\(\int \frac {(e+f x) \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) [347]

Optimal result

Integrand size = 34, antiderivative size = 641 \[ \int \frac {(e+f x) \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {b f x}{4 a^2 d}-\frac {\left (a^2-b^2\right ) f x}{4 a^2 b d}+\frac {i b (e+f x)^2}{2 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^2}{2 a^2 b^3 f}-\frac {f \text {arctanh}(\cos (c+d x))}{a d^2}-\frac {f \cos (c+d x)}{a d^2}-\frac {\left (a^2-b^2\right ) f \cos (c+d x)}{a b^2 d^2}-\frac {(e+f x) \csc (c+d x)}{a d}+\frac {\left (a^2-b^2\right )^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d}+\frac {\left (a^2-b^2\right )^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d}-\frac {b (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}-\frac {i \left (a^2-b^2\right )^2 f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}-\frac {i \left (a^2-b^2\right )^2 f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}+\frac {i b f \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{2 a^2 d^2}-\frac {(e+f x) \sin (c+d x)}{a d}-\frac {\left (a^2-b^2\right ) (e+f x) \sin (c+d x)}{a b^2 d}+\frac {b f \cos (c+d x) \sin (c+d x)}{4 a^2 d^2}+\frac {\left (a^2-b^2\right ) f \cos (c+d x) \sin (c+d x)}{4 a^2 b d^2}+\frac {b (e+f x) \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (a^2-b^2\right ) (e+f x) \sin ^2(c+d x)}{2 a^2 b d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 641, normalized size of antiderivative = 1.00, number of steps used = 45, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4639, 4493, 3391, 3377, 2718, 4495, 3855, 4490, 2715, 8, 4489, 3798, 2221, 2317, 2438, 4621, 4615} \[ \int \frac {(e+f x) \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {f \left (a^2-b^2\right ) \cos (c+d x)}{a b^2 d^2}+\frac {f \left (a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{4 a^2 b d^2}+\frac {\left (a^2-b^2\right ) (e+f x) \sin ^2(c+d x)}{2 a^2 b d}-\frac {\left (a^2-b^2\right ) (e+f x) \sin (c+d x)}{a b^2 d}-\frac {f x \left (a^2-b^2\right )}{4 a^2 b d}-\frac {i f \left (a^2-b^2\right )^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}-\frac {i f \left (a^2-b^2\right )^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}+\frac {\left (a^2-b^2\right )^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d}+\frac {\left (a^2-b^2\right )^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a^2 b^3 d}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^2}{2 a^2 b^3 f}+\frac {i b f \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{2 a^2 d^2}+\frac {b f \sin (c+d x) \cos (c+d x)}{4 a^2 d^2}-\frac {b (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {b (e+f x) \sin ^2(c+d x)}{2 a^2 d}-\frac {b f x}{4 a^2 d}+\frac {i b (e+f x)^2}{2 a^2 f}-\frac {f \text {arctanh}(\cos (c+d x))}{a d^2}-\frac {f \cos (c+d x)}{a d^2}-\frac {(e+f x) \sin (c+d x)}{a d}-\frac {(e+f x) \csc (c+d x)}{a d} \]

[In]

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

[Out]

-1/4*(b*f*x)/(a^2*d) - ((a^2 - b^2)*f*x)/(4*a^2*b*d) + ((I/2)*b*(e + f*x)^2)/(a^2*f) - ((I/2)*(a^2 - b^2)^2*(e
+ f*x)^2)/(a^2*b^3*f) - (f*ArcTanh[Cos[c + d*x]])/(a*d^2) - (f*Cos[c + d*x])/(a*d^2) - ((a^2 - b^2)*f*Cos[c +
d*x])/(a*b^2*d^2) - ((e + f*x)*Csc[c + d*x])/(a*d) + ((a^2 - b^2)^2*(e + f*x)*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)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^
2])])/(a^2*b^3*d) - (b*(e + f*x)*Log[1 - E^((2*I)*(c + d*x))])/(a^2*d) - (I*(a^2 - b^2)^2*f*PolyLog[2, (I*b*E^
(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^3*d^2) - (I*(a^2 - b^2)^2*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a
+ Sqrt[a^2 - b^2])])/(a^2*b^3*d^2) + ((I/2)*b*f*PolyLog[2, E^((2*I)*(c + d*x))])/(a^2*d^2) - ((e + f*x)*Sin[c
+ d*x])/(a*d) - ((a^2 - b^2)*(e + f*x)*Sin[c + d*x])/(a*b^2*d) + (b*f*Cos[c + d*x]*Sin[c + d*x])/(4*a^2*d^2) +
((a^2 - b^2)*f*Cos[c + d*x]*Sin[c + d*x])/(4*a^2*b*d^2) + (b*(e + f*x)*Sin[c + d*x]^2)/(2*a^2*d) + ((a^2 - b^
2)*(e + f*x)*Sin[c + d*x]^2)/(2*a^2*b*d)

Rule 8

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

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 2317

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

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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 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 3855

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

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x) \cos ^3(c+d x) \cot ^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx}{a} \\ & = -\frac {\int (e+f x) \cos ^3(c+d x) \, dx}{a}+\frac {\int (e+f x) \cos (c+d x) \cot ^2(c+d x) \, dx}{a}-\frac {b \int (e+f x) \cos ^4(c+d x) \cot (c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x) \cos ^5(c+d x)}{a+b \sin (c+d x)} \, dx}{a^2} \\ & = -\frac {f \cos ^3(c+d x)}{9 a d^2}-\frac {(e+f x) \cos ^2(c+d x) \sin (c+d x)}{3 a d}-\frac {2 \int (e+f x) \cos (c+d x) \, dx}{3 a}-\frac {\int (e+f x) \cos (c+d x) \, dx}{a}+\frac {\int (e+f x) \cos ^3(c+d x) \, dx}{a}+\frac {\int (e+f x) \cot (c+d x) \csc (c+d x) \, dx}{a}-\frac {b \int (e+f x) \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) \cos ^3(c+d x)}{a+b \sin (c+d x)} \, dx \\ & = -\frac {(e+f x) \csc (c+d x)}{a d}-\frac {5 (e+f x) \sin (c+d x)}{3 a d}+\frac {2 \int (e+f x) \cos (c+d x) \, dx}{3 a}-\frac {b \int (e+f x) \cot (c+d x) \, dx}{a^2}+\frac {b \int (e+f x) \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) \cos (c+d x) \, dx}{b^2}-\frac {\left (-1+\frac {b^2}{a^2}\right ) \int (e+f x) \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) \cos (c+d x)}{a+b \sin (c+d x)} \, dx}{b^2}+\frac {(2 f) \int \sin (c+d x) \, dx}{3 a d}+\frac {f \int \csc (c+d x) \, dx}{a d}+\frac {f \int \sin (c+d x) \, dx}{a d} \\ & = \frac {i b (e+f x)^2}{2 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^2}{2 a^2 b^3 f}-\frac {f \text {arctanh}(\cos (c+d x))}{a d^2}-\frac {5 f \cos (c+d x)}{3 a d^2}-\frac {(e+f x) \csc (c+d x)}{a d}-\frac {(e+f x) \sin (c+d x)}{a d}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x) \sin (c+d x)}{b^2 d}+\frac {b (e+f x) \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \sin ^2(c+d x)}{2 b d}+\frac {(2 i b) \int \frac {e^{2 i (c+d x)} (e+f x)}{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)}{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)}{a+\sqrt {a^2-b^2}-i b e^{i (c+d x)}} \, dx}{b^2}-\frac {(2 f) \int \sin (c+d x) \, dx}{3 a d}-\frac {(b f) \int \sin ^2(c+d x) \, dx}{2 a^2 d}+\frac {\left (a \left (1-\frac {b^2}{a^2}\right ) f\right ) \int \sin (c+d x) \, dx}{b^2 d}-\frac {\left (\left (1-\frac {b^2}{a^2}\right ) f\right ) \int \sin ^2(c+d x) \, dx}{2 b d} \\ & = \frac {i b (e+f x)^2}{2 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^2}{2 a^2 b^3 f}-\frac {f \text {arctanh}(\cos (c+d x))}{a d^2}-\frac {f \cos (c+d x)}{a d^2}-\frac {a \left (1-\frac {b^2}{a^2}\right ) f \cos (c+d x)}{b^2 d^2}-\frac {(e+f x) \csc (c+d x)}{a d}+\frac {\left (a^2-b^2\right )^2 (e+f x) \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) \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) \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}-\frac {(e+f x) \sin (c+d x)}{a d}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x) \sin (c+d x)}{b^2 d}+\frac {b f \cos (c+d x) \sin (c+d x)}{4 a^2 d^2}+\frac {\left (1-\frac {b^2}{a^2}\right ) f \cos (c+d x) \sin (c+d x)}{4 b d^2}+\frac {b (e+f x) \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \sin ^2(c+d x)}{2 b d}-\frac {(b f) \int 1 \, dx}{4 a^2 d}+\frac {(b f) \int \log \left (1-e^{2 i (c+d x)}\right ) \, dx}{a^2 d}-\frac {\left (\left (a^2-b^2\right )^2 f\right ) \int \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 (\left (a^2-b^2\right )^2 f\right ) \int \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 (\left (1-\frac {b^2}{a^2}\right ) f\right ) \int 1 \, dx}{4 b d} \\ & = -\frac {b f x}{4 a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) f x}{4 b d}+\frac {i b (e+f x)^2}{2 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^2}{2 a^2 b^3 f}-\frac {f \text {arctanh}(\cos (c+d x))}{a d^2}-\frac {f \cos (c+d x)}{a d^2}-\frac {a \left (1-\frac {b^2}{a^2}\right ) f \cos (c+d x)}{b^2 d^2}-\frac {(e+f x) \csc (c+d x)}{a d}+\frac {\left (a^2-b^2\right )^2 (e+f x) \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) \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) \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}-\frac {(e+f x) \sin (c+d x)}{a d}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x) \sin (c+d x)}{b^2 d}+\frac {b f \cos (c+d x) \sin (c+d x)}{4 a^2 d^2}+\frac {\left (1-\frac {b^2}{a^2}\right ) f \cos (c+d x) \sin (c+d x)}{4 b d^2}+\frac {b (e+f x) \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \sin ^2(c+d x)}{2 b d}-\frac {(i b f) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i (c+d x)}\right )}{2 a^2 d^2}+\frac {\left (i \left (a^2-b^2\right )^2 f\right ) \text {Subst}\left (\int \frac {\log \left (1-\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^2}+\frac {\left (i \left (a^2-b^2\right )^2 f\right ) \text {Subst}\left (\int \frac {\log \left (1-\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^2} \\ & = -\frac {b f x}{4 a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) f x}{4 b d}+\frac {i b (e+f x)^2}{2 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^2}{2 a^2 b^3 f}-\frac {f \text {arctanh}(\cos (c+d x))}{a d^2}-\frac {f \cos (c+d x)}{a d^2}-\frac {a \left (1-\frac {b^2}{a^2}\right ) f \cos (c+d x)}{b^2 d^2}-\frac {(e+f x) \csc (c+d x)}{a d}+\frac {\left (a^2-b^2\right )^2 (e+f x) \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) \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) \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}-\frac {i \left (a^2-b^2\right )^2 f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}-\frac {i \left (a^2-b^2\right )^2 f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}+\frac {i b f \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{2 a^2 d^2}-\frac {(e+f x) \sin (c+d x)}{a d}-\frac {a \left (1-\frac {b^2}{a^2}\right ) (e+f x) \sin (c+d x)}{b^2 d}+\frac {b f \cos (c+d x) \sin (c+d x)}{4 a^2 d^2}+\frac {\left (1-\frac {b^2}{a^2}\right ) f \cos (c+d x) \sin (c+d x)}{4 b d^2}+\frac {b (e+f x) \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \sin ^2(c+d x)}{2 b d} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1666\) vs. \(2(641)=1282\).

Time = 8.52 (sec) , antiderivative size = 1666, normalized size of antiderivative = 2.60 \[ \int \frac {(e+f x) \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {a f \cos (c+d x)}{b^2 d^2}-\frac {(d e-c f+f (c+d x)) \cos (2 (c+d x))}{4 b d^2}+\frac {\left (-d e \cos \left (\frac {1}{2} (c+d x)\right )+c f \cos \left (\frac {1}{2} (c+d x)\right )-f (c+d x) \cos \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right )}{2 a d^2}+\frac {a^2 e \log \left (1+\frac {b \sin (c+d x)}{a}\right )}{b^3 d}-\frac {2 e \log \left (1+\frac {b \sin (c+d x)}{a}\right )}{b d}+\frac {b e \log \left (1+\frac {b \sin (c+d x)}{a}\right )}{a^2 d}-\frac {a^2 c f \log \left (1+\frac {b \sin (c+d x)}{a}\right )}{b^3 d^2}+\frac {2 c f \log \left (1+\frac {b \sin (c+d x)}{a}\right )}{b d^2}-\frac {b c f \log \left (1+\frac {b \sin (c+d x)}{a}\right )}{a^2 d^2}+\frac {f \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d^2}-\frac {b e (\log (\cos (c+d x))+\log (\tan (c+d x)))}{a^2 d}+\frac {b c f (\log (\cos (c+d x))+\log (\tan (c+d x)))}{a^2 d^2}-\frac {2 f \left (\frac {(c+d x) \log (a+b \sin (c+d x))}{b}-\frac {-\frac {1}{2} i \left (-c+\frac {\pi }{2}-d x\right )^2+4 i \arcsin \left (\frac {\sqrt {\frac {a+b}{b}}}{\sqrt {2}}\right ) \arctan \left (\frac {(a-b) \tan \left (\frac {1}{2} \left (-c+\frac {\pi }{2}-d x\right )\right )}{\sqrt {a^2-b^2}}\right )+\left (-c+\frac {\pi }{2}-d x+2 \arcsin \left (\frac {\sqrt {\frac {a+b}{b}}}{\sqrt {2}}\right )\right ) \log \left (1+\frac {\left (a-\sqrt {a^2-b^2}\right ) e^{i \left (-c+\frac {\pi }{2}-d x\right )}}{b}\right )+\left (-c+\frac {\pi }{2}-d x-2 \arcsin \left (\frac {\sqrt {\frac {a+b}{b}}}{\sqrt {2}}\right )\right ) \log \left (1+\frac {\left (a+\sqrt {a^2-b^2}\right ) e^{i \left (-c+\frac {\pi }{2}-d x\right )}}{b}\right )-\left (-c+\frac {\pi }{2}-d x\right ) \log (a+b \sin (c+d x))-i \left (\operatorname {PolyLog}\left (2,\frac {\left (-a-\sqrt {a^2-b^2}\right ) e^{i \left (-c+\frac {\pi }{2}-d x\right )}}{b}\right )+\operatorname {PolyLog}\left (2,\frac {\left (-a+\sqrt {a^2-b^2}\right ) e^{i \left (-c+\frac {\pi }{2}-d x\right )}}{b}\right )\right )}{b}\right )}{d^2}+\frac {a^2 f \left (\frac {(c+d x) \log (a+b \sin (c+d x))}{b}-\frac {-\frac {1}{2} i \left (-c+\frac {\pi }{2}-d x\right )^2+4 i \arcsin \left (\frac {\sqrt {\frac {a+b}{b}}}{\sqrt {2}}\right ) \arctan \left (\frac {(a-b) \tan \left (\frac {1}{2} \left (-c+\frac {\pi }{2}-d x\right )\right )}{\sqrt {a^2-b^2}}\right )+\left (-c+\frac {\pi }{2}-d x+2 \arcsin \left (\frac {\sqrt {\frac {a+b}{b}}}{\sqrt {2}}\right )\right ) \log \left (1+\frac {\left (a-\sqrt {a^2-b^2}\right ) e^{i \left (-c+\frac {\pi }{2}-d x\right )}}{b}\right )+\left (-c+\frac {\pi }{2}-d x-2 \arcsin \left (\frac {\sqrt {\frac {a+b}{b}}}{\sqrt {2}}\right )\right ) \log \left (1+\frac {\left (a+\sqrt {a^2-b^2}\right ) e^{i \left (-c+\frac {\pi }{2}-d x\right )}}{b}\right )-\left (-c+\frac {\pi }{2}-d x\right ) \log (a+b \sin (c+d x))-i \left (\operatorname {PolyLog}\left (2,\frac {\left (-a-\sqrt {a^2-b^2}\right ) e^{i \left (-c+\frac {\pi }{2}-d x\right )}}{b}\right )+\operatorname {PolyLog}\left (2,\frac {\left (-a+\sqrt {a^2-b^2}\right ) e^{i \left (-c+\frac {\pi }{2}-d x\right )}}{b}\right )\right )}{b}\right )}{b^2 d^2}+\frac {b^2 f \left (\frac {(c+d x) \log (a+b \sin (c+d x))}{b}-\frac {-\frac {1}{2} i \left (-c+\frac {\pi }{2}-d x\right )^2+4 i \arcsin \left (\frac {\sqrt {\frac {a+b}{b}}}{\sqrt {2}}\right ) \arctan \left (\frac {(a-b) \tan \left (\frac {1}{2} \left (-c+\frac {\pi }{2}-d x\right )\right )}{\sqrt {a^2-b^2}}\right )+\left (-c+\frac {\pi }{2}-d x+2 \arcsin \left (\frac {\sqrt {\frac {a+b}{b}}}{\sqrt {2}}\right )\right ) \log \left (1+\frac {\left (a-\sqrt {a^2-b^2}\right ) e^{i \left (-c+\frac {\pi }{2}-d x\right )}}{b}\right )+\left (-c+\frac {\pi }{2}-d x-2 \arcsin \left (\frac {\sqrt {\frac {a+b}{b}}}{\sqrt {2}}\right )\right ) \log \left (1+\frac {\left (a+\sqrt {a^2-b^2}\right ) e^{i \left (-c+\frac {\pi }{2}-d x\right )}}{b}\right )-\left (-c+\frac {\pi }{2}-d x\right ) \log (a+b \sin (c+d x))-i \left (\operatorname {PolyLog}\left (2,\frac {\left (-a-\sqrt {a^2-b^2}\right ) e^{i \left (-c+\frac {\pi }{2}-d x\right )}}{b}\right )+\operatorname {PolyLog}\left (2,\frac {\left (-a+\sqrt {a^2-b^2}\right ) e^{i \left (-c+\frac {\pi }{2}-d x\right )}}{b}\right )\right )}{b}\right )}{a^2 d^2}-\frac {b f \left ((c+d x) \log \left (1-e^{2 i (c+d x)}\right )-\frac {1}{2} i \left ((c+d x)^2+\operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )\right )\right )}{a^2 d^2}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (-d e \sin \left (\frac {1}{2} (c+d x)\right )+c f \sin \left (\frac {1}{2} (c+d x)\right )-f (c+d x) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a d^2}-\frac {a (d e-c f+f (c+d x)) \sin (c+d x)}{b^2 d^2}+\frac {f \sin (2 (c+d x))}{8 b d^2} \]

[In]

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

[Out]

-((a*f*Cos[c + d*x])/(b^2*d^2)) - ((d*e - c*f + f*(c + d*x))*Cos[2*(c + d*x)])/(4*b*d^2) + ((-(d*e*Cos[(c + d*
x)/2]) + c*f*Cos[(c + d*x)/2] - f*(c + d*x)*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(2*a*d^2) + (a^2*e*Log[1 + (b*
Sin[c + d*x])/a])/(b^3*d) - (2*e*Log[1 + (b*Sin[c + d*x])/a])/(b*d) + (b*e*Log[1 + (b*Sin[c + d*x])/a])/(a^2*d
) - (a^2*c*f*Log[1 + (b*Sin[c + d*x])/a])/(b^3*d^2) + (2*c*f*Log[1 + (b*Sin[c + d*x])/a])/(b*d^2) - (b*c*f*Log
[1 + (b*Sin[c + d*x])/a])/(a^2*d^2) + (f*Log[Tan[(c + d*x)/2]])/(a*d^2) - (b*e*(Log[Cos[c + d*x]] + Log[Tan[c
+ d*x]]))/(a^2*d) + (b*c*f*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/(a^2*d^2) - (2*f*(((c + d*x)*Log[a + b*Sin
[c + d*x]])/b - ((-1/2*I)*(-c + Pi/2 - d*x)^2 + (4*I)*ArcSin[Sqrt[(a + b)/b]/Sqrt[2]]*ArcTan[((a - b)*Tan[(-c
+ Pi/2 - d*x)/2])/Sqrt[a^2 - b^2]] + (-c + Pi/2 - d*x + 2*ArcSin[Sqrt[(a + b)/b]/Sqrt[2]])*Log[1 + ((a - Sqrt[
a^2 - b^2])*E^(I*(-c + Pi/2 - d*x)))/b] + (-c + Pi/2 - d*x - 2*ArcSin[Sqrt[(a + b)/b]/Sqrt[2]])*Log[1 + ((a +
Sqrt[a^2 - b^2])*E^(I*(-c + Pi/2 - d*x)))/b] - (-c + Pi/2 - d*x)*Log[a + b*Sin[c + d*x]] - I*(PolyLog[2, ((-a
- Sqrt[a^2 - b^2])*E^(I*(-c + Pi/2 - d*x)))/b] + PolyLog[2, ((-a + Sqrt[a^2 - b^2])*E^(I*(-c + Pi/2 - d*x)))/b
]))/b))/d^2 + (a^2*f*(((c + d*x)*Log[a + b*Sin[c + d*x]])/b - ((-1/2*I)*(-c + Pi/2 - d*x)^2 + (4*I)*ArcSin[Sqr
t[(a + b)/b]/Sqrt[2]]*ArcTan[((a - b)*Tan[(-c + Pi/2 - d*x)/2])/Sqrt[a^2 - b^2]] + (-c + Pi/2 - d*x + 2*ArcSin
[Sqrt[(a + b)/b]/Sqrt[2]])*Log[1 + ((a - Sqrt[a^2 - b^2])*E^(I*(-c + Pi/2 - d*x)))/b] + (-c + Pi/2 - d*x - 2*A
rcSin[Sqrt[(a + b)/b]/Sqrt[2]])*Log[1 + ((a + Sqrt[a^2 - b^2])*E^(I*(-c + Pi/2 - d*x)))/b] - (-c + Pi/2 - d*x)
*Log[a + b*Sin[c + d*x]] - I*(PolyLog[2, ((-a - Sqrt[a^2 - b^2])*E^(I*(-c + Pi/2 - d*x)))/b] + PolyLog[2, ((-a
+ Sqrt[a^2 - b^2])*E^(I*(-c + Pi/2 - d*x)))/b]))/b))/(b^2*d^2) + (b^2*f*(((c + d*x)*Log[a + b*Sin[c + d*x]])/
b - ((-1/2*I)*(-c + Pi/2 - d*x)^2 + (4*I)*ArcSin[Sqrt[(a + b)/b]/Sqrt[2]]*ArcTan[((a - b)*Tan[(-c + Pi/2 - d*x
)/2])/Sqrt[a^2 - b^2]] + (-c + Pi/2 - d*x + 2*ArcSin[Sqrt[(a + b)/b]/Sqrt[2]])*Log[1 + ((a - Sqrt[a^2 - b^2])*
E^(I*(-c + Pi/2 - d*x)))/b] + (-c + Pi/2 - d*x - 2*ArcSin[Sqrt[(a + b)/b]/Sqrt[2]])*Log[1 + ((a + Sqrt[a^2 - b
^2])*E^(I*(-c + Pi/2 - d*x)))/b] - (-c + Pi/2 - d*x)*Log[a + b*Sin[c + d*x]] - I*(PolyLog[2, ((-a - Sqrt[a^2 -
b^2])*E^(I*(-c + Pi/2 - d*x)))/b] + PolyLog[2, ((-a + Sqrt[a^2 - b^2])*E^(I*(-c + Pi/2 - d*x)))/b]))/b))/(a^2
*d^2) - (b*f*((c + d*x)*Log[1 - E^((2*I)*(c + d*x))] - (I/2)*((c + d*x)^2 + PolyLog[2, E^((2*I)*(c + d*x))])))
/(a^2*d^2) + (Sec[(c + d*x)/2]*(-(d*e*Sin[(c + d*x)/2]) + c*f*Sin[(c + d*x)/2] - f*(c + d*x)*Sin[(c + d*x)/2])
)/(2*a*d^2) - (a*(d*e - c*f + f*(c + d*x))*Sin[c + d*x])/(b^2*d^2) + (f*Sin[2*(c + d*x)])/(8*b*d^2)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 5155 vs. \(2 (594 ) = 1188\).

Time = 6.46 (sec) , antiderivative size = 5156, normalized size of antiderivative = 8.04

method	result	size
risch	\(\text {Expression too large to display}\)	\(5156\)

[In]

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

[Out]

result too large to display

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1707 vs. \(2 (582) = 1164\).

Time = 0.56 (sec) , antiderivative size = 1707, normalized size of antiderivative = 2.66 \[ \int \frac {(e+f x) \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/4*(a^2*b^2*f*cos(d*x + c)^3 - 2*I*b^4*f*dilog(cos(d*x + c) + I*sin(d*x + c))*sin(d*x + c) + 2*I*b^4*f*dilog
(cos(d*x + c) - I*sin(d*x + c))*sin(d*x + c) + 2*I*b^4*f*dilog(-cos(d*x + c) + I*sin(d*x + c))*sin(d*x + c) -
2*I*b^4*f*dilog(-cos(d*x + c) - I*sin(d*x + c))*sin(d*x + c) - a^2*b^2*f*cos(d*x + c) + 4*(a^3*b + a*b^3)*d*f*
x + 2*I*(a^4 - 2*a^2*b^2 + b^4)*f*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) + 2*I*(a^4 - 2*a^2*b^2 + b^4)*f*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) - 2*I*(a^4
- 2*a^2*b^2 + b^4)*f*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) - 2*I*(a^4 - 2*a^2*b^2 + b^4)*f*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) + 4*(a^3*b + a*b^3)*
d*e - 4*(a^3*b*d*f*x + a^3*b*d*e)*cos(d*x + c)^2 - 2*((a^4 - 2*a^2*b^2 + b^4)*d*e - (a^4 - 2*a^2*b^2 + b^4)*c*
f)*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) - 2*((a^4 - 2*
a^2*b^2 + b^4)*d*e - (a^4 - 2*a^2*b^2 + b^4)*c*f)*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) - 2*((a^4 - 2*a^2*b^2 + b^4)*d*e - (a^4 - 2*a^2*b^2 + b^4)*c*f)*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) - 2*((a^4 - 2*a^2*b^2 + b^4)*
d*e - (a^4 - 2*a^2*b^2 + b^4)*c*f)*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) - 2*((a^4 - 2*a^2*b^2 + b^4)*d*f*x + (a^4 - 2*a^2*b^2 + b^4)*c*f)*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) - 2*((a^4 - 2
*a^2*b^2 + b^4)*d*f*x + (a^4 - 2*a^2*b^2 + b^4)*c*f)*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) - 2*((a^4 - 2*a^2*b^2 + b^4)*d*f*x + (a^4 - 2
*a^2*b^2 + b^4)*c*f)*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) - 2*((a^4 - 2*a^2*b^2 + b^4)*d*f*x + (a^4 - 2*a^2*b^2 + b^4)*c*f)*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) + 2*(b^4*d*f*x + b^4*d*e + a*b^3*f)*log(cos(d*x + c) + I*sin(d*x + c) + 1)*sin(d*x + c) + 2*(b^4*d*f*x + b^
4*d*e + a*b^3*f)*log(cos(d*x + c) - I*sin(d*x + c) + 1)*sin(d*x + c) + 2*(b^4*d*e - (b^4*c + a*b^3)*f)*log(-1/
2*cos(d*x + c) + 1/2*I*sin(d*x + c) + 1/2)*sin(d*x + c) + 2*(b^4*d*e - (b^4*c + a*b^3)*f)*log(-1/2*cos(d*x + c
) - 1/2*I*sin(d*x + c) + 1/2)*sin(d*x + c) + 2*(b^4*d*f*x + b^4*c*f)*log(-cos(d*x + c) + I*sin(d*x + c) + 1)*s
in(d*x + c) + 2*(b^4*d*f*x + b^4*c*f)*log(-cos(d*x + c) - I*sin(d*x + c) + 1)*sin(d*x + c) - (a^2*b^2*d*f*x +
a^2*b^2*d*e - 4*a^3*b*f*cos(d*x + c) - 2*(a^2*b^2*d*f*x + a^2*b^2*d*e)*cos(d*x + c)^2)*sin(d*x + c))/(a^2*b^3*
d^2*sin(d*x + c))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x) \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((f*x+e)*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.

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x) \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Hanged} \]

[In]

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

[Out]

\text{Hanged}