Optimal. Leaf size=103 \[ -\frac {F_1\left (-\frac {3}{2};-\frac {3}{2},-p;-\frac {1}{2};-\sinh ^2(c+d x),-\frac {b \sinh ^2(c+d x)}{a}\right ) \sqrt {\cosh ^2(c+d x)} \text {csch}^3(c+d x) \text {sech}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^p \left (1+\frac {b \sinh ^2(c+d x)}{a}\right )^{-p}}{3 d} \]
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Rubi [A]
time = 0.07, 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 = {3275, 525, 524}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 524
Rule 525
Rule 3275
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 ) \text {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 ) \text {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 = 23.97, size = 0, normalized size = 0.00 \begin {gather*} \int \coth ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^p \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 1.43, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.43, size = 25, normalized size = 0.24 \begin {gather*} {\rm integral}\left ({\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{p} \coth \left (d x + c\right )^{4}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\mathrm {coth}\left (c+d\,x\right )}^4\,{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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