Optimal. Leaf size=63 \[ \frac{c^2 \cos \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \sec ^{-1}(c x)\right )}{2 b}-\frac{c^2 \sin \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 a}{b}+2 \sec ^{-1}(c x)\right )}{2 b} \]
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Rubi [A] time = 0.135768, antiderivative size = 63, normalized size of antiderivative = 1., 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 = {5222, 4406, 12, 3303, 3299, 3302} \[ \frac{c^2 \cos \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \sec ^{-1}(c x)\right )}{2 b}-\frac{c^2 \sin \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 a}{b}+2 \sec ^{-1}(c x)\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 5222
Rule 4406
Rule 12
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a+b \sec ^{-1}(c x)\right )} \, dx &=c^2 \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )\\ &=c^2 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 (a+b x)} \, dx,x,\sec ^{-1}(c x)\right )\\ &=\frac{1}{2} c^2 \operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [A] time = 0.0716536, size = 56, normalized size = 0.89 \[ \frac{c^2 \left (\cos \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \sec ^{-1}(c x)\right )-\sin \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 a}{b}+2 \sec ^{-1}(c x)\right )\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.24, size = 58, normalized size = 0.9 \begin{align*}{c}^{2} \left ({\frac{1}{2\,b}{\it Si} \left ( 2\,{\frac{a}{b}}+2\,{\rm arcsec} \left (cx\right ) \right ) \cos \left ( 2\,{\frac{a}{b}} \right ) }-{\frac{1}{2\,b}{\it Ci} \left ( 2\,{\frac{a}{b}}+2\,{\rm arcsec} \left (cx\right ) \right ) \sin \left ( 2\,{\frac{a}{b}} \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{arcsec}\left (c x\right ) + a\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b x^{3} \operatorname{arcsec}\left (c x\right ) + a x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \left (a + b \operatorname{asec}{\left (c x \right )}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{arcsec}\left (c x\right ) + a\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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