3.146 \(\int \frac{x}{(a+a \cosh (x))^{3/2}} \, dx\)

Optimal. Leaf size=140 \[ -\frac{i \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{i \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{1}{a \sqrt{a \cosh (x)+a}}+\frac{x \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac{x}{2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{x \tanh \left (\frac{x}{2}\right )}{2 a \sqrt{a \cosh (x)+a}} \]

[Out]

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

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Rubi [A]  time = 0.101197, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {3319, 4185, 4180, 2279, 2391} \[ -\frac{i \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{i \cosh \left (\frac{x}{2}\right ) \text{PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{1}{a \sqrt{a \cosh (x)+a}}+\frac{x \tan ^{-1}\left (e^{x/2}\right ) \cosh \left (\frac{x}{2}\right )}{a \sqrt{a \cosh (x)+a}}+\frac{x \tanh \left (\frac{x}{2}\right )}{2 a \sqrt{a \cosh (x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + a*Cosh[x])^(3/2),x]

[Out]

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

Rule 3319

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 4185

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> -Simp[(b^2*(c + d*x)*Cot[e + f*x]*
(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2
), x], x] - Simp[(b^2*d*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && G
tQ[n, 1] && NeQ[n, 2]

Rule 4180

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 2279

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

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

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

Mathematica [A]  time = 0.101678, size = 137, normalized size = 0.98 \[ \frac{2 \cosh ^3\left (\frac{x}{2}\right ) \left (-i \left (\text{PolyLog}\left (2,-i e^{-x/2}\right )-\text{PolyLog}\left (2,i e^{-x/2}\right )\right )-\frac{1}{2} i x \left (\log \left (1-i e^{-x/2}\right )-\log \left (1+i e^{-x/2}\right )\right )\right )}{(a (\cosh (x)+1))^{3/2}}+\frac{2 \cosh ^2\left (\frac{x}{2}\right )}{(a (\cosh (x)+1))^{3/2}}+\frac{x \sinh \left (\frac{x}{2}\right ) \cosh \left (\frac{x}{2}\right )}{(a (\cosh (x)+1))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + a*Cosh[x])^(3/2),x]

[Out]

(2*Cosh[x/2]^2)/(a*(1 + Cosh[x]))^(3/2) + (2*Cosh[x/2]^3*((-I/2)*x*(Log[1 - I/E^(x/2)] - Log[1 + I/E^(x/2)]) -
I*(PolyLog[2, (-I)/E^(x/2)] - PolyLog[2, I/E^(x/2)])))/(a*(1 + Cosh[x]))^(3/2) + (x*Cosh[x/2]*Sinh[x/2])/(a*(
1 + Cosh[x]))^(3/2)

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Maple [F]  time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{x \left ( a+a\cosh \left ( x \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{9} \, \sqrt{2}{\left (\frac{3 \, e^{\left (\frac{5}{2} \, x\right )} + 8 \, e^{\left (\frac{3}{2} \, x\right )} - 3 \, e^{\left (\frac{1}{2} \, x\right )}}{a^{\frac{3}{2}} e^{\left (3 \, x\right )} + 3 \, a^{\frac{3}{2}} e^{\left (2 \, x\right )} + 3 \, a^{\frac{3}{2}} e^{x} + a^{\frac{3}{2}}} + \frac{3 \, \arctan \left (e^{\left (\frac{1}{2} \, x\right )}\right )}{a^{\frac{3}{2}}}\right )} + 12 \, \sqrt{2} \int \frac{x e^{\left (\frac{3}{2} \, x\right )}}{3 \,{\left (a^{\frac{3}{2}} e^{\left (4 \, x\right )} + 4 \, a^{\frac{3}{2}} e^{\left (3 \, x\right )} + 6 \, a^{\frac{3}{2}} e^{\left (2 \, x\right )} + 4 \, a^{\frac{3}{2}} e^{x} + a^{\frac{3}{2}}\right )}}\,{d x} - \frac{4 \,{\left (3 \, \sqrt{2} \sqrt{a} x + 2 \, \sqrt{2} \sqrt{a}\right )} e^{\left (\frac{3}{2} \, x\right )}}{9 \,{\left (a^{2} e^{\left (3 \, x\right )} + 3 \, a^{2} e^{\left (2 \, x\right )} + 3 \, a^{2} e^{x} + a^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*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)) + 12*sqrt(2)*integrate(1/3*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) - 4/9*(3*sqrt(2)*sqrt(a)*x + 2*sqrt(2)*sqrt(a))*e^(3/
2*x)/(a^2*e^(3*x) + 3*a^2*e^(2*x) + 3*a^2*e^x + a^2)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a \cosh \left (x\right ) + a} x}{a^{2} \cosh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (a \cosh \left (x\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/(a*cosh(x) + a)^(3/2), x)