Optimal. Leaf size=96 \[ \frac {b^2 (6 a+b) \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^{7/2}}+\frac {(a+3 b) \tan (x)}{(a+b)^3}+\frac {\tan ^3(x)}{3 (a+b)^2}+\frac {b^3 \tan (x)}{2 a (a+b)^3 \left (a+(a+b) \tan ^2(x)\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 96, 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^2 (6 a+b) \text {ArcTan}\left (\frac {\sqrt {a+b} \tan (x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^{7/2}}+\frac {b^3 \tan (x)}{2 a (a+b)^3 \left ((a+b) \tan ^2(x)+a\right )}+\frac {\tan ^3(x)}{3 (a+b)^2}+\frac {(a+3 b) \tan (x)}{(a+b)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 393
Rule 398
Rule 3270
Rubi steps
\begin {align*} \int \frac {\sec ^4(x)}{\left (a+b \sin ^2(x)\right )^2} \, dx &=\text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{\left (a+(a+b) x^2\right )^2} \, dx,x,\tan (x)\right )\\ &=\text {Subst}\left (\int \left (\frac {a+3 b}{(a+b)^3}+\frac {x^2}{(a+b)^2}+\frac {b^2 (3 a+b)+3 b^2 (a+b) x^2}{(a+b)^3 \left (a+(a+b) x^2\right )^2}\right ) \, dx,x,\tan (x)\right )\\ &=\frac {(a+3 b) \tan (x)}{(a+b)^3}+\frac {\tan ^3(x)}{3 (a+b)^2}+\frac {\text {Subst}\left (\int \frac {b^2 (3 a+b)+3 b^2 (a+b) x^2}{\left (a+(a+b) x^2\right )^2} \, dx,x,\tan (x)\right )}{(a+b)^3}\\ &=\frac {(a+3 b) \tan (x)}{(a+b)^3}+\frac {\tan ^3(x)}{3 (a+b)^2}+\frac {b^3 \tan (x)}{2 a (a+b)^3 \left (a+(a+b) \tan ^2(x)\right )}+\frac {\left (b^2 (6 a+b)\right ) \text {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\tan (x)\right )}{2 a (a+b)^3}\\ &=\frac {b^2 (6 a+b) \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^{7/2}}+\frac {(a+3 b) \tan (x)}{(a+b)^3}+\frac {\tan ^3(x)}{3 (a+b)^2}+\frac {b^3 \tan (x)}{2 a (a+b)^3 \left (a+(a+b) \tan ^2(x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.70, size = 97, normalized size = 1.01 \begin {gather*} \frac {1}{6} \left (\frac {3 b^2 (6 a+b) \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (x)}{\sqrt {a}}\right )}{a^{3/2} (a+b)^{7/2}}+\frac {\frac {3 b^3 \sin (2 x)}{a (2 a+b-b \cos (2 x))}+4 a \tan (x)+16 b \tan (x)+2 (a+b) \sec ^2(x) \tan (x)}{(a+b)^3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.37, size = 110, normalized size = 1.15
method | result | size |
default | \(\frac {\frac {a \left (\tan ^{3}\left (x \right )\right )}{3}+\frac {b \left (\tan ^{3}\left (x \right )\right )}{3}+\tan \left (x \right ) a +3 \tan \left (x \right ) b}{\left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right )}+\frac {b^{2} \left (\frac {b \tan \left (x \right )}{2 a \left (a \left (\tan ^{2}\left (x \right )\right )+b \left (\tan ^{2}\left (x \right )\right )+a \right )}+\frac {\left (6 a +b \right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (x \right )}{\sqrt {a \left (a +b \right )}}\right )}{2 a \sqrt {a \left (a +b \right )}}\right )}{\left (a +b \right )^{3}}\) | \(110\) |
risch | \(\frac {i \left (-18 a \,b^{2} {\mathrm e}^{8 i x}-3 b^{3} {\mathrm e}^{8 i x}+36 a^{2} b \,{\mathrm e}^{6 i x}-30 a \,b^{2} {\mathrm e}^{6 i x}-6 b^{3} {\mathrm e}^{6 i x}+48 a^{3} {\mathrm e}^{4 i x}+164 a^{2} b \,{\mathrm e}^{4 i x}+26 a \,b^{2} {\mathrm e}^{4 i x}+16 a^{3} {\mathrm e}^{2 i x}+60 a^{2} b \,{\mathrm e}^{2 i x}-10 a \,b^{2} {\mathrm e}^{2 i x}+6 b^{3} {\mathrm e}^{2 i x}-4 a^{2} b -16 a \,b^{2}+3 b^{3}\right )}{3 \left ({\mathrm e}^{2 i x}+1\right )^{3} \left (a +b \right )^{3} a \left (-b \,{\mathrm e}^{4 i x}+4 a \,{\mathrm e}^{2 i x}+2 b \,{\mathrm e}^{2 i x}-b \right )}-\frac {3 b^{2} \ln \left ({\mathrm e}^{2 i x}-\frac {2 i a^{2}+2 i a b +2 a \sqrt {-a^{2}-a b}+b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3}}-\frac {b^{3} \ln \left ({\mathrm e}^{2 i x}-\frac {2 i a^{2}+2 i a b +2 a \sqrt {-a^{2}-a b}+b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3} a}+\frac {3 b^{2} \ln \left ({\mathrm e}^{2 i x}-\frac {-2 i a^{2}-2 i a b +2 a \sqrt {-a^{2}-a b}+b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3}}+\frac {b^{3} \ln \left ({\mathrm e}^{2 i x}-\frac {-2 i a^{2}-2 i a b +2 a \sqrt {-a^{2}-a b}+b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3} a}\) | \(551\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 170 vs.
\(2 (82) = 164\).
time = 0.49, size = 170, normalized size = 1.77 \begin {gather*} \frac {b^{3} \tan \left (x\right )}{2 \, {\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3} + {\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} \tan \left (x\right )^{2}\right )}} + \frac {{\left (6 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {{\left (a + b\right )} \tan \left (x\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{2 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \sqrt {{\left (a + b\right )} a}} + \frac {{\left (a + b\right )} \tan \left (x\right )^{3} + 3 \, {\left (a + 3 \, b\right )} \tan \left (x\right )}{3 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 284 vs.
\(2 (82) = 164\).
time = 0.44, size = 653, normalized size = 6.80 \begin {gather*} \left [-\frac {3 \, {\left ({\left (6 \, a b^{3} + b^{4}\right )} \cos \left (x\right )^{5} - {\left (6 \, a^{2} b^{2} + 7 \, a b^{3} + b^{4}\right )} \cos \left (x\right )^{3}\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^{5} + 6 \, a^{4} b + 6 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - {\left (4 \, a^{4} b + 20 \, a^{3} b^{2} + 13 \, a^{2} b^{3} - 3 \, a b^{4}\right )} \cos \left (x\right )^{4} + 2 \, {\left (2 \, a^{5} + 11 \, a^{4} b + 16 \, a^{3} b^{2} + 7 \, a^{2} b^{3}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{24 \, {\left ({\left (a^{6} b + 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 4 \, a^{3} b^{4} + a^{2} b^{5}\right )} \cos \left (x\right )^{5} - {\left (a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right )} \cos \left (x\right )^{3}\right )}}, -\frac {3 \, {\left ({\left (6 \, a b^{3} + b^{4}\right )} \cos \left (x\right )^{5} - {\left (6 \, a^{2} b^{2} + 7 \, a b^{3} + b^{4}\right )} \cos \left (x\right )^{3}\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^{5} + 6 \, a^{4} b + 6 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - {\left (4 \, a^{4} b + 20 \, a^{3} b^{2} + 13 \, a^{2} b^{3} - 3 \, a b^{4}\right )} \cos \left (x\right )^{4} + 2 \, {\left (2 \, a^{5} + 11 \, a^{4} b + 16 \, a^{3} b^{2} + 7 \, a^{2} b^{3}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{12 \, {\left ({\left (a^{6} b + 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 4 \, a^{3} b^{4} + a^{2} b^{5}\right )} \cos \left (x\right )^{5} - {\left (a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right )} \cos \left (x\right )^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sec ^{4}{\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.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 270 vs.
\(2 (82) = 164\).
time = 0.48, size = 270, normalized size = 2.81 \begin {gather*} \frac {b^{3} \tan \left (x\right )}{2 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} {\left (a \tan \left (x\right )^{2} + b \tan \left (x\right )^{2} + a\right )}} + \frac {{\left (6 \, a b^{2} + b^{3}\right )} {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (x\right ) + b \tan \left (x\right )}{\sqrt {a^{2} + a b}}\right )\right )}}{2 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \sqrt {a^{2} + a b}} + \frac {a^{4} \tan \left (x\right )^{3} + 4 \, a^{3} b \tan \left (x\right )^{3} + 6 \, a^{2} b^{2} \tan \left (x\right )^{3} + 4 \, a b^{3} \tan \left (x\right )^{3} + b^{4} \tan \left (x\right )^{3} + 3 \, a^{4} \tan \left (x\right ) + 18 \, a^{3} b \tan \left (x\right ) + 36 \, a^{2} b^{2} \tan \left (x\right ) + 30 \, a b^{3} \tan \left (x\right ) + 9 \, b^{4} \tan \left (x\right )}{3 \, {\left (a^{6} + 6 \, a^{5} b + 15 \, a^{4} b^{2} + 20 \, a^{3} b^{3} + 15 \, a^{2} b^{4} + 6 \, a b^{5} + b^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 14.30, size = 176, normalized size = 1.83 \begin {gather*} \frac {{\mathrm {tan}\left (x\right )}^3}{3\,{\left (a+b\right )}^2}-\mathrm {tan}\left (x\right )\,\left (\frac {2\,a}{{\left (a+b\right )}^3}-\frac {3}{{\left (a+b\right )}^2}\right )+\frac {b^3\,\mathrm {tan}\left (x\right )}{2\,a\,\left ({\mathrm {tan}\left (x\right )}^2\,\left (a^4+4\,a^3\,b+6\,a^2\,b^2+4\,a\,b^3+b^4\right )+a\,b^3+3\,a^3\,b+a^4+3\,a^2\,b^2\right )}+\frac {b^2\,\mathrm {atan}\left (\frac {b^2\,\mathrm {tan}\left (x\right )\,\left (6\,a+b\right )\,\left (2\,a+2\,b\right )\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}{2\,\sqrt {a}\,{\left (a+b\right )}^{7/2}\,\left (b^3+6\,a\,b^2\right )}\right )\,\left (6\,a+b\right )}{2\,a^{3/2}\,{\left (a+b\right )}^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________