3.2.45 \(\int \frac {x^2}{(a+a \cosh (x))^{3/2}} \, dx\) [145]

Optimal. Leaf size=248 \[ \frac {2 x}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \text {ArcTan}\left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {4 \text {ArcTan}\left (\sinh \left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {2 i x \cosh \left (\frac {x}{2}\right ) \text {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {2 i x \cosh \left (\frac {x}{2}\right ) \text {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {4 i \cosh \left (\frac {x}{2}\right ) \text {PolyLog}\left (3,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {4 i \cosh \left (\frac {x}{2}\right ) \text {PolyLog}\left (3,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}} \]

[Out]

2*x/a/(a+a*cosh(x))^(1/2)+x^2*arctan(exp(1/2*x))*cosh(1/2*x)/a/(a+a*cosh(x))^(1/2)-4*arctan(sinh(1/2*x))*cosh(
1/2*x)/a/(a+a*cosh(x))^(1/2)-2*I*x*cosh(1/2*x)*polylog(2,-I*exp(1/2*x))/a/(a+a*cosh(x))^(1/2)+2*I*x*cosh(1/2*x
)*polylog(2,I*exp(1/2*x))/a/(a+a*cosh(x))^(1/2)+4*I*cosh(1/2*x)*polylog(3,-I*exp(1/2*x))/a/(a+a*cosh(x))^(1/2)
-4*I*cosh(1/2*x)*polylog(3,I*exp(1/2*x))/a/(a+a*cosh(x))^(1/2)+1/2*x^2*tanh(1/2*x)/a/(a+a*cosh(x))^(1/2)

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Rubi [A]
time = 0.13, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3400, 4271, 3855, 4265, 2611, 2320, 6724} \begin {gather*} \frac {x^2 \text {ArcTan}\left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}-\frac {4 \cosh \left (\frac {x}{2}\right ) \text {ArcTan}\left (\sinh \left (\frac {x}{2}\right )\right )}{a \sqrt {a \cosh (x)+a}}-\frac {2 i x \text {Li}_2\left (-i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {2 i x \text {Li}_2\left (i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {4 i \text {Li}_3\left (-i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}-\frac {4 i \text {Li}_3\left (i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {x^2 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a \cosh (x)+a}}+\frac {2 x}{a \sqrt {a \cosh (x)+a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(a + a*Cosh[x])^(3/2),x]

[Out]

(2*x)/(a*Sqrt[a + a*Cosh[x]]) + (x^2*ArcTan[E^(x/2)]*Cosh[x/2])/(a*Sqrt[a + a*Cosh[x]]) - (4*ArcTan[Sinh[x/2]]
*Cosh[x/2])/(a*Sqrt[a + a*Cosh[x]]) - ((2*I)*x*Cosh[x/2]*PolyLog[2, (-I)*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) + (
(2*I)*x*Cosh[x/2]*PolyLog[2, I*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) + ((4*I)*Cosh[x/2]*PolyLog[3, (-I)*E^(x/2)])/
(a*Sqrt[a + a*Cosh[x]]) - ((4*I)*Cosh[x/2]*PolyLog[3, I*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) + (x^2*Tanh[x/2])/(2
*a*Sqrt[a + a*Cosh[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 3855

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

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 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*} \int \frac {x^2}{(a+a \cosh (x))^{3/2}} \, dx &=\frac {\cosh \left (\frac {x}{2}\right ) \int x^2 \text {sech}^3\left (\frac {x}{2}\right ) \, dx}{2 a \sqrt {a+a \cosh (x)}}\\ &=\frac {2 x}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}}+\frac {\cosh \left (\frac {x}{2}\right ) \int x^2 \text {sech}\left (\frac {x}{2}\right ) \, dx}{4 a \sqrt {a+a \cosh (x)}}-\frac {\left (2 \cosh \left (\frac {x}{2}\right )\right ) \int \text {sech}\left (\frac {x}{2}\right ) \, dx}{a \sqrt {a+a \cosh (x)}}\\ &=\frac {2 x}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {4 \tan ^{-1}\left (\sinh \left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}}-\frac {\left (i \cosh \left (\frac {x}{2}\right )\right ) \int x \log \left (1-i e^{x/2}\right ) \, dx}{a \sqrt {a+a \cosh (x)}}+\frac {\left (i \cosh \left (\frac {x}{2}\right )\right ) \int x \log \left (1+i e^{x/2}\right ) \, dx}{a \sqrt {a+a \cosh (x)}}\\ &=\frac {2 x}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {4 \tan ^{-1}\left (\sinh \left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {2 i x \cosh \left (\frac {x}{2}\right ) \text {Li}_2\left (-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {2 i x \cosh \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}}+\frac {\left (2 i \cosh \left (\frac {x}{2}\right )\right ) \int \text {Li}_2\left (-i e^{x/2}\right ) \, dx}{a \sqrt {a+a \cosh (x)}}-\frac {\left (2 i \cosh \left (\frac {x}{2}\right )\right ) \int \text {Li}_2\left (i e^{x/2}\right ) \, dx}{a \sqrt {a+a \cosh (x)}}\\ &=\frac {2 x}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {4 \tan ^{-1}\left (\sinh \left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {2 i x \cosh \left (\frac {x}{2}\right ) \text {Li}_2\left (-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {2 i x \cosh \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}}+\frac {\left (4 i \cosh \left (\frac {x}{2}\right )\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {\left (4 i \cosh \left (\frac {x}{2}\right )\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}\\ &=\frac {2 x}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {4 \tan ^{-1}\left (\sinh \left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {2 i x \cosh \left (\frac {x}{2}\right ) \text {Li}_2\left (-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {2 i x \cosh \left (\frac {x}{2}\right ) \text {Li}_2\left (i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {4 i \cosh \left (\frac {x}{2}\right ) \text {Li}_3\left (-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {4 i \cosh \left (\frac {x}{2}\right ) \text {Li}_3\left (i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x^2 \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}}\\ \end {align*}

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Mathematica [A]
time = 0.69, size = 214, normalized size = 0.86 \begin {gather*} \frac {\cosh \left (\frac {x}{2}\right ) \left (4 x \cosh \left (\frac {x}{2}\right )-16 \text {ArcTan}\left (\cosh \left (\frac {x}{2}\right )+\sinh \left (\frac {x}{2}\right )\right ) \cosh ^2\left (\frac {x}{2}\right )+2 x^2 \text {ArcTan}\left (\cosh \left (\frac {x}{2}\right )+\sinh \left (\frac {x}{2}\right )\right ) \cosh ^2\left (\frac {x}{2}\right )-4 i x \cosh ^2\left (\frac {x}{2}\right ) \text {PolyLog}\left (2,-i \left (\cosh \left (\frac {x}{2}\right )+\sinh \left (\frac {x}{2}\right )\right )\right )+4 i x \cosh ^2\left (\frac {x}{2}\right ) \text {PolyLog}\left (2,i \left (\cosh \left (\frac {x}{2}\right )+\sinh \left (\frac {x}{2}\right )\right )\right )+8 i \cosh ^2\left (\frac {x}{2}\right ) \text {PolyLog}\left (3,-i \left (\cosh \left (\frac {x}{2}\right )+\sinh \left (\frac {x}{2}\right )\right )\right )-8 i \cosh ^2\left (\frac {x}{2}\right ) \text {PolyLog}\left (3,i \left (\cosh \left (\frac {x}{2}\right )+\sinh \left (\frac {x}{2}\right )\right )\right )+x^2 \sinh \left (\frac {x}{2}\right )\right )}{(a (1+\cosh (x)))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + a*Cosh[x])^(3/2),x]

[Out]

(Cosh[x/2]*(4*x*Cosh[x/2] - 16*ArcTan[Cosh[x/2] + Sinh[x/2]]*Cosh[x/2]^2 + 2*x^2*ArcTan[Cosh[x/2] + Sinh[x/2]]
*Cosh[x/2]^2 - (4*I)*x*Cosh[x/2]^2*PolyLog[2, (-I)*(Cosh[x/2] + Sinh[x/2])] + (4*I)*x*Cosh[x/2]^2*PolyLog[2, I
*(Cosh[x/2] + Sinh[x/2])] + (8*I)*Cosh[x/2]^2*PolyLog[3, (-I)*(Cosh[x/2] + Sinh[x/2])] - (8*I)*Cosh[x/2]^2*Pol
yLog[3, I*(Cosh[x/2] + Sinh[x/2])] + x^2*Sinh[x/2]))/(a*(1 + Cosh[x]))^(3/2)

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Maple [F]
time = 0.33, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (a +a \cosh \left (x \right )\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/27*sqrt(2)*((3*e^(5/2*x) + 8*e^(3/2*x) - 3*e^(1/2*x))/(a^(3/2)*e^(3*x) + 3*a^(3/2)*e^(2*x) + 3*a^(3/2)*e^x +
a^(3/2)) + 3*arctan(e^(1/2*x))/a^(3/2)) + 36*sqrt(2)*integrate(1/9*x^2*e^(3/2*x)/(a^(3/2)*e^(4*x) + 4*a^(3/2)
*e^(3*x) + 6*a^(3/2)*e^(2*x) + 4*a^(3/2)*e^x + a^(3/2)), x) + 48*sqrt(2)*integrate(1/9*x*e^(3/2*x)/(a^(3/2)*e^
(4*x) + 4*a^(3/2)*e^(3*x) + 6*a^(3/2)*e^(2*x) + 4*a^(3/2)*e^x + a^(3/2)), x) - 4/27*(9*sqrt(2)*x^2 + 12*sqrt(2
)*x + 8*sqrt(2))*e^(3/2*x)/(a^(3/2)*e^(3*x) + 3*a^(3/2)*e^(2*x) + 3*a^(3/2)*e^x + a^(3/2))

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\left (a \left (\cosh {\left (x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/(a*(cosh(x) + 1))**(3/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/(a*cosh(x) + a)^(3/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2}{{\left (a+a\,\mathrm {cosh}\left (x\right )\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + a*cosh(x))^(3/2),x)

[Out]

int(x^2/(a + a*cosh(x))^(3/2), x)

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