Optimal. Leaf size=294 \[ -\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{d x}+\frac {b e x \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac {b c \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{d \sqrt {-c^2 x^2}}-\frac {b c^2 x \sqrt {d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{d \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac {b c^3 x^2 \sqrt {d+e x^2}}{d \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1}} \]
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Rubi [A] time = 0.25, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {264, 6302, 12, 475, 21, 422, 418, 492, 411} \[ -\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{d x}+\frac {b e x \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac {b c^3 x^2 \sqrt {d+e x^2}}{d \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1}}+\frac {b c \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{d \sqrt {-c^2 x^2}}-\frac {b c^2 x \sqrt {d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{d \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 21
Rule 264
Rule 411
Rule 418
Rule 422
Rule 475
Rule 492
Rule 6302
Rubi steps
\begin {align*} \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx &=-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{d x}+\frac {(b c x) \int \frac {\sqrt {d+e x^2}}{d x^2 \sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{d x}+\frac {(b c x) \int \frac {\sqrt {d+e x^2}}{x^2 \sqrt {-1-c^2 x^2}} \, dx}{d \sqrt {-c^2 x^2}}\\ &=\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{d \sqrt {-c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{d x}-\frac {(b c x) \int \frac {-e-c^2 e x^2}{\sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{d \sqrt {-c^2 x^2}}\\ &=\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{d \sqrt {-c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{d x}-\frac {(b c e x) \int \frac {\sqrt {-1-c^2 x^2}}{\sqrt {d+e x^2}} \, dx}{d \sqrt {-c^2 x^2}}\\ &=\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{d \sqrt {-c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{d x}+\frac {(b c e x) \int \frac {1}{\sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{d \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 \sqrt {-c^2 x^2}}\\ &=\frac {b c^3 x^2 \sqrt {d+e x^2}}{d \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 \sqrt {-c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{d x}+\frac {b e x \sqrt {d+e x^2} F\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 {\left (b c^3 x\right ) \int \frac {\sqrt {d+e x^2}}{\left (-1-c^2 x^2\right )^{3/2}} \, dx}{d \sqrt {-c^2 x^2}}\\ &=\frac {b c^3 x^2 \sqrt {d+e x^2}}{d \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 \sqrt {-c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{d x}-\frac {b c^2 x \sqrt {d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{d \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 {b e x \sqrt {d+e x^2} F\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 )}}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 139, normalized size = 0.47 \[ \frac {\sqrt {d+e x^2} \left (-a+b c x \sqrt {\frac {1}{c^2 x^2}+1}-b \text {csch}^{-1}(c x)\right )}{d x}-\frac {b c e x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {\frac {e x^2}{d}+1} E\left (\sin ^{-1}\left (\sqrt {-\frac {e}{d}} x\right )|\frac {c^2 d}{e}\right )}{d \sqrt {c^2 x^2+1} \sqrt {-\frac {e}{d}} \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{e x^{4} + d x^{2}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arcsch}\left (c x\right ) + a}{\sqrt {e x^{2} + d} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.51, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arccsch}\left (c x \right )}{x^{2} \sqrt {e \,x^{2}+d}}\, dx \]
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 \[ -b {\left (\frac {\sqrt {e x^{2} + d} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{d x} + \int \frac {c^{2} e x^{2} + c^{2} d}{{\left (c^{2} d x^{2} + d\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {e x^{2} + d} + {\left (c^{2} d x^{2} + d\right )} \sqrt {e x^{2} + d}}\,{d x} + \int -\frac {c^{2} e x^{4} - {\left (d \log \relax (c) - d\right )} c^{2} x^{2} - d \log \relax (c) - {\left (c^{2} d x^{2} + d\right )} \log \relax (x)}{{\left (c^{2} d x^{4} + d x^{2}\right )} \sqrt {e x^{2} + d}}\,{d x}\right )} - \frac {\sqrt {e x^{2} + d} a}{d x} \]
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 \[ \int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{x^2\,\sqrt {e\,x^2+d}} \,d x \]
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 \[ \int \frac {a + b \operatorname {acsch}{\left (c x \right )}}{x^{2} \sqrt {d + e x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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