Optimal. Leaf size=118 \[ \frac{6 \cosh (c+d x)}{5 b^3 d \sqrt{b \sinh (c+d x)}}+\frac{6 i E\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{b \sinh (c+d x)}}{5 b^4 d \sqrt{i \sinh (c+d x)}}-\frac{2 \cosh (c+d x)}{5 b d (b \sinh (c+d x))^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0560453, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2636, 2640, 2639} \[ \frac{6 \cosh (c+d x)}{5 b^3 d \sqrt{b \sinh (c+d x)}}+\frac{6 i E\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{b \sinh (c+d x)}}{5 b^4 d \sqrt{i \sinh (c+d x)}}-\frac{2 \cosh (c+d x)}{5 b d (b \sinh (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2636
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{(b \sinh (c+d x))^{7/2}} \, dx &=-\frac{2 \cosh (c+d x)}{5 b d (b \sinh (c+d x))^{5/2}}-\frac{3 \int \frac{1}{(b \sinh (c+d x))^{3/2}} \, dx}{5 b^2}\\ &=-\frac{2 \cosh (c+d x)}{5 b d (b \sinh (c+d x))^{5/2}}+\frac{6 \cosh (c+d x)}{5 b^3 d \sqrt{b \sinh (c+d x)}}-\frac{3 \int \sqrt{b \sinh (c+d x)} \, dx}{5 b^4}\\ &=-\frac{2 \cosh (c+d x)}{5 b d (b \sinh (c+d x))^{5/2}}+\frac{6 \cosh (c+d x)}{5 b^3 d \sqrt{b \sinh (c+d x)}}-\frac{\left (3 \sqrt{b \sinh (c+d x)}\right ) \int \sqrt{i \sinh (c+d x)} \, dx}{5 b^4 \sqrt{i \sinh (c+d x)}}\\ &=-\frac{2 \cosh (c+d x)}{5 b d (b \sinh (c+d x))^{5/2}}+\frac{6 \cosh (c+d x)}{5 b^3 d \sqrt{b \sinh (c+d x)}}+\frac{6 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)}}{5 b^4 d \sqrt{i \sinh (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.163265, size = 79, normalized size = 0.67 \[ -\frac{2 \left (-3 \cosh (c+d x)+\coth (c+d x) \text{csch}(c+d x)+3 \sqrt{i \sinh (c+d x)} E\left (\left .\frac{1}{4} (-2 i c-2 i d x+\pi )\right |2\right )\right )}{5 b^3 d \sqrt{b \sinh (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.049, size = 205, normalized size = 1.7 \begin{align*} -{\frac{1}{5\,{b}^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}\cosh \left ( dx+c \right ) d} \left ( 6\,\sqrt{-i \left ( \sinh \left ( dx+c \right ) +i \right ) }\sqrt{2}\sqrt{-i \left ( i-\sinh \left ( dx+c \right ) \right ) }\sqrt{i\sinh \left ( dx+c \right ) } \left ( \sinh \left ( dx+c \right ) \right ) ^{2}{\it EllipticE} \left ( \sqrt{-i \left ( \sinh \left ( dx+c \right ) +i \right ) },1/2\,\sqrt{2} \right ) -3\,\sqrt{-i \left ( \sinh \left ( dx+c \right ) +i \right ) }\sqrt{2}\sqrt{-i \left ( i-\sinh \left ( dx+c \right ) \right ) }\sqrt{i\sinh \left ( dx+c \right ) } \left ( \sinh \left ( dx+c \right ) \right ) ^{2}{\it EllipticF} \left ( \sqrt{-i \left ( \sinh \left ( dx+c \right ) +i \right ) },1/2\,\sqrt{2} \right ) -6\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}-4\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}+2 \right ){\frac{1}{\sqrt{b\sinh \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \sinh \left (d x + c\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \sinh \left (d x + c\right )}}{b^{4} \sinh \left (d x + c\right )^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \sinh \left (d x + c\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]