\(\int \frac {(c+d x)^3}{a-a \sin (e+f x)} \, dx\) [117]

Optimal result

Integrand size = 21, antiderivative size = 147 \[ \int \frac {(c+d x)^3}{a-a \sin (e+f x)} \, dx=-\frac {i (c+d x)^3}{a f}+\frac {6 d (c+d x)^2 \log \left (1+i e^{i (e+f x)}\right )}{a f^2}-\frac {12 i d^2 (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{a f^3}+\frac {12 d^3 \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right )}{a f^4}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3399, 4269, 3798, 2221, 2611, 2320, 6724} \[ \int \frac {(c+d x)^3}{a-a \sin (e+f x)} \, dx=-\frac {12 i d^2 (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{a f^3}+\frac {6 d (c+d x)^2 \log \left (1+i e^{i (e+f x)}\right )}{a f^2}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{a f}-\frac {i (c+d x)^3}{a f}+\frac {12 d^3 \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right )}{a f^4} \]

[In]

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

[Out]

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

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*} \text {integral}& = \frac {\int (c+d x)^3 \csc ^2\left (\frac {1}{2} \left (e-\frac {\pi }{2}\right )+\frac {f x}{2}\right ) \, dx}{2 a} \\ & = \frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f}+\frac {(3 d) \int (c+d x)^2 \cot \left (\frac {e}{2}-\frac {\pi }{4}+\frac {f x}{2}\right ) \, dx}{a f} \\ & = -\frac {i (c+d x)^3}{a f}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f}-\frac {(6 d) \int \frac {e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )} (c+d x)^2}{1+i e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}} \, dx}{a f} \\ & = -\frac {i (c+d x)^3}{a f}+\frac {6 d (c+d x)^2 \log \left (1+i e^{i (e+f x)}\right )}{a f^2}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f}-\frac {\left (12 d^2\right ) \int (c+d x) \log \left (1+i 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 {6 d (c+d x)^2 \log \left (1+i e^{i (e+f x)}\right )}{a f^2}-\frac {12 i d^2 (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{a f^3}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f}+\frac {\left (12 i d^3\right ) \int \operatorname {PolyLog}\left (2,-i 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 {6 d (c+d x)^2 \log \left (1+i e^{i (e+f x)}\right )}{a f^2}-\frac {12 i d^2 (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{a f^3}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f}+\frac {\left (12 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i 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 {6 d (c+d x)^2 \log \left (1+i e^{i (e+f x)}\right )}{a f^2}-\frac {12 i d^2 (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{a f^3}+\frac {12 d^3 \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right )}{a f^4}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.84 \[ \int \frac {(c+d x)^3}{a-a \sin (e+f x)} \, dx=\frac {-12 i d^2 f (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )+12 d^3 \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right )+f^2 (c+d x)^2 \left (-i f (c+d x)+6 d \log \left (1+i e^{i (e+f x)}\right )+f (c+d x) \tan \left (\frac {1}{4} (2 e+\pi +2 f x)\right )\right )}{a f^4} \]

[In]

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

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 483 vs. \(2 (130 ) = 260\).

Time = 0.25 (sec) , antiderivative size = 484, normalized size of antiderivative = 3.29

method	result	size
risch	\(\frac {2 d^{3} x^{3}+6 c \,d^{2} x^{2}+6 c^{2} d x +2 c^{3}}{f a \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}-\frac {6 \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right ) c^{2} d}{a \,f^{2}}+\frac {12 e c \,d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{3}}+\frac {12 c \,d^{2} \ln \left (1+i {\mathrm e}^{i \left (f x +e \right )}\right ) e}{a \,f^{3}}+\frac {6 e^{2} d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{a \,f^{4}}-\frac {6 e^{2} d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{4}}-\frac {12 i c \,d^{2} e x}{a \,f^{2}}+\frac {12 c \,d^{2} \ln \left (1+i {\mathrm e}^{i \left (f x +e \right )}\right ) x}{a \,f^{2}}+\frac {6 d^{3} \ln \left (1+i {\mathrm e}^{i \left (f x +e \right )}\right ) x^{2}}{a \,f^{2}}-\frac {12 i d^{3} \operatorname {Li}_{2}\left (-i {\mathrm e}^{i \left (f x +e \right )}\right ) x}{a \,f^{3}}+\frac {4 i d^{3} e^{3}}{a \,f^{4}}+\frac {12 d^{3} \operatorname {Li}_{3}\left (-i {\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{4}}-\frac {12 e c \,d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{a \,f^{3}}-\frac {12 i c \,d^{2} \operatorname {Li}_{2}\left (-i {\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{3}}-\frac {6 i c \,d^{2} e^{2}}{a \,f^{3}}-\frac {6 e^{2} d^{3} \ln \left (1+i {\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{4}}+\frac {6 i d^{3} e^{2} x}{a \,f^{3}}+\frac {6 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) c^{2} d}{a \,f^{2}}-\frac {2 i d^{3} x^{3}}{a f}-\frac {6 i c \,d^{2} x^{2}}{a f}\)	\(484\)

[In]

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

[Out]

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

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. 916 vs. \(2 (124) = 248\).

Time = 0.37 (sec) , antiderivative size = 916, normalized size of antiderivative = 6.23 \[ \int \frac {(c+d x)^3}{a-a \sin (e+f x)} \, dx=\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} + {\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 ) + 6 \, {\left (i \, d^{3} f x + i \, c d^{2} f + {\left (i \, d^{3} f x + i \, c d^{2} f\right )} \cos \left (f x + e\right ) + {\left (-i \, d^{3} f x - i \, c d^{2} f\right )} \sin \left (f x + e\right )\right )} {\rm Li}_2\left (i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right )\right ) + 6 \, {\left (-i \, d^{3} f x - i \, c d^{2} f + {\left (-i \, d^{3} f x - i \, c d^{2} f\right )} \cos \left (f x + e\right ) + {\left (i \, d^{3} f x + i \, c d^{2} f\right )} \sin \left (f x + e\right )\right )} {\rm Li}_2\left (-i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right )\right ) + 3 \, {\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2} + {\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2}\right )} \cos \left (f x + e\right ) - {\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2}\right )} \sin \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ) + i\right ) + 3 \, {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x - d^{3} e^{2} + 2 \, c d^{2} e f + {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x - d^{3} e^{2} + 2 \, c d^{2} e f\right )} \cos \left (f x + e\right ) - {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x - d^{3} e^{2} + 2 \, c d^{2} e f\right )} \sin \left (f x + e\right )\right )} \log \left (i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right ) + 3 \, {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x - d^{3} e^{2} + 2 \, c d^{2} e f + {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x - d^{3} e^{2} + 2 \, c d^{2} e f\right )} \cos \left (f x + e\right ) - {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x - d^{3} e^{2} + 2 \, c d^{2} e f\right )} \sin \left (f x + e\right )\right )} \log \left (-i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right ) + 3 \, {\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2} + {\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2}\right )} \cos \left (f x + e\right ) - {\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2}\right )} \sin \left (f x + e\right )\right )} \log \left (-\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ) + i\right ) + 6 \, {\left (d^{3} \cos \left (f x + e\right ) - d^{3} \sin \left (f x + e\right ) + d^{3}\right )} {\rm polylog}\left (3, i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right )\right ) + 6 \, {\left (d^{3} \cos \left (f x + e\right ) - d^{3} \sin \left (f x + e\right ) + d^{3}\right )} {\rm polylog}\left (3, -i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right )\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 )} \sin \left (f x + e\right )}{a f^{4} \cos \left (f x + e\right ) - a f^{4} \sin \left (f x + e\right ) + a f^{4}} \]

[In]

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

Sympy [F]

\[ \int \frac {(c+d x)^3}{a-a \sin (e+f x)} \, dx=- \frac {\int \frac {c^{3}}{\sin {\left (e + f x \right )} - 1}\, dx + \int \frac {d^{3} x^{3}}{\sin {\left (e + f x \right )} - 1}\, dx + \int \frac {3 c d^{2} x^{2}}{\sin {\left (e + f x \right )} - 1}\, dx + \int \frac {3 c^{2} d x}{\sin {\left (e + f x \right )} - 1}\, dx}{a} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 984 vs. \(2 (124) = 248\).

Time = 0.31 (sec) , antiderivative size = 984, normalized size of antiderivative = 6.69 \[ \int \frac {(c+d x)^3}{a-a \sin (e+f x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^3}{a-a \sin (e+f x)} \, dx=\int \frac {{\left (c+d\,x\right )}^3}{a-a\,\sin \left (e+f\,x\right )} \,d x \]

[In]

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

[Out]

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