Optimal. Leaf size=110 \[ \frac{\text{sech}^2(c+d x) \left (a+b \sinh ^2(c+d x)\right )^{p+1}}{2 d (a-b)}-\frac{(a-b (p+1)) \left (a+b \sinh ^2(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \sinh ^2(c+d x)+a}{a-b}\right )}{2 d (p+1) (a-b)^2} \]
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Rubi [A] time = 0.120445, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3194, 78, 68} \[ \frac{\text{sech}^2(c+d x) \left (a+b \sinh ^2(c+d x)\right )^{p+1}}{2 d (a-b)}-\frac{(a-b (p+1)) \left (a+b \sinh ^2(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \sinh ^2(c+d x)+a}{a-b}\right )}{2 d (p+1) (a-b)^2} \]
Antiderivative was successfully verified.
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Rule 3194
Rule 78
Rule 68
Rubi steps
\begin{align*} \int \left (a+b \sinh ^2(c+d x)\right )^p \tanh ^3(c+d x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x (a+b x)^p}{(1+x)^2} \, dx,x,\sinh ^2(c+d x)\right )}{2 d}\\ &=\frac{\text{sech}^2(c+d x) \left (a+b \sinh ^2(c+d x)\right )^{1+p}}{2 (a-b) d}-\frac{(a-b (1+p)) \operatorname{Subst}\left (\int \frac{(a+b x)^p}{1+x} \, dx,x,\sinh ^2(c+d x)\right )}{2 (-a+b) d}\\ &=-\frac{(a-b (1+p)) \, _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)^2 d (1+p)}+\frac{\text{sech}^2(c+d x) \left (a+b \sinh ^2(c+d x)\right )^{1+p}}{2 (a-b) d}\\ \end{align*}
Mathematica [A] time = 0.238538, size = 90, normalized size = 0.82 \[ \frac{\left (a+b \sinh ^2(c+d x)\right )^{p+1} \left ((-a+b p+b) \, _2F_1\left (1,p+1;p+2;\frac{b \sinh ^2(c+d x)+a}{a-b}\right )+(p+1) (a-b) \text{sech}^2(c+d x)\right )}{2 d (p+1) (a-b)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.36, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ( \sinh \left ( dx+c \right ) \right ) ^{2} \right ) ^{p} \left ( \tanh \left ( dx+c \right ) \right ) ^{3}\, 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 )^{3}\,{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 )^{3}, 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 )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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