Integrand size = 22, antiderivative size = 561 \[ \int \frac {x^{3/2}}{a+b \text {csch}\left (c+d \sqrt {x}\right )} \, dx=\frac {2 x^{5/2}}{5 a}-\frac {2 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {2 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}-\frac {8 b x^{3/2} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {8 b x^{3/2} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {24 b x \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {24 b x \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {48 b \sqrt {x} \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}+\frac {48 b \sqrt {x} \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}+\frac {48 b \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^5}-\frac {48 b \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^5} \]
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Time = 0.70 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {5545, 4276, 3403, 2296, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {x^{3/2}}{a+b \text {csch}\left (c+d \sqrt {x}\right )} \, dx=\frac {48 b \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a d^5 \sqrt {a^2+b^2}}-\frac {48 b \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a d^5 \sqrt {a^2+b^2}}-\frac {48 b \sqrt {x} \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a d^4 \sqrt {a^2+b^2}}+\frac {48 b \sqrt {x} \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a d^4 \sqrt {a^2+b^2}}+\frac {24 b x \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a d^3 \sqrt {a^2+b^2}}-\frac {24 b x \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a d^3 \sqrt {a^2+b^2}}-\frac {8 b x^{3/2} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a d^2 \sqrt {a^2+b^2}}+\frac {8 b x^{3/2} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a d^2 \sqrt {a^2+b^2}}-\frac {2 b x^2 \log \left (\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}+1\right )}{a d \sqrt {a^2+b^2}}+\frac {2 b x^2 \log \left (\frac {a e^{c+d \sqrt {x}}}{\sqrt {a^2+b^2}+b}+1\right )}{a d \sqrt {a^2+b^2}}+\frac {2 x^{5/2}}{5 a} \]
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Rule 2221
Rule 2296
Rule 2320
Rule 2611
Rule 3403
Rule 4276
Rule 5545
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^4}{a+b \text {csch}(c+d x)} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {x^4}{a}-\frac {b x^4}{a (b+a \sinh (c+d x))}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 x^{5/2}}{5 a}-\frac {(2 b) \text {Subst}\left (\int \frac {x^4}{b+a \sinh (c+d x)} \, dx,x,\sqrt {x}\right )}{a} \\ & = \frac {2 x^{5/2}}{5 a}-\frac {(4 b) \text {Subst}\left (\int \frac {e^{c+d x} x^4}{-a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,\sqrt {x}\right )}{a} \\ & = \frac {2 x^{5/2}}{5 a}-\frac {(4 b) \text {Subst}\left (\int \frac {e^{c+d x} x^4}{2 b-2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2+b^2}}+\frac {(4 b) \text {Subst}\left (\int \frac {e^{c+d x} x^4}{2 b+2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2+b^2}} \\ & = \frac {2 x^{5/2}}{5 a}-\frac {2 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {2 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {(8 b) \text {Subst}\left (\int x^3 \log \left (1+\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2} d}-\frac {(8 b) \text {Subst}\left (\int x^3 \log \left (1+\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2} d} \\ & = \frac {2 x^{5/2}}{5 a}-\frac {2 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {2 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}-\frac {8 b x^{3/2} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {8 b x^{3/2} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {(24 b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (2,-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2} d^2}-\frac {(24 b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (2,-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2} d^2} \\ & = \frac {2 x^{5/2}}{5 a}-\frac {2 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {2 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}-\frac {8 b x^{3/2} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {8 b x^{3/2} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {24 b x \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {24 b x \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {(48 b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (3,-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2} d^3}+\frac {(48 b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (3,-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2} d^3} \\ & = \frac {2 x^{5/2}}{5 a}-\frac {2 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {2 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}-\frac {8 b x^{3/2} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {8 b x^{3/2} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {24 b x \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {24 b x \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {48 b \sqrt {x} \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}+\frac {48 b \sqrt {x} \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}+\frac {(48 b) \text {Subst}\left (\int \operatorname {PolyLog}\left (4,-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2} d^4}-\frac {(48 b) \text {Subst}\left (\int \operatorname {PolyLog}\left (4,-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2} d^4} \\ & = \frac {2 x^{5/2}}{5 a}-\frac {2 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {2 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}-\frac {8 b x^{3/2} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {8 b x^{3/2} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {24 b x \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {24 b x \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {48 b \sqrt {x} \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}+\frac {48 b \sqrt {x} \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}+\frac {(48 b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (4,\frac {a x}{-b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{a \sqrt {a^2+b^2} d^5}-\frac {(48 b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (4,-\frac {a x}{b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{a \sqrt {a^2+b^2} d^5} \\ & = \frac {2 x^{5/2}}{5 a}-\frac {2 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {2 b x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}-\frac {8 b x^{3/2} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {8 b x^{3/2} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {24 b x \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {24 b x \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {48 b \sqrt {x} \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}+\frac {48 b \sqrt {x} \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}+\frac {48 b \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^5}-\frac {48 b \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^5} \\ \end{align*}
Time = 0.80 (sec) , antiderivative size = 436, normalized size of antiderivative = 0.78 \[ \int \frac {x^{3/2}}{a+b \text {csch}\left (c+d \sqrt {x}\right )} \, dx=\frac {2 \left (\sqrt {a^2+b^2} d^5 x^{5/2}-5 b d^4 x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )+5 b d^4 x^2 \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )-20 b d^3 x^{3/2} \operatorname {PolyLog}\left (2,\frac {a e^{c+d \sqrt {x}}}{-b+\sqrt {a^2+b^2}}\right )+20 b d^3 x^{3/2} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )+60 b d^2 x \operatorname {PolyLog}\left (3,\frac {a e^{c+d \sqrt {x}}}{-b+\sqrt {a^2+b^2}}\right )-60 b d^2 x \operatorname {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )-120 b d \sqrt {x} \operatorname {PolyLog}\left (4,\frac {a e^{c+d \sqrt {x}}}{-b+\sqrt {a^2+b^2}}\right )+120 b d \sqrt {x} \operatorname {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )+120 b \operatorname {PolyLog}\left (5,\frac {a e^{c+d \sqrt {x}}}{-b+\sqrt {a^2+b^2}}\right )-120 b \operatorname {PolyLog}\left (5,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )\right )}{5 a \sqrt {a^2+b^2} d^5} \]
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\[\int \frac {x^{\frac {3}{2}}}{a +b \,\operatorname {csch}\left (c +d \sqrt {x}\right )}d x\]
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\[ \int \frac {x^{3/2}}{a+b \text {csch}\left (c+d \sqrt {x}\right )} \, dx=\int { \frac {x^{\frac {3}{2}}}{b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a} \,d x } \]
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\[ \int \frac {x^{3/2}}{a+b \text {csch}\left (c+d \sqrt {x}\right )} \, dx=\int \frac {x^{\frac {3}{2}}}{a + b \operatorname {csch}{\left (c + d \sqrt {x} \right )}}\, dx \]
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\[ \int \frac {x^{3/2}}{a+b \text {csch}\left (c+d \sqrt {x}\right )} \, dx=\int { \frac {x^{\frac {3}{2}}}{b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a} \,d x } \]
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\[ \int \frac {x^{3/2}}{a+b \text {csch}\left (c+d \sqrt {x}\right )} \, dx=\int { \frac {x^{\frac {3}{2}}}{b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {x^{3/2}}{a+b \text {csch}\left (c+d \sqrt {x}\right )} \, dx=\int \frac {x^{3/2}}{a+\frac {b}{\mathrm {sinh}\left (c+d\,\sqrt {x}\right )}} \,d x \]
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