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

Optimal. Leaf size=309 \[ -\frac{4 i d^2 (c+d x) \text{PolyLog}\left (2,i e^{i (e+f x)}\right )}{a^2 f^3}+\frac{4 d^3 \text{PolyLog}\left (3,i e^{i (e+f x)}\right )}{a^2 f^4}-\frac{2 d^2 (c+d x) \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )}{a^2 f^3}+\frac{2 d (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right )}{a^2 f^2}-\frac{d (c+d x)^2 \csc ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )}{2 a^2 f^2}-\frac{(c+d x)^3 \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )}{3 a^2 f}-\frac{(c+d x)^3 \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \csc ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )}{6 a^2 f}-\frac{i (c+d x)^3}{3 a^2 f}+\frac{4 d^3 \log \left (\sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )\right )}{a^2 f^4} \]

[Out]

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

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Rubi [A]  time = 0.376733, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {3318, 4186, 4184, 3475, 3717, 2190, 2531, 2282, 6589} \[ -\frac{4 i d^2 (c+d x) \text{PolyLog}\left (2,i e^{i (e+f x)}\right )}{a^2 f^3}+\frac{4 d^3 \text{PolyLog}\left (3,i e^{i (e+f x)}\right )}{a^2 f^4}-\frac{2 d^2 (c+d x) \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )}{a^2 f^3}+\frac{2 d (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right )}{a^2 f^2}-\frac{d (c+d x)^2 \csc ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )}{2 a^2 f^2}-\frac{(c+d x)^3 \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )}{3 a^2 f}-\frac{(c+d x)^3 \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \csc ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )}{6 a^2 f}-\frac{i (c+d x)^3}{3 a^2 f}+\frac{4 d^3 \log \left (\sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )\right )}{a^2 f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3318

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*(e + (Pi*a)/(2*b)))/2 + (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 4186

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

Rule 4184

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 3475

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

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 2531

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 2282

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 6589

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

Mathematica [A]  time = 1.90441, size = 257, normalized size = 0.83 \[ \frac{\frac{24 d^2 \left (d \text{PolyLog}\left (3,i e^{i (e+f x)}\right )-i f (c+d x) \text{PolyLog}\left (2,i e^{i (e+f x)}\right )\right )}{f^2}+\frac{12 d^2 (c+d x) \tan \left (\frac{1}{4} (2 e+2 f x-\pi )\right )}{f}+12 d (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right )+2 f (c+d x)^3 \tan \left (\frac{1}{4} (2 e+2 f x-\pi )\right )-3 d (c+d x)^2 \sec ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )+f (c+d x)^3 \tan \left (\frac{1}{4} (2 e+2 f x-\pi )\right ) \sec ^2\left (\frac{1}{4} (2 e+2 f x-\pi )\right )-2 i f (c+d x)^3+\frac{24 d^3 \log \left (\cos \left (\frac{1}{4} (2 e+2 f x-\pi )\right )\right )}{f^2}}{6 a^2 f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*I)*f*(c + d*x)^3 + 12*d*(c + d*x)^2*Log[1 - I*E^(I*(e + f*x))] + (24*d^3*Log[Cos[(2*e - Pi + 2*f*x)/4]])/
f^2 + (24*d^2*((-I)*f*(c + d*x)*PolyLog[2, I*E^(I*(e + f*x))] + d*PolyLog[3, I*E^(I*(e + f*x))]))/f^2 - 3*d*(c
+ d*x)^2*Sec[(2*e - Pi + 2*f*x)/4]^2 + (12*d^2*(c + d*x)*Tan[(2*e - Pi + 2*f*x)/4])/f + 2*f*(c + d*x)^3*Tan[(
2*e - Pi + 2*f*x)/4] + f*(c + d*x)^3*Sec[(2*e - Pi + 2*f*x)/4]^2*Tan[(2*e - Pi + 2*f*x)/4])/(6*a^2*f^2)

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Maple [B]  time = 0.641, size = 807, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*I/f^3/a^2*polylog(2,I*exp(I*(f*x+e)))*d^3*x-4*I/f^2/a^2*c*d^2*e*x+2/f^4/a^2*ln(exp(I*(f*x+e))+I)*d^3*e^2-2/
f^4/a^2*ln(1-I*exp(I*(f*x+e)))*d^3*e^2-2/f^4/a^2*ln(exp(I*(f*x+e)))*d^3*e^2+2/f^2/a^2*ln(exp(I*(f*x+e))+I)*c^2
*d-2/f^2/a^2*ln(exp(I*(f*x+e)))*c^2*d+2*I/f^3/a^2*d^3*e^2*x-2*I/f^3/a^2*c*d^2*e^2-4/f^3/a^2*ln(exp(I*(f*x+e))+
I)*c*d^2*e+4/f^3/a^2*ln(exp(I*(f*x+e)))*c*d^2*e+4/f^2/a^2*ln(1-I*exp(I*(f*x+e)))*c*d^2*x+4/f^3/a^2*ln(1-I*exp(
I*(f*x+e)))*c*d^2*e+2/f^2/a^2*ln(1-I*exp(I*(f*x+e)))*d^3*x^2-2/3*I/f/a^2*d^3*x^3-2*I/f/a^2*c*d^2*x^2+4/3*I/f^4
/a^2*d^3*e^3-2/3*I*(6*I*c*d^2+3*d^3*f^2*x^3*exp(I*(f*x+e))+I*c^3*f^2+3*I*c^2*d*f^2*x+9*c*d^2*f^2*x^2*exp(I*(f*
x+e))+3*f*d^3*x^2*exp(2*I*(f*x+e))+3*I*c*d^2*f^2*x^2-6*I*d^3*x*exp(2*I*(f*x+e))+I*d^3*f^2*x^3+9*c^2*d*f^2*x*ex
p(I*(f*x+e))+6*f*c*d^2*x*exp(2*I*(f*x+e))+3*I*f*c^2*d*exp(I*(f*x+e))+6*I*d^3*x+3*I*f*d^3*x^2*exp(I*(f*x+e))+3*
c^3*f^2*exp(I*(f*x+e))+3*f*c^2*d*exp(2*I*(f*x+e))+6*I*f*c*d^2*x*exp(I*(f*x+e))+12*d^3*x*exp(I*(f*x+e))-6*I*c*d
^2*exp(2*I*(f*x+e))+12*c*d^2*exp(I*(f*x+e)))/(exp(I*(f*x+e))+I)^3/f^3/a^2-4*I/f^3/a^2*c*d^2*polylog(2,I*exp(I*
(f*x+e)))+4*d^3*polylog(3,I*exp(I*(f*x+e)))/a^2/f^4+4/f^4/a^2*ln(exp(I*(f*x+e))+I)*d^3-4/f^4/a^2*ln(exp(I*(f*x
+e)))*d^3

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Maxima [B]  time = 4.04143, size = 4834, normalized size = 15.64 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(6*c*d^2*e^2*(3*sin(f*x + e)/(cos(f*x + e) + 1) + 3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 2)/(a^2*f^2 + 3
*a^2*f^2*sin(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*f^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*f^2*sin(f*x + e
)^3/(cos(f*x + e) + 1)^3) + 6*(2*(f*x + 3*(f*x + e)*sin(f*x + e) + e + cos(f*x + e) + sin(2*f*x + 2*e))*cos(3*
f*x + 3*e) - 2*(9*(f*x + e)*cos(f*x + e) - 6*sin(f*x + e) - 1)*cos(2*f*x + 2*e) - 6*cos(2*f*x + 2*e)^2 - 6*cos
(f*x + e)^2 - (6*(cos(f*x + e) + sin(2*f*x + 2*e))*cos(3*f*x + 3*e) - cos(3*f*x + 3*e)^2 + 6*(3*sin(f*x + e) +
1)*cos(2*f*x + 2*e) - 9*cos(2*f*x + 2*e)^2 - 9*cos(f*x + e)^2 - 2*(3*cos(2*f*x + 2*e) - 3*sin(f*x + e) - 1)*s
in(3*f*x + 3*e) - sin(3*f*x + 3*e)^2 - 18*cos(f*x + e)*sin(2*f*x + 2*e) - 9*sin(2*f*x + 2*e)^2 - 9*sin(f*x + e
)^2 - 6*sin(f*x + e) - 1)*log(cos(f*x + e)^2 + sin(f*x + e)^2 + 2*sin(f*x + e) + 1) - 2*(3*(f*x + e)*cos(f*x +
e) + cos(2*f*x + 2*e) - sin(f*x + e))*sin(3*f*x + 3*e) - 6*(f*x + 3*(f*x + e)*sin(f*x + e) + e + 2*cos(f*x +
e))*sin(2*f*x + 2*e) - 6*sin(2*f*x + 2*e)^2 - 6*sin(f*x + e)^2 - 2*sin(f*x + e))*c*d^2*e/(a^2*f^2*cos(3*f*x +
3*e)^2 + 9*a^2*f^2*cos(2*f*x + 2*e)^2 + 9*a^2*f^2*cos(f*x + e)^2 + a^2*f^2*sin(3*f*x + 3*e)^2 + 18*a^2*f^2*cos
(f*x + e)*sin(2*f*x + 2*e) + 9*a^2*f^2*sin(2*f*x + 2*e)^2 + 9*a^2*f^2*sin(f*x + e)^2 + 6*a^2*f^2*sin(f*x + e)
+ a^2*f^2 - 6*(a^2*f^2*cos(f*x + e) + a^2*f^2*sin(2*f*x + 2*e))*cos(3*f*x + 3*e) - 6*(3*a^2*f^2*sin(f*x + e) +
a^2*f^2)*cos(2*f*x + 2*e) + 2*(3*a^2*f^2*cos(2*f*x + 2*e) - 3*a^2*f^2*sin(f*x + e) - a^2*f^2)*sin(3*f*x + 3*e
)) - 6*c^2*d*e*(3*sin(f*x + e)/(cos(f*x + e) + 1) + 3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 2)/(a^2*f + 3*a^2*
f*sin(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*f*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*f*sin(f*x + e)^3/(cos(f*
x + e) + 1)^3) - 3*(2*(f*x + 3*(f*x + e)*sin(f*x + e) + e + cos(f*x + e) + sin(2*f*x + 2*e))*cos(3*f*x + 3*e)
- 2*(9*(f*x + e)*cos(f*x + e) - 6*sin(f*x + e) - 1)*cos(2*f*x + 2*e) - 6*cos(2*f*x + 2*e)^2 - 6*cos(f*x + e)^2
- (6*(cos(f*x + e) + sin(2*f*x + 2*e))*cos(3*f*x + 3*e) - cos(3*f*x + 3*e)^2 + 6*(3*sin(f*x + e) + 1)*cos(2*f
*x + 2*e) - 9*cos(2*f*x + 2*e)^2 - 9*cos(f*x + e)^2 - 2*(3*cos(2*f*x + 2*e) - 3*sin(f*x + e) - 1)*sin(3*f*x +
3*e) - sin(3*f*x + 3*e)^2 - 18*cos(f*x + e)*sin(2*f*x + 2*e) - 9*sin(2*f*x + 2*e)^2 - 9*sin(f*x + e)^2 - 6*sin
(f*x + e) - 1)*log(cos(f*x + e)^2 + sin(f*x + e)^2 + 2*sin(f*x + e) + 1) - 2*(3*(f*x + e)*cos(f*x + e) + cos(2
*f*x + 2*e) - sin(f*x + e))*sin(3*f*x + 3*e) - 6*(f*x + 3*(f*x + e)*sin(f*x + e) + e + 2*cos(f*x + e))*sin(2*f
*x + 2*e) - 6*sin(2*f*x + 2*e)^2 - 6*sin(f*x + e)^2 - 2*sin(f*x + e))*c^2*d/(a^2*f*cos(3*f*x + 3*e)^2 + 9*a^2*
f*cos(2*f*x + 2*e)^2 + 9*a^2*f*cos(f*x + e)^2 + a^2*f*sin(3*f*x + 3*e)^2 + 18*a^2*f*cos(f*x + e)*sin(2*f*x + 2
*e) + 9*a^2*f*sin(2*f*x + 2*e)^2 + 9*a^2*f*sin(f*x + e)^2 + 6*a^2*f*sin(f*x + e) + a^2*f - 6*(a^2*f*cos(f*x +
e) + a^2*f*sin(2*f*x + 2*e))*cos(3*f*x + 3*e) - 6*(3*a^2*f*sin(f*x + e) + a^2*f)*cos(2*f*x + 2*e) + 2*(3*a^2*f
*cos(2*f*x + 2*e) - 3*a^2*f*sin(f*x + e) - a^2*f)*sin(3*f*x + 3*e)) + 2*c^3*(3*sin(f*x + e)/(cos(f*x + e) + 1)
+ 3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 2)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*sin(f*x + e
)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3) - 3*(2*I*d^3*e^3 + 12*I*d^3*e - 12*I*c*d^2
*f + (-6*I*d^3*e^2 - 12*I*d^3 + 6*(d^3*e^2 + 2*d^3)*cos(3*f*x + 3*e) + (18*I*d^3*e^2 + 36*I*d^3)*cos(2*f*x + 2
*e) - 18*(d^3*e^2 + 2*d^3)*cos(f*x + e) + (6*I*d^3*e^2 + 12*I*d^3)*sin(3*f*x + 3*e) - 18*(d^3*e^2 + 2*d^3)*sin
(2*f*x + 2*e) + (-18*I*d^3*e^2 - 36*I*d^3)*sin(f*x + e))*arctan2(sin(f*x + e) + 1, cos(f*x + e)) + (6*I*(f*x +
e)^2*d^3 + (-12*I*d^3*e + 12*I*c*d^2*f)*(f*x + e) - 6*((f*x + e)^2*d^3 - 2*(d^3*e - c*d^2*f)*(f*x + e))*cos(3
*f*x + 3*e) + (-18*I*(f*x + e)^2*d^3 + (36*I*d^3*e - 36*I*c*d^2*f)*(f*x + e))*cos(2*f*x + 2*e) + 18*((f*x + e)
^2*d^3 - 2*(d^3*e - c*d^2*f)*(f*x + e))*cos(f*x + e) + (-6*I*(f*x + e)^2*d^3 + (12*I*d^3*e - 12*I*c*d^2*f)*(f*
x + e))*sin(3*f*x + 3*e) + 18*((f*x + e)^2*d^3 - 2*(d^3*e - c*d^2*f)*(f*x + e))*sin(2*f*x + 2*e) + (18*I*(f*x
+ e)^2*d^3 + (-36*I*d^3*e + 36*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*(d^3*e - c*d^2*f)*(f*x + e)^2 + 3*(d^3*e^2 + 2*d^3)*(f*x + e))*cos(3*f*x + 3*e) + (-6*I*
(f*x + e)^3*d^3 - 6*d^3*e^2 - 12*I*d^3*e + 12*I*c*d^2*f + (18*I*d^3*e - 18*I*c*d^2*f - 6*d^3)*(f*x + e)^2 + (-
18*I*d^3*e^2 + 12*d^3*e - 12*c*d^2*f - 24*I*d^3)*(f*x + e))*cos(2*f*x + 2*e) + (6*d^3*e^3 - 6*I*(f*x + e)^2*d^
3 - 6*I*d^3*e^2 + 24*d^3*e - 24*c*d^2*f + (12*I*d^3*e - 12*I*c*d^2*f + 12*d^3)*(f*x + e))*cos(f*x + e) + (12*I
*(f*x + e)*d^3 - 12*I*d^3*e + 12*I*c*d^2*f - 12*((f*x + e)*d^3 - d^3*e + c*d^2*f)*cos(3*f*x + 3*e) + (-36*I*(f
*x + e)*d^3 + 36*I*d^3*e - 36*I*c*d^2*f)*cos(2*f*x + 2*e) + 36*((f*x + e)*d^3 - d^3*e + c*d^2*f)*cos(f*x + e)
+ (-12*I*(f*x + e)*d^3 + 12*I*d^3*e - 12*I*c*d^2*f)*sin(3*f*x + 3*e) + 36*((f*x + e)*d^3 - d^3*e + c*d^2*f)*si
n(2*f*x + 2*e) + (36*I*(f*x + e)*d^3 - 36*I*d^3*e + 36*I*c*d^2*f)*sin(f*x + e))*dilog(I*e^(I*f*x + I*e)) - (3*
(f*x + e)^2*d^3 + 3*d^3*e^2 + 6*d^3 - 6*(d^3*e - c*d^2*f)*(f*x + e) - (-3*I*(f*x + e)^2*d^3 - 3*I*d^3*e^2 - 6*
I*d^3 + (6*I*d^3*e - 6*I*c*d^2*f)*(f*x + e))*cos(3*f*x + 3*e) - 9*((f*x + e)^2*d^3 + d^3*e^2 + 2*d^3 - 2*(d^3*
e - c*d^2*f)*(f*x + e))*cos(2*f*x + 2*e) - (9*I*(f*x + e)^2*d^3 + 9*I*d^3*e^2 + 18*I*d^3 + (-18*I*d^3*e + 18*I
*c*d^2*f)*(f*x + e))*cos(f*x + e) - 3*((f*x + e)^2*d^3 + d^3*e^2 + 2*d^3 - 2*(d^3*e - c*d^2*f)*(f*x + e))*sin(
3*f*x + 3*e) - (9*I*(f*x + e)^2*d^3 + 9*I*d^3*e^2 + 18*I*d^3 + (-18*I*d^3*e + 18*I*c*d^2*f)*(f*x + e))*sin(2*f
*x + 2*e) + 9*((f*x + e)^2*d^3 + d^3*e^2 + 2*d^3 - 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(3*f*x + 3*e) + 36*d^3*cos(2*f*x + 2*e) + 36*I*d^3
*cos(f*x + e) + 12*d^3*sin(3*f*x + 3*e) + 36*I*d^3*sin(2*f*x + 2*e) - 36*d^3*sin(f*x + e) - 12*d^3)*polylog(3,
I*e^(I*f*x + I*e)) + (-2*I*(f*x + e)^3*d^3 + (6*I*d^3*e - 6*I*c*d^2*f)*(f*x + e)^2 + (-6*I*d^3*e^2 - 12*I*d^3
)*(f*x + e))*sin(3*f*x + 3*e) + (6*(f*x + e)^3*d^3 - 6*I*d^3*e^2 + 12*d^3*e - 12*c*d^2*f - (18*d^3*e - 18*c*d^
2*f + 6*I*d^3)*(f*x + e)^2 + (18*d^3*e^2 + 12*I*d^3*e - 12*I*c*d^2*f + 24*d^3)*(f*x + e))*sin(2*f*x + 2*e) + (
6*I*d^3*e^3 + 6*(f*x + e)^2*d^3 + 6*d^3*e^2 + 24*I*d^3*e - 24*I*c*d^2*f - (12*d^3*e - 12*c*d^2*f - 12*I*d^3)*(
f*x + e))*sin(f*x + e))/(-3*I*a^2*f^3*cos(3*f*x + 3*e) + 9*a^2*f^3*cos(2*f*x + 2*e) + 9*I*a^2*f^3*cos(f*x + e)
+ 3*a^2*f^3*sin(3*f*x + 3*e) + 9*I*a^2*f^3*sin(2*f*x + 2*e) - 9*a^2*f^3*sin(f*x + e) - 3*a^2*f^3))/f

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Fricas [C]  time = 2.69471, size = 3872, normalized size = 12.53 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(d^3*f^3*x^3 + c^3*f^3 + 3*c^2*d*f^2 + 3*(c*d^2*f^3 + d^3*f^2)*x^2 + (d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + c^3*
f^3 + 6*c*d^2*f + 3*(c^2*d*f^3 + 2*d^3*f)*x)*cos(f*x + e)^2 + 3*(c^2*d*f^3 + 2*c*d^2*f^2)*x + (2*d^3*f^3*x^3 +
2*c^3*f^3 + 3*c^2*d*f^2 + 6*c*d^2*f + 3*(2*c*d^2*f^3 + d^3*f^2)*x^2 + 6*(c^2*d*f^3 + c*d^2*f^2 + d^3*f)*x)*co
s(f*x + e) - (-12*I*d^3*f*x - 12*I*c*d^2*f + (6*I*d^3*f*x + 6*I*c*d^2*f)*cos(f*x + e)^2 + (-6*I*d^3*f*x - 6*I*
c*d^2*f)*cos(f*x + e) + (-12*I*d^3*f*x - 12*I*c*d^2*f + (-6*I*d^3*f*x - 6*I*c*d^2*f)*cos(f*x + e))*sin(f*x + e
))*dilog(I*cos(f*x + e) - sin(f*x + e)) - (12*I*d^3*f*x + 12*I*c*d^2*f + (-6*I*d^3*f*x - 6*I*c*d^2*f)*cos(f*x
+ e)^2 + (6*I*d^3*f*x + 6*I*c*d^2*f)*cos(f*x + e) + (12*I*d^3*f*x + 12*I*c*d^2*f + (6*I*d^3*f*x + 6*I*c*d^2*f)
*cos(f*x + e))*sin(f*x + e))*dilog(-I*cos(f*x + e) - sin(f*x + e)) - 3*(2*d^3*e^2 - 4*c*d^2*e*f + 2*c^2*d*f^2
+ 4*d^3 - (d^3*e^2 - 2*c*d^2*e*f + c^2*d*f^2 + 2*d^3)*cos(f*x + e)^2 + (d^3*e^2 - 2*c*d^2*e*f + c^2*d*f^2 + 2*
d^3)*cos(f*x + e) + (2*d^3*e^2 - 4*c*d^2*e*f + 2*c^2*d*f^2 + 4*d^3 + (d^3*e^2 - 2*c*d^2*e*f + c^2*d*f^2 + 2*d^
3)*cos(f*x + e))*sin(f*x + e))*log(cos(f*x + e) + I*sin(f*x + e) + I) - 3*(2*d^3*f^2*x^2 + 4*c*d^2*f^2*x - 2*d
^3*e^2 + 4*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)^2 + (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) + (2*d^3*f^2*x^2 + 4*c*d^2*f^2*x - 2*d^3*e^2 + 4*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))*sin(f*x + e))*log(I*cos(f*x + e) + sin(f
*x + e) + 1) - 3*(2*d^3*f^2*x^2 + 4*c*d^2*f^2*x - 2*d^3*e^2 + 4*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)^2 + (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) + (2*d
^3*f^2*x^2 + 4*c*d^2*f^2*x - 2*d^3*e^2 + 4*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)*c
os(f*x + e))*sin(f*x + e))*log(-I*cos(f*x + e) + sin(f*x + e) + 1) - 3*(2*d^3*e^2 - 4*c*d^2*e*f + 2*c^2*d*f^2
+ 4*d^3 - (d^3*e^2 - 2*c*d^2*e*f + c^2*d*f^2 + 2*d^3)*cos(f*x + e)^2 + (d^3*e^2 - 2*c*d^2*e*f + c^2*d*f^2 + 2*
d^3)*cos(f*x + e) + (2*d^3*e^2 - 4*c*d^2*e*f + 2*c^2*d*f^2 + 4*d^3 + (d^3*e^2 - 2*c*d^2*e*f + c^2*d*f^2 + 2*d^
3)*cos(f*x + e))*sin(f*x + e))*log(-cos(f*x + e) + I*sin(f*x + e) + I) + 6*(d^3*cos(f*x + e)^2 - d^3*cos(f*x +
e) - 2*d^3 - (d^3*cos(f*x + e) + 2*d^3)*sin(f*x + e))*polylog(3, I*cos(f*x + e) - sin(f*x + e)) + 6*(d^3*cos(
f*x + e)^2 - d^3*cos(f*x + e) - 2*d^3 - (d^3*cos(f*x + e) + 2*d^3)*sin(f*x + e))*polylog(3, -I*cos(f*x + e) -
sin(f*x + e)) - (d^3*f^3*x^3 + c^3*f^3 - 3*c^2*d*f^2 + 3*(c*d^2*f^3 - d^3*f^2)*x^2 + 3*(c^2*d*f^3 - 2*c*d^2*f^
2)*x - (d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + c^3*f^3 + 6*c*d^2*f + 3*(c^2*d*f^3 + 2*d^3*f)*x)*cos(f*x + e))*sin(f*x
+ e))/(a^2*f^4*cos(f*x + e)^2 - a^2*f^4*cos(f*x + e) - 2*a^2*f^4 - (a^2*f^4*cos(f*x + e) + 2*a^2*f^4)*sin(f*x
+ e))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{3}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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