Optimal. Leaf size=63 \[ -\frac {\, _2F_1\left (1,1+p;2+p;\frac {a+b \sinh ^2(c+d x)}{a-b}\right ) \left (a+b \sinh ^2(c+d x)\right )^{1+p}}{2 (a-b) d (1+p)} \]
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Rubi [A]
time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3273, 70}
\begin {gather*} -\frac {\left (a+b \sinh ^2(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b \sinh ^2(c+d x)+a}{a-b}\right )}{2 d (p+1) (a-b)} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 3273
Rubi steps
\begin {align*} \int \left (a+b \sinh ^2(c+d x)\right )^p \tanh (c+d x) \, dx &=\frac {\text {Subst}\left (\int \frac {(a+b x)^p}{1+x} \, dx,x,\sinh ^2(c+d x)\right )}{2 d}\\ &=-\frac {\, _2F_1\left (1,1+p;2+p;\frac {a+b \sinh ^2(c+d x)}{a-b}\right ) \left (a+b \sinh ^2(c+d x)\right )^{1+p}}{2 (a-b) d (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 65, normalized size = 1.03 \begin {gather*} -\frac {\left (a-b+b \cosh ^2(c+d x)\right )^{1+p} \, _2F_1\left (1,1+p;2+p;1+\frac {b \cosh ^2(c+d x)}{a-b}\right )}{2 (a-b) d (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.02, size = 0, normalized size = 0.00 \[\int \left (a +b \left (\sinh ^{2}\left (d x +c \right )\right )\right )^{p} \tanh \left (d x +c \right )\, dx\]
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 = 0.41, size = 23, normalized size = 0.37 \begin {gather*} {\rm integral}\left ({\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{p} \tanh \left (d x + c\right ), x\right ) \end {gather*}
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 \begin {gather*} \int \left (a + b \sinh ^{2}{\left (c + d x \right )}\right )^{p} \tanh {\left (c + d x \right )}\, dx \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 = -0.02 \begin {gather*} \int \mathrm {tanh}\left (c+d\,x\right )\,{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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