3.2.38 \(\int \frac {(c+d x)^3}{a-a \cos (e+f x)} \, dx\) [138]

Optimal. Leaf size=133 \[ -\frac {i (c+d x)^3}{a f}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {6 d (c+d x)^2 \log \left (1-e^{i (e+f x)}\right )}{a f^2}-\frac {12 i d^2 (c+d x) \text {PolyLog}\left (2,e^{i (e+f x)}\right )}{a f^3}+\frac {12 d^3 \text {PolyLog}\left (3,e^{i (e+f x)}\right )}{a f^4} \]

[Out]

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

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Rubi [A]
time = 0.18, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3399, 4269, 3798, 2221, 2611, 2320, 6724} \begin {gather*} -\frac {12 i d^2 (c+d x) \text {Li}_2\left (e^{i (e+f x)}\right )}{a f^3}+\frac {6 d (c+d x)^2 \log \left (1-e^{i (e+f x)}\right )}{a f^2}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}-\frac {i (c+d x)^3}{a f}+\frac {12 d^3 \text {Li}_3\left (e^{i (e+f x)}\right )}{a f^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*(c + d*x)^3)/(a*f) - ((c + d*x)^3*Cot[e/2 + (f*x)/2])/(a*f) + (6*d*(c + d*x)^2*Log[1 - E^(I*(e + f*x))])
/(a*f^2) - ((12*I)*d^2*(c + d*x)*PolyLog[2, E^(I*(e + f*x))])/(a*f^3) + (12*d^3*PolyLog[3, E^(I*(e + f*x))])/(
a*f^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 3399

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

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 4269

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

Rubi steps

\begin {align*} \int \frac {(c+d x)^3}{a-a \cos (e+f x)} \, dx &=\frac {\int (c+d x)^3 \csc ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{2 a}\\ &=-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {(3 d) \int (c+d x)^2 \cot \left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{a f}\\ &=-\frac {i (c+d x)^3}{a f}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}-\frac {(6 i d) \int \frac {e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )} (c+d x)^2}{1-e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}} \, dx}{a f}\\ &=-\frac {i (c+d x)^3}{a f}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {6 d (c+d x)^2 \log \left (1-e^{i (e+f x)}\right )}{a f^2}-\frac {\left (12 d^2\right ) \int (c+d x) \log \left (1-e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a f^2}\\ &=-\frac {i (c+d x)^3}{a f}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {6 d (c+d x)^2 \log \left (1-e^{i (e+f x)}\right )}{a f^2}-\frac {12 i d^2 (c+d x) \text {Li}_2\left (e^{i (e+f x)}\right )}{a f^3}+\frac {\left (12 i d^3\right ) \int \text {Li}_2\left (e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a f^3}\\ &=-\frac {i (c+d x)^3}{a f}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {6 d (c+d x)^2 \log \left (1-e^{i (e+f x)}\right )}{a f^2}-\frac {12 i d^2 (c+d x) \text {Li}_2\left (e^{i (e+f x)}\right )}{a f^3}+\frac {\left (12 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right )}{a f^4}\\ &=-\frac {i (c+d x)^3}{a f}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {6 d (c+d x)^2 \log \left (1-e^{i (e+f x)}\right )}{a f^2}-\frac {12 i d^2 (c+d x) \text {Li}_2\left (e^{i (e+f x)}\right )}{a f^3}+\frac {12 d^3 \text {Li}_3\left (e^{i (e+f x)}\right )}{a f^4}\\ \end {align*}

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Mathematica [A]
time = 0.77, size = 198, normalized size = 1.49 \begin {gather*} \frac {2 \sin \left (\frac {1}{2} (e+f x)\right ) \left (f^3 (c+d x)^3 \csc \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right )+2 d \left (3 f^2 (c+d x)^2 \log (1-\cos (e+f x)-i \sin (e+f x))-6 i d f (c+d x) \text {PolyLog}(2,\cos (e+f x)+i \sin (e+f x))+6 d^2 \text {PolyLog}(3,\cos (e+f x)+i \sin (e+f x))+\frac {f^3 x \left (3 c^2+3 c d x+d^2 x^2\right ) (-i \cos (e)+\sin (e))}{-1+\cos (e)+i \sin (e)}\right ) \sin \left (\frac {1}{2} (e+f x)\right )\right )}{f^4 (a-a \cos (e+f x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 467 vs. \(2 (121 ) = 242\).
time = 0.14, size = 468, normalized size = 3.52

method	result	size
risch	\(-\frac {2 i \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )}{f a \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}-\frac {6 d \,c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{2}}+\frac {6 d \,c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{a \,f^{2}}+\frac {6 d^{3} e^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{a \,f^{4}}-\frac {6 d^{3} e^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{4}}-\frac {12 i d^{3} \polylog \left (2, {\mathrm e}^{i \left (f x +e \right )}\right ) x}{a \,f^{3}}+\frac {6 i d^{3} e^{2} x}{a \,f^{3}}+\frac {4 i d^{3} e^{3}}{a \,f^{4}}-\frac {12 i d^{2} c e x}{a \,f^{2}}+\frac {6 d^{3} \ln \left (1-{\mathrm e}^{i \left (f x +e \right )}\right ) x^{2}}{a \,f^{2}}-\frac {6 d^{3} \ln \left (1-{\mathrm e}^{i \left (f x +e \right )}\right ) e^{2}}{a \,f^{4}}-\frac {6 i d^{2} c \,e^{2}}{a \,f^{3}}+\frac {12 d^{3} \polylog \left (3, {\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{4}}-\frac {12 d^{2} c e \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{a \,f^{3}}+\frac {12 d^{2} c e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{3}}-\frac {12 i d^{2} c \polylog \left (2, {\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{3}}-\frac {6 i d^{2} c \,x^{2}}{a f}-\frac {2 i d^{3} x^{3}}{a f}+\frac {12 d^{2} c \ln \left (1-{\mathrm e}^{i \left (f x +e \right )}\right ) x}{a \,f^{2}}+\frac {12 d^{2} c \ln \left (1-{\mathrm e}^{i \left (f x +e \right )}\right ) e}{a \,f^{3}}\)	\(468\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1042 vs. \(2 (122) = 244\).
time = 0.39, size = 1042, normalized size = 7.83 \begin {gather*} -\frac {\frac {6 \, {\left ({\left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) + 1\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) + 1\right ) - 2 \, {\left (f x + e\right )} \sin \left (f x + e\right )\right )} c d^{2} e}{a f^{2} \cos \left (f x + e\right )^{2} + a f^{2} \sin \left (f x + e\right )^{2} - 2 \, a f^{2} \cos \left (f x + e\right ) + a f^{2}} - \frac {3 \, {\left ({\left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) + 1\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) + 1\right ) - 2 \, {\left (f x + e\right )} \sin \left (f x + e\right )\right )} c^{2} d}{a f \cos \left (f x + e\right )^{2} + a f \sin \left (f x + e\right )^{2} - 2 \, a f \cos \left (f x + e\right ) + a f} + \frac {c^{3} {\left (\cos \left (f x + e\right ) + 1\right )}}{a \sin \left (f x + e\right )} - \frac {3 \, c^{2} d {\left (\cos \left (f x + e\right ) + 1\right )} e}{a f \sin \left (f x + e\right )} + \frac {3 \, c d^{2} {\left (\cos \left (f x + e\right ) + 1\right )} e^{2}}{a f^{2} \sin \left (f x + e\right )} - \frac {2 \, d^{3} e^{3} + 6 \, {\left (d^{3} \cos \left (f x + e\right ) e^{2} + i \, d^{3} e^{2} \sin \left (f x + e\right ) - d^{3} e^{2}\right )} \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right ) - 1\right ) + 6 \, {\left ({\left (f x + e\right )}^{2} d^{3} + 2 \, {\left (c d^{2} f - d^{3} e\right )} {\left (f x + e\right )} - {\left ({\left (f x + e\right )}^{2} d^{3} + 2 \, {\left (c d^{2} f - d^{3} e\right )} {\left (f x + e\right )}\right )} \cos \left (f x + e\right ) - {\left (i \, {\left (f x + e\right )}^{2} d^{3} + 2 \, {\left (i \, c d^{2} f - i \, d^{3} e\right )} {\left (f x + e\right )}\right )} \sin \left (f x + e\right )\right )} \arctan \left (\sin \left (f x + e\right ), -\cos \left (f x + e\right ) + 1\right ) - 2 \, {\left ({\left (f x + e\right )}^{3} d^{3} + 3 \, {\left (f x + e\right )} d^{3} e^{2} + 3 \, {\left (c d^{2} f - d^{3} e\right )} {\left (f x + e\right )}^{2}\right )} \cos \left (f x + e\right ) + 12 \, {\left ({\left (f x + e\right )} d^{3} + c d^{2} f - d^{3} e - {\left ({\left (f x + e\right )} d^{3} + c d^{2} f - d^{3} e\right )} \cos \left (f x + e\right ) - {\left (i \, {\left (f x + e\right )} d^{3} + i \, c d^{2} f - i \, d^{3} e\right )} \sin \left (f x + e\right )\right )} {\rm Li}_2\left (e^{\left (i \, f x + i \, e\right )}\right ) - 3 \, {\left (-i \, {\left (f x + e\right )}^{2} d^{3} - i \, d^{3} e^{2} + 2 \, {\left (-i \, c d^{2} f + i \, d^{3} e\right )} {\left (f x + e\right )} + {\left (i \, {\left (f x + e\right )}^{2} d^{3} + i \, d^{3} e^{2} + 2 \, {\left (i \, c d^{2} f - i \, d^{3} e\right )} {\left (f x + e\right )}\right )} \cos \left (f x + e\right ) - {\left ({\left (f x + e\right )}^{2} d^{3} + d^{3} e^{2} + 2 \, {\left (c d^{2} f - d^{3} e\right )} {\left (f x + e\right )}\right )} \sin \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) + 1\right ) - 12 \, {\left (i \, d^{3} \cos \left (f x + e\right ) - d^{3} \sin \left (f x + e\right ) - i \, d^{3}\right )} {\rm Li}_{3}(e^{\left (i \, f x + i \, e\right )}) - 2 \, {\left (i \, {\left (f x + e\right )}^{3} d^{3} + 3 i \, {\left (f x + e\right )} d^{3} e^{2} + 3 \, {\left (i \, c d^{2} f - i \, d^{3} e\right )} {\left (f x + e\right )}^{2}\right )} \sin \left (f x + e\right )}{-i \, a f^{3} \cos \left (f x + e\right ) + a f^{3} \sin \left (f x + e\right ) + i \, a f^{3}}}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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. 493 vs. \(2 (122) = 244\).
time = 0.41, size = 493, normalized size = 3.71 \begin {gather*} -\frac {d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2} + 3 \, c^{2} d f^{3} x + c^{3} f^{3} - 6 \, d^{3} {\rm polylog}\left (3, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) \sin \left (f x + e\right ) - 6 \, d^{3} {\rm polylog}\left (3, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) \sin \left (f x + e\right ) + 6 \, {\left (i \, d^{3} f x + i \, c d^{2} f\right )} {\rm Li}_2\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) \sin \left (f x + e\right ) + 6 \, {\left (-i \, d^{3} f x - i \, c d^{2} f\right )} {\rm Li}_2\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) \sin \left (f x + e\right ) - 3 \, {\left (c^{2} d f^{2} - 2 \, c d^{2} f e + d^{3} e^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2} i \, \sin \left (f x + e\right ) + \frac {1}{2}\right ) \sin \left (f x + e\right ) - 3 \, {\left (c^{2} d f^{2} - 2 \, c d^{2} f e + d^{3} e^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) - \frac {1}{2} i \, \sin \left (f x + e\right ) + \frac {1}{2}\right ) \sin \left (f x + e\right ) - 3 \, {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + 2 \, c d^{2} f e - d^{3} e^{2}\right )} \log \left (-\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ) + 1\right ) \sin \left (f x + e\right ) - 3 \, {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + 2 \, c d^{2} f e - d^{3} e^{2}\right )} \log \left (-\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ) + 1\right ) \sin \left (f x + e\right ) + {\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2} + 3 \, c^{2} d f^{3} x + c^{3} f^{3}\right )} \cos \left (f x + e\right )}{a f^{4} \sin \left (f x + e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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