+f x)^3 \csc (c+d x)}{a d}+\frac {\left (a^2-b^2\right )^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d}+\frac {\left (a^2-b^2\right )^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d}-\frac {b (e+f x)^3 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^3}+\frac {10 f^2 (e+f x) \sin (c+d x)}{a d^3}-\frac {(e+f x)^3 \sin (c+d x)}{a d}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \sin (c+d x)}{b^2 d}+\frac {3 b f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 a^2 d^2}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 b d^2}-\frac {3 b f^2 (e+f x) \sin ^2(c+d x)}{4 a^2 d^3}-\frac {3 \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \sin ^2(c+d x)}{4 b d^3}+\frac {b (e+f x)^3 \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \sin ^2(c+d x)}{2 b d}-\frac {(3 b f) \int (e+f x)^2 \, dx}{4 a^2 d}+\frac {(3 b f) \int (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right ) \, dx}{a^2 d}-\frac {\left (3 \left (a^2-b^2\right )^2 f\right ) \int (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^3 d}-\frac {\left (3 \left (a^2-b^2\right )^2 f\right ) \int (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^3 d}-\frac {\left (3 \left (1-\frac {b^2}{a^2}\right ) f\right ) \int (e+f x)^2 \, dx}{4 b d}-\frac {\left (4 f^2\right ) \int (e+f x) \cos (c+d x) \, dx}{a d^2}+\frac {\left (6 a \left (1-\frac {b^2}{a^2}\right ) f^2\right ) \int (e+f x) \cos (c+d x) \, dx}{b^2 d^2}-\frac {\left (6 i f^3\right ) \int \text {Li}_2\left (-e^{i (c+d x)}\right ) \, dx}{a d^3}+\frac {\left (6 i f^3\right ) \int \text {Li}_2\left (e^{i (c+d x)}\right ) \, dx}{a d^3}+\frac {\left (4 f^3\right ) \int \sin (c+d x) \, dx}{9 a d^3}-\frac {\left (4 f^3\right ) \int \sin (c+d x) \, dx}{a d^3}-\frac {\left (6 f^3\right ) \int \sin (c+d x) \, dx}{a d^3}+\frac {\left (3 b f^3\right ) \int \sin ^2(c+d x) \, dx}{4 a^2 d^3}+\frac {\left (3 \left (1-\frac {b^2}{a^2}\right ) f^3\right ) \int \sin ^2(c+d x) \, dx}{4 b d^3}\\ &=-\frac {b (e+f x)^3}{4 a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3}{4 b d}+\frac {i b (e+f x)^4}{4 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^4}{4 a^2 b^3 f}-\frac {6 f (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^2}+\frac {10 f^3 \cos (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \cos (c+d x)}{a d^2}-\frac {3 a \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \cos (c+d x)}{b^2 d^2}-\frac {(e+f x)^3 \csc (c+d x)}{a d}+\frac {\left (a^2-b^2\right )^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d}+\frac {\left (a^2-b^2\right )^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d}-\frac {b (e+f x)^3 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 i \left (a^2-b^2\right )^2 f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}-\frac {3 i \left (a^2-b^2\right )^2 f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (e^{2 i (c+d x)}\right )}{2 a^2 d^2}+\frac {6 f^2 (e+f x) \sin (c+d x)}{a d^3}+\frac {6 a \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \sin (c+d x)}{b^2 d^3}-\frac {(e+f x)^3 \sin (c+d x)}{a d}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \sin (c+d x)}{b^2 d}-\frac {3 b f^3 \cos (c+d x) \sin (c+d x)}{8 a^2 d^4}-\frac {3 \left (1-\frac {b^2}{a^2}\right ) f^3 \cos (c+d x) \sin (c+d x)}{8 b d^4}+\frac {3 b f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 a^2 d^2}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 b d^2}-\frac {3 b f^2 (e+f x) \sin ^2(c+d x)}{4 a^2 d^3}-\frac {3 \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \sin ^2(c+d x)}{4 b d^3}+\frac {b (e+f x)^3 \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \sin ^2(c+d x)}{2 b d}-\frac {\left (3 i b f^2\right ) \int (e+f x) \text {Li}_2\left (e^{2 i (c+d x)}\right ) \, dx}{a^2 d^2}+\frac {\left (6 i \left (a^2-b^2\right )^2 f^2\right ) \int (e+f x) \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^3 d^2}+\frac {\left (6 i \left (a^2-b^2\right )^2 f^2\right ) \int (e+f x) \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^3 d^2}-\frac {\left (6 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}+\frac {\left (6 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}+\frac {\left (4 f^3\right ) \int \sin (c+d x) \, dx}{a d^3}+\frac {\left (3 b f^3\right ) \int 1 \, dx}{8 a^2 d^3}-\frac {\left (6 a \left (1-\frac {b^2}{a^2}\right ) f^3\right ) \int \sin (c+d x) \, dx}{b^2 d^3}+\frac {\left (3 \left (1-\frac {b^2}{a^2}\right ) f^3\right ) \int 1 \, dx}{8 b d^3}\\ &=\frac {3 b f^3 x}{8 a^2 d^3}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f^3 x}{8 b d^3}-\frac {b (e+f x)^3}{4 a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3}{4 b d}+\frac {i b (e+f x)^4}{4 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^4}{4 a^2 b^3 f}-\frac {6 f (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^2}+\frac {6 f^3 \cos (c+d x)}{a d^4}+\frac {6 a \left (1-\frac {b^2}{a^2}\right ) f^3 \cos (c+d x)}{b^2 d^4}-\frac {3 f (e+f x)^2 \cos (c+d x)}{a d^2}-\frac {3 a \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \cos (c+d x)}{b^2 d^2}-\frac {(e+f x)^3 \csc (c+d x)}{a d}+\frac {\left (a^2-b^2\right )^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d}+\frac {\left (a^2-b^2\right )^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d}-\frac {b (e+f x)^3 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 i \left (a^2-b^2\right )^2 f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}-\frac {3 i \left (a^2-b^2\right )^2 f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (e^{2 i (c+d x)}\right )}{2 a^2 d^2}-\frac {6 f^3 \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {6 f^3 \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^4}+\frac {6 \left (a^2-b^2\right )^2 f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^3}+\frac {6 \left (a^2-b^2\right )^2 f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^3}-\frac {3 b f^2 (e+f x) \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a^2 d^3}+\frac {6 f^2 (e+f x) \sin (c+d x)}{a d^3}+\frac {6 a \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \sin (c+d x)}{b^2 d^3}-\frac {(e+f x)^3 \sin (c+d x)}{a d}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \sin (c+d x)}{b^2 d}-\frac {3 b f^3 \cos (c+d x) \sin (c+d x)}{8 a^2 d^4}-\frac {3 \left (1-\frac {b^2}{a^2}\right ) f^3 \cos (c+d x) \sin (c+d x)}{8 b d^4}+\frac {3 b f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 a^2 d^2}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 b d^2}-\frac {3 b f^2 (e+f x) \sin ^2(c+d x)}{4 a^2 d^3}-\frac {3 \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \sin ^2(c+d x)}{4 b d^3}+\frac {b (e+f x)^3 \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \sin ^2(c+d x)}{2 b d}+\frac {\left (3 b f^3\right ) \int \text {Li}_3\left (e^{2 i (c+d x)}\right ) \, dx}{2 a^2 d^3}-\frac {\left (6 \left (a^2-b^2\right )^2 f^3\right ) \int \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^3 d^3}-\frac {\left (6 \left (a^2-b^2\right )^2 f^3\right ) \int \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^3 d^3}\\ &=\frac {3 b f^3 x}{8 a^2 d^3}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f^3 x}{8 b d^3}-\frac {b (e+f x)^3}{4 a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3}{4 b d}+\frac {i b (e+f x)^4}{4 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^4}{4 a^2 b^3 f}-\frac {6 f (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^2}+\frac {6 f^3 \cos (c+d x)}{a d^4}+\frac {6 a \left (1-\frac {b^2}{a^2}\right ) f^3 \cos (c+d x)}{b^2 d^4}-\frac {3 f (e+f x)^2 \cos (c+d x)}{a d^2}-\frac {3 a \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \cos (c+d x)}{b^2 d^2}-\frac {(e+f x)^3 \csc (c+d x)}{a d}+\frac {\left (a^2-b^2\right )^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d}+\frac {\left (a^2-b^2\right )^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d}-\frac {b (e+f x)^3 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 i \left (a^2-b^2\right )^2 f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}-\frac {3 i \left (a^2-b^2\right )^2 f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (e^{2 i (c+d x)}\right )}{2 a^2 d^2}-\frac {6 f^3 \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {6 f^3 \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^4}+\frac {6 \left (a^2-b^2\right )^2 f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^3}+\frac {6 \left (a^2-b^2\right )^2 f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^3}-\frac {3 b f^2 (e+f x) \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a^2 d^3}+\frac {6 f^2 (e+f x) \sin (c+d x)}{a d^3}+\frac {6 a \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \sin (c+d x)}{b^2 d^3}-\frac {(e+f x)^3 \sin (c+d x)}{a d}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \sin (c+d x)}{b^2 d}-\frac {3 b f^3 \cos (c+d x) \sin (c+d x)}{8 a^2 d^4}-\frac {3 \left (1-\frac {b^2}{a^2}\right ) f^3 \cos (c+d x) \sin (c+d x)}{8 b d^4}+\frac {3 b f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 a^2 d^2}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 b d^2}-\frac {3 b f^2 (e+f x) \sin ^2(c+d x)}{4 a^2 d^3}-\frac {3 \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \sin ^2(c+d x)}{4 b d^3}+\frac {b (e+f x)^3 \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \sin ^2(c+d x)}{2 b d}-\frac {\left (3 i b f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{2 i (c+d x)}\right )}{4 a^2 d^4}+\frac {\left (6 i \left (a^2-b^2\right )^2 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {i b x}{a-\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 b^3 d^4}+\frac {\left (6 i \left (a^2-b^2\right )^2 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {i b x}{a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 b^3 d^4}\\ &=\frac {3 b f^3 x}{8 a^2 d^3}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f^3 x}{8 b d^3}-\frac {b (e+f x)^3}{4 a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3}{4 b d}+\frac {i b (e+f x)^4}{4 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^4}{4 a^2 b^3 f}-\frac {6 f (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^2}+\frac {6 f^3 \cos (c+d x)}{a d^4}+\frac {6 a \left (1-\frac {b^2}{a^2}\right ) f^3 \cos (c+d x)}{b^2 d^4}-\frac {3 f (e+f x)^2 \cos (c+d x)}{a d^2}-\frac {3 a \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \cos (c+d x)}{b^2 d^2}-\frac {(e+f x)^3 \csc (c+d x)}{a d}+\frac {\left (a^2-b^2\right )^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d}+\frac {\left (a^2-b^2\right )^2 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d}-\frac {b (e+f x)^3 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 i \left (a^2-b^2\right )^2 f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}-\frac {3 i \left (a^2-b^2\right )^2 f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}+\frac {3 i b f (e+f x)^2 \text {Li}_2\left (e^{2 i (c+d x)}\right )}{2 a^2 d^2}-\frac {6 f^3 \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {6 f^3 \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^4}+\frac {6 \left (a^2-b^2\right )^2 f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^3}+\frac {6 \left (a^2-b^2\right )^2 f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^3}-\frac {3 b f^2 (e+f x) \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a^2 d^3}+\frac {6 i \left (a^2-b^2\right )^2 f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^4}+\frac {6 i \left (a^2-b^2\right )^2 f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^4}-\frac {3 i b f^3 \text {Li}_4\left (e^{2 i (c+d x)}\right )}{4 a^2 d^4}+\frac {6 f^2 (e+f x) \sin (c+d x)}{a d^3}+\frac {6 a \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \sin (c+d x)}{b^2 d^3}-\frac {(e+f x)^3 \sin (c+d x)}{a d}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \sin (c+d x)}{b^2 d}-\frac {3 b f^3 \cos (c+d x) \sin (c+d x)}{8 a^2 d^4}-\frac {3 \left (1-\frac {b^2}{a^2}\right ) f^3 \cos (c+d x) \sin (c+d x)}{8 b d^4}+\frac {3 b f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 a^2 d^2}+\frac {3 \left (1-\frac {b^2}{a^2}\right ) f (e+f x)^2 \cos (c+d x) \sin (c+d x)}{4 b d^2}-\frac {3 b f^2 (e+f x) \sin ^2(c+d x)}{4 a^2 d^3}-\frac {3 \left (1-\frac {b^2}{a^2}\right ) f^2 (e+f x) \sin ^2(c+d x)}{4 b d^3}+\frac {b (e+f x)^3 \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x)^3 \sin ^2(c+d x)}{2 b d}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(4084\) vs. \(2(1432)=2864\).
time = 30.10, size = 4084, normalized size = 2.85 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((e + f*x)^3*Cos[c + d*x]^3*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]

[Out]

((-e^3 - 3*e^2*f*x - 3*e*f^2*x^2 - f^3*x^3)*Csc[c + d*x])/(a*d) + (3*e^2*f*Log[Tan[(c + d*x)/2]])/(a*d^2) + (6
*e*f^2*((c + d*x)*(Log[1 - E^(I*(c + d*x))] - Log[1 + E^(I*(c + d*x))]) - c*Log[Tan[(c + d*x)/2]] + I*(PolyLog
[2, -E^(I*(c + d*x))] - PolyLog[2, E^(I*(c + d*x))])))/(a*d^3) + (b*e*f^2*Csc[c]*(2*d^2*x^2*(2*d*E^((2*I)*c)*x
+ (3*I)*(-1 + E^((2*I)*c))*Log[1 - E^((2*I)*(c + d*x))]) + 6*d*(-1 + E^((2*I)*c))*x*PolyLog[2, E^((2*I)*(c +
d*x))] + (3*I)*(-1 + E^((2*I)*c))*PolyLog[3, E^((2*I)*(c + d*x))]))/(4*a^2*d^3*E^(I*c)) - (6*f^3*(d^2*x^2*ArcT
anh[Cos[c + d*x] + I*Sin[c + d*x]] - I*d*x*PolyLog[2, -Cos[c + d*x] - I*Sin[c + d*x]] + I*d*x*PolyLog[2, Cos[c
+ d*x] + I*Sin[c + d*x]] + PolyLog[3, -Cos[c + d*x] - I*Sin[c + d*x]] - PolyLog[3, Cos[c + d*x] + I*Sin[c + d
*x]]))/(a*d^4) + (b*E^(I*c)*f^3*Csc[c]*(x^4 + (-1 + E^((-2*I)*c))*x^4 + ((-1 + E^((2*I)*c))*(2*d^4*x^4 + (4*I)
*d^3*x^3*Log[1 - E^((2*I)*(c + d*x))] + 6*d^2*x^2*PolyLog[2, E^((2*I)*(c + d*x))] + (6*I)*d*x*PolyLog[3, E^((2
*I)*(c + d*x))] - 3*PolyLog[4, E^((2*I)*(c + d*x))]))/(2*d^4*E^((2*I)*c))))/(4*a^2) + ((a^2 - b^2)^2*((-4*I)*d
^4*e^3*E^((2*I)*c)*x - (6*I)*d^4*e^2*E^((2*I)*c)*f*x^2 - (4*I)*d^4*e*E^((2*I)*c)*f^2*x^3 - I*d^4*E^((2*I)*c)*f
^3*x^4 - (2*I)*d^3*e^3*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))] + (2*I)*d^3*e^3*E^((2*I)*c
)*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x))))] - d^3*e^3*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(
-1 + E^((2*I)*(c + d*x)))^2] + d^3*e^3*E^((2*I)*c)*Log[4*a^2*E^((2*I)*(c + d*x)) + b^2*(-1 + E^((2*I)*(c + d*x
)))^2] - 6*d^3*e^2*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e
^2*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^3*e*f^2
*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e*E^((2*I)*c)*f^2*x
^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 2*d^3*f^3*x^3*Log[1 + (b*E^
(I*(2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 2*d^3*E^((2*I)*c)*f^3*x^3*Log[1 + (b*E^(I*(
2*c + d*x)))/(I*a*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^3*e^2*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*
a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e^2*E^((2*I)*c)*f*x*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^
(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 6*d^3*e*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(
-a^2 + b^2)*E^((2*I)*c)])] + 6*d^3*e*E^((2*I)*c)*f^2*x^2*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a
^2 + b^2)*E^((2*I)*c)])] - 2*d^3*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*
I)*c)])] + 2*d^3*E^((2*I)*c)*f^3*x^3*Log[1 + (b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c
)])] - (6*I)*d^2*(-1 + E^((2*I)*c))*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2
+ b^2)*E^((2*I)*c)])] - (6*I)*d^2*(-1 + E^((2*I)*c))*f*(e + f*x)^2*PolyLog[2, -((b*E^(I*(2*c + d*x)))/(I*a*E^
(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 12*d*e*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[
(-a^2 + b^2)*E^((2*I)*c)])] + 12*d*e*E^((2*I)*c)*f^2*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-
a^2 + b^2)*E^((2*I)*c)])] - 12*d*f^3*x*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^(
(2*I)*c)])] + 12*d*E^((2*I)*c)*f^3*x*PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2
*I)*c)])] - 12*d*e*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 1
2*d*e*E^((2*I)*c)*f^2*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 12
*d*f^3*x*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 12*d*E^((2*I)*c
)*f^3*x*PolyLog[3, -((b*E^(I*(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (12*I)*f^3*PolyL
og[4, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (12*I)*E^((2*I)*c)*f^3*PolyLog
[4, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + I*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (12*I)*f^3*PolyLog[4, -((b*E^(I*
(2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (12*I)*E^((2*I)*c)*f^3*PolyLog[4, -((b*E^(I*(
2*c + d*x)))/(I*a*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))]))/(2*a^2*b^3*d^4*(-1 + E^((2*I)*c))) - (b*e^3*Cs
c[c]*(-(d*x*Cos[c]) + Log[Cos[d*x]*Sin[c] + Cos[c]*Sin[d*x]]*Sin[c]))/(a^2*d*(Cos[c]^2 + Sin[c]^2)) - (I*(-a^2
+ 2*b^2)*e^3*x*(1 + Cos[2*c] + I*Sin[2*c]))/(b^3*(-1 + Cos[2*c] + I*Sin[2*c])) - (((3*I)/2)*(-a^2 + 2*b^2)*e^
2*f*x^2*(1 + Cos[2*c] + I*Sin[2*c]))/(b^3*(-1 + Cos[2*c] + I*Sin[2*c])) - (I*(-a^2 + 2*b^2)*e*f^2*x^3*(1 + Cos
[2*c] + I*Sin[2*c]))/(b^3*(-1 + Cos[2*c] + I*Sin[2*c])) - ((I/4)*(-a^2 + 2*b^2)*f^3*x^4*(1 + Cos[2*c] + I*Sin[
2*c]))/(b^3*(-1 + Cos[2*c] + I*Sin[2*c])) + (((-1/2*I)*a*f^3*x^3*Cos[c])/(b^2*d) - (a*f^3*x^3*Sin[c])/(2*b^2*d
) + ((-I)*d^3*e^3 - 3*d^2*e^2*f + (6*I)*d*e*f^2 + 6*f^3)*((a*Cos[c])/(2*b^2*d^4) - ((I/2)*a*Sin[c])/(b^2*d^4))
+ (a*d^2*e^2*f - (2*I)*a*d*e*f^2 - 2*a*f^3)*((...

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Maple [F]
time = 1.00, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{3} \left (\cos ^{3}\left (d x +c \right )\right ) \left (\cot ^{2}\left (d x +c \right )\right )}{a +b \sin \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^3*cos(d*x+c)^3*cot(d*x+c)^2/(a+b*sin(d*x+c)),x)

[Out]

int((f*x+e)^3*cos(d*x+c)^3*cot(d*x+c)^2/(a+b*sin(d*x+c)),x)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*cos(d*x+c)^3*cot(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4903 vs. \(2 (1336) = 2672\).
time = 1.03, size = 4903, normalized size = 3.42 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*cos(d*x+c)^3*cot(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/8*(8*(a^3*b + a*b^3)*d^3*f^3*x^3 - 48*a^3*b*d*f^3*x + 24*I*b^4*f^3*polylog(4, cos(d*x + c) + I*sin(d*x + c)
)*sin(d*x + c) - 24*I*b^4*f^3*polylog(4, cos(d*x + c) - I*sin(d*x + c))*sin(d*x + c) - 24*I*b^4*f^3*polylog(4,
-cos(d*x + c) + I*sin(d*x + c))*sin(d*x + c) + 24*I*b^4*f^3*polylog(4, -cos(d*x + c) - I*sin(d*x + c))*sin(d*
x + c) + 24*(a^3*b + a*b^3)*d^3*f*x*e^2 + 24*I*(a^4 - 2*a^2*b^2 + b^4)*f^3*polylog(4, -(I*a*cos(d*x + c) + a*s
in(d*x + c) + (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b)*sin(d*x + c) + 24*I*(a^4 - 2*a^2*
b^2 + b^4)*f^3*polylog(4, -(I*a*cos(d*x + c) + a*sin(d*x + c) - (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2
- b^2)/b^2))/b)*sin(d*x + c) - 24*I*(a^4 - 2*a^2*b^2 + b^4)*f^3*polylog(4, -(-I*a*cos(d*x + c) + a*sin(d*x +
c) + (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b)*sin(d*x + c) - 24*I*(a^4 - 2*a^2*b^2 + b^4
)*f^3*polylog(4, -(-I*a*cos(d*x + c) + a*sin(d*x + c) - (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/
b^2))/b)*sin(d*x + c) + 8*(a^3*b + a*b^3)*d^3*e^3 + 3*(2*a^2*b^2*d^2*f^3*x^2 + 4*a^2*b^2*d^2*f^2*x*e + 2*a^2*b
^2*d^2*f*e^2 - a^2*b^2*f^3)*cos(d*x + c)^3 - 8*(a^3*b*d^3*f^3*x^3 + 3*a^3*b*d^3*f*x*e^2 - 6*a^3*b*d*f^3*x + a^
3*b*d^3*e^3 + 3*(a^3*b*d^3*f^2*x^2 - 2*a^3*b*d*f^2)*e)*cos(d*x + c)^2 + 12*(I*(a^4 - 2*a^2*b^2 + b^4)*d^2*f^3*
x^2 + 2*I*(a^4 - 2*a^2*b^2 + b^4)*d^2*f^2*x*e + I*(a^4 - 2*a^2*b^2 + b^4)*d^2*f*e^2)*dilog((I*a*cos(d*x + c) -
a*sin(d*x + c) + (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1)*sin(d*x + c) + 12*(I*
(a^4 - 2*a^2*b^2 + b^4)*d^2*f^3*x^2 + 2*I*(a^4 - 2*a^2*b^2 + b^4)*d^2*f^2*x*e + I*(a^4 - 2*a^2*b^2 + b^4)*d^2*
f*e^2)*dilog((I*a*cos(d*x + c) - a*sin(d*x + c) - (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) -
b)/b + 1)*sin(d*x + c) + 12*(-I*(a^4 - 2*a^2*b^2 + b^4)*d^2*f^3*x^2 - 2*I*(a^4 - 2*a^2*b^2 + b^4)*d^2*f^2*x*e
- I*(a^4 - 2*a^2*b^2 + b^4)*d^2*f*e^2)*dilog((-I*a*cos(d*x + c) - a*sin(d*x + c) + (b*cos(d*x + c) - I*b*sin(
d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1)*sin(d*x + c) + 12*(-I*(a^4 - 2*a^2*b^2 + b^4)*d^2*f^3*x^2 - 2*I*(
a^4 - 2*a^2*b^2 + b^4)*d^2*f^2*x*e - I*(a^4 - 2*a^2*b^2 + b^4)*d^2*f*e^2)*dilog((-I*a*cos(d*x + c) - a*sin(d*x
+ c) - (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1)*sin(d*x + c) + 12*(-I*b^4*d^2*f
^3*x^2 + 2*I*a*b^3*d*f^3*x - I*b^4*d^2*f*e^2 - 2*I*(b^4*d^2*f^2*x - a*b^3*d*f^2)*e)*dilog(cos(d*x + c) + I*sin
(d*x + c))*sin(d*x + c) + 12*(I*b^4*d^2*f^3*x^2 - 2*I*a*b^3*d*f^3*x + I*b^4*d^2*f*e^2 + 2*I*(b^4*d^2*f^2*x - a
*b^3*d*f^2)*e)*dilog(cos(d*x + c) - I*sin(d*x + c))*sin(d*x + c) + 12*(I*b^4*d^2*f^3*x^2 + 2*I*a*b^3*d*f^3*x +
I*b^4*d^2*f*e^2 + 2*I*(b^4*d^2*f^2*x + a*b^3*d*f^2)*e)*dilog(-cos(d*x + c) + I*sin(d*x + c))*sin(d*x + c) + 1
2*(-I*b^4*d^2*f^3*x^2 - 2*I*a*b^3*d*f^3*x - I*b^4*d^2*f*e^2 - 2*I*(b^4*d^2*f^2*x + a*b^3*d*f^2)*e)*dilog(-cos(
d*x + c) - I*sin(d*x + c))*sin(d*x + c) + 4*((a^4 - 2*a^2*b^2 + b^4)*c^3*f^3 - 3*(a^4 - 2*a^2*b^2 + b^4)*c^2*d
*f^2*e + 3*(a^4 - 2*a^2*b^2 + b^4)*c*d^2*f*e^2 - (a^4 - 2*a^2*b^2 + b^4)*d^3*e^3)*log(2*b*cos(d*x + c) + 2*I*b
*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) + 2*I*a)*sin(d*x + c) + 4*((a^4 - 2*a^2*b^2 + b^4)*c^3*f^3 - 3*(a^4
- 2*a^2*b^2 + b^4)*c^2*d*f^2*e + 3*(a^4 - 2*a^2*b^2 + b^4)*c*d^2*f*e^2 - (a^4 - 2*a^2*b^2 + b^4)*d^3*e^3)*log
(2*b*cos(d*x + c) - 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) - 2*I*a)*sin(d*x + c) + 4*((a^4 - 2*a^2*b^
2 + b^4)*c^3*f^3 - 3*(a^4 - 2*a^2*b^2 + b^4)*c^2*d*f^2*e + 3*(a^4 - 2*a^2*b^2 + b^4)*c*d^2*f*e^2 - (a^4 - 2*a^
2*b^2 + b^4)*d^3*e^3)*log(-2*b*cos(d*x + c) + 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) + 2*I*a)*sin(d*x
+ c) + 4*((a^4 - 2*a^2*b^2 + b^4)*c^3*f^3 - 3*(a^4 - 2*a^2*b^2 + b^4)*c^2*d*f^2*e + 3*(a^4 - 2*a^2*b^2 + b^4)
*c*d^2*f*e^2 - (a^4 - 2*a^2*b^2 + b^4)*d^3*e^3)*log(-2*b*cos(d*x + c) - 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 -
b^2)/b^2) - 2*I*a)*sin(d*x + c) - 4*((a^4 - 2*a^2*b^2 + b^4)*d^3*f^3*x^3 + (a^4 - 2*a^2*b^2 + b^4)*c^3*f^3 + 3
*((a^4 - 2*a^2*b^2 + b^4)*d^3*f*x + (a^4 - 2*a^2*b^2 + b^4)*c*d^2*f)*e^2 + 3*((a^4 - 2*a^2*b^2 + b^4)*d^3*f^2*
x^2 - (a^4 - 2*a^2*b^2 + b^4)*c^2*d*f^2)*e)*log(-(I*a*cos(d*x + c) - a*sin(d*x + c) + (b*cos(d*x + c) + I*b*si
n(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b)*sin(d*x + c) - 4*((a^4 - 2*a^2*b^2 + b^4)*d^3*f^3*x^3 + (a^4 - 2*a^
2*b^2 + b^4)*c^3*f^3 + 3*((a^4 - 2*a^2*b^2 + b^4)*d^3*f*x + (a^4 - 2*a^2*b^2 + b^4)*c*d^2*f)*e^2 + 3*((a^4 - 2
*a^2*b^2 + b^4)*d^3*f^2*x^2 - (a^4 - 2*a^2*b^2 + b^4)*c^2*d*f^2)*e)*log(-(I*a*cos(d*x + c) - a*sin(d*x + c) -
(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b)*sin(d*x + c) - 4*((a^4 - 2*a^2*b^2 + b^4)*d
^3*f^3*x^3 + (a^4 - 2*a^2*b^2 + b^4)*c^3*f^3 + 3*((a^4 - 2*a^2*b^2 + b^4)*d^3*f*x + (a^4 - 2*a^2*b^2 + b^4)*c*
d^2*f)*e^2 + 3*((a^4 - 2*a^2*b^2 + b^4)*d^3*f^2*x^2 - (a^4 - 2*a^2*b^2 + b^4)*c^2*d*f^2)*e)*log(-(-I*a*cos(d*x
+ c) - a*sin(d*x + c) + (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b)*sin(d*x + c) - 4*(
(a^4 - 2*a^2*b^2 + b^4)*d^3*f^3*x^3 + (a^4 - 2*...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**3*cos(d*x+c)**3*cot(d*x+c)**2/(a+b*sin(d*x+c)),x)

[Out]

Timed out

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*cos(d*x+c)^3*cot(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

Timed out

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^3*cot(c + d*x)^2*(e + f*x)^3)/(a + b*sin(c + d*x)),x)

[Out]

\text{Hanged}

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