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 = 0.04, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2636, 2640, 2639} \[ -\frac {2 \cosh (a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}+\frac {6 \cosh (a+b x)}{5 b \sqrt {\sinh (a+b x)}}+\frac {6 i \sqrt {\sinh (a+b x)} E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{5 b \sqrt {i \sinh (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 2640
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*}
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Mathematica [A] time = 0.17, 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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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sinh \left (b x + a\right )^{\frac {7}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sinh \left (b x + a\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 192, normalized size = 1.86 \[ -\frac {6 \sqrt {-i \left (\sinh \left (b x +a \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (b x +a \right )+i\right )}\, \sqrt {i \sinh \left (b x +a \right )}\, \left (\sinh ^{2}\left (b x +a \right )\right ) \EllipticE \left (\sqrt {-i \left (\sinh \left (b x +a \right )+i\right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {-i \left (\sinh \left (b x +a \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (b x +a \right )+i\right )}\, \sqrt {i \sinh \left (b x +a \right )}\, \left (\sinh ^{2}\left (b x +a \right )\right ) \EllipticF \left (\sqrt {-i \left (\sinh \left (b x +a \right )+i\right )}, \frac {\sqrt {2}}{2}\right )-6 \left (\sinh ^{4}\left (b x +a \right )\right )-4 \left (\sinh ^{2}\left (b x +a \right )\right )+2}{5 \sinh \left (b x +a \right )^{\frac {5}{2}} \cosh \left (b x +a \right ) b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sinh \left (b x + a\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\mathrm {sinh}\left (a+b\,x\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sinh ^{\frac {7}{2}}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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