Integrand size = 16, antiderivative size = 126 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^{3/2} \, dx=\frac {a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a-b+b \coth ^2(c+d x)}}\right )}{d}-\frac {(3 a-b) \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \coth (c+d x)}{\sqrt {a-b+b \coth ^2(c+d x)}}\right )}{2 d}-\frac {b \coth (c+d x) \sqrt {a-b+b \coth ^2(c+d x)}}{2 d} \]
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Time = 0.08 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4213, 427, 537, 223, 212, 385} \[ \int \left (a+b \text {csch}^2(c+d x)\right )^{3/2} \, dx=\frac {a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+b \coth ^2(c+d x)-b}}\right )}{d}-\frac {\sqrt {b} (3 a-b) \text {arctanh}\left (\frac {\sqrt {b} \coth (c+d x)}{\sqrt {a+b \coth ^2(c+d x)-b}}\right )}{2 d}-\frac {b \coth (c+d x) \sqrt {a+b \coth ^2(c+d x)-b}}{2 d} \]
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Rule 212
Rule 223
Rule 385
Rule 427
Rule 537
Rule 4213
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a-b+b x^2\right )^{3/2}}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d} \\ & = -\frac {b \coth (c+d x) \sqrt {a-b+b \coth ^2(c+d x)}}{2 d}-\frac {\text {Subst}\left (\int \frac {-((a-b) (2 a-b))-(3 a-b) b x^2}{\left (1-x^2\right ) \sqrt {a-b+b x^2}} \, dx,x,\coth (c+d x)\right )}{2 d} \\ & = -\frac {b \coth (c+d x) \sqrt {a-b+b \coth ^2(c+d x)}}{2 d}+\frac {a^2 \text {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 {((3 a-b) b) \text {Subst}\left (\int \frac {1}{\sqrt {a-b+b x^2}} \, dx,x,\coth (c+d x)\right )}{2 d} \\ & = -\frac {b \coth (c+d x) \sqrt {a-b+b \coth ^2(c+d x)}}{2 d}+\frac {a^2 \text {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 {((3 a-b) b) \text {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 )}{2 d} \\ & = \frac {a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a-b+b \coth ^2(c+d x)}}\right )}{d}-\frac {(3 a-b) \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \coth (c+d x)}{\sqrt {a-b+b \coth ^2(c+d x)}}\right )}{2 d}-\frac {b \coth (c+d x) \sqrt {a-b+b \coth ^2(c+d x)}}{2 d} \\ \end{align*}
Time = 0.65 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.53 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^{3/2} \, dx=\frac {\left (a+b \text {csch}^2(c+d x)\right )^{3/2} \left (\sqrt {2} \sqrt {b} (-3 a+b) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {b} \cosh (c+d x)}{\sqrt {-a+2 b+a \cosh (2 (c+d x))}}\right )-b \sqrt {-a+2 b+a \cosh (2 (c+d x))} \coth (c+d x) \text {csch}(c+d x)+2 \sqrt {2} a^{3/2} \log \left (\sqrt {2} \sqrt {a} \cosh (c+d x)+\sqrt {-a+2 b+a \cosh (2 (c+d x))}\right )\right ) \sinh ^3(c+d x)}{d (-a+2 b+a \cosh (2 (c+d x)))^{3/2}} \]
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\[\int \left (a +b \operatorname {csch}\left (d x +c \right )^{2}\right )^{\frac {3}{2}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 1272 vs. \(2 (108) = 216\).
Time = 0.45 (sec) , antiderivative size = 6645, normalized size of antiderivative = 52.74 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^{3/2} \, dx=\text {Too large to display} \]
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\[ \int \left (a+b \text {csch}^2(c+d x)\right )^{3/2} \, dx=\int \left (a + b \operatorname {csch}^{2}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \]
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\[ \int \left (a+b \text {csch}^2(c+d x)\right )^{3/2} \, dx=\int { {\left (b \operatorname {csch}\left (d x + c\right )^{2} + a\right )}^{\frac {3}{2}} \,d x } \]
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Exception generated. \[ \int \left (a+b \text {csch}^2(c+d x)\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (a+b \text {csch}^2(c+d x)\right )^{3/2} \, dx=\int {\left (a+\frac {b}{{\mathrm {sinh}\left (c+d\,x\right )}^2}\right )}^{3/2} \,d x \]
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