Optimal. Leaf size=249 \[ \frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{d^2 x}-\frac {a+b \text {sech}^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \text {sech}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}+\frac {b c \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x^2} E\left (\text {ArcSin}(c x)\left |-\frac {e}{c^2 d}\right .\right )}{d^2 \sqrt {1+\frac {e x^2}{d}}}-\frac {b \left (c^2 d+2 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1+\frac {e x^2}{d}} F\left (\text {ArcSin}(c x)\left |-\frac {e}{c^2 d}\right .\right )}{c d^2 \sqrt {d+e x^2}} \]
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Rubi [A]
time = 0.19, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {277, 197,
6436, 12, 597, 538, 437, 435, 432, 430} \begin {gather*} -\frac {2 e x \left (a+b \text {sech}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \text {sech}^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (c^2 d+2 e\right ) \sqrt {\frac {e x^2}{d}+1} F\left (\text {ArcSin}(c x)\left |-\frac {e}{c^2 d}\right .\right )}{c d^2 \sqrt {d+e x^2}}+\frac {b c \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {d+e x^2} E\left (\text {ArcSin}(c x)\left |-\frac {e}{c^2 d}\right .\right )}{d^2 \sqrt {\frac {e x^2}{d}+1}}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{d^2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 197
Rule 277
Rule 430
Rule 432
Rule 435
Rule 437
Rule 538
Rule 597
Rule 6436
Rubi steps
\begin {align*} \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx &=-\frac {a+b \text {sech}^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \text {sech}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-d-2 e x^2}{d^2 x^2 \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \, dx\\ &=-\frac {a+b \text {sech}^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \text {sech}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-d-2 e x^2}{x^2 \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{d^2}\\ &=\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{d^2 x}-\frac {a+b \text {sech}^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \text {sech}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \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}\\ &=\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{d^2 x}-\frac {a+b \text {sech}^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \text {sech}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}+\frac {\left (b c^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}} \, dx}{d^2}-\frac {\left (b \left (c^2 d+2 e\right ) \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^2}\\ &=\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{d^2 x}-\frac {a+b \text {sech}^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \text {sech}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}+\frac {\left (b c^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x^2}\right ) \int \frac {\sqrt {1+\frac {e x^2}{d}}}{\sqrt {1-c^2 x^2}} \, dx}{d^2 \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b \left (c^2 d+2 e\right ) \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^2 \sqrt {d+e x^2}}\\ &=\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{d^2 x}-\frac {a+b \text {sech}^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \text {sech}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}+\frac {b c \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{d^2 \sqrt {1+\frac {e x^2}{d}}}-\frac {b \left (c^2 d+2 e\right ) \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^2 \sqrt {d+e x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 22.40, size = 501, normalized size = 2.01 \begin {gather*} \frac {\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (d+e x^2\right )}{x}-\frac {a \left (d+2 e x^2\right )}{x}-\frac {b \left (d+2 e x^2\right ) \text {sech}^{-1}(c x)}{x}+\frac {b \sqrt {\frac {1-c x}{1+c x}} \left (-c^2 \left (d+e x^2\right )+\frac {(1+c x) \sqrt {\frac {c \left (\sqrt {d}-i \sqrt {e} x\right )}{\left (c \sqrt {d}-i \sqrt {e}\right ) (1+c x)}} \sqrt {\frac {c \left (\sqrt {d}+i \sqrt {e} x\right )}{\left (c \sqrt {d}+i \sqrt {e}\right ) (1+c x)}} \left (-i \left (c \sqrt {d}-i \sqrt {e}\right )^2 E\left (i \sinh ^{-1}\left (\sqrt {\frac {\left (c^2 d+e\right ) (1-c x)}{\left (c \sqrt {d}+i \sqrt {e}\right )^2 (1+c x)}}\right )|\frac {\left (c \sqrt {d}+i \sqrt {e}\right )^2}{\left (c \sqrt {d}-i \sqrt {e}\right )^2}\right )+2 \left (c \sqrt {d}-2 i \sqrt {e}\right ) \sqrt {e} F\left (i \sinh ^{-1}\left (\sqrt {\frac {\left (c^2 d+e\right ) (1-c x)}{\left (c \sqrt {d}+i \sqrt {e}\right )^2 (1+c x)}}\right )|\frac {\left (c \sqrt {d}+i \sqrt {e}\right )^2}{\left (c \sqrt {d}-i \sqrt {e}\right )^2}\right )\right )}{\sqrt {-\frac {\left (c \sqrt {d}-i \sqrt {e}\right ) (-1+c x)}{\left (c \sqrt {d}+i \sqrt {e}\right ) (1+c x)}}}\right )}{c}}{d^2 \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.40, size = 0, normalized size = 0.00 \[\int \frac {a +b \,\mathrm {arcsech}\left (c x \right )}{x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx\]
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 {asech}{\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 {acosh}\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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