3.3.51 \(\int \frac {(e+f x)^3 \cos (c+d x)}{a+a \sin (c+d x)} \, dx\) [251]

Optimal. Leaf size=151 \[ -\frac {i (e+f x)^4}{4 a f}+\frac {2 (e+f x)^3 \log \left (1-i e^{i (c+d x)}\right )}{a d}-\frac {6 i f (e+f x)^2 \text {Li}_2\left (i e^{i (c+d x)}\right )}{a d^2}+\frac {12 f^2 (e+f x) \text {Li}_3\left (i e^{i (c+d x)}\right )}{a d^3}+\frac {12 i f^3 \text {Li}_4\left (i e^{i (c+d x)}\right )}{a d^4} \]

[Out]

-1/4*I*(f*x+e)^4/a/f+2*(f*x+e)^3*ln(1-I*exp(I*(d*x+c)))/a/d-6*I*f*(f*x+e)^2*polylog(2,I*exp(I*(d*x+c)))/a/d^2+
12*f^2*(f*x+e)*polylog(3,I*exp(I*(d*x+c)))/a/d^3+12*I*f^3*polylog(4,I*exp(I*(d*x+c)))/a/d^4

________________________________________________________________________________________

Rubi [A]
time = 0.15, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {4613, 2221, 2611, 6744, 2320, 6724} \begin {gather*} \frac {12 i f^3 \text {PolyLog}\left (4,i e^{i (c+d x)}\right )}{a d^4}+\frac {12 f^2 (e+f x) \text {PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^3}-\frac {6 i f (e+f x)^2 \text {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^2}+\frac {2 (e+f x)^3 \log \left (1-i e^{i (c+d x)}\right )}{a d}-\frac {i (e+f x)^4}{4 a f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1/4*I)*(e + f*x)^4)/(a*f) + (2*(e + f*x)^3*Log[1 - I*E^(I*(c + d*x))])/(a*d) - ((6*I)*f*(e + f*x)^2*PolyLog
[2, I*E^(I*(c + d*x))])/(a*d^2) + (12*f^2*(e + f*x)*PolyLog[3, I*E^(I*(c + d*x))])/(a*d^3) + ((12*I)*f^3*PolyL
og[4, I*E^(I*(c + d*x))])/(a*d^4)

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 4613

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] + Dist[2, Int[(e + f*x)^m*(E^(I*(c + d*x))/(a - I*b*E^(I*(c +
d*x)))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && EqQ[a^2 - b^2, 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 (c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac {i (e+f x)^4}{4 a f}+2 \int \frac {e^{i (c+d x)} (e+f x)^3}{a-i a e^{i (c+d x)}} \, dx\\ &=-\frac {i (e+f x)^4}{4 a f}+\frac {2 (e+f x)^3 \log \left (1-i e^{i (c+d x)}\right )}{a d}-\frac {(6 f) \int (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right ) \, dx}{a d}\\ &=-\frac {i (e+f x)^4}{4 a f}+\frac {2 (e+f x)^3 \log \left (1-i e^{i (c+d x)}\right )}{a d}-\frac {6 i f (e+f x)^2 \text {Li}_2\left (i e^{i (c+d x)}\right )}{a d^2}+\frac {\left (12 i f^2\right ) \int (e+f x) \text {Li}_2\left (i e^{i (c+d x)}\right ) \, dx}{a d^2}\\ &=-\frac {i (e+f x)^4}{4 a f}+\frac {2 (e+f x)^3 \log \left (1-i e^{i (c+d x)}\right )}{a d}-\frac {6 i f (e+f x)^2 \text {Li}_2\left (i e^{i (c+d x)}\right )}{a d^2}+\frac {12 f^2 (e+f x) \text {Li}_3\left (i e^{i (c+d x)}\right )}{a d^3}-\frac {\left (12 f^3\right ) \int \text {Li}_3\left (i e^{i (c+d x)}\right ) \, dx}{a d^3}\\ &=-\frac {i (e+f x)^4}{4 a f}+\frac {2 (e+f x)^3 \log \left (1-i e^{i (c+d x)}\right )}{a d}-\frac {6 i f (e+f x)^2 \text {Li}_2\left (i e^{i (c+d x)}\right )}{a d^2}+\frac {12 f^2 (e+f x) \text {Li}_3\left (i e^{i (c+d x)}\right )}{a d^3}+\frac {\left (12 i f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}\\ &=-\frac {i (e+f x)^4}{4 a f}+\frac {2 (e+f x)^3 \log \left (1-i e^{i (c+d x)}\right )}{a d}-\frac {6 i f (e+f x)^2 \text {Li}_2\left (i e^{i (c+d x)}\right )}{a d^2}+\frac {12 f^2 (e+f x) \text {Li}_3\left (i e^{i (c+d x)}\right )}{a d^3}+\frac {12 i f^3 \text {Li}_4\left (i e^{i (c+d x)}\right )}{a d^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.05, size = 275, normalized size = 1.82 \begin {gather*} -\frac {i \left (4 d^4 e^3 x+6 d^4 e^2 f x^2+4 d^4 e f^2 x^3+d^4 f^3 x^4+8 i d^3 e^3 \log (1-i \cos (c+d x)+\sin (c+d x))+24 i d^3 e^2 f x \log (1-i \cos (c+d x)+\sin (c+d x))+24 i d^3 e f^2 x^2 \log (1-i \cos (c+d x)+\sin (c+d x))+8 i d^3 f^3 x^3 \log (1-i \cos (c+d x)+\sin (c+d x))+24 d^2 f (e+f x)^2 \text {Li}_2(i \cos (c+d x)-\sin (c+d x))+48 i d f^2 (e+f x) \text {Li}_3(i \cos (c+d x)-\sin (c+d x))-48 f^3 \text {Li}_4(i \cos (c+d x)-\sin (c+d x))\right )}{4 a d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1/4*I)*(4*d^4*e^3*x + 6*d^4*e^2*f*x^2 + 4*d^4*e*f^2*x^3 + d^4*f^3*x^4 + (8*I)*d^3*e^3*Log[1 - I*Cos[c + d*x
] + Sin[c + d*x]] + (24*I)*d^3*e^2*f*x*Log[1 - I*Cos[c + d*x] + Sin[c + d*x]] + (24*I)*d^3*e*f^2*x^2*Log[1 - I
*Cos[c + d*x] + Sin[c + d*x]] + (8*I)*d^3*f^3*x^3*Log[1 - I*Cos[c + d*x] + Sin[c + d*x]] + 24*d^2*f*(e + f*x)^
2*PolyLog[2, I*Cos[c + d*x] - Sin[c + d*x]] + (48*I)*d*f^2*(e + f*x)*PolyLog[3, I*Cos[c + d*x] - Sin[c + d*x]]
- 48*f^3*PolyLog[4, I*Cos[c + d*x] - Sin[c + d*x]]))/(a*d^4)

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 690 vs. \(2 (134 ) = 268\).
time = 0.18, size = 691, normalized size = 4.58

method	result	size
risch	\(\frac {6 i e \,f^{2} c^{2} x}{d^{2} a}-\frac {12 i e \,f^{2} \polylog \left (2, i {\mathrm e}^{i \left (d x +c \right )}\right ) x}{d^{2} a}+\frac {12 e \,f^{2} \polylog \left (3, i {\mathrm e}^{i \left (d x +c \right )}\right )}{d^{3} a}-\frac {i f^{2} e \,x^{3}}{a}-\frac {3 i f \,e^{2} x^{2}}{2 a}+\frac {12 i f^{3} \polylog \left (4, i {\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{4}}+\frac {6 e \,f^{2} \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) x^{2}}{d a}-\frac {6 e \,f^{2} \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) c^{2}}{d^{3} a}-\frac {6 i e^{2} f \polylog \left (2, i {\mathrm e}^{i \left (d x +c \right )}\right )}{d^{2} a}-\frac {6 i e^{2} f c x}{d a}+\frac {2 f^{3} \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) x^{3}}{d a}+\frac {2 f^{3} \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) c^{3}}{d^{4} a}+\frac {2 f^{3} c^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{d^{4} a}-\frac {2 f^{3} c^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d^{4} a}-\frac {3 i f^{3} c^{4}}{2 d^{4} a}+\frac {12 f^{3} \polylog \left (3, i {\mathrm e}^{i \left (d x +c \right )}\right ) x}{d^{3} a}-\frac {6 i f^{3} \polylog \left (2, i {\mathrm e}^{i \left (d x +c \right )}\right ) x^{2}}{d^{2} a}+\frac {6 e^{2} f \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) x}{d a}+\frac {6 e^{2} f \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) c}{d^{2} a}-\frac {3 i e^{2} f \,c^{2}}{d^{2} a}-\frac {2 i f^{3} c^{3} x}{d^{3} a}+\frac {4 i e \,f^{2} c^{3}}{d^{3} a}-\frac {6 e^{2} f c \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d^{2} a}+\frac {6 e \,f^{2} c^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d^{3} a}-\frac {6 e \,f^{2} c^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{d^{3} a}+\frac {6 e^{2} f c \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{d^{2} a}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) e^{3}}{d a}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right ) e^{3}}{d a}-\frac {i f^{3} x^{4}}{4 a}+\frac {i e^{3} x}{a}+\frac {i e^{4}}{4 f a}\)	\(691\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*f^2/a*e*x^3-3/2*I*f/a*e^2*x^2+2/d/a*ln(exp(I*(d*x+c))+I)*e^3-2/d/a*ln(exp(I*(d*x+c)))*e^3-1/4*I*f^3/a*x^4+6
/d/a*e*f^2*ln(1-I*exp(I*(d*x+c)))*x^2-6/d^3/a*e*f^2*ln(1-I*exp(I*(d*x+c)))*c^2-6*I/d/a*e^2*f*c*x+6*I/d^2/a*e*f
^2*c^2*x-12*I/d^2/a*e*f^2*polylog(2,I*exp(I*(d*x+c)))*x+2/d/a*f^3*ln(1-I*exp(I*(d*x+c)))*x^3+2/d^4/a*f^3*ln(1-
I*exp(I*(d*x+c)))*c^3+6/d/a*e^2*f*ln(1-I*exp(I*(d*x+c)))*x+6/d^2/a*e^2*f*ln(1-I*exp(I*(d*x+c)))*c-6*I/d^2/a*e^
2*f*polylog(2,I*exp(I*(d*x+c)))-3*I/d^2/a*e^2*f*c^2-2*I/d^3/a*f^3*c^3*x+4*I/d^3/a*e*f^2*c^3-6*I/d^2/a*f^3*poly
log(2,I*exp(I*(d*x+c)))*x^2-6/d^2/a*e^2*f*c*ln(exp(I*(d*x+c))+I)+6/d^3/a*e*f^2*c^2*ln(exp(I*(d*x+c))+I)-6/d^3/
a*e*f^2*c^2*ln(exp(I*(d*x+c)))+6/d^2/a*e^2*f*c*ln(exp(I*(d*x+c)))+I/a*e^3*x+1/4*I/f/a*e^4+12/d^3/a*e*f^2*polyl
og(3,I*exp(I*(d*x+c)))+2/d^4/a*f^3*c^3*ln(exp(I*(d*x+c)))-2/d^4/a*f^3*c^3*ln(exp(I*(d*x+c))+I)+12/d^3/a*f^3*po
lylog(3,I*exp(I*(d*x+c)))*x-3/2*I/d^4/a*f^3*c^4+12*I*f^3*polylog(4,I*exp(I*(d*x+c)))/a/d^4

________________________________________________________________________________________

Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 519 vs. \(2 (126) = 252\).
time = 0.35, size = 519, normalized size = 3.44 \begin {gather*} -\frac {\frac {12 \, c e^{2} f \log \left (a d \sin \left (d x + c\right ) + a d\right )}{a d} - \frac {4 \, e^{3} \log \left (a \sin \left (d x + c\right ) + a\right )}{a} - \frac {-i \, {\left (d x + c\right )}^{4} f^{3} - 4 \, {\left (i \, d e f^{2} - i \, c f^{3}\right )} {\left (d x + c\right )}^{3} + 48 i \, f^{3} {\rm Li}_{4}(i \, e^{\left (i \, d x + i \, c\right )}) - 6 \, {\left (i \, d^{2} e^{2} f - 2 i \, c d e f^{2} + i \, c^{2} f^{3}\right )} {\left (d x + c\right )}^{2} - 4 \, {\left (3 i \, c^{2} d e f^{2} - i \, c^{3} f^{3}\right )} {\left (d x + c\right )} - 8 \, {\left (-3 i \, c^{2} d e f^{2} + i \, c^{3} f^{3}\right )} \arctan \left (\sin \left (d x + c\right ) + 1, \cos \left (d x + c\right )\right ) - 8 \, {\left (i \, {\left (d x + c\right )}^{3} f^{3} + 3 \, {\left (i \, d e f^{2} - i \, c f^{3}\right )} {\left (d x + c\right )}^{2} + 3 \, {\left (i \, d^{2} e^{2} f - 2 i \, c d e f^{2} + i \, c^{2} f^{3}\right )} {\left (d x + c\right )}\right )} \arctan \left (\cos \left (d x + c\right ), \sin \left (d x + c\right ) + 1\right ) - 24 \, {\left (i \, d^{2} e^{2} f - 2 i \, c d e f^{2} + i \, {\left (d x + c\right )}^{2} f^{3} + i \, c^{2} f^{3} + 2 \, {\left (i \, d e f^{2} - i \, c f^{3}\right )} {\left (d x + c\right )}\right )} {\rm Li}_2\left (i \, e^{\left (i \, d x + i \, c\right )}\right ) + 4 \, {\left (3 \, c^{2} d e f^{2} + {\left (d x + c\right )}^{3} f^{3} - c^{3} f^{3} + 3 \, {\left (d e f^{2} - c f^{3}\right )} {\left (d x + c\right )}^{2} + 3 \, {\left (d^{2} e^{2} f - 2 \, c d e f^{2} + c^{2} f^{3}\right )} {\left (d x + c\right )}\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right ) + 48 \, {\left (d e f^{2} + {\left (d x + c\right )} f^{3} - c f^{3}\right )} {\rm Li}_{3}(i \, e^{\left (i \, d x + i \, c\right )})}{a d^{3}}}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(12*c*e^2*f*log(a*d*sin(d*x + c) + a*d)/(a*d) - 4*e^3*log(a*sin(d*x + c) + a)/a - (-I*(d*x + c)^4*f^3 - 4
*(I*d*e*f^2 - I*c*f^3)*(d*x + c)^3 + 48*I*f^3*polylog(4, I*e^(I*d*x + I*c)) - 6*(I*d^2*e^2*f - 2*I*c*d*e*f^2 +
I*c^2*f^3)*(d*x + c)^2 - 4*(3*I*c^2*d*e*f^2 - I*c^3*f^3)*(d*x + c) - 8*(-3*I*c^2*d*e*f^2 + I*c^3*f^3)*arctan2
(sin(d*x + c) + 1, cos(d*x + c)) - 8*(I*(d*x + c)^3*f^3 + 3*(I*d*e*f^2 - I*c*f^3)*(d*x + c)^2 + 3*(I*d^2*e^2*f
- 2*I*c*d*e*f^2 + I*c^2*f^3)*(d*x + c))*arctan2(cos(d*x + c), sin(d*x + c) + 1) - 24*(I*d^2*e^2*f - 2*I*c*d*e
*f^2 + I*(d*x + c)^2*f^3 + I*c^2*f^3 + 2*(I*d*e*f^2 - I*c*f^3)*(d*x + c))*dilog(I*e^(I*d*x + I*c)) + 4*(3*c^2*
d*e*f^2 + (d*x + c)^3*f^3 - c^3*f^3 + 3*(d*e*f^2 - c*f^3)*(d*x + c)^2 + 3*(d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*
(d*x + c))*log(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*sin(d*x + c) + 1) + 48*(d*e*f^2 + (d*x + c)*f^3 - c*f^3)*po
lylog(3, I*e^(I*d*x + I*c)))/(a*d^3))/d

________________________________________________________________________________________

Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 492 vs. \(2 (130) = 260\).
time = 0.38, size = 492, normalized size = 3.26 \begin {gather*} \frac {6 i \, f^{3} {\rm polylog}\left (4, i \, \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right ) - 6 i \, f^{3} {\rm polylog}\left (4, -i \, \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right ) - 3 \, {\left (i \, d^{2} f^{3} x^{2} + 2 i \, d^{2} f^{2} x e + i \, d^{2} f e^{2}\right )} {\rm Li}_2\left (i \, \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right ) - 3 \, {\left (-i \, d^{2} f^{3} x^{2} - 2 i \, d^{2} f^{2} x e - i \, d^{2} f e^{2}\right )} {\rm Li}_2\left (-i \, \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right ) - {\left (c^{3} f^{3} - 3 \, c^{2} d f^{2} e + 3 \, c d^{2} f e^{2} - d^{3} e^{3}\right )} \log \left (\cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + i\right ) + {\left (d^{3} f^{3} x^{3} + c^{3} f^{3} + 3 \, {\left (d^{3} f x + c d^{2} f\right )} e^{2} + 3 \, {\left (d^{3} f^{2} x^{2} - c^{2} d f^{2}\right )} e\right )} \log \left (i \, \cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right ) + {\left (d^{3} f^{3} x^{3} + c^{3} f^{3} + 3 \, {\left (d^{3} f x + c d^{2} f\right )} e^{2} + 3 \, {\left (d^{3} f^{2} x^{2} - c^{2} d f^{2}\right )} e\right )} \log \left (-i \, \cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right ) - {\left (c^{3} f^{3} - 3 \, c^{2} d f^{2} e + 3 \, c d^{2} f e^{2} - d^{3} e^{3}\right )} \log \left (-\cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + i\right ) + 6 \, {\left (d f^{3} x + d f^{2} e\right )} {\rm polylog}\left (3, i \, \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right ) + 6 \, {\left (d f^{3} x + d f^{2} e\right )} {\rm polylog}\left (3, -i \, \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )}{a d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6*I*f^3*polylog(4, I*cos(d*x + c) - sin(d*x + c)) - 6*I*f^3*polylog(4, -I*cos(d*x + c) - sin(d*x + c)) - 3*(I
*d^2*f^3*x^2 + 2*I*d^2*f^2*x*e + I*d^2*f*e^2)*dilog(I*cos(d*x + c) - sin(d*x + c)) - 3*(-I*d^2*f^3*x^2 - 2*I*d
^2*f^2*x*e - I*d^2*f*e^2)*dilog(-I*cos(d*x + c) - sin(d*x + c)) - (c^3*f^3 - 3*c^2*d*f^2*e + 3*c*d^2*f*e^2 - d
^3*e^3)*log(cos(d*x + c) + I*sin(d*x + c) + I) + (d^3*f^3*x^3 + c^3*f^3 + 3*(d^3*f*x + c*d^2*f)*e^2 + 3*(d^3*f
^2*x^2 - c^2*d*f^2)*e)*log(I*cos(d*x + c) + sin(d*x + c) + 1) + (d^3*f^3*x^3 + c^3*f^3 + 3*(d^3*f*x + c*d^2*f)
*e^2 + 3*(d^3*f^2*x^2 - c^2*d*f^2)*e)*log(-I*cos(d*x + c) + sin(d*x + c) + 1) - (c^3*f^3 - 3*c^2*d*f^2*e + 3*c
*d^2*f*e^2 - d^3*e^3)*log(-cos(d*x + c) + I*sin(d*x + c) + I) + 6*(d*f^3*x + d*f^2*e)*polylog(3, I*cos(d*x + c
) - sin(d*x + c)) + 6*(d*f^3*x + d*f^2*e)*polylog(3, -I*cos(d*x + c) - sin(d*x + c)))/(a*d^4)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {e^{3} \cos {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f^{3} x^{3} \cos {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e f^{2} x^{2} \cos {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e^{2} f x \cos {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(e**3*cos(c + d*x)/(sin(c + d*x) + 1), x) + Integral(f**3*x**3*cos(c + d*x)/(sin(c + d*x) + 1), x) +
Integral(3*e*f**2*x**2*cos(c + d*x)/(sin(c + d*x) + 1), x) + Integral(3*e**2*f*x*cos(c + d*x)/(sin(c + d*x) +
1), x))/a

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\cos \left (c+d\,x\right )\,{\left (e+f\,x\right )}^3}{a+a\,\sin \left (c+d\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________