Optimal. Leaf size=118 \[ \frac {6 i E\left (\left .\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {b \sinh (c+d x)}}{5 b^4 d \sqrt {i \sinh (c+d x)}}+\frac {6 \cosh (c+d x)}{5 b^3 d \sqrt {b \sinh (c+d x)}}-\frac {2 \cosh (c+d x)}{5 b d (b \sinh (c+d x))^{5/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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.
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Rule 2636
Rule 2639
Rule 2640
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*}
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Mathematica [A] time = 0.17, 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.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sinh \left (d x + c\right )}}{b^{4} \sinh \left (d x + c\right )^{4}}, x\right ) \]
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 \[ \int \frac {1}{\left (b \sinh \left (d x + c\right )\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 205, normalized size = 1.74 \[ -\frac {6 \sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (d x +c \right )+i\right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \left (\sinh ^{2}\left (d x +c \right )\right ) \EllipticE \left (\sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (d x +c \right )+i\right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \left (\sinh ^{2}\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}, \frac {\sqrt {2}}{2}\right )-6 \left (\sinh ^{4}\left (d x +c \right )\right )-4 \left (\sinh ^{2}\left (d x +c \right )\right )+2}{5 b^{3} \sinh \left (d x +c \right )^{2} \cosh \left (d x +c \right ) \sqrt {b \sinh \left (d x +c \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \sinh \left (d x + c\right )\right )^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (b\,\mathrm {sinh}\left (c+d\,x\right )\right )}^{7/2}} \,d x \]
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 \[ \int \frac {1}{\left (b \sinh {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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