Optimal. Leaf size=79 \[ \frac {3}{4} a \text {Shi}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}+\frac {3}{4} a \text {Shi}\left (\frac {3 x}{2}\right ) \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}-\frac {2 a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3319, 3313, 3298} \[ \frac {3}{4} a \text {Shi}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}+\frac {3}{4} a \text {Shi}\left (\frac {3 x}{2}\right ) \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}-\frac {2 a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3298
Rule 3313
Rule 3319
Rubi steps
\begin {align*} \int \frac {(a+a \cosh (x))^{3/2}}{x^2} \, dx &=\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] time = 0.09, size = 53, normalized size = 0.67 \[ -\frac {a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a (\cosh (x)+1)} \left (-3 x \text {Shi}\left (\frac {x}{2}\right )-3 x \text {Shi}\left (\frac {3 x}{2}\right )+8 \cosh ^3\left (\frac {x}{2}\right )\right )}{4 x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 112, normalized size = 1.42 \[ \frac {1}{8} \, \sqrt {2} {\left (\frac {3 \, a^{\frac {3}{2}} x {\rm Ei}\left (\frac {3}{2} \, x\right ) + 3 \, a^{\frac {3}{2}} x {\rm Ei}\left (\frac {1}{2} \, x\right ) - a^{\frac {3}{2}} x {\rm Ei}\left (-\frac {1}{2} \, x\right ) - 2 \, a^{\frac {3}{2}} e^{\left (\frac {3}{2} \, x\right )} - 6 \, a^{\frac {3}{2}} e^{\left (\frac {1}{2} \, x\right )} - 2 \, a^{\frac {3}{2}} e^{\left (-\frac {1}{2} \, x\right )}}{x} - \frac {2 \, a^{\frac {3}{2}} x {\rm Ei}\left (-\frac {1}{2} \, x\right ) + 3 \, a^{\frac {3}{2}} x {\rm Ei}\left (-\frac {3}{2} \, x\right ) + 4 \, a^{\frac {3}{2}} e^{\left (-\frac {1}{2} \, x\right )} + 2 \, a^{\frac {3}{2}} e^{\left (-\frac {3}{2} \, x\right )}}{x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +a \cosh \relax (x )\right )^{\frac {3}{2}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \cosh \relax (x) + a\right )}^{\frac {3}{2}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+a\,\mathrm {cosh}\relax (x)\right )}^{3/2}}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a \left (\cosh {\relax (x )} + 1\right )\right )^{\frac {3}{2}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________