Optimal. Leaf size=70 \[ \frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} (b c-a d)}-\frac{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} (b c-a d)} \]
[Out]
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Rubi [A] time = 0.0685471, 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 \[ \frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} (b c-a d)}-\frac{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x^2)*(c + d*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 15.6197, size = 60, normalized size = 0.86 \[ \frac{\sqrt{d} \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{\sqrt{c} \left (a d - b c\right )} - \frac{\sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{\sqrt{a} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**2+a)/(d*x**2+c),x)
[Out]
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Mathematica [A] time = 0.0718551, 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.
[In] Integrate[1/((a + b*x^2)*(c + d*x^2)),x]
[Out]
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Maple [A] time = 0., size = 55, normalized size = 0.8 \[{\frac{d}{ad-bc}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{b}{ad-bc}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^2+a)/(d*x^2+c),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.251557, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{2} + 2 \, c x \sqrt{-\frac{d}{c}} - c}{d x^{2} + c}\right )}{2 \,{\left (b c - a d\right )}}, -\frac{2 \, \sqrt{\frac{d}{c}} \arctan \left (\frac{d x}{c \sqrt{\frac{d}{c}}}\right ) + \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{2 \,{\left (b c - a d\right )}}, \frac{2 \, \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) - \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{2} + 2 \, c x \sqrt{-\frac{d}{c}} - c}{d x^{2} + c}\right )}{2 \,{\left (b c - a d\right )}}, \frac{\sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) - \sqrt{\frac{d}{c}} \arctan \left (\frac{d x}{c \sqrt{\frac{d}{c}}}\right )}{b c - a d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.12987, size = 712, normalized size = 10.17 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**2+a)/(d*x**2+c),x)
[Out]
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GIAC/XCAS [A] time = 0.25289, size = 257, normalized size = 3.67 \[ -\frac{2 \, \sqrt{c d} b{\left | d \right |} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} x}{\sqrt{\frac{b c + a d + \sqrt{-4 \, a b c d +{\left (b c + a d\right )}^{2}}}{b d}}}\right )}{b c d{\left | b c - a d \right |} + a d^{2}{\left | b c - a d \right |} +{\left (b c - a d\right )}^{2} d} + \frac{2 \, \sqrt{a b} d{\left | b \right |} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} x}{\sqrt{\frac{b c + a d - \sqrt{-4 \, a b c d +{\left (b c + a d\right )}^{2}}}{b d}}}\right )}{b^{2} c{\left | b c - a d \right |} + a b d{\left | b c - a d \right |} -{\left (b c - a d\right )}^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)),x, algorithm="giac")
[Out]