Integrand size = 16, antiderivative size = 174 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^{5/2} \, dx=\frac {a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a-b+b \coth ^2(c+d x)}}\right )}{d}-\frac {\sqrt {b} \left (15 a^2-10 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \coth (c+d x)}{\sqrt {a-b+b \coth ^2(c+d x)}}\right )}{8 d}-\frac {(7 a-3 b) b \coth (c+d x) \sqrt {a-b+b \coth ^2(c+d x)}}{8 d}-\frac {b \coth (c+d x) \left (a-b+b \coth ^2(c+d x)\right )^{3/2}}{4 d} \]
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Time = 0.14 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {4213, 427, 542, 537, 223, 212, 385} \[ \int \left (a+b \text {csch}^2(c+d x)\right )^{5/2} \, dx=\frac {a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+b \coth ^2(c+d x)-b}}\right )}{d}-\frac {\sqrt {b} \left (15 a^2-10 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \coth (c+d x)}{\sqrt {a+b \coth ^2(c+d x)-b}}\right )}{8 d}-\frac {b \coth (c+d x) \left (a+b \coth ^2(c+d x)-b\right )^{3/2}}{4 d}-\frac {b (7 a-3 b) \coth (c+d x) \sqrt {a+b \coth ^2(c+d x)-b}}{8 d} \]
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Rule 212
Rule 223
Rule 385
Rule 427
Rule 537
Rule 542
Rule 4213
Rubi steps \begin{align*} \text {integral}& = \frac {\text {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 {\text {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 {\text {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 \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 {\left (b \left (15 a^2-10 a b+3 b^2\right )\right ) \text {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 \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 {\left (b \left (15 a^2-10 a b+3 b^2\right )\right ) \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 )}{8 d} \\ & = \frac {a^{5/2} \text {arctanh}\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 ) \text {arctanh}\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*}
Time = 3.39 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.33 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^{5/2} \, dx=\frac {\left (a+b \text {csch}^2(c+d x)\right )^{5/2} \left (-2 \sqrt {2} \sqrt {b} \left (15 a^2-10 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {b} \cosh (c+d x)}{\sqrt {-a+2 b+a \cosh (2 (c+d x))}}\right )+b \sqrt {-a+2 b+a \cosh (2 (c+d x))} (9 a-7 b+(-9 a+3 b) \cosh (2 (c+d x))) \coth (c+d x) \text {csch}^3(c+d x)+16 \sqrt {2} a^{5/2} \log \left (\sqrt {2} \sqrt {a} \cosh (c+d x)+\sqrt {-a+2 b+a \cosh (2 (c+d x))}\right )\right ) \sinh ^5(c+d x)}{4 d (-a+2 b+a \cosh (2 (c+d x)))^{5/2}} \]
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\[\int \left (a +b \operatorname {csch}\left (d x +c \right )^{2}\right )^{\frac {5}{2}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 2759 vs. \(2 (152) = 304\).
Time = 0.73 (sec) , antiderivative size = 12590, normalized size of antiderivative = 72.36 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^{5/2} \, dx=\text {Too large to display} \]
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\[ \int \left (a+b \text {csch}^2(c+d x)\right )^{5/2} \, dx=\int \left (a + b \operatorname {csch}^{2}{\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \]
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\[ \int \left (a+b \text {csch}^2(c+d x)\right )^{5/2} \, dx=\int { {\left (b \operatorname {csch}\left (d x + c\right )^{2} + a\right )}^{\frac {5}{2}} \,d x } \]
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Exception generated. \[ \int \left (a+b \text {csch}^2(c+d x)\right )^{5/2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (a+b \text {csch}^2(c+d x)\right )^{5/2} \, dx=\int {\left (a+\frac {b}{{\mathrm {sinh}\left (c+d\,x\right )}^2}\right )}^{5/2} \,d x \]
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