Optimal. Leaf size=76 \[ -\frac {2 \cosh (a+b x)}{b \sqrt {\sinh (a+b x)}}-\frac {2 i \sqrt {\sinh (a+b x)} E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {i \sinh (a+b x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2636, 2640, 2639} \[ -\frac {2 \cosh (a+b x)}{b \sqrt {\sinh (a+b x)}}-\frac {2 i \sqrt {\sinh (a+b x)} E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {i \sinh (a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2636
Rule 2639
Rule 2640
Rubi steps
\begin {align*} \int \frac {1}{\sinh ^{\frac {3}{2}}(a+b x)} \, dx &=-\frac {2 \cosh (a+b x)}{b \sqrt {\sinh (a+b x)}}+\int \sqrt {\sinh (a+b x)} \, dx\\ &=-\frac {2 \cosh (a+b x)}{b \sqrt {\sinh (a+b x)}}+\frac {\sqrt {\sinh (a+b x)} \int \sqrt {i \sinh (a+b x)} \, dx}{\sqrt {i \sinh (a+b x)}}\\ &=-\frac {2 \cosh (a+b x)}{b \sqrt {\sinh (a+b x)}}-\frac {2 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {\sinh (a+b x)}}{b \sqrt {i \sinh (a+b x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 57, normalized size = 0.75 \[ -\frac {2 \left (\cosh (a+b x)-\sqrt {i \sinh (a+b x)} E\left (\left .\frac {1}{4} (-2 i a-2 i b x+\pi )\right |2\right )\right )}{b \sqrt {\sinh (a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sinh \left (b x + a\right )^{\frac {3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sinh \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 154, normalized size = 2.03 \[ \frac {2 \sqrt {1-i \sinh \left (b x +a \right )}\, \sqrt {2}\, \sqrt {i \sinh \left (b x +a \right )+1}\, \sqrt {i \sinh \left (b x +a \right )}\, \EllipticE \left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1-i \sinh \left (b x +a \right )}\, \sqrt {2}\, \sqrt {i \sinh \left (b x +a \right )+1}\, \sqrt {i \sinh \left (b x +a \right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-2 \left (\cosh ^{2}\left (b x +a \right )\right )}{\cosh \left (b x +a \right ) \sqrt {\sinh \left (b x +a \right )}\, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sinh \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\mathrm {sinh}\left (a+b\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sinh ^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________