Optimal. Leaf size=321 \[ -\frac {2 e x \left (a+b \text {csch}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{d x \sqrt {d+e x^2}}+\frac {2 b e x \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{d^3 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac {b c \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{d^2 \sqrt {-c^2 x^2}}-\frac {b c^2 x \sqrt {d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac {b c^3 x^2 \sqrt {d+e x^2}}{d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1}} \]
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Rubi [A] time = 0.32, antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {271, 191, 6302, 12, 583, 531, 418, 492, 411} \[ -\frac {2 e x \left (a+b \text {csch}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{d x \sqrt {d+e x^2}}+\frac {b c^3 x^2 \sqrt {d+e x^2}}{d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1}}+\frac {b c \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{d^2 \sqrt {-c^2 x^2}}+\frac {2 b e x \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{d^3 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}-\frac {b c^2 x \sqrt {d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 191
Rule 271
Rule 411
Rule 418
Rule 492
Rule 531
Rule 583
Rule 6302
Rubi steps
\begin {align*} \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx &=-\frac {a+b \text {csch}^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \text {csch}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {-d-2 e x^2}{d^2 x^2 \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=-\frac {a+b \text {csch}^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \text {csch}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {-d-2 e x^2}{x^2 \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{d^2 \sqrt {-c^2 x^2}}\\ &=\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{d^2 \sqrt {-c^2 x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \text {csch}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {-2 d e-c^2 d e x^2}{\sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{d^3 \sqrt {-c^2 x^2}}\\ &=\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{d^2 \sqrt {-c^2 x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \text {csch}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}+\frac {(2 b c e x) \int \frac {1}{\sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{d^2 \sqrt {-c^2 x^2}}+\frac {\left (b c^3 e x\right ) \int \frac {x^2}{\sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{d^2 \sqrt {-c^2 x^2}}\\ &=\frac {b c^3 x^2 \sqrt {d+e x^2}}{d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{d^2 \sqrt {-c^2 x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \text {csch}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}+\frac {2 b e x \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{d^3 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}}+\frac {\left (b c^3 x\right ) \int \frac {\sqrt {d+e x^2}}{\left (-1-c^2 x^2\right )^{3/2}} \, dx}{d^2 \sqrt {-c^2 x^2}}\\ &=\frac {b c^3 x^2 \sqrt {d+e x^2}}{d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{d^2 \sqrt {-c^2 x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \text {csch}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {b c^2 x \sqrt {d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}}+\frac {2 b e x \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{d^3 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}}\\ \end {align*}
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Mathematica [C] time = 0.46, size = 201, normalized size = 0.63 \[ \frac {-a \left (d+2 e x^2\right )+b c x \sqrt {\frac {1}{c^2 x^2}+1} \left (d+e x^2\right )-b \text {csch}^{-1}(c x) \left (d+2 e x^2\right )}{d^2 x \sqrt {d+e x^2}}+\frac {i b c x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {\frac {e x^2}{d}+1} \left (\left (2 e-c^2 d\right ) F\left (i \sinh ^{-1}\left (\sqrt {c^2} x\right )|\frac {e}{c^2 d}\right )+c^2 d E\left (i \sinh ^{-1}\left (\sqrt {c^2} x\right )|\frac {e}{c^2 d}\right )\right )}{\sqrt {c^2} d^2 \sqrt {c^2 x^2+1} \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{e^{2} x^{6} + 2 \, d e x^{4} + d^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.45, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arccsch}\left (c x \right )}{x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]
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 \[ -a {\left (\frac {2 \, e x}{\sqrt {e x^{2} + d} d^{2}} + \frac {1}{\sqrt {e x^{2} + d} d x}\right )} - b {\left (\frac {{\left (2 \, e x^{2} + d\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{\sqrt {e x^{2} + d} d^{2} x} + \int \frac {2 \, c^{2} e x^{2} + c^{2} d}{{\left (c^{2} d^{2} x^{2} + d^{2}\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {e x^{2} + d} + {\left (c^{2} d^{2} x^{2} + d^{2}\right )} \sqrt {e x^{2} + d}}\,{d x} + \int -\frac {2 \, c^{2} e^{2} x^{6} + 3 \, c^{2} d e x^{4} - {\left (d^{2} \log \relax (c) - d^{2}\right )} c^{2} x^{2} - d^{2} \log \relax (c) - {\left (c^{2} d^{2} x^{2} + d^{2}\right )} \log \relax (x)}{{\left (c^{2} d^{2} e x^{6} + d^{3} x^{2} + {\left (c^{2} d^{3} + d^{2} e\right )} x^{4}\right )} \sqrt {e x^{2} + d}}\,{d x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{x^2\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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