\(\int \frac {x^2}{\sqrt {a+a \cosh (c+d x)}} \, dx\) [140]

Optimal result

Integrand size = 18, antiderivative size = 269 \[ \int \frac {x^2}{\sqrt {a+a \cosh (c+d x)}} \, dx=\frac {4 x^2 \arctan \left (e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a+a \cosh (c+d x)}}-\frac {8 i x \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2 \sqrt {a+a \cosh (c+d x)}}+\frac {8 i x \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^2 \sqrt {a+a \cosh (c+d x)}}+\frac {16 i \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,-i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^3 \sqrt {a+a \cosh (c+d x)}}-\frac {16 i \cosh \left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,i e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^3 \sqrt {a+a \cosh (c+d x)}} \]

[Out]

4*x^2*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)-8*I*x*cosh(1/2*d*x+1/2*c)*polyl
og(2,-I*exp(1/2*d*x+1/2*c))/d^2/(a+a*cosh(d*x+c))^(1/2)+8*I*x*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)+16*I*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)-16*I*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)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3400, 4265, 2611, 2320, 6724} \[ \int \frac {x^2}{\sqrt {a+a \cosh (c+d x)}} \, dx=\frac {4 x^2 \arctan \left (e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d \sqrt {a \cosh (c+d x)+a}}+\frac {16 i \operatorname {PolyLog}\left (3,-i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^3 \sqrt {a \cosh (c+d x)+a}}-\frac {16 i \operatorname {PolyLog}\left (3,i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^3 \sqrt {a \cosh (c+d x)+a}}-\frac {8 i x \operatorname {PolyLog}\left (2,-i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^2 \sqrt {a \cosh (c+d x)+a}}+\frac {8 i x \operatorname {PolyLog}\left (2,i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^2 \sqrt {a \cosh (c+d x)+a}} \]

[In]

Int[x^2/Sqrt[a + a*Cosh[c + d*x]],x]

[Out]

(4*x^2*ArcTan[E^(c/2 + (d*x)/2)]*Cosh[c/2 + (d*x)/2])/(d*Sqrt[a + a*Cosh[c + d*x]]) - ((8*I)*x*Cosh[c/2 + (d*x
)/2]*PolyLog[2, (-I)*E^(c/2 + (d*x)/2)])/(d^2*Sqrt[a + a*Cosh[c + d*x]]) + ((8*I)*x*Cosh[c/2 + (d*x)/2]*PolyLo
g[2, I*E^(c/2 + (d*x)/2)])/(d^2*Sqrt[a + a*Cosh[c + d*x]]) + ((16*I)*Cosh[c/2 + (d*x)/2]*PolyLog[3, (-I)*E^(c/
2 + (d*x)/2)])/(d^3*Sqrt[a + a*Cosh[c + d*x]]) - ((16*I)*Cosh[c/2 + (d*x)/2]*PolyLog[3, I*E^(c/2 + (d*x)/2)])/
(d^3*Sqrt[a + a*Cosh[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]

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^2 \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^2 \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 (4 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 \log \left (1-i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \, dx}{d \sqrt {a+a \cosh (c+d x)}}+\frac {\left (4 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 \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^2 \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 {8 i x \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 {8 i x \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 (8 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 (2,-i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \, dx}{d^2 \sqrt {a+a \cosh (c+d x)}}-\frac {\left (8 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 (2,i e^{\frac {c}{2}+\frac {d x}{2}}\right ) \, dx}{d^2 \sqrt {a+a \cosh (c+d x)}} \\ & = \frac {4 x^2 \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 {8 i x \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 {8 i x \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 (16 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}(2,-i x)}{x} \, dx,x,e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^3 \sqrt {a+a \cosh (c+d x)}}-\frac {\left (16 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}(2,i x)}{x} \, dx,x,e^{\frac {c}{2}+\frac {d x}{2}}\right )}{d^3 \sqrt {a+a \cosh (c+d x)}} \\ & = \frac {4 x^2 \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 {8 i x \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 {8 i x \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 {16 i \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 {16 i \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)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.61 \[ \int \frac {x^2}{\sqrt {a+a \cosh (c+d x)}} \, dx=\frac {2 i \cosh \left (\frac {1}{2} (c+d x)\right ) \left (d^2 x^2 \log \left (1-i e^{\frac {1}{2} (c+d x)}\right )-d^2 x^2 \log \left (1+i e^{\frac {1}{2} (c+d x)}\right )-4 d x \operatorname {PolyLog}\left (2,-i e^{\frac {1}{2} (c+d x)}\right )+4 d x \operatorname {PolyLog}\left (2,i e^{\frac {1}{2} (c+d x)}\right )+8 \operatorname {PolyLog}\left (3,-i e^{\frac {1}{2} (c+d x)}\right )-8 \operatorname {PolyLog}\left (3,i e^{\frac {1}{2} (c+d x)}\right )\right )}{d^3 \sqrt {a (1+\cosh (c+d x))}} \]

[In]

Integrate[x^2/Sqrt[a + a*Cosh[c + d*x]],x]

[Out]

((2*I)*Cosh[(c + d*x)/2]*(d^2*x^2*Log[1 - I*E^((c + d*x)/2)] - d^2*x^2*Log[1 + I*E^((c + d*x)/2)] - 4*d*x*Poly
Log[2, (-I)*E^((c + d*x)/2)] + 4*d*x*PolyLog[2, I*E^((c + d*x)/2)] + 8*PolyLog[3, (-I)*E^((c + d*x)/2)] - 8*Po
lyLog[3, I*E^((c + d*x)/2)]))/(d^3*Sqrt[a*(1 + Cosh[c + d*x])])

Maple [F]

\[\int \frac {x^{2}}{\sqrt {a +a \cosh \left (d x +c \right )}}d x\]

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(x^2/sqrt(a*cosh(d*x + c) + a), x)

Sympy [F]

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

[In]

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

[Out]

Integral(x**2/sqrt(a*(cosh(c + d*x) + 1)), x)

Maxima [F]

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

[In]

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

[Out]

2*sqrt(2)*d^2*integrate(x^2*e^(1/2*d*x + 1/2*c)/(sqrt(a)*d^2*e^(2*d*x + 2*c) + 2*sqrt(a)*d^2*e^(d*x + c) + sqr
t(a)*d^2), x) + 8*sqrt(2)*d*integrate(x*e^(1/2*d*x + 1/2*c)/(sqrt(a)*d^2*e^(2*d*x + 2*c) + 2*sqrt(a)*d^2*e^(d*
x + c) + sqrt(a)*d^2), x) + 16*sqrt(2)*(e^(1/2*d*x + 1/2*c)/((sqrt(a)*d^2*e^(d*x + c) + sqrt(a)*d^2)*d) + arct
an(e^(1/2*d*x + 1/2*c))/(sqrt(a)*d^3)) - 2*(sqrt(2)*d^2*x^2*e^(1/2*c) + 4*sqrt(2)*d*x*e^(1/2*c) + 8*sqrt(2)*e^
(1/2*c))*e^(1/2*d*x)/(sqrt(a)*d^3*e^(d*x + c) + sqrt(a)*d^3)

Giac [F]

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

[In]

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

[Out]

integrate(x^2/sqrt(a*cosh(d*x + c) + a), x)

Mupad [F(-1)]

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

[In]

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

[Out]

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