Optimal. Leaf size=63 \[ -\frac{\left (a+b \sinh ^2(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \sinh ^2(c+d x)+a}{a-b}\right )}{2 d (p+1) (a-b)} \]
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Rubi [A] time = 0.0547016, antiderivative size = 63, normalized size of antiderivative = 1., 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 = {3194, 68} \[ -\frac{\left (a+b \sinh ^2(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \sinh ^2(c+d x)+a}{a-b}\right )}{2 d (p+1) (a-b)} \]
Antiderivative was successfully verified.
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Rule 3194
Rule 68
Rubi steps
\begin{align*} \int \left (a+b \sinh ^2(c+d x)\right )^p \tanh (c+d x) \, dx &=\frac{\operatorname{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*}
Mathematica [A] time = 0.070719, size = 65, normalized size = 1.03 \[ -\frac{\left (a+b \cosh ^2(c+d x)-b\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \cosh ^2(c+d x)}{a-b}+1\right )}{2 d (p+1) (a-b)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.255, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ( \sinh \left ( dx+c \right ) \right ) ^{2} \right ) ^{p}\tanh \left ( dx+c \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{p} \tanh \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{p} \tanh \left (d x + c\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{p} \tanh \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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