Optimal. Leaf size=86 \[ \frac{\sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{b^2 c}-\frac{\cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{b^2 c}-\frac{\sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.172826, antiderivative size = 82, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4621, 4723, 3303, 3299, 3302} \[ \frac{\sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{b^2 c}-\frac{\cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{b^2 c}-\frac{\sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 4621
Rule 4723
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=-\frac{\sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{c \int \frac{x}{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b}\\ &=-\frac{\sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c}\\ &=-\frac{\sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{\cos \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c}+\frac{\sin \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c}\\ &=-\frac{\sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{\text{Ci}\left (\frac{a}{b}+\sin ^{-1}(c x)\right ) \sin \left (\frac{a}{b}\right )}{b^2 c}-\frac{\cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{b^2 c}\\ \end{align*}
Mathematica [A] time = 0.226707, size = 72, normalized size = 0.84 \[ \frac{-\frac{b \sqrt{1-c^2 x^2}}{a+b \sin ^{-1}(c x)}+\sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )-\cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{b^2 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 76, normalized size = 0.9 \begin{align*}{\frac{1}{c} \left ( -{\frac{1}{ \left ( a+b\arcsin \left ( cx \right ) \right ) b}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{1}{{b}^{2}} \left ({\it Si} \left ( \arcsin \left ( cx \right ) +{\frac{a}{b}} \right ) \cos \left ({\frac{a}{b}} \right ) -{\it Ci} \left ( \arcsin \left ( cx \right ) +{\frac{a}{b}} \right ) \sin \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{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}{\left (a + b \operatorname{asin}{\left (c x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.33835, size = 259, normalized size = 3.01 \begin{align*} \frac{b \arcsin \left (c x\right ) \operatorname{Ci}\left (\frac{a}{b} + \arcsin \left (c x\right )\right ) \sin \left (\frac{a}{b}\right )}{b^{3} c \arcsin \left (c x\right ) + a b^{2} c} - \frac{b \arcsin \left (c x\right ) \cos \left (\frac{a}{b}\right ) \operatorname{Si}\left (\frac{a}{b} + \arcsin \left (c x\right )\right )}{b^{3} c \arcsin \left (c x\right ) + a b^{2} c} + \frac{a \operatorname{Ci}\left (\frac{a}{b} + \arcsin \left (c x\right )\right ) \sin \left (\frac{a}{b}\right )}{b^{3} c \arcsin \left (c x\right ) + a b^{2} c} - \frac{a \cos \left (\frac{a}{b}\right ) \operatorname{Si}\left (\frac{a}{b} + \arcsin \left (c x\right )\right )}{b^{3} c \arcsin \left (c x\right ) + a b^{2} c} - \frac{\sqrt{-c^{2} x^{2} + 1} b}{b^{3} c \arcsin \left (c x\right ) + a b^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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