\(\int \frac {(c+d x)^3}{(a+a \cosh (e+f x))^2} \, dx\) [116]

Optimal result

Integrand size = 20, antiderivative size = 255 \[ \int \frac {(c+d x)^3}{(a+a \cosh (e+f x))^2} \, dx=\frac {(c+d x)^3}{3 a^2 f}-\frac {2 d (c+d x)^2 \log \left (1+e^{e+f x}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {4 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{e+f x}\right )}{a^2 f^3}+\frac {4 d^3 \operatorname {PolyLog}\left (3,-e^{e+f x}\right )}{a^2 f^4}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3399, 4271, 4269, 3556, 3799, 2221, 2611, 2320, 6724} \[ \int \frac {(c+d x)^3}{(a+a \cosh (e+f x))^2} \, dx=-\frac {4 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{e+f x}\right )}{a^2 f^3}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}-\frac {2 d (c+d x)^2 \log \left (e^{e+f x}+1\right )}{a^2 f^2}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {(c+d x)^3}{3 a^2 f}+\frac {4 d^3 \operatorname {PolyLog}\left (3,-e^{e+f x}\right )}{a^2 f^4}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4} \]

[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 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 3556

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

Rule 3799

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 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 4271

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]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

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 ^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) \operatorname {PolyLog}\left (2,-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 \operatorname {PolyLog}\left (2,-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) \operatorname {PolyLog}\left (2,-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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(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) \operatorname {PolyLog}\left (2,-e^{e+f x}\right )}{a^2 f^3}+\frac {4 d^3 \operatorname {PolyLog}\left (3,-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] (verified)

Time = 2.01 (sec) , antiderivative size = 506, normalized size of antiderivative = 1.98 \[ \int \frac {(c+d x)^3}{(a+a \cosh (e+f x))^2} \, dx=\frac {\cosh \left (\frac {1}{2} (e+f x)\right ) \left (-\frac {8 d \cosh ^3\left (\frac {1}{2} (e+f x)\right ) \left (6 d^2 e^e f x-3 c^2 e^e f^3 x+3 c d f^3 x^2+d^2 f^3 x^3+6 c d f^2 x \log \left (1+e^{-e-f x}\right )+6 c d e^e f^2 x \log \left (1+e^{-e-f x}\right )+3 d^2 f^2 x^2 \log \left (1+e^{-e-f x}\right )+3 d^2 e^e f^2 x^2 \log \left (1+e^{-e-f x}\right )-6 d^2 \log \left (1+e^{e+f x}\right )-6 d^2 e^e \log \left (1+e^{e+f x}\right )+3 c^2 f^2 \log \left (1+e^{e+f x}\right )+3 c^2 e^e f^2 \log \left (1+e^{e+f x}\right )-6 d \left (1+e^e\right ) f (c+d x) \operatorname {PolyLog}\left (2,-e^{-e-f x}\right )-6 d^2 \left (1+e^e\right ) \operatorname {PolyLog}\left (3,-e^{-e-f x}\right )\right )}{\left (1+e^e\right ) f}+(c+d x) \text {sech}\left (\frac {e}{2}\right ) \left (3 d f (c+d x) \cosh \left (\frac {f x}{2}\right )+3 d f (c+d x) \cosh \left (e+\frac {f x}{2}\right )-12 d^2 \sinh \left (\frac {f x}{2}\right )+3 c^2 f^2 \sinh \left (\frac {f x}{2}\right )+6 c d f^2 x \sinh \left (\frac {f x}{2}\right )+3 d^2 f^2 x^2 \sinh \left (\frac {f x}{2}\right )+6 d^2 \sinh \left (e+\frac {f x}{2}\right )-6 d^2 \sinh \left (e+\frac {3 f x}{2}\right )+c^2 f^2 \sinh \left (e+\frac {3 f x}{2}\right )+2 c d f^2 x \sinh \left (e+\frac {3 f x}{2}\right )+d^2 f^2 x^2 \sinh \left (e+\frac {3 f x}{2}\right )\right )\right )}{3 a^2 f^3 (1+\cosh (e+f x))^2} \]

[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*E^e*f*x - 3*c^2*E^e*f^3*x + 3*c*d*f^3*x^2 + d^2*f^3*x^3 +
6*c*d*f^2*x*Log[1 + E^(-e - f*x)] + 6*c*d*E^e*f^2*x*Log[1 + E^(-e - f*x)] + 3*d^2*f^2*x^2*Log[1 + E^(-e - f*x
)] + 3*d^2*E^e*f^2*x^2*Log[1 + E^(-e - f*x)] - 6*d^2*Log[1 + E^(e + f*x)] - 6*d^2*E^e*Log[1 + E^(e + f*x)] + 3
*c^2*f^2*Log[1 + E^(e + f*x)] + 3*c^2*E^e*f^2*Log[1 + E^(e + f*x)] - 6*d*(1 + E^e)*f*(c + d*x)*PolyLog[2, -E^(
-e - f*x)] - 6*d^2*(1 + E^e)*PolyLog[3, -E^(-e - f*x)]))/((1 + E^e)*f) + (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 + Co
sh[e + f*x])^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(599\) vs. \(2(220)=440\).

Time = 0.23 (sec) , antiderivative size = 600, normalized size of antiderivative = 2.35

method	result	size
risch	\(-\frac {2 \left (3 d^{3} f^{2} x^{3} {\mathrm e}^{f x +e}+9 c \,d^{2} f^{2} x^{2} {\mathrm e}^{f x +e}+d^{3} f^{2} x^{3}-3 d^{3} f \,x^{2} {\mathrm e}^{2 f x +2 e}+9 c^{2} d \,f^{2} x \,{\mathrm e}^{f x +e}+3 c \,d^{2} f^{2} x^{2}-6 c \,d^{2} f x \,{\mathrm e}^{2 f x +2 e}-3 d^{3} f \,x^{2} {\mathrm e}^{f x +e}+3 c^{3} f^{2} {\mathrm e}^{f x +e}+3 c^{2} d \,f^{2} x -3 c^{2} d f \,{\mathrm e}^{2 f x +2 e}-6 c \,d^{2} f x \,{\mathrm e}^{f x +e}-6 d^{3} x \,{\mathrm e}^{2 f x +2 e}+c^{3} f^{2}-3 c^{2} d f \,{\mathrm e}^{f x +e}-6 c \,d^{2} {\mathrm e}^{2 f x +2 e}-12 d^{3} x \,{\mathrm e}^{f x +e}-12 c \,d^{2} {\mathrm e}^{f x +e}-6 d^{3} x -6 d^{2} c \right )}{3 f^{3} a^{2} \left (1+{\mathrm e}^{f x +e}\right )^{3}}+\frac {2 d \,c^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{a^{2} f^{2}}-\frac {2 d \,c^{2} \ln \left (1+{\mathrm e}^{f x +e}\right )}{a^{2} f^{2}}+\frac {2 d^{3} e^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{a^{2} f^{4}}+\frac {2 d^{2} c \,x^{2}}{a^{2} f}+\frac {2 d^{3} x^{3}}{3 a^{2} f}-\frac {4 d^{3} e^{3}}{3 a^{2} f^{4}}+\frac {4 d^{3} \operatorname {polylog}\left (3, -{\mathrm e}^{f x +e}\right )}{a^{2} f^{4}}-\frac {4 d^{3} \ln \left ({\mathrm e}^{f x +e}\right )}{a^{2} f^{4}}+\frac {4 d^{3} \ln \left (1+{\mathrm e}^{f x +e}\right )}{a^{2} f^{4}}+\frac {4 d^{2} c e x}{a^{2} f^{2}}-\frac {4 d^{2} c \ln \left (1+{\mathrm e}^{f x +e}\right ) x}{a^{2} f^{2}}-\frac {4 d^{2} c e \ln \left ({\mathrm e}^{f x +e}\right )}{a^{2} f^{3}}-\frac {2 d^{3} e^{2} x}{a^{2} f^{3}}-\frac {2 d^{3} \ln \left (1+{\mathrm e}^{f x +e}\right ) x^{2}}{a^{2} f^{2}}-\frac {4 d^{3} \operatorname {polylog}\left (2, -{\mathrm e}^{f x +e}\right ) x}{a^{2} f^{3}}+\frac {2 d^{2} c \,e^{2}}{a^{2} f^{3}}-\frac {4 d^{2} c \operatorname {polylog}\left (2, -{\mathrm e}^{f x +e}\right )}{a^{2} f^{3}}\)	\(600\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1863 vs. \(2 (219) = 438\).

Time = 0.27 (sec) , antiderivative size = 1863, normalized size of antiderivative = 7.31 \[ \int \frac {(c+d x)^3}{(a+a \cosh (e+f x))^2} \, dx=\text {Too large to display} \]

[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))

Sympy [F]

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

[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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 610 vs. \(2 (219) = 438\).

Time = 0.36 (sec) , antiderivative size = 610, normalized size of antiderivative = 2.39 \[ \int \frac {(c+d x)^3}{(a+a \cosh (e+f x))^2} \, dx=2 \, c^{2} d {\left (\frac {f x e^{\left (3 \, f x + 3 \, e\right )} + {\left (3 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} + e^{\left (f x + e\right )}}{a^{2} f^{2} e^{\left (3 \, f x + 3 \, e\right )} + 3 \, a^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} + 3 \, a^{2} f^{2} e^{\left (f x + e\right )} + a^{2} f^{2}} - \frac {\log \left ({\left (e^{\left (f x + e\right )} + 1\right )} e^{\left (-e\right )}\right )}{a^{2} f^{2}}\right )} + \frac {2}{3} \, c^{3} {\left (\frac {3 \, e^{\left (-f x - e\right )}}{{\left (3 \, a^{2} e^{\left (-f x - e\right )} + 3 \, a^{2} e^{\left (-2 \, f x - 2 \, e\right )} + a^{2} e^{\left (-3 \, f x - 3 \, e\right )} + a^{2}\right )} f} + \frac {1}{{\left (3 \, a^{2} e^{\left (-f x - e\right )} + 3 \, a^{2} e^{\left (-2 \, f x - 2 \, e\right )} + a^{2} e^{\left (-3 \, f x - 3 \, e\right )} + a^{2}\right )} f}\right )} - \frac {2 \, {\left (d^{3} f^{2} x^{3} + 3 \, c d^{2} f^{2} x^{2} - 6 \, d^{3} x - 6 \, c d^{2} - 3 \, {\left (d^{3} f x^{2} e^{\left (2 \, e\right )} + 2 \, c d^{2} e^{\left (2 \, e\right )} + 2 \, {\left (c d^{2} f e^{\left (2 \, e\right )} + d^{3} e^{\left (2 \, e\right )}\right )} x\right )} e^{\left (2 \, f x\right )} + 3 \, {\left (d^{3} f^{2} x^{3} e^{e} - 4 \, c d^{2} e^{e} + {\left (3 \, c d^{2} f^{2} e^{e} - d^{3} f e^{e}\right )} x^{2} - 2 \, {\left (c d^{2} f e^{e} + 2 \, d^{3} e^{e}\right )} x\right )} e^{\left (f x\right )}\right )}}{3 \, {\left (a^{2} f^{3} e^{\left (3 \, f x + 3 \, e\right )} + 3 \, a^{2} f^{3} e^{\left (2 \, f x + 2 \, e\right )} + 3 \, a^{2} f^{3} e^{\left (f x + e\right )} + a^{2} f^{3}\right )}} - \frac {4 \, {\left (f x \log \left (e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (f x + e\right )}\right )\right )} c d^{2}}{a^{2} f^{3}} - \frac {4 \, d^{3} x}{a^{2} f^{3}} - \frac {2 \, {\left (f^{2} x^{2} \log \left (e^{\left (f x + e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (-e^{\left (f x + e\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (f x + e\right )})\right )} d^{3}}{a^{2} f^{4}} + \frac {4 \, d^{3} \log \left (e^{\left (f x + e\right )} + 1\right )}{a^{2} f^{4}} + \frac {2 \, {\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2}\right )}}{3 \, a^{2} f^{4}} \]

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^3}{(a+a \cosh (e+f x))^2} \, dx=\int \frac {{\left (c+d\,x\right )}^3}{{\left (a+a\,\mathrm {cosh}\left (e+f\,x\right )\right )}^2} \,d x \]

[In]

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

[Out]

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