Optimal. Leaf size=92 \[ \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{d \sqrt {d+e x^2}}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1+\frac {e x^2}{d}} F\left (\text {ArcSin}(c x)\left |-\frac {e}{c^2 d}\right .\right )}{c d \sqrt {d+e x^2}} \]
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Rubi [A]
time = 0.05, 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 = {197, 6426, 12,
432, 430} \begin {gather*} \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 (\text {ArcSin}(c x)\left |-\frac {e}{c^2 d}\right .\right )}{c d \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 197
Rule 430
Rule 432
Rule 6426
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] Result contains complex when optimal does not.
time = 46.62, size = 334, normalized size = 3.63 \begin {gather*} \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}{1+c x}} \sqrt {\frac {\left (c \sqrt {d}+i \sqrt {e}\right ) (1+c x)}{\left (c \sqrt {d}-i \sqrt {e}\right ) (-1+c x)}} \left (-i \sqrt {d}+\sqrt {e} x\right ) \sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{1-c x}} F\left (\text {ArcSin}\left (\sqrt {\frac {1+\frac {i c \sqrt {d}}{\sqrt {e}}-c x+\frac {i \sqrt {e} x}{\sqrt {d}}}{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 {\frac {1+\frac {i c \sqrt {d}}{\sqrt {e}}-c x+\frac {i \sqrt {e} x}{\sqrt {d}}}{1-c x}} \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.70, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 143 vs.
\(2 (71) = 142\).
time = 0.14, size = 143, normalized size = 1.55 \begin {gather*} \frac {\sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} b c d x \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} a c d x + {\left (b x^{2} \cosh \left (1\right ) + b x^{2} \sinh \left (1\right ) + b d\right )} \sqrt {d} {\rm ellipticF}\left (c x, -\frac {\cosh \left (1\right ) + \sinh \left (1\right )}{c^{2} d}\right )}{c d^{2} x^{2} \cosh \left (1\right ) + c d^{2} x^{2} \sinh \left (1\right ) + c d^{3}} \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 {asech}{\left (c x \right )}}{\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 = -0.01 \begin {gather*} \int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{{\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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