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

Optimal. Leaf size=641 \[ -\frac{i f \left (a^2-b^2\right )^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a^2 b^3 d^2}-\frac{i f \left (a^2-b^2\right )^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a^2 b^3 d^2}+\frac{i b f \text{PolyLog}\left (2,e^{2 i (c+d x)}\right )}{2 a^2 d^2}-\frac{f \left (a^2-b^2\right ) \cos (c+d x)}{a b^2 d^2}+\frac{f \left (a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{4 a^2 b d^2}+\frac{\left (a^2-b^2\right )^2 (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a^2 b^3 d}+\frac{\left (a^2-b^2\right )^2 (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a^2 b^3 d}+\frac{\left (a^2-b^2\right ) (e+f x) \sin ^2(c+d x)}{2 a^2 b d}-\frac{\left (a^2-b^2\right ) (e+f x) \sin (c+d x)}{a b^2 d}-\frac{f x \left (a^2-b^2\right )}{4 a^2 b d}-\frac{i \left (a^2-b^2\right )^2 (e+f x)^2}{2 a^2 b^3 f}+\frac{b f \sin (c+d x) \cos (c+d x)}{4 a^2 d^2}-\frac{b (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac{b (e+f x) \sin ^2(c+d x)}{2 a^2 d}-\frac{b f x}{4 a^2 d}+\frac{i b (e+f x)^2}{2 a^2 f}-\frac{f \cos (c+d x)}{a d^2}-\frac{f \tanh ^{-1}(\cos (c+d x))}{a d^2}-\frac{(e+f x) \sin (c+d x)}{a d}-\frac{(e+f x) \csc (c+d x)}{a d} \]

[Out]

-(b*f*x)/(4*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)

________________________________________________________________________________________

Rubi [A]  time = 1.22528, antiderivative size = 641, normalized size of antiderivative = 1., number of steps used = 45, number of rules used = 17, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4543, 4408, 3310, 3296, 2638, 4410, 3770, 4405, 2635, 8, 4404, 3717, 2190, 2279, 2391, 4525, 4519} \[ -\frac{i f \left (a^2-b^2\right )^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a^2 b^3 d^2}-\frac{i f \left (a^2-b^2\right )^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a^2 b^3 d^2}+\frac{i b f \text{PolyLog}\left (2,e^{2 i (c+d x)}\right )}{2 a^2 d^2}-\frac{f \left (a^2-b^2\right ) \cos (c+d x)}{a b^2 d^2}+\frac{f \left (a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{4 a^2 b d^2}+\frac{\left (a^2-b^2\right )^2 (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a^2 b^3 d}+\frac{\left (a^2-b^2\right )^2 (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a^2 b^3 d}+\frac{\left (a^2-b^2\right ) (e+f x) \sin ^2(c+d x)}{2 a^2 b d}-\frac{\left (a^2-b^2\right ) (e+f x) \sin (c+d x)}{a b^2 d}-\frac{f x \left (a^2-b^2\right )}{4 a^2 b d}-\frac{i \left (a^2-b^2\right )^2 (e+f x)^2}{2 a^2 b^3 f}+\frac{b f \sin (c+d x) \cos (c+d x)}{4 a^2 d^2}-\frac{b (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac{b (e+f x) \sin ^2(c+d x)}{2 a^2 d}-\frac{b f x}{4 a^2 d}+\frac{i b (e+f x)^2}{2 a^2 f}-\frac{f \cos (c+d x)}{a d^2}-\frac{f \tanh ^{-1}(\cos (c+d x))}{a d^2}-\frac{(e+f x) \sin (c+d x)}{a d}-\frac{(e+f x) \csc (c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*f*x)/(4*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 4543

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 4408

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 3310

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 3296

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 2638

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

Rule 4410

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] /; Fr
eeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 3770

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

Rule 4405

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 2635

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] && Integer
Q[2*n]

Rule 8

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

Rule 4404

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 3717

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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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 4525

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 4519

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]

Rubi steps

\begin{align*} \int \frac{(e+f x) \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=\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 \tanh ^{-1}(\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 \tanh ^{-1}(\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 \tanh ^{-1}(\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) \operatorname{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 ) \operatorname{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 ) \operatorname{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 \tanh ^{-1}(\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 \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{i \left (a^2-b^2\right )^2 f \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{i b f \text{Li}_2\left (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]  time = 15.2167, size = 2504, normalized size = 3.91 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[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) - (b*e*Log[Sin[c + d
*x]])/(a^2*d) + (b*c*f*Log[Sin[c + d*x]])/(a^2*d^2) + (f*Log[Tan[(c + d*x)/2]])/(a*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) + ((f*(c + d*x)^2 + (2*I)*d*e*Log[Sec[
(c + d*x)/2]^2] - (2*I)*c*f*Log[Sec[(c + d*x)/2]^2] - (2*I)*d*e*Log[Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])] +
(2*I)*c*f*Log[Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])] - (4*I)*f*(c + d*x)*Log[(-2*I)/(-I + Tan[(c + d*x)/2])
] - 2*f*Log[1 + I*Tan[(c + d*x)/2]]*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2
])] + 2*f*Log[1 - I*Tan[(c + d*x)/2]]*Log[-((b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 +
b^2]))] + 2*f*Log[1 - I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[
-a^2 + b^2])] - 2*f*Log[1 + I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqr
t[-a^2 + b^2])] + 4*f*PolyLog[2, -Cos[c + d*x] + I*Sin[c + d*x]] + 2*f*PolyLog[2, (a*(1 - I*Tan[(c + d*x)/2]))
/(a + I*(b + Sqrt[-a^2 + b^2]))] - 2*f*PolyLog[2, (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))]
+ 2*f*PolyLog[2, (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])] - 2*f*PolyLog[2, (a + I*a*Tan[(c +
d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))])*((-2*e*Cos[c + d*x])/(a + b*Sin[c + d*x]) + (a^2*e*Cos[c + d*x])/(b
^2*(a + b*Sin[c + d*x])) + (b^2*e*Cos[c + d*x])/(a^2*(a + b*Sin[c + d*x])) + (2*c*f*Cos[c + d*x])/(d*(a + b*Si
n[c + d*x])) - (a^2*c*f*Cos[c + d*x])/(b^2*d*(a + b*Sin[c + d*x])) - (b^2*c*f*Cos[c + d*x])/(a^2*d*(a + b*Sin[
c + d*x])) - (2*f*(c + d*x)*Cos[c + d*x])/(d*(a + b*Sin[c + d*x])) + (a^2*f*(c + d*x)*Cos[c + d*x])/(b^2*d*(a
+ b*Sin[c + d*x])) + (b^2*f*(c + d*x)*Cos[c + d*x])/(a^2*d*(a + b*Sin[c + d*x]))))/(d*(2*f*(c + d*x) - (4*I)*f
*Log[(-2*I)/(-I + Tan[(c + d*x)/2])] - (4*f*Log[1 + Cos[c + d*x] - I*Sin[c + d*x]]*(I*Cos[c + d*x] + Sin[c + d
*x]))/(-Cos[c + d*x] + I*Sin[c + d*x]) + (I*f*Log[1 - (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2
]))]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) - (I*f*Log[-((b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*
a - b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) - (I*f*Log[(b + Sqrt[-a^2 + b^2] + a*
Tan[(c + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 - I*Tan[(c + d*x)/2]) + (I*f*Log[1 -
(a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) - (
I*f*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])]*Sec[(c + d*x)/2]^2)/(1 + I*T
an[(c + d*x)/2]) - (I*f*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])]*Sec[(c +
d*x)/2]^2)/(1 + I*Tan[(c + d*x)/2]) + (2*I)*d*e*Tan[(c + d*x)/2] - (2*I)*c*f*Tan[(c + d*x)/2] + ((2*I)*f*(c +
d*x)*Sec[(c + d*x)/2]^2)/(-I + Tan[(c + d*x)/2]) - (f*Log[1 - (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2
+ b^2])]*Sec[(c + d*x)/2]^2)/(I + Tan[(c + d*x)/2]) + (I*a*f*Log[1 - (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b +
Sqrt[-a^2 + b^2]))]*Sec[(c + d*x)/2]^2)/(a + I*a*Tan[(c + d*x)/2]) + (a*f*Log[1 - I*Tan[(c + d*x)/2]]*Sec[(c +
d*x)/2]^2)/(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]) - (a*f*Log[1 + I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)
/(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2]) + (a*f*Log[1 - I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(b + Sqrt[
-a^2 + b^2] + a*Tan[(c + d*x)/2]) - (a*f*Log[1 + I*Tan[(c + d*x)/2]]*Sec[(c + d*x)/2]^2)/(b + Sqrt[-a^2 + b^2]
+ a*Tan[(c + d*x)/2]) - ((2*I)*d*e*Cos[(c + d*x)/2]^2*(b*Cos[c + d*x]*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^2
*(a + b*Sin[c + d*x])*Tan[(c + d*x)/2]))/(a + b*Sin[c + d*x]) + ((2*I)*c*f*Cos[(c + d*x)/2]^2*(b*Cos[c + d*x]*
Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^2*(a + b*Sin[c + d*x])*Tan[(c + d*x)/2]))/(a + b*Sin[c + d*x])))

________________________________________________________________________________________

Maple [B]  time = 2.569, size = 2485, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b^3/d*a^2*e*ln(I*b*exp(2*I*(d*x+c))-2*a*exp(I*(d*x+c))-I*b)-2/b^3/d*a^2*e*ln(exp(I*(d*x+c)))-1/b^3/d*a^4*f/(
-a^2+b^2)*ln((I*a+b*exp(I*(d*x+c))-(-a^2+b^2)^(1/2))/(I*a-(-a^2+b^2)^(1/2)))*x-1/b^3/d^2*a^4*f/(-a^2+b^2)*ln((
I*a+b*exp(I*(d*x+c))-(-a^2+b^2)^(1/2))/(I*a-(-a^2+b^2)^(1/2)))*c-1/b^3/d*a^4*f/(-a^2+b^2)*ln((I*a+b*exp(I*(d*x
+c))+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))*x-1/b^3/d^2*a^4*f/(-a^2+b^2)*ln((I*a+b*exp(I*(d*x+c))+(-a^2+b^2
)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))*c-2*I*(f*x+e)*exp(I*(d*x+c))/d/a/(exp(2*I*(d*x+c))-1)+2/b/d^2*f*c*ln(I*b*exp(
2*I*(d*x+c))-2*a*exp(I*(d*x+c))-I*b)-4/b/d^2*f*c*ln(exp(I*(d*x+c)))+1/d*b/a^2*e*ln(I*b*exp(2*I*(d*x+c))-2*a*ex
p(I*(d*x+c))-I*b)+I/b^3*a^2*e*x-3*b/d*f/(-a^2+b^2)*ln((I*a+b*exp(I*(d*x+c))-(-a^2+b^2)^(1/2))/(I*a-(-a^2+b^2)^
(1/2)))*x-3*b/d^2*f/(-a^2+b^2)*ln((I*a+b*exp(I*(d*x+c))-(-a^2+b^2)^(1/2))/(I*a-(-a^2+b^2)^(1/2)))*c-3*b/d*f/(-
a^2+b^2)*ln((I*a+b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))*x-3*b/d^2*f/(-a^2+b^2)*ln((I*a+b*e
xp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))*c-2/b/d*e*ln(I*b*exp(2*I*(d*x+c))-2*a*exp(I*(d*x+c))-I
*b)+4/b/d*ln(exp(I*(d*x+c)))*e+1/d^2/a^2*b*f*c*ln(exp(I*(d*x+c))-1)-1/d/a^2*b*f*ln(exp(I*(d*x+c))+1)*x-I/d^2/a
^2*b*f*dilog(exp(I*(d*x+c)))+3/b/d*f/(-a^2+b^2)*ln((I*a+b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/
2)))*a^2*x+3/b/d^2*f/(-a^2+b^2)*ln((I*a+b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))*a^2*c+3/b/d
*f/(-a^2+b^2)*ln((I*a+b*exp(I*(d*x+c))-(-a^2+b^2)^(1/2))/(I*a-(-a^2+b^2)^(1/2)))*a^2*x+3/b/d^2*f/(-a^2+b^2)*ln
((I*a+b*exp(I*(d*x+c))-(-a^2+b^2)^(1/2))/(I*a-(-a^2+b^2)^(1/2)))*a^2*c-1/d^2*b/a^2*f*c*ln(I*b*exp(2*I*(d*x+c))
-2*a*exp(I*(d*x+c))-I*b)+I/d^2*b/a^2*f*dilog(exp(I*(d*x+c))+1)-1/d/a^2*b*e*ln(exp(I*(d*x+c))-1)-1/d/a^2*b*e*ln
(exp(I*(d*x+c))+1)+2/b^3/d^2*a^2*f*c*ln(exp(I*(d*x+c)))-1/b^3/d^2*a^2*f*c*ln(I*b*exp(2*I*(d*x+c))-2*a*exp(I*(d
*x+c))-I*b)+1/2*I*a*(d*f*x+I*f+d*e)/b^2/d^2*exp(I*(d*x+c))+1/d^2/a*f*ln(exp(I*(d*x+c))-1)-1/d^2/a*f*ln(exp(I*(
d*x+c))+1)-I/b^3/d^2*a^2*f*c^2+3*I/d^2*b*f/(-a^2+b^2)*dilog((I*a+b*exp(I*(d*x+c))-(-a^2+b^2)^(1/2))/(I*a-(-a^2
+b^2)^(1/2)))+3*I/d^2*b*f/(-a^2+b^2)*dilog((I*a+b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))+4*I
/d/b*c*f*x-1/16*(2*d*f*x+I*f+2*d*e)/b/d^2*exp(2*I*(d*x+c))-1/16*(2*d*f*x-I*f+2*d*e)/b/d^2*exp(-2*I*(d*x+c))+I/
b*f*x^2-2*I/b*e*x+1/d^2*b^3/a^2*f/(-a^2+b^2)*ln((I*a+b*exp(I*(d*x+c))-(-a^2+b^2)^(1/2))/(I*a-(-a^2+b^2)^(1/2))
)*c+1/d^2*b^3/a^2*f/(-a^2+b^2)*ln((I*a+b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))*c+1/d*b^3/a^
2*f/(-a^2+b^2)*ln((I*a+b*exp(I*(d*x+c))-(-a^2+b^2)^(1/2))/(I*a-(-a^2+b^2)^(1/2)))*x+1/d*b^3/a^2*f/(-a^2+b^2)*l
n((I*a+b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))*x-I/d^2*b^3/a^2*f/(-a^2+b^2)*dilog((I*a+b*ex
p(I*(d*x+c))-(-a^2+b^2)^(1/2))/(I*a-(-a^2+b^2)^(1/2)))-I/d^2*b^3/a^2*f/(-a^2+b^2)*dilog((I*a+b*exp(I*(d*x+c))+
(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))+I/d^2/b^3*a^4*f/(-a^2+b^2)*dilog((I*a+b*exp(I*(d*x+c))-(-a^2+b^2)^(1
/2))/(I*a-(-a^2+b^2)^(1/2)))+I/d^2/b^3*a^4*f/(-a^2+b^2)*dilog((I*a+b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(I*a+(-a
^2+b^2)^(1/2)))-3*I/b/d^2*f/(-a^2+b^2)*dilog((I*a+b*exp(I*(d*x+c))-(-a^2+b^2)^(1/2))/(I*a-(-a^2+b^2)^(1/2)))*a
^2-3*I/b/d^2*f/(-a^2+b^2)*dilog((I*a+b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))*a^2-2*I/d/b^3*
a^2*c*f*x-1/2*I/b^3*a^2*f*x^2-1/2*I*a*(d*f*x-I*f+d*e)/b^2/d^2*exp(-I*(d*x+c))+2*I/d^2/b*c^2*f

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

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 4.42835, size = 4313, normalized size = 6.73 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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(-1/2*(2*I*a*cos(d*x + c) + 2*a*sin(d*x + c) + 2*(b*cos(d*x + c) - I*b*
sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b + 1)*sin(d*x + c) - 2*I*(a^4 - 2*a^2*b^2 + b^4)*f*dilog(-1/2*(2*
I*a*cos(d*x + c) + 2*a*sin(d*x + c) - 2*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b +
1)*sin(d*x + c) + 2*I*(a^4 - 2*a^2*b^2 + b^4)*f*dilog(-1/2*(-2*I*a*cos(d*x + c) + 2*a*sin(d*x + c) + 2*(b*cos(
d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b + 1)*sin(d*x + c) + 2*I*(a^4 - 2*a^2*b^2 + b^4)*f
*dilog(-1/2*(-2*I*a*cos(d*x + c) + 2*a*sin(d*x + c) - 2*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/
b^2) + 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(1/2*(2*I*a*cos(d*x + c) + 2*a*sin(d*x + c) + 2*(b*cos(d*x + c) - I*b*sin(d*x +
c))*sqrt(-(a^2 - b^2)/b^2) + 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(1/2*(2*I*a*cos(d*x + c) + 2*a*sin(d*x + c) - 2*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)
/b^2) + 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(1/2*(-2*I*a
*cos(d*x + c) + 2*a*sin(d*x + c) + 2*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 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(1/2*(-2*I*a*cos(d*x + c) + 2*a*
sin(d*x + c) - 2*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 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)*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)*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))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

Timed out