Optimal. Leaf size=90 \[ -\frac{a+b \sin ^{-1}\left (c+d x^2\right )}{2 x^2}-\frac{b d \tanh ^{-1}\left (\frac{-c^2-c d x^2+1}{\sqrt{1-c^2} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}\right )}{2 \sqrt{1-c^2}} \]
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Rubi [A] time = 0.0924383, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {4842, 12, 1114, 724, 206} \[ -\frac{a+b \sin ^{-1}\left (c+d x^2\right )}{2 x^2}-\frac{b d \tanh ^{-1}\left (\frac{-c^2-c d x^2+1}{\sqrt{1-c^2} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}\right )}{2 \sqrt{1-c^2}} \]
Antiderivative was successfully verified.
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Rule 4842
Rule 12
Rule 1114
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}\left (c+d x^2\right )}{x^3} \, dx &=-\frac{a+b \sin ^{-1}\left (c+d x^2\right )}{2 x^2}+\frac{1}{2} b \int \frac{2 d}{x \sqrt{1-c^2-2 c d x^2-d^2 x^4}} \, dx\\ &=-\frac{a+b \sin ^{-1}\left (c+d x^2\right )}{2 x^2}+(b d) \int \frac{1}{x \sqrt{1-c^2-2 c d x^2-d^2 x^4}} \, dx\\ &=-\frac{a+b \sin ^{-1}\left (c+d x^2\right )}{2 x^2}+\frac{1}{2} (b d) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx,x,x^2\right )\\ &=-\frac{a+b \sin ^{-1}\left (c+d x^2\right )}{2 x^2}-(b d) \operatorname{Subst}\left (\int \frac{1}{4 \left (1-c^2\right )-x^2} \, dx,x,\frac{2 \left (1-c^2-c d x^2\right )}{\sqrt{1-c^2-2 c d x^2-d^2 x^4}}\right )\\ &=-\frac{a+b \sin ^{-1}\left (c+d x^2\right )}{2 x^2}-\frac{b d \tanh ^{-1}\left (\frac{1-c^2-c d x^2}{\sqrt{1-c^2} \sqrt{1-c^2-2 c d x^2-d^2 x^4}}\right )}{2 \sqrt{1-c^2}}\\ \end{align*}
Mathematica [A] time = 0.0773173, size = 81, normalized size = 0.9 \[ \frac{1}{2} \left (-\frac{a+b \sin ^{-1}\left (c+d x^2\right )}{x^2}-\frac{b d \tanh ^{-1}\left (\frac{-c^2-c d x^2+1}{\sqrt{1-c^2} \sqrt{1-\left (c+d x^2\right )^2}}\right )}{\sqrt{1-c^2}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 89, normalized size = 1. \begin{align*} -{\frac{a}{2\,{x}^{2}}}-{\frac{b\arcsin \left ( d{x}^{2}+c \right ) }{2\,{x}^{2}}}-{\frac{bd}{2}\ln \left ({\frac{1}{{x}^{2}} \left ( -2\,{c}^{2}+2-2\,cd{x}^{2}+2\,\sqrt{-{c}^{2}+1}\sqrt{-{d}^{2}{x}^{4}-2\,cd{x}^{2}-{c}^{2}+1} \right ) } \right ){\frac{1}{\sqrt{-{c}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.03477, size = 640, normalized size = 7.11 \begin{align*} \left [-\frac{\sqrt{-c^{2} + 1} b d x^{2} \log \left (\frac{{\left (2 \, c^{2} - 1\right )} d^{2} x^{4} + 2 \, c^{4} + 4 \,{\left (c^{3} - c\right )} d x^{2} + 2 \, \sqrt{-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1}{\left (c d x^{2} + c^{2} - 1\right )} \sqrt{-c^{2} + 1} - 4 \, c^{2} + 2}{x^{4}}\right ) + 2 \, a c^{2} + 2 \,{\left (b c^{2} - b\right )} \arcsin \left (d x^{2} + c\right ) - 2 \, a}{4 \,{\left (c^{2} - 1\right )} x^{2}}, \frac{\sqrt{c^{2} - 1} b d x^{2} \arctan \left (\frac{\sqrt{-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1}{\left (c d x^{2} + c^{2} - 1\right )} \sqrt{c^{2} - 1}}{{\left (c^{2} - 1\right )} d^{2} x^{4} + c^{4} + 2 \,{\left (c^{3} - c\right )} d x^{2} - 2 \, c^{2} + 1}\right ) - a c^{2} -{\left (b c^{2} - b\right )} \arcsin \left (d x^{2} + c\right ) + a}{2 \,{\left (c^{2} - 1\right )} x^{2}}\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{a + b \operatorname{asin}{\left (c + d x^{2} \right )}}{x^{3}}\, 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{b \arcsin \left (d x^{2} + c\right ) + a}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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