Integrand size = 22, antiderivative size = 209 \[ \int \sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=-\frac {2 b^2 x}{d}+\frac {2}{3} a^2 x^{3/2}-\frac {8 a b x \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x \coth \left (c+d \sqrt {x}\right )}{d}+\frac {4 b^2 \sqrt {x} \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {8 a b \sqrt {x} \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {8 a b \sqrt {x} \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {2 b^2 \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {8 a b \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {8 a b \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3} \]
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Time = 0.23 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5545, 4275, 4267, 2611, 2320, 6724, 4269, 3797, 2221, 2317, 2438} \[ \int \sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {2}{3} a^2 x^{3/2}-\frac {8 a b x \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}+\frac {8 a b \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {8 a b \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {8 a b \sqrt {x} \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {8 a b \sqrt {x} \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {2 b^2 \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {4 b^2 \sqrt {x} \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {2 b^2 x \coth \left (c+d \sqrt {x}\right )}{d}-\frac {2 b^2 x}{d} \]
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Rule 2221
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3797
Rule 4267
Rule 4269
Rule 4275
Rule 5545
Rule 6724
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x^2 (a+b \text {csch}(c+d x))^2 \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (a^2 x^2+2 a b x^2 \text {csch}(c+d x)+b^2 x^2 \text {csch}^2(c+d x)\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2}{3} a^2 x^{3/2}+(4 a b) \text {Subst}\left (\int x^2 \text {csch}(c+d x) \, dx,x,\sqrt {x}\right )+\left (2 b^2\right ) \text {Subst}\left (\int x^2 \text {csch}^2(c+d x) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2}{3} a^2 x^{3/2}-\frac {8 a b x \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x \coth \left (c+d \sqrt {x}\right )}{d}-\frac {(8 a b) \text {Subst}\left (\int x \log \left (1-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(8 a b) \text {Subst}\left (\int x \log \left (1+e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {\left (4 b^2\right ) \text {Subst}\left (\int x \coth (c+d x) \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {2 b^2 x}{d}+\frac {2}{3} a^2 x^{3/2}-\frac {8 a b x \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x \coth \left (c+d \sqrt {x}\right )}{d}-\frac {8 a b \sqrt {x} \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {8 a b \sqrt {x} \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {(8 a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(8 a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {\left (8 b^2\right ) \text {Subst}\left (\int \frac {e^{2 (c+d x)} x}{1-e^{2 (c+d x)}} \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {2 b^2 x}{d}+\frac {2}{3} a^2 x^{3/2}-\frac {8 a b x \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x \coth \left (c+d \sqrt {x}\right )}{d}+\frac {4 b^2 \sqrt {x} \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {8 a b \sqrt {x} \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {8 a b \sqrt {x} \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {(8 a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {(8 a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {\left (4 b^2\right ) \text {Subst}\left (\int \log \left (1-e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2} \\ & = -\frac {2 b^2 x}{d}+\frac {2}{3} a^2 x^{3/2}-\frac {8 a b x \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x \coth \left (c+d \sqrt {x}\right )}{d}+\frac {4 b^2 \sqrt {x} \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {8 a b \sqrt {x} \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {8 a b \sqrt {x} \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {8 a b \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {8 a b \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3} \\ & = -\frac {2 b^2 x}{d}+\frac {2}{3} a^2 x^{3/2}-\frac {8 a b x \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x \coth \left (c+d \sqrt {x}\right )}{d}+\frac {4 b^2 \sqrt {x} \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {8 a b \sqrt {x} \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {8 a b \sqrt {x} \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {2 b^2 \operatorname {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {8 a b \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {8 a b \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(637\) vs. \(2(209)=418\).
Time = 2.34 (sec) , antiderivative size = 637, normalized size of antiderivative = 3.05 \[ \int \sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=-\frac {12 b^2 d^2 x+2 a^2 d^3 x^{3/2}-2 a^2 d^3 e^{2 c} x^{3/2}+12 b^2 d \sqrt {x} \log \left (1-e^{-c-d \sqrt {x}}\right )-12 b^2 d e^{2 c} \sqrt {x} \log \left (1-e^{-c-d \sqrt {x}}\right )+12 a b d^2 x \log \left (1-e^{-c-d \sqrt {x}}\right )-12 a b d^2 e^{2 c} x \log \left (1-e^{-c-d \sqrt {x}}\right )+12 b^2 d \sqrt {x} \log \left (1+e^{-c-d \sqrt {x}}\right )-12 b^2 d e^{2 c} \sqrt {x} \log \left (1+e^{-c-d \sqrt {x}}\right )-12 a b d^2 x \log \left (1+e^{-c-d \sqrt {x}}\right )+12 a b d^2 e^{2 c} x \log \left (1+e^{-c-d \sqrt {x}}\right )+12 b \left (-1+e^{2 c}\right ) \left (b-2 a d \sqrt {x}\right ) \operatorname {PolyLog}\left (2,-e^{-c-d \sqrt {x}}\right )+12 b \left (-1+e^{2 c}\right ) \left (b+2 a d \sqrt {x}\right ) \operatorname {PolyLog}\left (2,e^{-c-d \sqrt {x}}\right )+24 a b \operatorname {PolyLog}\left (3,-e^{-c-d \sqrt {x}}\right )-24 a b e^{2 c} \operatorname {PolyLog}\left (3,-e^{-c-d \sqrt {x}}\right )-24 a b \operatorname {PolyLog}\left (3,e^{-c-d \sqrt {x}}\right )+24 a b e^{2 c} \operatorname {PolyLog}\left (3,e^{-c-d \sqrt {x}}\right )+3 b^2 d^2 x \text {csch}\left (\frac {c}{2}\right ) \text {csch}\left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right ) \sinh \left (\frac {d \sqrt {x}}{2}\right )-3 b^2 d^2 e^{2 c} x \text {csch}\left (\frac {c}{2}\right ) \text {csch}\left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right ) \sinh \left (\frac {d \sqrt {x}}{2}\right )-3 b^2 d^2 x \text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right ) \sinh \left (\frac {d \sqrt {x}}{2}\right )+3 b^2 d^2 e^{2 c} x \text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right ) \sinh \left (\frac {d \sqrt {x}}{2}\right )}{3 d^3 \left (-1+e^{2 c}\right )} \]
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\[\int \left (a +b \,\operatorname {csch}\left (c +d \sqrt {x}\right )\right )^{2} \sqrt {x}d x\]
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\[ \int \sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\int { {\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )}^{2} \sqrt {x} \,d x } \]
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\[ \int \sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\int \sqrt {x} \left (a + b \operatorname {csch}{\left (c + d \sqrt {x} \right )}\right )^{2}\, dx \]
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Time = 0.34 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.26 \[ \int \sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {2}{3} \, a^{2} x^{\frac {3}{2}} - \frac {4 \, b^{2} x}{d e^{\left (2 \, d \sqrt {x} + 2 \, c\right )} - d} - \frac {4 \, {\left (d^{2} x \log \left (e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 2 \, d \sqrt {x} {\rm Li}_2\left (-e^{\left (d \sqrt {x} + c\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (d \sqrt {x} + c\right )})\right )} a b}{d^{3}} + \frac {4 \, {\left (d^{2} x \log \left (-e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 2 \, d \sqrt {x} {\rm Li}_2\left (e^{\left (d \sqrt {x} + c\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (d \sqrt {x} + c\right )})\right )} a b}{d^{3}} + \frac {4 \, {\left (d \sqrt {x} \log \left (e^{\left (d \sqrt {x} + c\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (d \sqrt {x} + c\right )}\right )\right )} b^{2}}{d^{3}} + \frac {4 \, {\left (d \sqrt {x} \log \left (-e^{\left (d \sqrt {x} + c\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (d \sqrt {x} + c\right )}\right )\right )} b^{2}}{d^{3}} - \frac {2 \, {\left (2 \, a b d^{3} x^{\frac {3}{2}} + 3 \, b^{2} d^{2} x\right )}}{3 \, d^{3}} + \frac {2 \, {\left (2 \, a b d^{3} x^{\frac {3}{2}} - 3 \, b^{2} d^{2} x\right )}}{3 \, d^{3}} \]
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\[ \int \sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\int { {\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )}^{2} \sqrt {x} \,d x } \]
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Timed out. \[ \int \sqrt {x} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\int \sqrt {x}\,{\left (a+\frac {b}{\mathrm {sinh}\left (c+d\,\sqrt {x}\right )}\right )}^2 \,d x \]
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