Optimal. Leaf size=75 \[ -\frac {c \sqrt {1-\frac {1}{c^2 x^2}}}{b \left (a+b \sec ^{-1}(c x)\right )}+\frac {c \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{b^2}+\frac {c \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{b^2} \]
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Rubi [A]
time = 0.10, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5330, 3378,
3384, 3380, 3383} \begin {gather*} \frac {c \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{b^2}+\frac {c \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{b^2}-\frac {c \sqrt {1-\frac {1}{c^2 x^2}}}{b \left (a+b \sec ^{-1}(c x)\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
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 )^2} \, dx &=c \text {Subst}\left (\int \frac {\sin (x)}{(a+b x)^2} \, dx,x,\sec ^{-1}(c x)\right )\\ &=-\frac {c \sqrt {1-\frac {1}{c^2 x^2}}}{b \left (a+b \sec ^{-1}(c x)\right )}+\frac {c \text {Subst}\left (\int \frac {\cos (x)}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )}{b}\\ &=-\frac {c \sqrt {1-\frac {1}{c^2 x^2}}}{b \left (a+b \sec ^{-1}(c x)\right )}+\frac {\left (c \cos \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 )}{b}+\frac {\left (c \sin \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 )}{b}\\ &=-\frac {c \sqrt {1-\frac {1}{c^2 x^2}}}{b \left (a+b \sec ^{-1}(c x)\right )}+\frac {c \cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{b^2}+\frac {c \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )}{b^2}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 69, normalized size = 0.92 \begin {gather*} \frac {c \left (-\frac {b \sqrt {1-\frac {1}{c^2 x^2}}}{a+b \sec ^{-1}(c x)}+\cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sec ^{-1}(c x)\right )\right )}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 78, normalized size = 1.04
method | result | size |
derivativedivides | \(c \left (-\frac {\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{\left (a +b \,\mathrm {arcsec}\left (c x \right )\right ) b}+\frac {\sinIntegral \left (\frac {a}{b}+\mathrm {arcsec}\left (c x \right )\right ) \sin \left (\frac {a}{b}\right )+\cosineIntegral \left (\frac {a}{b}+\mathrm {arcsec}\left (c x \right )\right ) \cos \left (\frac {a}{b}\right )}{b^{2}}\right )\) | \(78\) |
default | \(c \left (-\frac {\sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{\left (a +b \,\mathrm {arcsec}\left (c x \right )\right ) b}+\frac {\sinIntegral \left (\frac {a}{b}+\mathrm {arcsec}\left (c x \right )\right ) \sin \left (\frac {a}{b}\right )+\cosineIntegral \left (\frac {a}{b}+\mathrm {arcsec}\left (c x \right )\right ) \cos \left (\frac {a}{b}\right )}{b^{2}}\right )\) | \(78\) |
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 )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 226 vs.
\(2 (73) = 146\).
time = 0.54, size = 226, normalized size = 3.01 \begin {gather*} {\left (\frac {b \arccos \left (\frac {1}{c x}\right ) \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arccos \left (\frac {1}{c x}\right )\right )}{b^{3} \arccos \left (\frac {1}{c x}\right ) + a b^{2}} + \frac {b \arccos \left (\frac {1}{c x}\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arccos \left (\frac {1}{c x}\right )\right )}{b^{3} \arccos \left (\frac {1}{c x}\right ) + a b^{2}} + \frac {a \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arccos \left (\frac {1}{c x}\right )\right )}{b^{3} \arccos \left (\frac {1}{c x}\right ) + a b^{2}} + \frac {a \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arccos \left (\frac {1}{c x}\right )\right )}{b^{3} \arccos \left (\frac {1}{c x}\right ) + a b^{2}} - \frac {b \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{b^{3} \arccos \left (\frac {1}{c x}\right ) + a b^{2}}\right )} c \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 {1}{x^2\,{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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