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 = 0.09, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, 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 65
Rule 78
Rule 3194
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*}
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Mathematica [A] time = 0.38, 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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fricas [F] time = 1.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{p} \coth \left (d x + c\right )^{3}, x\right ) \]
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 \[ \int {\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{p} \coth \left (d x + c\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.47, size = 0, normalized size = 0.00 \[ \int \left (\coth ^{3}\left (d x +c \right )\right ) \left (a +b \left (\sinh ^{2}\left (d x +c \right )\right )\right )^{p}\, dx \]
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 \[ \int {\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{p} \coth \left (d x + c\right )^{3}\,{d x} \]
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 \[ \int {\mathrm {coth}\left (c+d\,x\right )}^3\,{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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