Optimal. Leaf size=66 \[ -\frac{d \left (a+b \sin ^{-1}(c x)\right )}{x}+e x \left (a+b \sin ^{-1}(c x)\right )-b c d \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )+\frac{b e \sqrt{1-c^2 x^2}}{c} \]
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Rubi [A] time = 0.0773946, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {14, 4731, 446, 80, 63, 208} \[ -\frac{d \left (a+b \sin ^{-1}(c x)\right )}{x}+e x \left (a+b \sin ^{-1}(c x)\right )-b c d \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )+\frac{b e \sqrt{1-c^2 x^2}}{c} \]
Antiderivative was successfully verified.
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Rule 14
Rule 4731
Rule 446
Rule 80
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{x^2} \, dx &=-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{x}+e x \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac{-d+e x^2}{x \sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{x}+e x \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{2} (b c) \operatorname{Subst}\left (\int \frac{-d+e x}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac{b e \sqrt{1-c^2 x^2}}{c}-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{x}+e x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} (b c d) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac{b e \sqrt{1-c^2 x^2}}{c}-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{x}+e x \left (a+b \sin ^{-1}(c x)\right )-\frac{(b d) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c^2}-\frac{x^2}{c^2}} \, dx,x,\sqrt{1-c^2 x^2}\right )}{c}\\ &=\frac{b e \sqrt{1-c^2 x^2}}{c}-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{x}+e x \left (a+b \sin ^{-1}(c x)\right )-b c d \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0546412, size = 71, normalized size = 1.08 \[ -\frac{a d}{x}+a e x-b c d \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )+\frac{b e \sqrt{1-c^2 x^2}}{c}-\frac{b d \sin ^{-1}(c x)}{x}+b e x \sin ^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 79, normalized size = 1.2 \begin{align*} c \left ({\frac{a}{{c}^{2}} \left ( ecx-{\frac{dc}{x}} \right ) }+{\frac{b}{{c}^{2}} \left ( \arcsin \left ( cx \right ) ecx-{\frac{\arcsin \left ( cx \right ) cd}{x}}+e\sqrt{-{c}^{2}{x}^{2}+1}-{c}^{2}d{\it Artanh} \left ({\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}} \right ) \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43747, size = 107, normalized size = 1.62 \begin{align*} -{\left (c \log \left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{\arcsin \left (c x\right )}{x}\right )} b d + a e x + \frac{{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} b e}{c} - \frac{a d}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.09, size = 244, normalized size = 3.7 \begin{align*} -\frac{b c^{2} d x \log \left (\sqrt{-c^{2} x^{2} + 1} + 1\right ) - b c^{2} d x \log \left (\sqrt{-c^{2} x^{2} + 1} - 1\right ) - 2 \, a c e x^{2} - 2 \, \sqrt{-c^{2} x^{2} + 1} b e x + 2 \, a c d - 2 \,{\left (b c e x^{2} - b c d\right )} \arcsin \left (c x\right )}{2 \, c x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.11549, size = 75, normalized size = 1.14 \begin{align*} - \frac{a d}{x} + a e x + b c d \left (\begin{cases} - \operatorname{acosh}{\left (\frac{1}{c x} \right )} & \text{for}\: \frac{1}{\left |{c^{2} x^{2}}\right |} > 1 \\i \operatorname{asin}{\left (\frac{1}{c x} \right )} & \text{otherwise} \end{cases}\right ) - \frac{b d \operatorname{asin}{\left (c x \right )}}{x} + b e \left (\begin{cases} 0 & \text{for}\: c = 0 \\x \operatorname{asin}{\left (c x \right )} + \frac{\sqrt{- c^{2} x^{2} + 1}}{c} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.75348, size = 1399, normalized size = 21.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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