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

Optimal result

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

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)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.79 \[ \int \frac {x}{(a+a \cosh (x))^{3/2}} \, dx=\frac {\cosh \left (\frac {x}{2}\right ) \left (2 \cosh \left (\frac {x}{2}\right )+i \cosh ^2\left (\frac {x}{2}\right ) \left (x \left (\log \left (1-i e^{x/2}\right )-\log \left (1+i e^{x/2}\right )\right )-2 \operatorname {PolyLog}\left (2,-i e^{x/2}\right )+2 \operatorname {PolyLog}\left (2,i e^{x/2}\right )\right )+x \sinh \left (\frac {x}{2}\right )\right )}{(a (1+\cosh (x)))^{3/2}} \] Input:

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

Output:

(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*Sin 
h[x/2]))/(a*(1 + Cosh[x]))^(3/2)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.69, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3800, 3042, 4673, 3042, 4668, 2715, 2838}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {x}{\left (a+a \sin \left (\frac {\pi }{2}+i x\right )\right )^{3/2}}dx\)

\(\Big \downarrow \) 3800

\(\displaystyle \frac {\cosh \left (\frac {x}{2}\right ) \int x \text {sech}^3\left (\frac {x}{2}\right )dx}{2 a \sqrt {a \cosh (x)+a}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\cosh \left (\frac {x}{2}\right ) \int x \csc \left (\frac {i x}{2}+\frac {\pi }{2}\right )^3dx}{2 a \sqrt {a \cosh (x)+a}}\)

\(\Big \downarrow \) 4673

\(\displaystyle \frac {\cosh \left (\frac {x}{2}\right ) \left (\frac {1}{2} \int x \text {sech}\left (\frac {x}{2}\right )dx+2 \text {sech}\left (\frac {x}{2}\right )+x \tanh \left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )\right )}{2 a \sqrt {a \cosh (x)+a}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\cosh \left (\frac {x}{2}\right ) \left (\frac {1}{2} \int x \csc \left (\frac {i x}{2}+\frac {\pi }{2}\right )dx+2 \text {sech}\left (\frac {x}{2}\right )+x \tanh \left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )\right )}{2 a \sqrt {a \cosh (x)+a}}\)

\(\Big \downarrow \) 4668

\(\displaystyle \frac {\cosh \left (\frac {x}{2}\right ) \left (\frac {1}{2} \left (-2 i \int \log \left (1-i e^{x/2}\right )dx+2 i \int \log \left (1+i e^{x/2}\right )dx+4 x \arctan \left (e^{x/2}\right )\right )+2 \text {sech}\left (\frac {x}{2}\right )+x \tanh \left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )\right )}{2 a \sqrt {a \cosh (x)+a}}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {\cosh \left (\frac {x}{2}\right ) \left (\frac {1}{2} \left (-4 i \int e^{-x/2} \log \left (1-i e^{x/2}\right )de^{x/2}+4 i \int e^{-x/2} \log \left (1+i e^{x/2}\right )de^{x/2}+4 x \arctan \left (e^{x/2}\right )\right )+2 \text {sech}\left (\frac {x}{2}\right )+x \tanh \left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )\right )}{2 a \sqrt {a \cosh (x)+a}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {\cosh \left (\frac {x}{2}\right ) \left (\frac {1}{2} \left (4 x \arctan \left (e^{x/2}\right )-4 i \operatorname {PolyLog}\left (2,-i e^{x/2}\right )+4 i \operatorname {PolyLog}\left (2,i e^{x/2}\right )\right )+2 \text {sech}\left (\frac {x}{2}\right )+x \tanh \left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )\right )}{2 a \sqrt {a \cosh (x)+a}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[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 2838

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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), 
 x_Symbol] :> Simp[(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 4668

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] + (-Simp[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] + Simp[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 4673

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] + (-Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x] + S 
imp[b^2*((n - 2)/(n - 1))   Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x], x]) 
 /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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)*sq 
rt(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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(a)*int((sqrt(cosh(x) + 1)*x)/(cosh(x)**2 + 2*cosh(x) + 1),x))/a**2