\(\int \frac {x^3}{\sqrt {a+a \cosh (c+d x)}} \, dx\) [139]

Optimal result

Integrand size = 18, antiderivative size = 383 \[ \int \frac {x^3}{\sqrt {a+a \cosh (c+d x)}} \, dx=\frac {4 x^3 \arctan \left (e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a+a \cosh (c+d x)}}-\frac {12 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2 \sqrt {a+a \cosh (c+d x)}}+\frac {12 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2 \sqrt {a+a \cosh (c+d x)}}+\frac {48 i x \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^3 \sqrt {a+a \cosh (c+d x)}}-\frac {48 i x \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^3 \sqrt {a+a \cosh (c+d x)}}-\frac {96 i \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (4,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^4 \sqrt {a+a \cosh (c+d x)}}+\frac {96 i \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (4,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^4 \sqrt {a+a \cosh (c+d x)}} \]

[Out]

4*x^3*arctan(exp(1/2*d*x+1/2*c))*cosh(1/2*d*x+1/2*c)/d/(a+a*cosh(d*x+c))^(1/2)-12*I*x^2*cosh(1/2*d*x+1/2*c)*po
lylog(2,-I*exp(1/2*d*x+1/2*c))/d^2/(a+a*cosh(d*x+c))^(1/2)+12*I*x^2*cosh(1/2*d*x+1/2*c)*polylog(2,I*exp(1/2*d*
x+1/2*c))/d^2/(a+a*cosh(d*x+c))^(1/2)+48*I*x*cosh(1/2*d*x+1/2*c)*polylog(3,-I*exp(1/2*d*x+1/2*c))/d^3/(a+a*cos
h(d*x+c))^(1/2)-48*I*x*cosh(1/2*d*x+1/2*c)*polylog(3,I*exp(1/2*d*x+1/2*c))/d^3/(a+a*cosh(d*x+c))^(1/2)-96*I*co
sh(1/2*d*x+1/2*c)*polylog(4,-I*exp(1/2*d*x+1/2*c))/d^4/(a+a*cosh(d*x+c))^(1/2)+96*I*cosh(1/2*d*x+1/2*c)*polylo
g(4,I*exp(1/2*d*x+1/2*c))/d^4/(a+a*cosh(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3400, 4265, 2611, 6744, 2320, 6724} \[ \int \frac {x^3}{\sqrt {a+a \cosh (c+d x)}} \, dx=\frac {4 x^3 \arctan \left (e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a \cosh (c+d x)+a}}-\frac {96 i \operatorname {PolyLog}\left (4,-i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^4 \sqrt {a \cosh (c+d x)+a}}+\frac {96 i \operatorname {PolyLog}\left (4,i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^4 \sqrt {a \cosh (c+d x)+a}}+\frac {48 i x \operatorname {PolyLog}\left (3,-i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^3 \sqrt {a \cosh (c+d x)+a}}-\frac {48 i x \operatorname {PolyLog}\left (3,i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^3 \sqrt {a \cosh (c+d x)+a}}-\frac {12 i x^2 \operatorname {PolyLog}\left (2,-i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^2 \sqrt {a \cosh (c+d x)+a}}+\frac {12 i x^2 \operatorname {PolyLog}\left (2,i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^2 \sqrt {a \cosh (c+d x)+a}} \]

[In]

Int[x^3/Sqrt[a + a*Cosh[c + d*x]],x]

[Out]

(4*x^3*ArcTan[E^(c/2 + (d*x)/2)]*Cosh[c/2 + (d*x)/2])/(d*Sqrt[a + a*Cosh[c + d*x]]) - ((12*I)*x^2*Cosh[c/2 + (
d*x)/2]*PolyLog[2, (-I)*E^(c/2 + (d*x)/2)])/(d^2*Sqrt[a + a*Cosh[c + d*x]]) + ((12*I)*x^2*Cosh[c/2 + (d*x)/2]*
PolyLog[2, I*E^(c/2 + (d*x)/2)])/(d^2*Sqrt[a + a*Cosh[c + d*x]]) + ((48*I)*x*Cosh[c/2 + (d*x)/2]*PolyLog[3, (-
I)*E^(c/2 + (d*x)/2)])/(d^3*Sqrt[a + a*Cosh[c + d*x]]) - ((48*I)*x*Cosh[c/2 + (d*x)/2]*PolyLog[3, I*E^(c/2 + (
d*x)/2)])/(d^3*Sqrt[a + a*Cosh[c + d*x]]) - ((96*I)*Cosh[c/2 + (d*x)/2]*PolyLog[4, (-I)*E^(c/2 + (d*x)/2)])/(d
^4*Sqrt[a + a*Cosh[c + d*x]]) + ((96*I)*Cosh[c/2 + (d*x)/2]*PolyLog[4, I*E^(c/2 + (d*x)/2)])/(d^4*Sqrt[a + a*C
osh[c + d*x]])

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 3400

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(2*a)^IntPart[n]
*((a + b*Sin[e + f*x])^FracPart[n]/Sin[e/2 + a*(Pi/(4*b)) + f*(x/2)]^(2*FracPart[n])), Int[(c + d*x)^m*Sin[e/2
+ a*(Pi/(4*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
+ 1/2] && (GtQ[n, 0] || IGtQ[m, 0])

Rule 4265

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c +
d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^(I*k*Pi)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*
Log[1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[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]

Rule 6744

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Dist[f*(m/(b*c*p*Log[F])), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sin \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right ) \int x^3 \csc \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right ) \, dx}{\sqrt {a+a \cosh (c+d x)}} \\ & = \frac {4 x^3 \arctan \left (e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a+a \cosh (c+d x)}}-\frac {\left (6 i \sin \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right )\right ) \int x^2 \log \left (1-i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \, dx}{d \sqrt {a+a \cosh (c+d x)}}+\frac {\left (6 i \sin \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right )\right ) \int x^2 \log \left (1+i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \, dx}{d \sqrt {a+a \cosh (c+d x)}} \\ & = \frac {4 x^3 \arctan \left (e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a+a \cosh (c+d x)}}-\frac {12 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2 \sqrt {a+a \cosh (c+d x)}}+\frac {12 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2 \sqrt {a+a \cosh (c+d x)}}+\frac {\left (24 i \sin \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right )\right ) \int x \operatorname {PolyLog}\left (2,-i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \, dx}{d^2 \sqrt {a+a \cosh (c+d x)}}-\frac {\left (24 i \sin \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right )\right ) \int x \operatorname {PolyLog}\left (2,i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \, dx}{d^2 \sqrt {a+a \cosh (c+d x)}} \\ & = \frac {4 x^3 \arctan \left (e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a+a \cosh (c+d x)}}-\frac {12 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2 \sqrt {a+a \cosh (c+d x)}}+\frac {12 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2 \sqrt {a+a \cosh (c+d x)}}+\frac {48 i x \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^3 \sqrt {a+a \cosh (c+d x)}}-\frac {48 i x \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^3 \sqrt {a+a \cosh (c+d x)}}-\frac {\left (48 i \sin \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right )\right ) \int \operatorname {PolyLog}\left (3,-i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \, dx}{d^3 \sqrt {a+a \cosh (c+d x)}}+\frac {\left (48 i \sin \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right )\right ) \int \operatorname {PolyLog}\left (3,i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \, dx}{d^3 \sqrt {a+a \cosh (c+d x)}} \\ & = \frac {4 x^3 \arctan \left (e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a+a \cosh (c+d x)}}-\frac {12 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2 \sqrt {a+a \cosh (c+d x)}}+\frac {12 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2 \sqrt {a+a \cosh (c+d x)}}+\frac {48 i x \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^3 \sqrt {a+a \cosh (c+d x)}}-\frac {48 i x \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^3 \sqrt {a+a \cosh (c+d x)}}-\frac {\left (96 i \sin \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right )\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^4 \sqrt {a+a \cosh (c+d x)}}+\frac {\left (96 i \sin \left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {\pi }{4}+\frac {i d x}{2}\right )\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^4 \sqrt {a+a \cosh (c+d x)}} \\ & = \frac {4 x^3 \arctan \left (e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a+a \cosh (c+d x)}}-\frac {12 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2 \sqrt {a+a \cosh (c+d x)}}+\frac {12 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2 \sqrt {a+a \cosh (c+d x)}}+\frac {48 i x \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^3 \sqrt {a+a \cosh (c+d x)}}-\frac {48 i x \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^3 \sqrt {a+a \cosh (c+d x)}}-\frac {96 i \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (4,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^4 \sqrt {a+a \cosh (c+d x)}}+\frac {96 i \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (4,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^4 \sqrt {a+a \cosh (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.56 \[ \int \frac {x^3}{\sqrt {a+a \cosh (c+d x)}} \, dx=\frac {2 i \cosh \left (\frac {1}{2} (c+d x)\right ) \left (d^3 x^3 \log \left (1-i e^{\frac {1}{2} (c+d x)}\right )-d^3 x^3 \log \left (1+i e^{\frac {1}{2} (c+d x)}\right )-6 d^2 x^2 \operatorname {PolyLog}\left (2,-i e^{\frac {1}{2} (c+d x)}\right )+6 d^2 x^2 \operatorname {PolyLog}\left (2,i e^{\frac {1}{2} (c+d x)}\right )+24 d x \operatorname {PolyLog}\left (3,-i e^{\frac {1}{2} (c+d x)}\right )-24 d x \operatorname {PolyLog}\left (3,i e^{\frac {1}{2} (c+d x)}\right )-48 \operatorname {PolyLog}\left (4,-i e^{\frac {1}{2} (c+d x)}\right )+48 \operatorname {PolyLog}\left (4,i e^{\frac {1}{2} (c+d x)}\right )\right )}{d^4 \sqrt {a (1+\cosh (c+d x))}} \]

[In]

Integrate[x^3/Sqrt[a + a*Cosh[c + d*x]],x]

[Out]

((2*I)*Cosh[(c + d*x)/2]*(d^3*x^3*Log[1 - I*E^((c + d*x)/2)] - d^3*x^3*Log[1 + I*E^((c + d*x)/2)] - 6*d^2*x^2*
PolyLog[2, (-I)*E^((c + d*x)/2)] + 6*d^2*x^2*PolyLog[2, I*E^((c + d*x)/2)] + 24*d*x*PolyLog[3, (-I)*E^((c + d*
x)/2)] - 24*d*x*PolyLog[3, I*E^((c + d*x)/2)] - 48*PolyLog[4, (-I)*E^((c + d*x)/2)] + 48*PolyLog[4, I*E^((c +
d*x)/2)]))/(d^4*Sqrt[a*(1 + Cosh[c + d*x])])

Maple [F]

\[\int \frac {x^{3}}{\sqrt {a +a \cosh \left (d x +c \right )}}d x\]

[In]

int(x^3/(a+a*cosh(d*x+c))^(1/2),x)

[Out]

int(x^3/(a+a*cosh(d*x+c))^(1/2),x)

Fricas [F]

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

[In]

integrate(x^3/(a+a*cosh(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(x^3/sqrt(a*cosh(d*x + c) + a), x)

Sympy [F]

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

[In]

integrate(x**3/(a+a*cosh(d*x+c))**(1/2),x)

[Out]

Integral(x**3/sqrt(a*(cosh(c + d*x) + 1)), x)

Maxima [F]

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

[In]

integrate(x^3/(a+a*cosh(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(2)*d^3*integrate(x^3*e^(1/2*d*x + 1/2*c)/(sqrt(a)*d^3*e^(2*d*x + 2*c) + 2*sqrt(a)*d^3*e^(d*x + c) + sqr
t(a)*d^3), x) + 12*sqrt(2)*d^2*integrate(x^2*e^(1/2*d*x + 1/2*c)/(sqrt(a)*d^3*e^(2*d*x + 2*c) + 2*sqrt(a)*d^3*
e^(d*x + c) + sqrt(a)*d^3), x) + 48*sqrt(2)*d*integrate(x*e^(1/2*d*x + 1/2*c)/(sqrt(a)*d^3*e^(2*d*x + 2*c) + 2
*sqrt(a)*d^3*e^(d*x + c) + sqrt(a)*d^3), x) + 96*sqrt(2)*(e^(1/2*d*x + 1/2*c)/((sqrt(a)*d^3*e^(d*x + c) + sqrt
(a)*d^3)*d) + arctan(e^(1/2*d*x + 1/2*c))/(sqrt(a)*d^4)) - 2*(sqrt(2)*sqrt(a)*d^3*x^3*e^(1/2*c) + 6*sqrt(2)*sq
rt(a)*d^2*x^2*e^(1/2*c) + 24*sqrt(2)*sqrt(a)*d*x*e^(1/2*c) + 48*sqrt(2)*sqrt(a)*e^(1/2*c))*e^(1/2*d*x)/(a*d^4*
e^(d*x + c) + a*d^4)

Giac [F]

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

[In]

integrate(x^3/(a+a*cosh(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(x^3/sqrt(a*cosh(d*x + c) + a), x)

Mupad [F(-1)]

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

[In]

int(x^3/(a + a*cosh(c + d*x))^(1/2),x)

[Out]

int(x^3/(a + a*cosh(c + d*x))^(1/2), x)