Optimal. Leaf size=321 \[ \frac {b c^3 x^2 \sqrt {d+e x^2}}{d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{d^2 \sqrt {-c^2 x^2}}-\frac {a+b \text {csch}^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \text {csch}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {b c^2 x \sqrt {d+e x^2} E\left (\text {ArcTan}(c x)\left |1-\frac {e}{c^2 d}\right .\right )}{d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}}+\frac {2 b e x \sqrt {d+e x^2} F\left (\text {ArcTan}(c x)\left |1-\frac {e}{c^2 d}\right .\right )}{d^3 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}} \]
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Rubi [A]
time = 0.22, 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 = {277, 197, 6437,
12, 597, 545, 429, 506, 422} \begin {gather*} -\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 (\text {ArcTan}(c x)\left |1-\frac {e}{c^2 d}\right .\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 (\text {ArcTan}(c x)\left |1-\frac {e}{c^2 d}\right .\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 \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{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 {-c^2 x^2-1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 197
Rule 277
Rule 422
Rule 429
Rule 506
Rule 545
Rule 597
Rule 6437
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] Result contains complex when optimal does not.
time = 3.89, size = 201, normalized size = 0.63 \begin {gather*} \frac {b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (d+e x^2\right )-a \left (d+2 e x^2\right )-b \left (d+2 e x^2\right ) \text {csch}^{-1}(c x)}{d^2 x \sqrt {d+e x^2}}+\frac {i b c \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (c^2 d E\left (i \sinh ^{-1}\left (\sqrt {c^2} x\right )|\frac {e}{c^2 d}\right )+\left (-c^2 d+2 e\right ) F\left (i \sinh ^{-1}\left (\sqrt {c^2} x\right )|\frac {e}{c^2 d}\right )\right )}{\sqrt {c^2} d^2 \sqrt {1+c^2 x^2} \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.13, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {a + b \operatorname {acsch}{\left (c x \right )}}{x^{2} \left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{x^2\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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