Optimal. Leaf size=103 \[ -\frac{2 \cosh (a+b x)}{5 b \sinh ^{\frac{5}{2}}(a+b x)}+\frac{6 \cosh (a+b x)}{5 b \sqrt{\sinh (a+b x)}}+\frac{6 i \sqrt{\sinh (a+b x)} E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{5 b \sqrt{i \sinh (a+b x)}} \]
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Rubi [A] time = 0.0442112, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2636, 2640, 2639} \[ -\frac{2 \cosh (a+b x)}{5 b \sinh ^{\frac{5}{2}}(a+b x)}+\frac{6 \cosh (a+b x)}{5 b \sqrt{\sinh (a+b x)}}+\frac{6 i \sqrt{\sinh (a+b x)} E\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{5 b \sqrt{i \sinh (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{\sinh ^{\frac{7}{2}}(a+b x)} \, dx &=-\frac{2 \cosh (a+b x)}{5 b \sinh ^{\frac{5}{2}}(a+b x)}-\frac{3}{5} \int \frac{1}{\sinh ^{\frac{3}{2}}(a+b x)} \, dx\\ &=-\frac{2 \cosh (a+b x)}{5 b \sinh ^{\frac{5}{2}}(a+b x)}+\frac{6 \cosh (a+b x)}{5 b \sqrt{\sinh (a+b x)}}-\frac{3}{5} \int \sqrt{\sinh (a+b x)} \, dx\\ &=-\frac{2 \cosh (a+b x)}{5 b \sinh ^{\frac{5}{2}}(a+b x)}+\frac{6 \cosh (a+b x)}{5 b \sqrt{\sinh (a+b x)}}-\frac{\left (3 \sqrt{\sinh (a+b x)}\right ) \int \sqrt{i \sinh (a+b x)} \, dx}{5 \sqrt{i \sinh (a+b x)}}\\ &=-\frac{2 \cosh (a+b x)}{5 b \sinh ^{\frac{5}{2}}(a+b x)}+\frac{6 \cosh (a+b x)}{5 b \sqrt{\sinh (a+b x)}}+\frac{6 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)}}{5 b \sqrt{i \sinh (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.16145, size = 73, normalized size = 0.71 \[ \frac{3 \sinh (2 (a+b x))-2 \coth (a+b x)+6 i (i \sinh (a+b x))^{3/2} E\left (\left .\frac{1}{4} (-2 i a-2 i b x+\pi )\right |2\right )}{5 b \sinh ^{\frac{3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 192, normalized size = 1.9 \begin{align*} -{\frac{1}{5\,b\cosh \left ( bx+a \right ) } \left ( 6\,\sqrt{-i \left ( \sinh \left ( bx+a \right ) +i \right ) }\sqrt{2}\sqrt{-i \left ( -\sinh \left ( bx+a \right ) +i \right ) }\sqrt{i\sinh \left ( bx+a \right ) } \left ( \sinh \left ( bx+a \right ) \right ) ^{2}{\it EllipticE} \left ( \sqrt{-i \left ( \sinh \left ( bx+a \right ) +i \right ) },1/2\,\sqrt{2} \right ) -3\,\sqrt{-i \left ( \sinh \left ( bx+a \right ) +i \right ) }\sqrt{2}\sqrt{-i \left ( -\sinh \left ( bx+a \right ) +i \right ) }\sqrt{i\sinh \left ( bx+a \right ) } \left ( \sinh \left ( bx+a \right ) \right ) ^{2}{\it EllipticF} \left ( \sqrt{-i \left ( \sinh \left ( bx+a \right ) +i \right ) },1/2\,\sqrt{2} \right ) -6\, \left ( \sinh \left ( bx+a \right ) \right ) ^{4}-4\, \left ( \sinh \left ( bx+a \right ) \right ) ^{2}+2 \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sinh \left (b x + a\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sinh \left (b x + a\right )^{\frac{7}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sinh \left (b x + a\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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