\(\int \frac {x}{(a+a \cosh (x))^{3/2}} \, dx\) [146]

Optimal result

Integrand size = 12, antiderivative size = 140 \[ \int \frac {x}{(a+a \cosh (x))^{3/2}} \, dx=\frac {1}{a \sqrt {a+a \cosh (x)}}+\frac {x \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a+a \cosh (x)}}-\frac {i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {x \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a+a \cosh (x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3400, 4270, 4265, 2317, 2438} \[ \int \frac {x}{(a+a \cosh (x))^{3/2}} \, dx=\frac {x \arctan \left (e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}-\frac {i \operatorname {PolyLog}\left (2,-i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {i \operatorname {PolyLog}\left (2,i e^{x/2}\right ) \cosh \left (\frac {x}{2}\right )}{a \sqrt {a \cosh (x)+a}}+\frac {1}{a \sqrt {a \cosh (x)+a}}+\frac {x \tanh \left (\frac {x}{2}\right )}{2 a \sqrt {a \cosh (x)+a}} \]

[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 2317

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 2438

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]

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 4270

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] &&
GtQ[n, 1] && NeQ[n, 2]

Rubi steps \begin{align*} \text {integral}& = \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 \arctan \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 \arctan \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 ) \text {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 ) \text {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 \arctan \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 ) \operatorname {PolyLog}\left (2,-i e^{x/2}\right )}{a \sqrt {a+a \cosh (x)}}+\frac {i \cosh \left (\frac {x}{2}\right ) \operatorname {PolyLog}\left (2,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] (verified)

Time = 0.15 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.79 \[ \int \frac {x}{(a+a \cosh (x))^{3/2}} \, dx=\frac {\cosh \left (\frac {x}{2}\right ) \left (2 \cosh \left (\frac {x}{2}\right )+i \cosh ^2\left (\frac {x}{2}\right ) \left (x \left (\log \left (1-i e^{x/2}\right )-\log \left (1+i e^{x/2}\right )\right )-2 \operatorname {PolyLog}\left (2,-i e^{x/2}\right )+2 \operatorname {PolyLog}\left (2,i e^{x/2}\right )\right )+x \sinh \left (\frac {x}{2}\right )\right )}{(a (1+\cosh (x)))^{3/2}} \]

[In]

Integrate[x/(a + a*Cosh[x])^(3/2),x]

[Out]

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

Maple [F]

\[\int \frac {x}{\left (a +a \cosh \left (x \right )\right )^{\frac {3}{2}}}d x\]

[In]

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

[Out]

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

Fricas [F]

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

[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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[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)

Giac [F]

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

[In]

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

[Out]

integrate(x/(a*cosh(x) + a)^(3/2), x)

Mupad [F(-1)]

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

[In]

int(x/(a + a*cosh(x))^(3/2),x)

[Out]

int(x/(a + a*cosh(x))^(3/2), x)