Optimal. Leaf size=174 \[ \frac {a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+b \coth ^2(c+d x)-b}}\right )}{d}-\frac {\sqrt {b} \left (15 a^2-10 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \coth (c+d x)}{\sqrt {a+b \coth ^2(c+d x)-b}}\right )}{8 d}-\frac {b \coth (c+d x) \left (a+b \coth ^2(c+d x)-b\right )^{3/2}}{4 d}-\frac {b (7 a-3 b) \coth (c+d x) \sqrt {a+b \coth ^2(c+d x)-b}}{8 d} \]
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Rubi [A] time = 0.19, antiderivative size = 174, normalized size of antiderivative = 1.00, 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 206
Rule 217
Rule 377
Rule 416
Rule 523
Rule 528
Rule 4128
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*}
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Mathematica [A] time = 4.90, size = 231, normalized size = 1.33 \[ \frac {\sinh ^5(c+d x) \left (a+b \text {csch}^2(c+d x)\right )^{5/2} \left (16 \sqrt {2} a^{5/2} \log \left (\sqrt {a \cosh (2 (c+d x))-a+2 b}+\sqrt {2} \sqrt {a} \cosh (c+d x)\right )-2 \sqrt {2} \sqrt {b} \left (15 a^2-10 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {b} \cosh (c+d x)}{\sqrt {a \cosh (2 (c+d x))-a+2 b}}\right )+b \coth (c+d x) \text {csch}^3(c+d x) \sqrt {a \cosh (2 (c+d x))-a+2 b} ((3 b-9 a) \cosh (2 (c+d x))+9 a-7 b)\right )}{4 d (a \cosh (2 (c+d x))-a+2 b)^{5/2}} \]
Antiderivative was successfully verified.
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fricas [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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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.62, size = 0, normalized size = 0.00 \[ \int \left (a +b \mathrm {csch}\left (d x +c \right )^{2}\right )^{\frac {5}{2}}\, dx \]
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 \[ \int {\left (b \operatorname {csch}\left (d x + c\right )^{2} + a\right )}^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (a+\frac {b}{{\mathrm {sinh}\left (c+d\,x\right )}^2}\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {csch}^{2}{\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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