Optimal. Leaf size=75 \[ \frac{c \cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sec ^{-1}(c x)\right )}{b^2}+\frac{c \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sec ^{-1}(c x)\right )}{b^2}-\frac{c \sqrt{1-\frac{1}{c^2 x^2}}}{b \left (a+b \sec ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.128973, antiderivative size = 75, normalized size of antiderivative = 1., 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 = {5222, 3297, 3303, 3299, 3302} \[ \frac{c \cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sec ^{-1}(c x)\right )}{b^2}+\frac{c \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sec ^{-1}(c x)\right )}{b^2}-\frac{c \sqrt{1-\frac{1}{c^2 x^2}}}{b \left (a+b \sec ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 5222
Rule 3297
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a+b \sec ^{-1}(c x)\right )^2} \, dx &=c \operatorname{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 \operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [A] time = 0.256383, size = 69, normalized size = 0.92 \[ \frac{c \left (-\frac{b \sqrt{1-\frac{1}{c^2 x^2}}}{a+b \sec ^{-1}(c x)}+\cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sec ^{-1}(c x)\right )+\sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sec ^{-1}(c x)\right )\right )}{b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.241, size = 78, normalized size = 1. \begin{align*} c \left ( -{\frac{1}{ \left ( a+b{\rm arcsec} \left (cx\right ) \right ) b}\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}+{\frac{1}{{b}^{2}} \left ({\it Si} \left ({\frac{a}{b}}+{\rm arcsec} \left (cx\right ) \right ) \sin \left ({\frac{a}{b}} \right ) +{\it Ci} \left ({\frac{a}{b}}+{\rm arcsec} \left (cx\right ) \right ) \cos \left ({\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*} -\frac{4 \, \sqrt{c x + 1} \sqrt{c x - 1}{\left (b \arctan \left (\sqrt{c x + 1} \sqrt{c x - 1}\right ) + a\right )} - 4 \,{\left (4 \, b^{3} x \arctan \left (\sqrt{c x + 1} \sqrt{c x - 1}\right )^{2} + b^{3} x \log \left (c^{2} x^{2}\right )^{2} + 8 \, b^{3} x \log \left (c\right ) \log \left (x\right ) + 4 \, b^{3} x \log \left (x\right )^{2} + 8 \, a b^{2} x \arctan \left (\sqrt{c x + 1} \sqrt{c x - 1}\right ) + 4 \,{\left (b^{3} \log \left (c\right )^{2} + a^{2} b\right )} x - 4 \,{\left (b^{3} x \log \left (c\right ) + b^{3} x \log \left (x\right )\right )} \log \left (c^{2} x^{2}\right )\right )} \int \frac{\sqrt{c x + 1} \sqrt{c x - 1}{\left (b \arctan \left (\sqrt{c x + 1} \sqrt{c x - 1}\right ) + a\right )}}{4 \,{\left (b^{3} c^{2} \log \left (c\right )^{2} + a^{2} b c^{2}\right )} x^{4} - 4 \,{\left (b^{3} \log \left (c\right )^{2} + a^{2} b\right )} x^{2} + 4 \,{\left (b^{3} c^{2} x^{4} - b^{3} x^{2}\right )} \arctan \left (\sqrt{c x + 1} \sqrt{c x - 1}\right )^{2} +{\left (b^{3} c^{2} x^{4} - b^{3} x^{2}\right )} \log \left (c^{2} x^{2}\right )^{2} + 4 \,{\left (b^{3} c^{2} x^{4} - b^{3} x^{2}\right )} \log \left (x\right )^{2} + 8 \,{\left (a b^{2} c^{2} x^{4} - a b^{2} x^{2}\right )} \arctan \left (\sqrt{c x + 1} \sqrt{c x - 1}\right ) - 4 \,{\left (b^{3} c^{2} x^{4} \log \left (c\right ) - b^{3} x^{2} \log \left (c\right ) +{\left (b^{3} c^{2} x^{4} - b^{3} x^{2}\right )} \log \left (x\right )\right )} \log \left (c^{2} x^{2}\right ) + 8 \,{\left (b^{3} c^{2} x^{4} \log \left (c\right ) - b^{3} x^{2} \log \left (c\right )\right )} \log \left (x\right )}\,{d x}}{4 \, b^{3} x \arctan \left (\sqrt{c x + 1} \sqrt{c x - 1}\right )^{2} + b^{3} x \log \left (c^{2} x^{2}\right )^{2} + 8 \, b^{3} x \log \left (c\right ) \log \left (x\right ) + 4 \, b^{3} x \log \left (x\right )^{2} + 8 \, a b^{2} x \arctan \left (\sqrt{c x + 1} \sqrt{c x - 1}\right ) + 4 \,{\left (b^{3} \log \left (c\right )^{2} + a^{2} b\right )} x - 4 \,{\left (b^{3} x \log \left (c\right ) + b^{3} x \log \left (x\right )\right )} \log \left (c^{2} x^{2}\right )} \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^{2} \operatorname{arcsec}\left (c x\right )^{2} + 2 \, a b x^{2} \operatorname{arcsec}\left (c x\right ) + a^{2} x^{2}}, 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^{2} \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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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