Optimal. Leaf size=122 \[ \frac {F_1\left (\frac {1+m}{2};\frac {1+m}{2},-p;\frac {3+m}{2};-\sinh ^2(e+f x),-\frac {b \sinh ^2(e+f x)}{a}\right ) \cosh ^2(e+f x)^{\frac {1+m}{2}} \left (a+b \sinh ^2(e+f x)\right )^p \left (1+\frac {b \sinh ^2(e+f x)}{a}\right )^{-p} (d \tanh (e+f x))^{1+m}}{d f (1+m)} \]
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Rubi [A]
time = 0.10, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3276, 525, 524}
\begin {gather*} \frac {\cosh ^2(e+f x)^{\frac {m+1}{2}} (d \tanh (e+f x))^{m+1} \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 {m+1}{2};\frac {m+1}{2},-p;\frac {m+3}{2};-\sinh ^2(e+f x),-\frac {b \sinh ^2(e+f x)}{a}\right )}{d f (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 524
Rule 525
Rule 3276
Rubi steps
\begin {align*} \int \left (a+b \sinh ^2(e+f x)\right )^p (d \tanh (e+f x))^m \, dx &=\frac {\left (i \cosh ^2(e+f x)^{\frac {1+m}{2}} (i \sinh (e+f x))^{-1-m} (d \tanh (e+f x))^{1+m}\right ) \text {Subst}\left (\int (i x)^m \left (1+x^2\right )^{\frac {1}{2} (-1-m)} \left (a+b x^2\right )^p \, dx,x,\sinh (e+f x)\right )}{d f}\\ &=\frac {\left (i \cosh ^2(e+f x)^{\frac {1+m}{2}} (i \sinh (e+f x))^{-1-m} \left (a+b \sinh ^2(e+f x)\right )^p \left (1+\frac {b \sinh ^2(e+f x)}{a}\right )^{-p} (d \tanh (e+f x))^{1+m}\right ) \text {Subst}\left (\int (i x)^m \left (1+x^2\right )^{\frac {1}{2} (-1-m)} \left (1+\frac {b x^2}{a}\right )^p \, dx,x,\sinh (e+f x)\right )}{d f}\\ &=\frac {F_1\left (\frac {1+m}{2};\frac {1+m}{2},-p;\frac {3+m}{2};-\sinh ^2(e+f x),-\frac {b \sinh ^2(e+f x)}{a}\right ) \cosh ^2(e+f x)^{\frac {1+m}{2}} \left (a+b \sinh ^2(e+f x)\right )^p \left (1+\frac {b \sinh ^2(e+f x)}{a}\right )^{-p} (d \tanh (e+f x))^{1+m}}{d f (1+m)}\\ \end {align*}
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Mathematica [F]
time = 7.84, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a+b \sinh ^2(e+f x)\right )^p (d \tanh (e+f x))^m \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 2.73, size = 0, normalized size = 0.00 \[\int \left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right )^{p} \left (d \tanh \left (f x +e \right )\right )^{m}\, dx\]
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 = 0.56, size = 27, normalized size = 0.22 \begin {gather*} {\rm integral}\left ({\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{p} \left (d \tanh \left (f x + e\right )\right )^{m}, 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 {\left (d\,\mathrm {tanh}\left (e+f\,x\right )\right )}^m\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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