Optimal. Leaf size=84 \[ \frac{c^2 \cos \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 a}{b}+2 \sec ^{-1}(c x)\right )}{b^2}+\frac{c^2 \sin \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \sec ^{-1}(c x)\right )}{b^2}-\frac{c^2 \sin \left (2 \sec ^{-1}(c x)\right )}{2 b \left (a+b \sec ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.149351, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5222, 4406, 12, 3297, 3303, 3299, 3302} \[ \frac{c^2 \cos \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 a}{b}+2 \sec ^{-1}(c x)\right )}{b^2}+\frac{c^2 \sin \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \sec ^{-1}(c x)\right )}{b^2}-\frac{c^2 \sin \left (2 \sec ^{-1}(c x)\right )}{2 b \left (a+b \sec ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 5222
Rule 4406
Rule 12
Rule 3297
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a+b \sec ^{-1}(c x)\right )^2} \, dx &=c^2 \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{(a+b x)^2} \, dx,x,\sec ^{-1}(c x)\right )\\ &=c^2 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 (a+b x)^2} \, dx,x,\sec ^{-1}(c x)\right )\\ &=\frac{1}{2} c^2 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{(a+b x)^2} \, dx,x,\sec ^{-1}(c x)\right )\\ &=-\frac{c^2 \sin \left (2 \sec ^{-1}(c x)\right )}{2 b \left (a+b \sec ^{-1}(c x)\right )}+\frac{c^2 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{a+b x} \, dx,x,\sec ^{-1}(c x)\right )}{b}\\ &=-\frac{c^2 \sin \left (2 \sec ^{-1}(c x)\right )}{2 b \left (a+b \sec ^{-1}(c x)\right )}+\frac{\left (c^2 \cos \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 )}{b}+\frac{\left (c^2 \sin \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 )}{b}\\ &=\frac{c^2 \cos \left (\frac{2 a}{b}\right ) \text{Ci}\left (\frac{2 a}{b}+2 \sec ^{-1}(c x)\right )}{b^2}-\frac{c^2 \sin \left (2 \sec ^{-1}(c x)\right )}{2 b \left (a+b \sec ^{-1}(c x)\right )}+\frac{c^2 \sin \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \sec ^{-1}(c x)\right )}{b^2}\\ \end{align*}
Mathematica [A] time = 0.350875, size = 80, normalized size = 0.95 \[ \frac{c \left (-\frac{b \sqrt{1-\frac{1}{c^2 x^2}}}{a x+b x \sec ^{-1}(c x)}+c \cos \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (2 \left (\frac{a}{b}+\sec ^{-1}(c x)\right )\right )+c \sin \left (\frac{2 a}{b}\right ) \text{Si}\left (2 \left (\frac{a}{b}+\sec ^{-1}(c x)\right )\right )\right )}{b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.243, size = 77, normalized size = 0.9 \begin{align*}{c}^{2} \left ( -{\frac{\sin \left ( 2\,{\rm arcsec} \left (cx\right ) \right ) }{ \left ( 2\,a+2\,b{\rm arcsec} \left (cx\right ) \right ) b}}+{\frac{1}{{b}^{2}} \left ({\it Si} \left ( 2\,{\frac{a}{b}}+2\,{\rm arcsec} \left (cx\right ) \right ) \sin \left ( 2\,{\frac{a}{b}} \right ) +{\it Ci} \left ( 2\,{\frac{a}{b}}+2\,{\rm arcsec} \left (cx\right ) \right ) \cos \left ( 2\,{\frac{a}{b}} \right ) \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*} \text{result too large to display} \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^{2} x^{3} \operatorname{arcsec}\left (c x\right )^{2} + 2 \, a b x^{3} \operatorname{arcsec}\left (c x\right ) + a^{2} 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 )^{2}}\, 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 )}^{2} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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