Optimal. Leaf size=86 \[ -\frac {2 \cosh (c+d x)}{b d \sqrt {b \sinh (c+d x)}}-\frac {2 i E\left (\left .\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {b \sinh (c+d x)}}{b^2 d \sqrt {i \sinh (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2636, 2640, 2639} \[ -\frac {2 \cosh (c+d x)}{b d \sqrt {b \sinh (c+d x)}}-\frac {2 i E\left (\left .\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {b \sinh (c+d x)}}{b^2 d \sqrt {i \sinh (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2636
Rule 2639
Rule 2640
Rubi steps
\begin {align*} \int \frac {1}{(b \sinh (c+d x))^{3/2}} \, dx &=-\frac {2 \cosh (c+d x)}{b d \sqrt {b \sinh (c+d x)}}+\frac {\int \sqrt {b \sinh (c+d x)} \, dx}{b^2}\\ &=-\frac {2 \cosh (c+d x)}{b d \sqrt {b \sinh (c+d x)}}+\frac {\sqrt {b \sinh (c+d x)} \int \sqrt {i \sinh (c+d x)} \, dx}{b^2 \sqrt {i \sinh (c+d x)}}\\ &=-\frac {2 \cosh (c+d x)}{b d \sqrt {b \sinh (c+d x)}}-\frac {2 i E\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right ) \sqrt {b \sinh (c+d x)}}{b^2 d \sqrt {i \sinh (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 62, normalized size = 0.72 \[ -\frac {2 \left (\cosh (c+d x)-\sqrt {i \sinh (c+d x)} E\left (\left .\frac {1}{4} (-2 i c-2 i d x+\pi )\right |2\right )\right )}{b d \sqrt {b \sinh (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sinh \left (d x + c\right )}}{b^{2} \sinh \left (d x + c\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \sinh \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 159, normalized size = 1.85 \[ \frac {2 \sqrt {1-i \sinh \left (d x +c \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (d x +c \right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \EllipticE \left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1-i \sinh \left (d x +c \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (d x +c \right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-2 \left (\cosh ^{2}\left (d x +c \right )\right )}{b \cosh \left (d x +c \right ) \sqrt {b \sinh \left (d x +c \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \sinh \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (b\,\mathrm {sinh}\left (c+d\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \sinh {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________