\(\int \frac {(a+a \cosh (x))^{3/2}}{x^2} \, dx\) [137]

Optimal result

Integrand size = 14, antiderivative size = 79 \[ \int \frac {(a+a \cosh (x))^{3/2}}{x^2} \, dx=-\frac {2 a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}}{x}+\frac {3}{4} a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right ) \text {Shi}\left (\frac {x}{2}\right )+\frac {3}{4} a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right ) \text {Shi}\left (\frac {3 x}{2}\right ) \]

[Out]

-2*a*cosh(1/2*x)^2*(a+a*cosh(x))^(1/2)/x+3/4*a*sech(1/2*x)*Shi(1/2*x)*(a+a*cosh(x))^(1/2)+3/4*a*sech(1/2*x)*Sh
i(3/2*x)*(a+a*cosh(x))^(1/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3400, 3394, 3379} \[ \int \frac {(a+a \cosh (x))^{3/2}}{x^2} \, dx=\frac {3}{4} a \text {Shi}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}+\frac {3}{4} a \text {Shi}\left (\frac {3 x}{2}\right ) \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}-\frac {2 a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}}{x} \]

[In]

Int[(a + a*Cosh[x])^(3/2)/x^2,x]

[Out]

(-2*a*Cosh[x/2]^2*Sqrt[a + a*Cosh[x]])/x + (3*a*Sqrt[a + a*Cosh[x]]*Sech[x/2]*SinhIntegral[x/2])/4 + (3*a*Sqrt
[a + a*Cosh[x]]*Sech[x/2]*SinhIntegral[(3*x)/2])/4

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3394

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]^
n/(d*(m + 1))), x] - Dist[f*(n/(d*(m + 1))), Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]
^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] && LtQ[m, -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])

Rubi steps \begin{align*} \text {integral}& = \left (2 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \frac {\cosh ^3\left (\frac {x}{2}\right )}{x^2} \, dx \\ & = -\frac {2 a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}}{x}+\left (3 i a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \left (-\frac {i \sinh \left (\frac {x}{2}\right )}{4 x}-\frac {i \sinh \left (\frac {3 x}{2}\right )}{4 x}\right ) \, dx \\ & = -\frac {2 a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}}{x}+\frac {1}{4} \left (3 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \frac {\sinh \left (\frac {x}{2}\right )}{x} \, dx+\frac {1}{4} \left (3 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \frac {\sinh \left (\frac {3 x}{2}\right )}{x} \, dx \\ & = -\frac {2 a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}}{x}+\frac {3}{4} a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right ) \text {Shi}\left (\frac {x}{2}\right )+\frac {3}{4} a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right ) \text {Shi}\left (\frac {3 x}{2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.67 \[ \int \frac {(a+a \cosh (x))^{3/2}}{x^2} \, dx=-\frac {a \sqrt {a (1+\cosh (x))} \text {sech}\left (\frac {x}{2}\right ) \left (8 \cosh ^3\left (\frac {x}{2}\right )-3 x \text {Shi}\left (\frac {x}{2}\right )-3 x \text {Shi}\left (\frac {3 x}{2}\right )\right )}{4 x} \]

[In]

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

[Out]

-1/4*(a*Sqrt[a*(1 + Cosh[x])]*Sech[x/2]*(8*Cosh[x/2]^3 - 3*x*SinhIntegral[x/2] - 3*x*SinhIntegral[(3*x)/2]))/x

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {(a+a \cosh (x))^{3/2}}{x^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.42 \[ \int \frac {(a+a \cosh (x))^{3/2}}{x^2} \, dx=\frac {1}{8} \, \sqrt {2} {\left (\frac {3 \, a^{\frac {3}{2}} x {\rm Ei}\left (\frac {3}{2} \, x\right ) + 3 \, a^{\frac {3}{2}} x {\rm Ei}\left (\frac {1}{2} \, x\right ) - a^{\frac {3}{2}} x {\rm Ei}\left (-\frac {1}{2} \, x\right ) - 2 \, a^{\frac {3}{2}} e^{\left (\frac {3}{2} \, x\right )} - 6 \, a^{\frac {3}{2}} e^{\left (\frac {1}{2} \, x\right )} - 2 \, a^{\frac {3}{2}} e^{\left (-\frac {1}{2} \, x\right )}}{x} - \frac {2 \, a^{\frac {3}{2}} x {\rm Ei}\left (-\frac {1}{2} \, x\right ) + 3 \, a^{\frac {3}{2}} x {\rm Ei}\left (-\frac {3}{2} \, x\right ) + 4 \, a^{\frac {3}{2}} e^{\left (-\frac {1}{2} \, x\right )} + 2 \, a^{\frac {3}{2}} e^{\left (-\frac {3}{2} \, x\right )}}{x}\right )} \]

[In]

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

[Out]

1/8*sqrt(2)*((3*a^(3/2)*x*Ei(3/2*x) + 3*a^(3/2)*x*Ei(1/2*x) - a^(3/2)*x*Ei(-1/2*x) - 2*a^(3/2)*e^(3/2*x) - 6*a
^(3/2)*e^(1/2*x) - 2*a^(3/2)*e^(-1/2*x))/x - (2*a^(3/2)*x*Ei(-1/2*x) + 3*a^(3/2)*x*Ei(-3/2*x) + 4*a^(3/2)*e^(-
1/2*x) + 2*a^(3/2)*e^(-3/2*x))/x)

Mupad [F(-1)]

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

[In]

int((a + a*cosh(x))^(3/2)/x^2,x)

[Out]

int((a + a*cosh(x))^(3/2)/x^2, x)