Optimal. Leaf size=69 \[ \frac {6 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{5 b}+\frac {2 \sinh (a+b x)}{5 b \cosh ^{\frac {5}{2}}(a+b x)}+\frac {6 \sinh (a+b x)}{5 b \sqrt {\cosh (a+b x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2636, 2639} \[ \frac {6 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{5 b}+\frac {2 \sinh (a+b x)}{5 b \cosh ^{\frac {5}{2}}(a+b x)}+\frac {6 \sinh (a+b x)}{5 b \sqrt {\cosh (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rubi steps
\begin {align*} \int \frac {1}{\cosh ^{\frac {7}{2}}(a+b x)} \, dx &=\frac {2 \sinh (a+b x)}{5 b \cosh ^{\frac {5}{2}}(a+b x)}+\frac {3}{5} \int \frac {1}{\cosh ^{\frac {3}{2}}(a+b x)} \, dx\\ &=\frac {2 \sinh (a+b x)}{5 b \cosh ^{\frac {5}{2}}(a+b x)}+\frac {6 \sinh (a+b x)}{5 b \sqrt {\cosh (a+b x)}}-\frac {3}{5} \int \sqrt {\cosh (a+b x)} \, dx\\ &=\frac {6 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{5 b}+\frac {2 \sinh (a+b x)}{5 b \cosh ^{\frac {5}{2}}(a+b x)}+\frac {6 \sinh (a+b x)}{5 b \sqrt {\cosh (a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 63, normalized size = 0.91 \[ \frac {3 \sinh (2 (a+b x))+2 \tanh (a+b x)+6 i \cosh ^{\frac {3}{2}}(a+b x) E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{5 b \cosh ^{\frac {3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\cosh \left (b x + a\right )^{\frac {7}{2}}}, x\right ) \]
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 \[ \int \frac {1}{\cosh \left (b x + a\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.63, size = 363, normalized size = 5.26 \[ \frac {2 \sqrt {\left (2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (12 \sqrt {-\left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticE \left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \left (\sinh ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+24 \left (\sinh ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )+12 \sqrt {-\left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticE \left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+24 \left (\sinh ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )+3 \sqrt {-2 \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticE \left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {-\left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}+8 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right ) \sqrt {2 \left (\sinh ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}}{5 \left (8 \left (\sinh ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+12 \left (\sinh ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+6 \left (\sinh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1\right ) \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{3} \sqrt {2 \left (\cosh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, b} \]
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 \[ \int \frac {1}{\cosh \left (b x + a\right )^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\mathrm {cosh}\left (a+b\,x\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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