Optimal. Leaf size=174 \[ -\frac{\sqrt{b} \left (15 a^2-10 a b+3 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \coth (c+d x)}{\sqrt{a+b \coth ^2(c+d x)-b}}\right )}{8 d}+\frac{a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a+b \coth ^2(c+d x)-b}}\right )}{d}-\frac{b \coth (c+d x) \left (a+b \coth ^2(c+d x)-b\right )^{3/2}}{4 d}-\frac{b (7 a-3 b) \coth (c+d x) \sqrt{a+b \coth ^2(c+d x)-b}}{8 d} \]
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Rubi [A] time = 0.188462, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {4128, 416, 528, 523, 217, 206, 377} \[ -\frac{\sqrt{b} \left (15 a^2-10 a b+3 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \coth (c+d x)}{\sqrt{a+b \coth ^2(c+d x)-b}}\right )}{8 d}+\frac{a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a+b \coth ^2(c+d x)-b}}\right )}{d}-\frac{b \coth (c+d x) \left (a+b \coth ^2(c+d x)-b\right )^{3/2}}{4 d}-\frac{b (7 a-3 b) \coth (c+d x) \sqrt{a+b \coth ^2(c+d x)-b}}{8 d} \]
Antiderivative was successfully verified.
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Rule 4128
Rule 416
Rule 528
Rule 523
Rule 217
Rule 206
Rule 377
Rubi steps
\begin{align*} \int \left (a+b \text{csch}^2(c+d x)\right )^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a-b+b x^2\right )^{5/2}}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=-\frac{b \coth (c+d x) \left (a-b+b \coth ^2(c+d x)\right )^{3/2}}{4 d}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a-b+b x^2} \left (-(4 a-3 b) (a-b)-(7 a-3 b) b x^2\right )}{1-x^2} \, dx,x,\coth (c+d x)\right )}{4 d}\\ &=-\frac{(7 a-3 b) b \coth (c+d x) \sqrt{a-b+b \coth ^2(c+d x)}}{8 d}-\frac{b \coth (c+d x) \left (a-b+b \coth ^2(c+d x)\right )^{3/2}}{4 d}+\frac{\operatorname{Subst}\left (\int \frac{(a-b) \left (8 a^2-7 a b+3 b^2\right )+b \left (15 a^2-10 a b+3 b^2\right ) x^2}{\left (1-x^2\right ) \sqrt{a-b+b x^2}} \, dx,x,\coth (c+d x)\right )}{8 d}\\ &=-\frac{(7 a-3 b) b \coth (c+d x) \sqrt{a-b+b \coth ^2(c+d x)}}{8 d}-\frac{b \coth (c+d x) \left (a-b+b \coth ^2(c+d x)\right )^{3/2}}{4 d}+\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{a-b+b x^2}} \, dx,x,\coth (c+d x)\right )}{d}-\frac{\left (b \left (15 a^2-10 a b+3 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-b+b x^2}} \, dx,x,\coth (c+d x)\right )}{8 d}\\ &=-\frac{(7 a-3 b) b \coth (c+d x) \sqrt{a-b+b \coth ^2(c+d x)}}{8 d}-\frac{b \coth (c+d x) \left (a-b+b \coth ^2(c+d x)\right )^{3/2}}{4 d}+\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{\coth (c+d x)}{\sqrt{a-b+b \coth ^2(c+d x)}}\right )}{d}-\frac{\left (b \left (15 a^2-10 a b+3 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\coth (c+d x)}{\sqrt{a-b+b \coth ^2(c+d x)}}\right )}{8 d}\\ &=\frac{a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a-b+b \coth ^2(c+d x)}}\right )}{d}-\frac{\sqrt{b} \left (15 a^2-10 a b+3 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \coth (c+d x)}{\sqrt{a-b+b \coth ^2(c+d x)}}\right )}{8 d}-\frac{(7 a-3 b) b \coth (c+d x) \sqrt{a-b+b \coth ^2(c+d x)}}{8 d}-\frac{b \coth (c+d x) \left (a-b+b \coth ^2(c+d x)\right )^{3/2}}{4 d}\\ \end{align*}
Mathematica [A] time = 4.88895, size = 231, normalized size = 1.33 \[ \frac{\sinh ^5(c+d x) \left (a+b \text{csch}^2(c+d x)\right )^{5/2} \left (-2 \sqrt{2} \sqrt{b} \left (15 a^2-10 a b+3 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{b} \cosh (c+d x)}{\sqrt{a \cosh (2 (c+d x))-a+2 b}}\right )+16 \sqrt{2} a^{5/2} \log \left (\sqrt{a \cosh (2 (c+d x))-a+2 b}+\sqrt{2} \sqrt{a} \cosh (c+d x)\right )+b \coth (c+d x) \text{csch}^3(c+d x) \sqrt{a \cosh (2 (c+d x))-a+2 b} ((3 b-9 a) \cosh (2 (c+d x))+9 a-7 b)\right )}{4 d (a \cosh (2 (c+d x))-a+2 b)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.187, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ({\rm csch} \left (dx+c\right ) \right ) ^{2} \right ) ^{{\frac{5}{2}}}\, 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 \operatorname{csch}\left (d x + c\right )^{2} + a\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{csch}^{2}{\left (c + d x \right )}\right )^{\frac{5}{2}}\, dx \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 \operatorname{csch}\left (d x + c\right )^{2} + a\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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