Optimal. Leaf size=92 \[ \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{d \sqrt {d+e x^2}}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{c d \sqrt {d+e x^2}} \]
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Rubi [A] time = 0.08, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {191, 6291, 12, 421, 419} \[ \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{d \sqrt {d+e x^2}}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{c d \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 191
Rule 419
Rule 421
Rule 6291
Rubi steps
\begin {align*} \int \frac {a+b \text {sech}^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx &=\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{d \sqrt {d+e x^2}}+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{d \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \, dx\\ &=\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{d \sqrt {d+e x^2}}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{d}\\ &=\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{d \sqrt {d+e x^2}}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{d \sqrt {d+e x^2}}\\ &=\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{d \sqrt {d+e x^2}}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1+\frac {e x^2}{d}} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{c d \sqrt {d+e x^2}}\\ \end {align*}
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Mathematica [C] time = 1.39, size = 334, normalized size = 3.63 \[ \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{d \sqrt {d+e x^2}}+\frac {2 i b \sqrt {\frac {1-c x}{c x+1}} \left (\sqrt {e} x-i \sqrt {d}\right ) \sqrt {\frac {(c x+1) \left (c \sqrt {d}+i \sqrt {e}\right )}{(c x-1) \left (c \sqrt {d}-i \sqrt {e}\right )}} \sqrt {-\frac {c \left (x+\frac {i \sqrt {d}}{\sqrt {e}}\right )+\frac {i \sqrt {e} x}{\sqrt {d}}-1}{1-c x}} F\left (\sin ^{-1}\left (\sqrt {\frac {-x c+\frac {i \sqrt {d} c}{\sqrt {e}}+\frac {i \sqrt {e} x}{\sqrt {d}}+1}{2-2 c x}}\right )|-\frac {4 i c \sqrt {d} \sqrt {e}}{\left (c \sqrt {d}-i \sqrt {e}\right )^2}\right )}{d \left (c \sqrt {d}+i \sqrt {e}\right ) \sqrt {d+e x^2} \sqrt {\frac {\frac {i c \sqrt {d}}{\sqrt {e}}+c (-x)+\frac {i \sqrt {e} x}{\sqrt {d}}+1}{1-c x}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.17, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arcsech}\left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]
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 \[ b \int \frac {\log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} + \frac {a x}{\sqrt {e x^{2} + d} d} \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
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 \[ \int \frac {a + b \operatorname {asech}{\left (c x \right )}}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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