Optimal. Leaf size=92 \[ \frac {F_1\left (\frac {1}{2};\frac {3}{2},-p;\frac {3}{2};-\sinh ^2(e+f x),-\frac {b \sinh ^2(e+f x)}{a}\right ) \sqrt {\cosh ^2(e+f x)} \left (a+b \sinh ^2(e+f x)\right )^p \left (1+\frac {b \sinh ^2(e+f x)}{a}\right )^{-p} \tanh (e+f x)}{f} \]
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Rubi [A]
time = 0.06, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3271, 441, 440}
\begin {gather*} \frac {\sqrt {\cosh ^2(e+f x)} \tanh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \left (\frac {b \sinh ^2(e+f x)}{a}+1\right )^{-p} F_1\left (\frac {1}{2};\frac {3}{2},-p;\frac {3}{2};-\sinh ^2(e+f x),-\frac {b \sinh ^2(e+f x)}{a}\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 440
Rule 441
Rule 3271
Rubi steps
\begin {align*} \int \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx &=\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {\left (a+b x^2\right )^p}{\left (1+x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \left (1+\frac {b \sinh ^2(e+f x)}{a}\right )^{-p}\right ) \text {Subst}\left (\int \frac {\left (1+\frac {b x^2}{a}\right )^p}{\left (1+x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=\frac {F_1\left (\frac {1}{2};\frac {3}{2},-p;\frac {3}{2};-\sinh ^2(e+f x),-\frac {b \sinh ^2(e+f x)}{a}\right ) \sqrt {\cosh ^2(e+f x)} \left (a+b \sinh ^2(e+f x)\right )^p \left (1+\frac {b \sinh ^2(e+f x)}{a}\right )^{-p} \tanh (e+f x)}{f}\\ \end {align*}
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Mathematica [F]
time = 3.22, size = 0, normalized size = 0.00 \begin {gather*} \int \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 1.01, size = 0, normalized size = 0.00 \[\int \mathrm {sech}\left (f x +e \right )^{2} \left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right )^{p}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.47, size = 25, normalized size = 0.27 \begin {gather*} {\rm integral}\left ({\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{p} \operatorname {sech}\left (f x + e\right )^{2}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^p}{{\mathrm {cosh}\left (e+f\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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