Optimal. Leaf size=63 \[ -\frac {c^2 \text {CosIntegral}\left (\frac {2 a}{b}+2 \sec ^{-1}(c x)\right ) \sin \left (\frac {2 a}{b}\right )}{2 b}+\frac {c^2 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \sec ^{-1}(c x)\right )}{2 b} \]
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Rubi [A]
time = 0.11, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5330, 4491, 12,
3384, 3380, 3383} \begin {gather*} \frac {c^2 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \sec ^{-1}(c x)\right )}{2 b}-\frac {c^2 \sin \left (\frac {2 a}{b}\right ) \text {CosIntegral}\left (\frac {2 a}{b}+2 \sec ^{-1}(c x)\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3380
Rule 3383
Rule 3384
Rule 4491
Rule 5330
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b \sec ^{-1}(c x)\right )} \, dx &=c^2 \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )\\ &=c^2 \text {Subst}\left (\int \frac {\sin (2 x)}{2 (a+b x)} \, dx,x,\sec ^{-1}(c x)\right )\\ &=\frac {1}{2} c^2 \text {Subst}\left (\int \frac {\sin (2 x)}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )\\ &=\frac {1}{2} \left (c^2 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )-\frac {1}{2} \left (c^2 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )\\ &=-\frac {c^2 \text {Ci}\left (\frac {2 a}{b}+2 \sec ^{-1}(c x)\right ) \sin \left (\frac {2 a}{b}\right )}{2 b}+\frac {c^2 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \sec ^{-1}(c x)\right )}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 56, normalized size = 0.89 \begin {gather*} \frac {c^2 \left (-\text {CosIntegral}\left (\frac {2 a}{b}+2 \sec ^{-1}(c x)\right ) \sin \left (\frac {2 a}{b}\right )+\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \sec ^{-1}(c x)\right )\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 58, normalized size = 0.92
method | result | size |
derivativedivides | \(c^{2} \left (\frac {\sinIntegral \left (\frac {2 a}{b}+2 \,\mathrm {arcsec}\left (c x \right )\right ) \cos \left (\frac {2 a}{b}\right )}{2 b}-\frac {\cosineIntegral \left (\frac {2 a}{b}+2 \,\mathrm {arcsec}\left (c x \right )\right ) \sin \left (\frac {2 a}{b}\right )}{2 b}\right )\) | \(58\) |
default | \(c^{2} \left (\frac {\sinIntegral \left (\frac {2 a}{b}+2 \,\mathrm {arcsec}\left (c x \right )\right ) \cos \left (\frac {2 a}{b}\right )}{2 b}-\frac {\cosineIntegral \left (\frac {2 a}{b}+2 \,\mathrm {arcsec}\left (c x \right )\right ) \sin \left (\frac {2 a}{b}\right )}{2 b}\right )\) | \(58\) |
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^{3} \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 = 0.43, size = 95, normalized size = 1.51 \begin {gather*} -\frac {1}{2} \, {\left (\frac {2 \, c \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arccos \left (\frac {1}{c x}\right )\right ) \sin \left (\frac {a}{b}\right )}{b} - \frac {2 \, c \cos \left (\frac {a}{b}\right )^{2} \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arccos \left (\frac {1}{c x}\right )\right )}{b} + \frac {c \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arccos \left (\frac {1}{c x}\right )\right )}{b}\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 = -0.02 \begin {gather*} \int \frac {1}{x^3\,\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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