Optimal. Leaf size=46 \[ \frac {c \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{b}-\frac {c \sin \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{b} \]
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Rubi [A] time = 0.11, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5222, 3303, 3299, 3302} \[ \frac {c \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{b}-\frac {c \sin \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{b} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 3302
Rule 3303
Rule 5222
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b \sec ^{-1}(c x)\right )} \, dx &=c \operatorname {Subst}\left (\int \frac {\sin (x)}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )\\ &=\left (c \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )-\left (c \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )\\ &=-\frac {c \text {Ci}\left (\frac {a}{b}+\sec ^{-1}(c x)\right ) \sin \left (\frac {a}{b}\right )}{b}+\frac {c \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 43, normalized size = 0.93 \[ \frac {c \left (\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )-\sin \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )\right )}{b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b x^{2} \operatorname {arcsec}\left (c x\right ) + a x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 55, normalized size = 1.20 \[ -c {\left (\frac {\operatorname {Ci}\left (\frac {a}{b} + \arccos \left (\frac {1}{c x}\right )\right ) \sin \left (\frac {a}{b}\right )}{b} - \frac {\cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arccos \left (\frac {1}{c x}\right )\right )}{b}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 47, normalized size = 1.02 \[ c \left (\frac {\Si \left (\frac {a}{b}+\mathrm {arcsec}\left (c x \right )\right ) \cos \left (\frac {a}{b}\right )}{b}-\frac {\Ci \left (\frac {a}{b}+\mathrm {arcsec}\left (c x \right )\right ) \sin \left (\frac {a}{b}\right )}{b}\right ) \]
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 \[ \int \frac {1}{{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^2\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )} \,d x \]
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 \[ \int \frac {1}{x^{2} \left (a + b \operatorname {asec}{\left (c x \right )}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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