Optimal. Leaf size=76 \[ \frac {b (4 a+b) \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^{5/2}}+\frac {\tan (x)}{(a+b)^2}+\frac {b^2 \tan (x)}{2 a (a+b)^2 \left (a+(a+b) \tan ^2(x)\right )} \]
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Rubi [A]
time = 0.08, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3270, 398, 393,
211} \begin {gather*} \frac {b (4 a+b) \text {ArcTan}\left (\frac {\sqrt {a+b} \tan (x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^{5/2}}+\frac {b^2 \tan (x)}{2 a (a+b)^2 \left ((a+b) \tan ^2(x)+a\right )}+\frac {\tan (x)}{(a+b)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 393
Rule 398
Rule 3270
Rubi steps
\begin {align*} \int \frac {\sec ^2(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx &=\text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{\left (a+(a+b) x^2\right )^2} \, dx,x,\tan (x)\right )\\ &=\text {Subst}\left (\int \left (\frac {1}{(a+b)^2}+\frac {b (2 a+b)+2 b (a+b) x^2}{(a+b)^2 \left (a+(a+b) x^2\right )^2}\right ) \, dx,x,\tan (x)\right )\\ &=\frac {\tan (x)}{(a+b)^2}+\frac {\text {Subst}\left (\int \frac {b (2 a+b)+2 b (a+b) x^2}{\left (a+(a+b) x^2\right )^2} \, dx,x,\tan (x)\right )}{(a+b)^2}\\ &=\frac {\tan (x)}{(a+b)^2}+\frac {b^2 \tan (x)}{2 a (a+b)^2 \left (a+(a+b) \tan ^2(x)\right )}+\frac {(b (4 a+b)) \text {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\tan (x)\right )}{2 a (a+b)^2}\\ &=\frac {b (4 a+b) \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^{5/2}}+\frac {\tan (x)}{(a+b)^2}+\frac {b^2 \tan (x)}{2 a (a+b)^2 \left (a+(a+b) \tan ^2(x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.37, size = 76, normalized size = 1.00 \begin {gather*} \frac {1}{2} \left (\frac {b (4 a+b) \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (x)}{\sqrt {a}}\right )}{a^{3/2} (a+b)^{5/2}}+\frac {\frac {b^2 \sin (2 x)}{a (2 a+b-b \cos (2 x))}+2 \tan (x)}{(a+b)^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 81, normalized size = 1.07
method | result | size |
default | \(\frac {\tan \left (x \right )}{a^{2}+2 a b +b^{2}}+\frac {b \left (\frac {b \tan \left (x \right )}{2 a \left (a \left (\tan ^{2}\left (x \right )\right )+b \left (\tan ^{2}\left (x \right )\right )+a \right )}+\frac {\left (4 a +b \right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (x \right )}{\sqrt {a \left (a +b \right )}}\right )}{2 a \sqrt {a \left (a +b \right )}}\right )}{\left (a +b \right )^{2}}\) | \(81\) |
risch | \(\frac {i \left (-4 a b \,{\mathrm e}^{4 i x}-b^{2} {\mathrm e}^{4 i x}+8 \,{\mathrm e}^{2 i x} a^{2}+2 b \,{\mathrm e}^{2 i x} a -2 a b +b^{2}\right )}{a \left (a +b \right )^{2} \left (-b \,{\mathrm e}^{4 i x}+4 a \,{\mathrm e}^{2 i x}+2 b \,{\mathrm e}^{2 i x}-b \right ) \left ({\mathrm e}^{2 i x}+1\right )}-\frac {b \ln \left ({\mathrm e}^{2 i x}-\frac {2 i a^{2}+2 i a b +2 a \sqrt {-a^{2}-a b}+b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{\sqrt {-a^{2}-a b}\, \left (a +b \right )^{2}}-\frac {b^{2} \ln \left ({\mathrm e}^{2 i x}-\frac {2 i a^{2}+2 i a b +2 a \sqrt {-a^{2}-a b}+b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} a}+\frac {b \ln \left ({\mathrm e}^{2 i x}-\frac {-2 i a^{2}-2 i a b +2 a \sqrt {-a^{2}-a b}+b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{\sqrt {-a^{2}-a b}\, \left (a +b \right )^{2}}+\frac {b^{2} \ln \left ({\mathrm e}^{2 i x}-\frac {-2 i a^{2}-2 i a b +2 a \sqrt {-a^{2}-a b}+b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} a}\) | \(447\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 119, normalized size = 1.57 \begin {gather*} \frac {b^{2} \tan \left (x\right )}{2 \, {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2} + {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \tan \left (x\right )^{2}\right )}} + \frac {{\left (4 \, a b + b^{2}\right )} \arctan \left (\frac {{\left (a + b\right )} \tan \left (x\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{2 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \sqrt {{\left (a + b\right )} a}} + \frac {\tan \left (x\right )}{a^{2} + 2 \, a b + b^{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. 210 vs.
\(2 (64) = 128\).
time = 0.44, size = 505, normalized size = 6.64 \begin {gather*} \left [-\frac {{\left ({\left (4 \, a b^{2} + b^{3}\right )} \cos \left (x\right )^{3} - {\left (4 \, a^{2} b + 5 \, a b^{2} + b^{3}\right )} \cos \left (x\right )\right )} \sqrt {-a^{2} - a b} \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (x\right )^{4} - 2 \, {\left (4 \, a^{2} + 5 \, a b + b^{2}\right )} \cos \left (x\right )^{2} + 4 \, {\left ({\left (2 \, a + b\right )} \cos \left (x\right )^{3} - {\left (a + b\right )} \cos \left (x\right )\right )} \sqrt {-a^{2} - a b} \sin \left (x\right ) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos \left (x\right )^{4} - 2 \, {\left (a b + b^{2}\right )} \cos \left (x\right )^{2} + a^{2} + 2 \, a b + b^{2}}\right ) + 4 \, {\left (2 \, a^{4} + 4 \, a^{3} b + 2 \, a^{2} b^{2} - {\left (2 \, a^{3} b + a^{2} b^{2} - a b^{3}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{8 \, {\left ({\left (a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right )} \cos \left (x\right )^{3} - {\left (a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right )} \cos \left (x\right )\right )}}, -\frac {{\left ({\left (4 \, a b^{2} + b^{3}\right )} \cos \left (x\right )^{3} - {\left (4 \, a^{2} b + 5 \, a b^{2} + b^{3}\right )} \cos \left (x\right )\right )} \sqrt {a^{2} + a b} \arctan \left (\frac {{\left (2 \, a + b\right )} \cos \left (x\right )^{2} - a - b}{2 \, \sqrt {a^{2} + a b} \cos \left (x\right ) \sin \left (x\right )}\right ) + 2 \, {\left (2 \, a^{4} + 4 \, a^{3} b + 2 \, a^{2} b^{2} - {\left (2 \, a^{3} b + a^{2} b^{2} - a b^{3}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{4 \, {\left ({\left (a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right )} \cos \left (x\right )^{3} - {\left (a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right )} \cos \left (x\right )\right )}}\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 ^{2}{\left (x \right )}}{\left (a + b \sin ^{2}{\left (x \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 113, normalized size = 1.49 \begin {gather*} \frac {b^{2} \tan \left (x\right )}{2 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} {\left (a \tan \left (x\right )^{2} + b \tan \left (x\right )^{2} + a\right )}} + \frac {{\left (4 \, a b + b^{2}\right )} \arctan \left (\frac {a \tan \left (x\right ) + b \tan \left (x\right )}{\sqrt {a^{2} + a b}}\right )}{2 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \sqrt {a^{2} + a b}} + \frac {\tan \left (x\right )}{a^{2} + 2 \, a b + b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 14.78, size = 123, normalized size = 1.62 \begin {gather*} \frac {\mathrm {tan}\left (x\right )}{{\left (a+b\right )}^2}+\frac {b^2\,\mathrm {tan}\left (x\right )}{2\,a\,\left (a\,b^2+2\,a^2\,b+{\mathrm {tan}\left (x\right )}^2\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )+a^3\right )}+\frac {b\,\mathrm {atan}\left (\frac {b\,\mathrm {tan}\left (x\right )\,\left (4\,a+b\right )\,\left (2\,a+2\,b\right )\,\left (a^2+2\,a\,b+b^2\right )}{2\,\sqrt {a}\,{\left (a+b\right )}^{5/2}\,\left (b^2+4\,a\,b\right )}\right )\,\left (4\,a+b\right )}{2\,a^{3/2}\,{\left (a+b\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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