Integrand size = 21, antiderivative size = 82 \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=-\frac {a+b \text {csch}^{-1}(c x)}{e \sqrt {d+e x^2}}-\frac {b c x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{\sqrt {d} e \sqrt {-c^2 x^2}} \]
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Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6435, 457, 95, 210} \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=-\frac {a+b \text {csch}^{-1}(c x)}{e \sqrt {d+e x^2}}-\frac {b c x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{\sqrt {d} e \sqrt {-c^2 x^2}} \]
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Rule 95
Rule 210
Rule 457
Rule 6435
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \text {csch}^{-1}(c x)}{e \sqrt {d+e x^2}}+\frac {(b c x) \int \frac {1}{x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{e \sqrt {-c^2 x^2}} \\ & = -\frac {a+b \text {csch}^{-1}(c x)}{e \sqrt {d+e x^2}}+\frac {(b c x) \text {Subst}\left (\int \frac {1}{x \sqrt {-1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{2 e \sqrt {-c^2 x^2}} \\ & = -\frac {a+b \text {csch}^{-1}(c x)}{e \sqrt {d+e x^2}}+\frac {(b c x) \text {Subst}\left (\int \frac {1}{-d-x^2} \, dx,x,\frac {\sqrt {d+e x^2}}{\sqrt {-1-c^2 x^2}}\right )}{e \sqrt {-c^2 x^2}} \\ & = -\frac {a+b \text {csch}^{-1}(c x)}{e \sqrt {d+e x^2}}-\frac {b c x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{\sqrt {d} e \sqrt {-c^2 x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.35 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.96 \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {b \sqrt {1+\frac {d}{e x^2}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,-\frac {1}{c^2 x^2},-\frac {d}{e x^2}\right )-2 c x \left (a+b \text {csch}^{-1}(c x)\right )}{2 c e x \sqrt {d+e x^2}} \]
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\[\int \frac {x \left (a +b \,\operatorname {arccsch}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 176 vs. \(2 (70) = 140\).
Time = 0.29 (sec) , antiderivative size = 368, normalized size of antiderivative = 4.49 \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\left [-\frac {4 \, \sqrt {e x^{2} + d} b d \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 4 \, \sqrt {e x^{2} + d} a d - {\left (b e x^{2} + b d\right )} \sqrt {d} \log \left (\frac {{\left (c^{4} d^{2} + 6 \, c^{2} d e + e^{2}\right )} x^{4} + 8 \, {\left (c^{2} d^{2} + d e\right )} x^{2} + 4 \, {\left ({\left (c^{3} d + c e\right )} x^{3} + 2 \, c d x\right )} \sqrt {e x^{2} + d} \sqrt {d} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 8 \, d^{2}}{x^{4}}\right )}{4 \, {\left (d e^{2} x^{2} + d^{2} e\right )}}, -\frac {2 \, \sqrt {e x^{2} + d} b d \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 2 \, \sqrt {e x^{2} + d} a d + {\left (b e x^{2} + b d\right )} \sqrt {-d} \arctan \left (\frac {{\left ({\left (c^{3} d + c e\right )} x^{3} + 2 \, c d x\right )} \sqrt {e x^{2} + d} \sqrt {-d} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{2 \, {\left (c^{2} d e x^{4} + {\left (c^{2} d^{2} + d e\right )} x^{2} + d^{2}\right )}}\right )}{2 \, {\left (d e^{2} x^{2} + d^{2} e\right )}}\right ] \]
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\[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
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