Optimal. Leaf size=70 \[ \frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} (b c-a d)}-\frac{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} (b c-a d)} \]
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Rubi [A] time = 0.0270253, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {391, 205} \[ \frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} (b c-a d)}-\frac{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 391
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx &=\frac{b \int \frac{1}{a+b x^2} \, dx}{b c-a d}-\frac{d \int \frac{1}{c+d x^2} \, dx}{b c-a d}\\ &=\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} (b c-a d)}-\frac{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} (b c-a d)}\\ \end{align*}
Mathematica [A] time = 0.0437391, size = 61, normalized size = 0.87 \[ \frac{\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c}}}{b c-a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 55, normalized size = 0.8 \begin{align*}{\frac{d}{ad-bc}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{b}{ad-bc}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \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 = 2.07481, size = 608, normalized size = 8.69 \begin{align*} \left [-\frac{\sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{2} + 2 \, c x \sqrt{-\frac{d}{c}} - c}{d x^{2} + c}\right )}{2 \,{\left (b c - a d\right )}}, -\frac{2 \, \sqrt{\frac{d}{c}} \arctan \left (x \sqrt{\frac{d}{c}}\right ) + \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{2 \,{\left (b c - a d\right )}}, \frac{2 \, \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) - \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{2} + 2 \, c x \sqrt{-\frac{d}{c}} - c}{d x^{2} + c}\right )}{2 \,{\left (b c - a d\right )}}, \frac{\sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) - \sqrt{\frac{d}{c}} \arctan \left (x \sqrt{\frac{d}{c}}\right )}{b c - a d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.2155, size = 712, normalized size = 10.17 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22475, size = 257, normalized size = 3.67 \begin{align*} -\frac{2 \, \sqrt{c d} b{\left | d \right |} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} x}{\sqrt{\frac{b c + a d + \sqrt{-4 \, a b c d +{\left (b c + a d\right )}^{2}}}{b d}}}\right )}{b c d{\left | b c - a d \right |} + a d^{2}{\left | b c - a d \right |} +{\left (b c - a d\right )}^{2} d} + \frac{2 \, \sqrt{a b} d{\left | b \right |} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} x}{\sqrt{\frac{b c + a d - \sqrt{-4 \, a b c d +{\left (b c + a d\right )}^{2}}}{b d}}}\right )}{b^{2} c{\left | b c - a d \right |} + a b d{\left | b c - a d \right |} -{\left (b c - a d\right )}^{2} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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