Optimal. Leaf size=103 \[ -\frac {\sqrt {\cosh ^2(c+d x)} \text {csch}^3(c+d x) \text {sech}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^p \left (\frac {b \sinh ^2(c+d x)}{a}+1\right )^{-p} F_1\left (-\frac {3}{2};-\frac {3}{2},-p;-\frac {1}{2};-\sinh ^2(c+d x),-\frac {b \sinh ^2(c+d x)}{a}\right )}{3 d} \]
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Rubi [A] time = 0.10, antiderivative size = 103, 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 = {3196, 511, 510} \[ -\frac {\sqrt {\cosh ^2(c+d x)} \text {csch}^3(c+d x) \text {sech}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^p \left (\frac {b \sinh ^2(c+d x)}{a}+1\right )^{-p} F_1\left (-\frac {3}{2};-\frac {3}{2},-p;-\frac {1}{2};-\sinh ^2(c+d x),-\frac {b \sinh ^2(c+d x)}{a}\right )}{3 d} \]
Antiderivative was successfully verified.
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Rule 510
Rule 511
Rule 3196
Rubi steps
\begin {align*} \int \coth ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^p \, dx &=\frac {\left (\sqrt {\cosh ^2(c+d x)} \text {sech}(c+d x)\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^{3/2} \left (a+b x^2\right )^p}{x^4} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\left (\sqrt {\cosh ^2(c+d x)} \text {sech}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^p \left (1+\frac {b \sinh ^2(c+d x)}{a}\right )^{-p}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^{3/2} \left (1+\frac {b x^2}{a}\right )^p}{x^4} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=-\frac {F_1\left (-\frac {3}{2};-\frac {3}{2},-p;-\frac {1}{2};-\sinh ^2(c+d x),-\frac {b \sinh ^2(c+d x)}{a}\right ) \sqrt {\cosh ^2(c+d x)} \text {csch}^3(c+d x) \text {sech}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^p \left (1+\frac {b \sinh ^2(c+d x)}{a}\right )^{-p}}{3 d}\\ \end {align*}
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Mathematica [F] time = 38.40, size = 0, normalized size = 0.00 \[ \int \coth ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^p \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.24, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{p} \coth \left (d x + c\right )^{4}, x\right ) \]
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 \[ \int {\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{p} \coth \left (d x + c\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.40, size = 0, normalized size = 0.00 \[ \int \left (\coth ^{4}\left (d x +c \right )\right ) \left (a +b \left (\sinh ^{2}\left (d x +c \right )\right )\right )^{p}\, dx \]
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 \[ \int {\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{p} \coth \left (d x + c\right )^{4}\,{d x} \]
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 \[ \int {\mathrm {coth}\left (c+d\,x\right )}^4\,{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^p \,d x \]
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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