Optimal. Leaf size=46 \[ -\frac {c \text {CosIntegral}\left (\frac {a}{b}+\sec ^{-1}(c x)\right ) \sin \left (\frac {a}{b}\right )}{b}+\frac {c \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{b} \]
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Rubi [A]
time = 0.08, 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 = {5330, 3384,
3380, 3383} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 3380
Rule 3383
Rule 3384
Rule 5330
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b \sec ^{-1}(c x)\right )} \, dx &=c \text {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 ) \text {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 ) \text {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 = 0.06, size = 43, normalized size = 0.93 \begin {gather*} \frac {c \left (-\text {CosIntegral}\left (\frac {a}{b}+\sec ^{-1}(c x)\right ) \sin \left (\frac {a}{b}\right )+\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 47, normalized size = 1.02
method | result | size |
derivativedivides | \(c \left (\frac {\sinIntegral \left (\frac {a}{b}+\mathrm {arcsec}\left (c x \right )\right ) \cos \left (\frac {a}{b}\right )}{b}-\frac {\cosineIntegral \left (\frac {a}{b}+\mathrm {arcsec}\left (c x \right )\right ) \sin \left (\frac {a}{b}\right )}{b}\right )\) | \(47\) |
default | \(c \left (\frac {\sinIntegral \left (\frac {a}{b}+\mathrm {arcsec}\left (c x \right )\right ) \cos \left (\frac {a}{b}\right )}{b}-\frac {\cosineIntegral \left (\frac {a}{b}+\mathrm {arcsec}\left (c x \right )\right ) \sin \left (\frac {a}{b}\right )}{b}\right )\) | \(47\) |
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]
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \left (a + b \operatorname {asec}{\left (c x \right )}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 55, normalized size = 1.20 \begin {gather*} -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 )} \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 = -0.02 \begin {gather*} \int \frac {1}{x^2\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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