Optimal. Leaf size=89 \[ -\frac {14 \cosh (x)}{45 a \sqrt {a \text {csch}^3(x)}}+\frac {2 \sinh ^2(x) \cosh (x)}{9 a \sqrt {a \text {csch}^3(x)}}+\frac {14 i \text {csch}(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{15 a \sqrt {i \sinh (x)} \sqrt {a \text {csch}^3(x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4123, 3769, 3771, 2639} \[ -\frac {14 \cosh (x)}{45 a \sqrt {a \text {csch}^3(x)}}+\frac {2 \sinh ^2(x) \cosh (x)}{9 a \sqrt {a \text {csch}^3(x)}}+\frac {14 i \text {csch}(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{15 a \sqrt {i \sinh (x)} \sqrt {a \text {csch}^3(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2639
Rule 3769
Rule 3771
Rule 4123
Rubi steps
\begin {align*} \int \frac {1}{\left (a \text {csch}^3(x)\right )^{3/2}} \, dx &=-\frac {\left (i (i \text {csch}(x))^{3/2}\right ) \int \frac {1}{(i \text {csch}(x))^{9/2}} \, dx}{a \sqrt {a \text {csch}^3(x)}}\\ &=\frac {2 \cosh (x) \sinh ^2(x)}{9 a \sqrt {a \text {csch}^3(x)}}-\frac {\left (7 i (i \text {csch}(x))^{3/2}\right ) \int \frac {1}{(i \text {csch}(x))^{5/2}} \, dx}{9 a \sqrt {a \text {csch}^3(x)}}\\ &=-\frac {14 \cosh (x)}{45 a \sqrt {a \text {csch}^3(x)}}+\frac {2 \cosh (x) \sinh ^2(x)}{9 a \sqrt {a \text {csch}^3(x)}}-\frac {\left (7 i (i \text {csch}(x))^{3/2}\right ) \int \frac {1}{\sqrt {i \text {csch}(x)}} \, dx}{15 a \sqrt {a \text {csch}^3(x)}}\\ &=-\frac {14 \cosh (x)}{45 a \sqrt {a \text {csch}^3(x)}}+\frac {2 \cosh (x) \sinh ^2(x)}{9 a \sqrt {a \text {csch}^3(x)}}+\frac {(7 \text {csch}(x)) \int \sqrt {i \sinh (x)} \, dx}{15 a \sqrt {a \text {csch}^3(x)} \sqrt {i \sinh (x)}}\\ &=-\frac {14 \cosh (x)}{45 a \sqrt {a \text {csch}^3(x)}}+\frac {14 i \text {csch}(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{15 a \sqrt {a \text {csch}^3(x)} \sqrt {i \sinh (x)}}+\frac {2 \cosh (x) \sinh ^2(x)}{9 a \sqrt {a \text {csch}^3(x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 57, normalized size = 0.64 \[ \frac {-33 \cosh (x)+5 \cosh (3 x)+84 \sqrt {i \sinh (x)} \text {csch}^2(x) E\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right )}{90 a \sqrt {a \text {csch}^3(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \operatorname {csch}\relax (x)^{3}}}{a^{2} \operatorname {csch}\relax (x)^{6}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \operatorname {csch}\relax (x)^{3}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \mathrm {csch}\relax (x )^{3}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \operatorname {csch}\relax (x)^{3}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (\frac {a}{{\mathrm {sinh}\relax (x)}^3}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \operatorname {csch}^{3}{\relax (x )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________