Optimal. Leaf size=94 \[ -\frac{(a+b p) \left (a+b \sinh ^2(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \sinh ^2(c+d x)}{a}+1\right )}{2 a^2 d (p+1)}-\frac{\text{csch}^2(c+d x) \left (a+b \sinh ^2(c+d x)\right )^{p+1}}{2 a d} \]
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Rubi [A] time = 0.0907626, antiderivative size = 94, 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, 65} \[ -\frac{(a+b p) \left (a+b \sinh ^2(c+d x)\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b \sinh ^2(c+d x)}{a}+1\right )}{2 a^2 d (p+1)}-\frac{\text{csch}^2(c+d x) \left (a+b \sinh ^2(c+d x)\right )^{p+1}}{2 a d} \]
Antiderivative was successfully verified.
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Rule 3194
Rule 78
Rule 65
Rubi steps
\begin{align*} \int \coth ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(1+x) (a+b x)^p}{x^2} \, dx,x,\sinh ^2(c+d x)\right )}{2 d}\\ &=-\frac{\text{csch}^2(c+d x) \left (a+b \sinh ^2(c+d x)\right )^{1+p}}{2 a d}+\frac{(a+b p) \operatorname{Subst}\left (\int \frac{(a+b x)^p}{x} \, dx,x,\sinh ^2(c+d x)\right )}{2 a d}\\ &=-\frac{\text{csch}^2(c+d x) \left (a+b \sinh ^2(c+d x)\right )^{1+p}}{2 a d}-\frac{(a+b p) \, _2F_1\left (1,1+p;2+p;1+\frac{b \sinh ^2(c+d x)}{a}\right ) \left (a+b \sinh ^2(c+d x)\right )^{1+p}}{2 a^2 d (1+p)}\\ \end{align*}
Mathematica [A] time = 0.379479, size = 71, normalized size = 0.76 \[ -\frac{\left (a+b \sinh ^2(c+d x)\right )^{p+1} \left (\frac{(a+b p) \, _2F_1\left (1,p+1;p+2;\frac{b \sinh ^2(c+d x)}{a}+1\right )}{p+1}+a \text{csch}^2(c+d x)\right )}{2 a^2 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.265, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm coth} \left (dx+c\right ) \right ) ^{3} \left ( a+b \left ( \sinh \left ( dx+c \right ) \right ) ^{2} \right ) ^{p}\, 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} \coth \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} \coth \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} \coth \left (d x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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