Optimal. Leaf size=56 \[ -\frac {b^2 \text {ArcTan}\left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{a^{5/2} \sqrt {a+b}}+\frac {(a-b) \tan (x)}{a^2}+\frac {\tan ^3(x)}{3 a} \]
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Rubi [A]
time = 0.06, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3266, 472, 211}
\begin {gather*} -\frac {b^2 \text {ArcTan}\left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{a^{5/2} \sqrt {a+b}}+\frac {(a-b) \tan (x)}{a^2}+\frac {\tan ^3(x)}{3 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 472
Rule 3266
Rubi steps
\begin {align*} \int \frac {\sec ^4(x)}{a+b \cos ^2(x)} \, dx &=-\text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^4 \left (a+(a+b) x^2\right )} \, dx,x,\cot (x)\right )\\ &=-\text {Subst}\left (\int \left (\frac {1}{a x^4}+\frac {a-b}{a^2 x^2}+\frac {b^2}{a^2 \left (a+(a+b) x^2\right )}\right ) \, dx,x,\cot (x)\right )\\ &=\frac {(a-b) \tan (x)}{a^2}+\frac {\tan ^3(x)}{3 a}-\frac {b^2 \text {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\cot (x)\right )}{a^2}\\ &=-\frac {b^2 \tan ^{-1}\left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{a^{5/2} \sqrt {a+b}}+\frac {(a-b) \tan (x)}{a^2}+\frac {\tan ^3(x)}{3 a}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 55, normalized size = 0.98 \begin {gather*} \frac {b^2 \text {ArcTan}\left (\frac {\sqrt {a} \tan (x)}{\sqrt {a+b}}\right )}{a^{5/2} \sqrt {a+b}}+\frac {\left (2 a-3 b+a \sec ^2(x)\right ) \tan (x)}{3 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 49, normalized size = 0.88
method | result | size |
default | \(\frac {\frac {a \left (\tan ^{3}\left (x \right )\right )}{3}+\tan \left (x \right ) a -\tan \left (x \right ) b}{a^{2}}+\frac {b^{2} \arctan \left (\frac {a \tan \left (x \right )}{\sqrt {\left (a +b \right ) a}}\right )}{a^{2} \sqrt {\left (a +b \right ) a}}\) | \(49\) |
risch | \(-\frac {2 i \left (3 b \,{\mathrm e}^{4 i x}-6 a \,{\mathrm e}^{2 i x}+6 b \,{\mathrm e}^{2 i x}-2 a +3 b \right )}{3 \left ({\mathrm e}^{2 i x}+1\right )^{3} a^{2}}-\frac {b^{2} \ln \left ({\mathrm e}^{2 i x}+\frac {2 i a^{2}+2 i a b +2 a \sqrt {-a^{2}-a b}+b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{2 \sqrt {-a^{2}-a b}\, a^{2}}+\frac {b^{2} \ln \left ({\mathrm e}^{2 i x}-\frac {2 i a^{2}+2 i a b -2 a \sqrt {-a^{2}-a b}-b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{2 \sqrt {-a^{2}-a b}\, a^{2}}\) | \(218\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 48, normalized size = 0.86 \begin {gather*} \frac {b^{2} \arctan \left (\frac {a \tan \left (x\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a} a^{2}} + \frac {a \tan \left (x\right )^{3} + 3 \, {\left (a - b\right )} \tan \left (x\right )}{3 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 106 vs.
\(2 (46) = 92\).
time = 0.46, size = 276, normalized size = 4.93 \begin {gather*} \left [-\frac {3 \, \sqrt {-a^{2} - a b} b^{2} \cos \left (x\right )^{3} \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (x\right )^{4} - 2 \, {\left (4 \, a^{2} + 3 \, a b\right )} \cos \left (x\right )^{2} + 4 \, {\left ({\left (2 \, a + b\right )} \cos \left (x\right )^{3} - a \cos \left (x\right )\right )} \sqrt {-a^{2} - a b} \sin \left (x\right ) + a^{2}}{b^{2} \cos \left (x\right )^{4} + 2 \, a b \cos \left (x\right )^{2} + a^{2}}\right ) - 4 \, {\left (a^{3} + a^{2} b + {\left (2 \, a^{3} - a^{2} b - 3 \, a b^{2}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{12 \, {\left (a^{4} + a^{3} b\right )} \cos \left (x\right )^{3}}, -\frac {3 \, \sqrt {a^{2} + a b} b^{2} \arctan \left (\frac {{\left (2 \, a + b\right )} \cos \left (x\right )^{2} - a}{2 \, \sqrt {a^{2} + a b} \cos \left (x\right ) \sin \left (x\right )}\right ) \cos \left (x\right )^{3} - 2 \, {\left (a^{3} + a^{2} b + {\left (2 \, a^{3} - a^{2} b - 3 \, a b^{2}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{6 \, {\left (a^{4} + a^{3} b\right )} \cos \left (x\right )^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sec ^{4}{\left (x \right )}}{a + b \cos ^{2}{\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 71, normalized size = 1.27 \begin {gather*} \frac {{\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (x\right )}{\sqrt {a^{2} + a b}}\right )\right )} b^{2}}{\sqrt {a^{2} + a b} a^{2}} + \frac {a^{2} \tan \left (x\right )^{3} + 3 \, a^{2} \tan \left (x\right ) - 3 \, a b \tan \left (x\right )}{3 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.31, size = 51, normalized size = 0.91 \begin {gather*} \frac {{\mathrm {tan}\left (x\right )}^3}{3\,a}-\mathrm {tan}\left (x\right )\,\left (\frac {a+b}{a^2}-\frac {2}{a}\right )+\frac {b^2\,\mathrm {atan}\left (\frac {\sqrt {a}\,\mathrm {tan}\left (x\right )}{\sqrt {a+b}}\right )}{a^{5/2}\,\sqrt {a+b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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