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

Optimal. Leaf size=255 \[ -\frac{4 d^2 (c+d x) \text{PolyLog}\left (2,-e^{e+f x}\right )}{a^2 f^3}+\frac{4 d^3 \text{PolyLog}\left (3,-e^{e+f x}\right )}{a^2 f^4}-\frac{2 d^2 (c+d x) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{a^2 f^3}-\frac{2 d (c+d x)^2 \log \left (e^{e+f x}+1\right )}{a^2 f^2}+\frac{d (c+d x)^2 \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{2 a^2 f^2}+\frac{(c+d x)^3 \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^3 \tanh \left (\frac{e}{2}+\frac{f x}{2}\right ) \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{(c+d x)^3}{3 a^2 f}+\frac{4 d^3 \log \left (\cosh \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{a^2 f^4} \]

[Out]

(c + d*x)^3/(3*a^2*f) - (2*d*(c + d*x)^2*Log[1 + E^(e + f*x)])/(a^2*f^2) + (4*d^3*Log[Cosh[e/2 + (f*x)/2]])/(a
^2*f^4) - (4*d^2*(c + d*x)*PolyLog[2, -E^(e + f*x)])/(a^2*f^3) + (4*d^3*PolyLog[3, -E^(e + f*x)])/(a^2*f^4) +
(d*(c + d*x)^2*Sech[e/2 + (f*x)/2]^2)/(2*a^2*f^2) - (2*d^2*(c + d*x)*Tanh[e/2 + (f*x)/2])/(a^2*f^3) + ((c + d*
x)^3*Tanh[e/2 + (f*x)/2])/(3*a^2*f) + ((c + d*x)^3*Sech[e/2 + (f*x)/2]^2*Tanh[e/2 + (f*x)/2])/(6*a^2*f)

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Rubi [A]  time = 0.363557, antiderivative size = 255, 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, 3718, 2190, 2531, 2282, 6589} \[ -\frac{4 d^2 (c+d x) \text{PolyLog}\left (2,-e^{e+f x}\right )}{a^2 f^3}+\frac{4 d^3 \text{PolyLog}\left (3,-e^{e+f x}\right )}{a^2 f^4}-\frac{2 d^2 (c+d x) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{a^2 f^3}-\frac{2 d (c+d x)^2 \log \left (e^{e+f x}+1\right )}{a^2 f^2}+\frac{d (c+d x)^2 \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{2 a^2 f^2}+\frac{(c+d x)^3 \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^3 \tanh \left (\frac{e}{2}+\frac{f x}{2}\right ) \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{(c+d x)^3}{3 a^2 f}+\frac{4 d^3 \log \left (\cosh \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{a^2 f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c + d*x)^3/(3*a^2*f) - (2*d*(c + d*x)^2*Log[1 + E^(e + f*x)])/(a^2*f^2) + (4*d^3*Log[Cosh[e/2 + (f*x)/2]])/(a
^2*f^4) - (4*d^2*(c + d*x)*PolyLog[2, -E^(e + f*x)])/(a^2*f^3) + (4*d^3*PolyLog[3, -E^(e + f*x)])/(a^2*f^4) +
(d*(c + d*x)^2*Sech[e/2 + (f*x)/2]^2)/(2*a^2*f^2) - (2*d^2*(c + d*x)*Tanh[e/2 + (f*x)/2])/(a^2*f^3) + ((c + d*
x)^3*Tanh[e/2 + (f*x)/2])/(3*a^2*f) + ((c + d*x)^3*Sech[e/2 + (f*x)/2]^2*Tanh[e/2 + (f*x)/2])/(6*a^2*f)

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 3718

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(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*e) + f*fz*x)))/(1 + E^(2*(-(I*e) + f*fz*x))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && 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 \cosh (e+f x))^2} \, dx &=\frac{\int (c+d x)^3 \csc ^4\left (\frac{1}{2} (i e+\pi )+\frac{i f x}{2}\right ) \, dx}{4 a^2}\\ &=\frac{d (c+d x)^2 \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{2 a^2 f^2}+\frac{(c+d x)^3 \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{\int (c+d x)^3 \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \, dx}{6 a^2}-\frac{d^2 \int (c+d x) \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \, dx}{a^2 f^2}\\ &=\frac{d (c+d x)^2 \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{2 a^2 f^2}-\frac{2 d^2 (c+d x) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{a^2 f^3}+\frac{(c+d x)^3 \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^3 \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{\left (2 d^3\right ) \int \tanh \left (\frac{e}{2}+\frac{f x}{2}\right ) \, dx}{a^2 f^3}-\frac{d \int (c+d x)^2 \tanh \left (\frac{e}{2}+\frac{f x}{2}\right ) \, dx}{a^2 f}\\ &=\frac{(c+d x)^3}{3 a^2 f}+\frac{4 d^3 \log \left (\cosh \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{a^2 f^4}+\frac{d (c+d x)^2 \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{2 a^2 f^2}-\frac{2 d^2 (c+d x) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{a^2 f^3}+\frac{(c+d x)^3 \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^3 \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}-\frac{(2 d) \int \frac{e^{2 \left (\frac{e}{2}+\frac{f x}{2}\right )} (c+d x)^2}{1+e^{2 \left (\frac{e}{2}+\frac{f x}{2}\right )}} \, dx}{a^2 f}\\ &=\frac{(c+d x)^3}{3 a^2 f}-\frac{2 d (c+d x)^2 \log \left (1+e^{e+f x}\right )}{a^2 f^2}+\frac{4 d^3 \log \left (\cosh \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{a^2 f^4}+\frac{d (c+d x)^2 \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{2 a^2 f^2}-\frac{2 d^2 (c+d x) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{a^2 f^3}+\frac{(c+d x)^3 \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^3 \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{\left (4 d^2\right ) \int (c+d x) \log \left (1+e^{2 \left (\frac{e}{2}+\frac{f x}{2}\right )}\right ) \, dx}{a^2 f^2}\\ &=\frac{(c+d x)^3}{3 a^2 f}-\frac{2 d (c+d x)^2 \log \left (1+e^{e+f x}\right )}{a^2 f^2}+\frac{4 d^3 \log \left (\cosh \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{a^2 f^4}-\frac{4 d^2 (c+d x) \text{Li}_2\left (-e^{e+f x}\right )}{a^2 f^3}+\frac{d (c+d x)^2 \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{2 a^2 f^2}-\frac{2 d^2 (c+d x) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{a^2 f^3}+\frac{(c+d x)^3 \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^3 \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{\left (4 d^3\right ) \int \text{Li}_2\left (-e^{2 \left (\frac{e}{2}+\frac{f x}{2}\right )}\right ) \, dx}{a^2 f^3}\\ &=\frac{(c+d x)^3}{3 a^2 f}-\frac{2 d (c+d x)^2 \log \left (1+e^{e+f x}\right )}{a^2 f^2}+\frac{4 d^3 \log \left (\cosh \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{a^2 f^4}-\frac{4 d^2 (c+d x) \text{Li}_2\left (-e^{e+f x}\right )}{a^2 f^3}+\frac{d (c+d x)^2 \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{2 a^2 f^2}-\frac{2 d^2 (c+d x) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{a^2 f^3}+\frac{(c+d x)^3 \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^3 \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{\left (4 d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 \left (\frac{e}{2}+\frac{f x}{2}\right )}\right )}{a^2 f^4}\\ &=\frac{(c+d x)^3}{3 a^2 f}-\frac{2 d (c+d x)^2 \log \left (1+e^{e+f x}\right )}{a^2 f^2}+\frac{4 d^3 \log \left (\cosh \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{a^2 f^4}-\frac{4 d^2 (c+d x) \text{Li}_2\left (-e^{e+f x}\right )}{a^2 f^3}+\frac{4 d^3 \text{Li}_3\left (-e^{e+f x}\right )}{a^2 f^4}+\frac{d (c+d x)^2 \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{2 a^2 f^2}-\frac{2 d^2 (c+d x) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{a^2 f^3}+\frac{(c+d x)^3 \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^3 \text{sech}^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \tanh \left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}\\ \end{align*}

Mathematica [A]  time = 3.43358, size = 462, normalized size = 1.81 \[ \frac{\cosh \left (\frac{1}{2} (e+f x)\right ) \left (\text{sech}\left (\frac{e}{2}\right ) (c+d x) \left (c^2 f^2 \sinh \left (e+\frac{3 f x}{2}\right )+3 c^2 f^2 \sinh \left (\frac{f x}{2}\right )+2 c d f^2 x \sinh \left (e+\frac{3 f x}{2}\right )+3 d f (c+d x) \cosh \left (e+\frac{f x}{2}\right )+6 c d f^2 x \sinh \left (\frac{f x}{2}\right )+3 d f (c+d x) \cosh \left (\frac{f x}{2}\right )+d^2 f^2 x^2 \sinh \left (e+\frac{3 f x}{2}\right )+6 d^2 \sinh \left (e+\frac{f x}{2}\right )-6 d^2 \sinh \left (e+\frac{3 f x}{2}\right )+3 d^2 f^2 x^2 \sinh \left (\frac{f x}{2}\right )-12 d^2 \sinh \left (\frac{f x}{2}\right )\right )-\frac{8 d \cosh ^3\left (\frac{1}{2} (e+f x)\right ) \left (-6 c d (\sinh (e)+\cosh (e)+1) \text{PolyLog}(2,\sinh (e+f x)-\cosh (e+f x))-\frac{6 d^2 (\sinh (e)+\cosh (e)+1) (f x \text{PolyLog}(2,\sinh (e+f x)-\cosh (e+f x))+\text{PolyLog}(3,\sinh (e+f x)-\cosh (e+f x)))}{f}-\frac{3 \left (c^2 f^2-2 d^2\right ) (\sinh (e)+\cosh (e)+1) (f x-\log (\sinh (e+f x)+\cosh (e+f x)+1))}{f}+3 c^2 f^2 x+6 c d f x (\sinh (e)+\cosh (e)+1) \log (-\sinh (e+f x)+\cosh (e+f x)+1)+3 c d f^2 x^2+3 d^2 f x^2 (\sinh (e)+\cosh (e)+1) \log (-\sinh (e+f x)+\cosh (e+f x)+1)+d^2 f^2 x^3-6 d^2 x\right )}{\sinh (e)+\cosh (e)+1}\right )}{3 a^2 f^3 (\cosh (e+f x)+1)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cosh[(e + f*x)/2]*((-8*d*Cosh[(e + f*x)/2]^3*(-6*d^2*x + 3*c^2*f^2*x + 3*c*d*f^2*x^2 + d^2*f^2*x^3 + 6*c*d*f*
x*Log[1 + Cosh[e + f*x] - Sinh[e + f*x]]*(1 + Cosh[e] + Sinh[e]) + 3*d^2*f*x^2*Log[1 + Cosh[e + f*x] - Sinh[e
+ f*x]]*(1 + Cosh[e] + Sinh[e]) - (3*(-2*d^2 + c^2*f^2)*(f*x - Log[1 + Cosh[e + f*x] + Sinh[e + f*x]])*(1 + Co
sh[e] + Sinh[e]))/f - 6*c*d*PolyLog[2, -Cosh[e + f*x] + Sinh[e + f*x]]*(1 + Cosh[e] + Sinh[e]) - (6*d^2*(f*x*P
olyLog[2, -Cosh[e + f*x] + Sinh[e + f*x]] + PolyLog[3, -Cosh[e + f*x] + Sinh[e + f*x]])*(1 + Cosh[e] + Sinh[e]
))/f))/(1 + Cosh[e] + Sinh[e]) + (c + d*x)*Sech[e/2]*(3*d*f*(c + d*x)*Cosh[(f*x)/2] + 3*d*f*(c + d*x)*Cosh[e +
(f*x)/2] - 12*d^2*Sinh[(f*x)/2] + 3*c^2*f^2*Sinh[(f*x)/2] + 6*c*d*f^2*x*Sinh[(f*x)/2] + 3*d^2*f^2*x^2*Sinh[(f
*x)/2] + 6*d^2*Sinh[e + (f*x)/2] - 6*d^2*Sinh[e + (3*f*x)/2] + c^2*f^2*Sinh[e + (3*f*x)/2] + 2*c*d*f^2*x*Sinh[
e + (3*f*x)/2] + d^2*f^2*x^2*Sinh[e + (3*f*x)/2])))/(3*a^2*f^3*(1 + Cosh[e + f*x])^2)

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Maple [B]  time = 0.116, size = 600, normalized size = 2.4 \begin{align*} -{\frac{6\,{f}^{2}{d}^{3}{x}^{3}{{\rm e}^{fx+e}}+18\,{f}^{2}c{d}^{2}{x}^{2}{{\rm e}^{fx+e}}+2\,{d}^{3}{f}^{2}{x}^{3}-6\,{d}^{3}f{x}^{2}{{\rm e}^{2\,fx+2\,e}}+18\,{f}^{2}{c}^{2}dx{{\rm e}^{fx+e}}+6\,c{d}^{2}{f}^{2}{x}^{2}-12\,c{d}^{2}fx{{\rm e}^{2\,fx+2\,e}}-6\,f{d}^{3}{x}^{2}{{\rm e}^{fx+e}}+6\,{f}^{2}{c}^{3}{{\rm e}^{fx+e}}+6\,{c}^{2}d{f}^{2}x-6\,{c}^{2}df{{\rm e}^{2\,fx+2\,e}}-12\,fc{d}^{2}x{{\rm e}^{fx+e}}-12\,{d}^{3}x{{\rm e}^{2\,fx+2\,e}}+2\,{c}^{3}{f}^{2}-6\,f{c}^{2}d{{\rm e}^{fx+e}}-12\,c{d}^{2}{{\rm e}^{2\,fx+2\,e}}-24\,{d}^{3}x{{\rm e}^{fx+e}}-24\,c{d}^{2}{{\rm e}^{fx+e}}-12\,{d}^{3}x-12\,c{d}^{2}}{3\,{a}^{2}{f}^{3} \left ({{\rm e}^{fx+e}}+1 \right ) ^{3}}}-2\,{\frac{{d}^{3}{e}^{2}x}{{a}^{2}{f}^{3}}}+2\,{\frac{c{d}^{2}{x}^{2}}{f{a}^{2}}}+2\,{\frac{c{d}^{2}{e}^{2}}{{a}^{2}{f}^{3}}}-2\,{\frac{{d}^{3}\ln \left ({{\rm e}^{fx+e}}+1 \right ){x}^{2}}{{a}^{2}{f}^{2}}}-4\,{\frac{{d}^{3}{\it polylog} \left ( 2,-{{\rm e}^{fx+e}} \right ) x}{{a}^{2}{f}^{3}}}-{\frac{4\,{d}^{3}{e}^{3}}{3\,{a}^{2}{f}^{4}}}+4\,{\frac{{d}^{3}{\it polylog} \left ( 3,-{{\rm e}^{fx+e}} \right ) }{{a}^{2}{f}^{4}}}+4\,{\frac{{d}^{3}\ln \left ({{\rm e}^{fx+e}}+1 \right ) }{{a}^{2}{f}^{4}}}-4\,{\frac{{d}^{3}\ln \left ({{\rm e}^{fx+e}} \right ) }{{a}^{2}{f}^{4}}}-4\,{\frac{c{d}^{2}{\it polylog} \left ( 2,-{{\rm e}^{fx+e}} \right ) }{{a}^{2}{f}^{3}}}+4\,{\frac{c{d}^{2}ex}{{a}^{2}{f}^{2}}}+{\frac{2\,{d}^{3}{x}^{3}}{3\,f{a}^{2}}}-2\,{\frac{{c}^{2}d\ln \left ({{\rm e}^{fx+e}}+1 \right ) }{{a}^{2}{f}^{2}}}+2\,{\frac{{c}^{2}d\ln \left ({{\rm e}^{fx+e}} \right ) }{{a}^{2}{f}^{2}}}+2\,{\frac{{d}^{3}{e}^{2}\ln \left ({{\rm e}^{fx+e}} \right ) }{{a}^{2}{f}^{4}}}-4\,{\frac{{d}^{2}\ln \left ({{\rm e}^{fx+e}}+1 \right ) cx}{{a}^{2}{f}^{2}}}-4\,{\frac{c{d}^{2}e\ln \left ({{\rm e}^{fx+e}} \right ) }{{a}^{2}{f}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.89542, size = 824, normalized size = 3.23 \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*cosh(f*x+e))^2,x, algorithm="maxima")

[Out]

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

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Fricas [C]  time = 2.32742, size = 4103, normalized size = 16.09 \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*cosh(f*x+e))^2,x, algorithm="fricas")

[Out]

2/3*(d^3*e^3 + 3*c^2*d*e*f^2 - c^3*f^3 - 6*d^3*e + (d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + d^3*e^3 - 3*c*d^2*e^2*f +
3*c^2*d*e*f^2 - 6*d^3*e + 3*(c^2*d*f^3 - 2*d^3*f)*x)*cosh(f*x + e)^3 + (d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + d^3*e^
3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 - 6*d^3*e + 3*(c^2*d*f^3 - 2*d^3*f)*x)*sinh(f*x + e)^3 + 3*(d^3*f^3*x^3 + d^
3*e^3 - 6*d^3*e + (3*c^2*d*e + c^2*d)*f^2 + (3*c*d^2*f^3 + d^3*f^2)*x^2 - (3*c*d^2*e^2 - 2*c*d^2)*f + (3*c^2*d
*f^3 + 2*c*d^2*f^2 - 4*d^3*f)*x)*cosh(f*x + e)^2 + 3*(d^3*f^3*x^3 + d^3*e^3 - 6*d^3*e + (3*c^2*d*e + c^2*d)*f^
2 + (3*c*d^2*f^3 + d^3*f^2)*x^2 - (3*c*d^2*e^2 - 2*c*d^2)*f + (3*c^2*d*f^3 + 2*c*d^2*f^2 - 4*d^3*f)*x + (d^3*f
^3*x^3 + 3*c*d^2*f^3*x^2 + d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 - 6*d^3*e + 3*(c^2*d*f^3 - 2*d^3*f)*x)*cosh
(f*x + e))*sinh(f*x + e)^2 - 3*(c*d^2*e^2 - 2*c*d^2)*f + 3*(d^3*f^2*x^2 + d^3*e^3 - c^3*f^3 - 6*d^3*e + (3*c^2
*d*e + c^2*d)*f^2 - (3*c*d^2*e^2 - 4*c*d^2)*f + 2*(c*d^2*f^2 - d^3*f)*x)*cosh(f*x + e) - 6*(d^3*f*x + c*d^2*f
+ (d^3*f*x + c*d^2*f)*cosh(f*x + e)^3 + (d^3*f*x + c*d^2*f)*sinh(f*x + e)^3 + 3*(d^3*f*x + c*d^2*f)*cosh(f*x +
e)^2 + 3*(d^3*f*x + c*d^2*f + (d^3*f*x + c*d^2*f)*cosh(f*x + e))*sinh(f*x + e)^2 + 3*(d^3*f*x + c*d^2*f)*cosh
(f*x + e) + 3*(d^3*f*x + c*d^2*f + (d^3*f*x + c*d^2*f)*cosh(f*x + e)^2 + 2*(d^3*f*x + c*d^2*f)*cosh(f*x + e))*
sinh(f*x + e))*dilog(-cosh(f*x + e) - sinh(f*x + e)) - 3*(d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 + (d^3*f^2*x
^2 + 2*c*d^2*f^2*x + c^2*d*f^2 - 2*d^3)*cosh(f*x + e)^3 + (d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 - 2*d^3)*si
nh(f*x + e)^3 - 2*d^3 + 3*(d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 - 2*d^3)*cosh(f*x + e)^2 + 3*(d^3*f^2*x^2 +
2*c*d^2*f^2*x + c^2*d*f^2 - 2*d^3 + (d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 - 2*d^3)*cosh(f*x + e))*sinh(f*x
+ e)^2 + 3*(d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 - 2*d^3)*cosh(f*x + e) + 3*(d^3*f^2*x^2 + 2*c*d^2*f^2*x +
c^2*d*f^2 - 2*d^3 + (d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 - 2*d^3)*cosh(f*x + e)^2 + 2*(d^3*f^2*x^2 + 2*c*
d^2*f^2*x + c^2*d*f^2 - 2*d^3)*cosh(f*x + e))*sinh(f*x + e))*log(cosh(f*x + e) + sinh(f*x + e) + 1) + 6*(d^3*c
osh(f*x + e)^3 + d^3*sinh(f*x + e)^3 + 3*d^3*cosh(f*x + e)^2 + 3*d^3*cosh(f*x + e) + d^3 + 3*(d^3*cosh(f*x + e
) + d^3)*sinh(f*x + e)^2 + 3*(d^3*cosh(f*x + e)^2 + 2*d^3*cosh(f*x + e) + d^3)*sinh(f*x + e))*polylog(3, -cosh
(f*x + e) - sinh(f*x + e)) + 3*(d^3*f^2*x^2 + d^3*e^3 - c^3*f^3 - 6*d^3*e + (3*c^2*d*e + c^2*d)*f^2 + (d^3*f^3
*x^3 + 3*c*d^2*f^3*x^2 + d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 - 6*d^3*e + 3*(c^2*d*f^3 - 2*d^3*f)*x)*cosh(f
*x + e)^2 - (3*c*d^2*e^2 - 4*c*d^2)*f + 2*(c*d^2*f^2 - d^3*f)*x + 2*(d^3*f^3*x^3 + d^3*e^3 - 6*d^3*e + (3*c^2*
d*e + c^2*d)*f^2 + (3*c*d^2*f^3 + d^3*f^2)*x^2 - (3*c*d^2*e^2 - 2*c*d^2)*f + (3*c^2*d*f^3 + 2*c*d^2*f^2 - 4*d^
3*f)*x)*cosh(f*x + e))*sinh(f*x + e))/(a^2*f^4*cosh(f*x + e)^3 + a^2*f^4*sinh(f*x + e)^3 + 3*a^2*f^4*cosh(f*x
+ e)^2 + 3*a^2*f^4*cosh(f*x + e) + a^2*f^4 + 3*(a^2*f^4*cosh(f*x + e) + a^2*f^4)*sinh(f*x + e)^2 + 3*(a^2*f^4*
cosh(f*x + e)^2 + 2*a^2*f^4*cosh(f*x + e) + a^2*f^4)*sinh(f*x + e))

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

[Out]

(Integral(c**3/(cosh(e + f*x)**2 + 2*cosh(e + f*x) + 1), x) + Integral(d**3*x**3/(cosh(e + f*x)**2 + 2*cosh(e
+ f*x) + 1), x) + Integral(3*c*d**2*x**2/(cosh(e + f*x)**2 + 2*cosh(e + f*x) + 1), x) + Integral(3*c**2*d*x/(c
osh(e + f*x)**2 + 2*cosh(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 \cosh \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*cosh(f*x+e))^2,x, algorithm="giac")

[Out]

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