Optimal. Leaf size=82 \[ -\frac {a+b \text {csch}^{-1}(c x)}{e \sqrt {d+e x^2}}-\frac {b c x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{\sqrt {d} e \sqrt {-c^2 x^2}} \]
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Rubi [A] time = 0.11, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6300, 446, 93, 204} \[ -\frac {a+b \text {csch}^{-1}(c x)}{e \sqrt {d+e x^2}}-\frac {b c x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{\sqrt {d} e \sqrt {-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 204
Rule 446
Rule 6300
Rubi steps
\begin {align*} \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) \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) \operatorname {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) \operatorname {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 \tan ^{-1}\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*}
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Mathematica [A] time = 0.16, size = 122, normalized size = 1.49 \[ \frac {b c x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {-d-e x^2} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {c^2 x^2+1}}{\sqrt {-d-e x^2}}\right )}{\sqrt {d} e \sqrt {c^2 x^2+1} \sqrt {d+e x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{e \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 368, normalized size = 4.49 \[ \left [-\frac {4 \, \sqrt {e x^{2} + d} b d \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 4 \, \sqrt {e x^{2} + d} a d - {\left (b e x^{2} + b d\right )} \sqrt {d} \log \left (\frac {{\left (c^{4} d^{2} + 6 \, c^{2} d e + e^{2}\right )} x^{4} + 8 \, {\left (c^{2} d^{2} + d e\right )} x^{2} + 4 \, {\left ({\left (c^{3} d + c e\right )} x^{3} + 2 \, c d x\right )} \sqrt {e x^{2} + d} \sqrt {d} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 8 \, d^{2}}{x^{4}}\right )}{4 \, {\left (d e^{2} x^{2} + d^{2} e\right )}}, -\frac {2 \, \sqrt {e x^{2} + d} b d \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 2 \, \sqrt {e x^{2} + d} a d + {\left (b e x^{2} + b d\right )} \sqrt {-d} \arctan \left (\frac {{\left ({\left (c^{3} d + c e\right )} x^{3} + 2 \, c d x\right )} \sqrt {e x^{2} + d} \sqrt {-d} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{2 \, {\left (c^{2} d e x^{4} + {\left (c^{2} d^{2} + d e\right )} x^{2} + d^{2}\right )}}\right )}{2 \, {\left (d e^{2} x^{2} + d^{2} e\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.44, size = 0, normalized size = 0.00 \[ \int \frac {x \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]
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 \[ -{\left (c^{2} \int \frac {x}{{\left (c^{2} e x^{2} + e\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {e x^{2} + d} + {\left (c^{2} e x^{2} + e\right )} \sqrt {e x^{2} + d}}\,{d x} + \frac {\log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{\sqrt {e x^{2} + d} e} + \int \frac {{\left (e \log \relax (c) - e\right )} c^{2} x^{3} - {\left (c^{2} d - e \log \relax (c)\right )} x + {\left (c^{2} e x^{3} + e x\right )} \log \relax (x)}{{\left (c^{2} e^{2} x^{4} + {\left (c^{2} d e + e^{2}\right )} x^{2} + d e\right )} \sqrt {e x^{2} + d}}\,{d x}\right )} b - \frac {a}{\sqrt {e x^{2} + d} e} \]
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 \[ \int \frac {x\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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