Optimal. Leaf size=87 \[ \frac {b^3 \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^{7/2}}+\frac {\left (a^2+3 a b+3 b^2\right ) \tan (x)}{(a+b)^3}+\frac {(2 a+3 b) \tan ^3(x)}{3 (a+b)^2}+\frac {\tan ^5(x)}{5 (a+b)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 87, 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 = {3270, 398, 211}
\begin {gather*} \frac {\left (a^2+3 a b+3 b^2\right ) \tan (x)}{(a+b)^3}+\frac {b^3 \text {ArcTan}\left (\frac {\sqrt {a+b} \tan (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^{7/2}}+\frac {\tan ^5(x)}{5 (a+b)}+\frac {(2 a+3 b) \tan ^3(x)}{3 (a+b)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 398
Rule 3270
Rubi steps
\begin {align*} \int \frac {\sec ^6(x)}{a+b \sin ^2(x)} \, dx &=\text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{a+(a+b) x^2} \, dx,x,\tan (x)\right )\\ &=\text {Subst}\left (\int \left (\frac {a^2+3 a b+3 b^2}{(a+b)^3}+\frac {(2 a+3 b) x^2}{(a+b)^2}+\frac {x^4}{a+b}+\frac {b^3}{(a+b)^3 \left (a+(a+b) x^2\right )}\right ) \, dx,x,\tan (x)\right )\\ &=\frac {\left (a^2+3 a b+3 b^2\right ) \tan (x)}{(a+b)^3}+\frac {(2 a+3 b) \tan ^3(x)}{3 (a+b)^2}+\frac {\tan ^5(x)}{5 (a+b)}+\frac {b^3 \text {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\tan (x)\right )}{(a+b)^3}\\ &=\frac {b^3 \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^{7/2}}+\frac {\left (a^2+3 a b+3 b^2\right ) \tan (x)}{(a+b)^3}+\frac {(2 a+3 b) \tan ^3(x)}{3 (a+b)^2}+\frac {\tan ^5(x)}{5 (a+b)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.28, size = 90, normalized size = 1.03 \begin {gather*} \frac {b^3 \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^{7/2}}+\frac {\left (8 a^2+26 a b+33 b^2+\left (4 a^2+13 a b+9 b^2\right ) \sec ^2(x)+3 (a+b)^2 \sec ^4(x)\right ) \tan (x)}{15 (a+b)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.35, size = 109, normalized size = 1.25
method | result | size |
default | \(\frac {\frac {a^{2} \left (\tan ^{5}\left (x \right )\right )}{5}+\frac {2 a b \left (\tan ^{5}\left (x \right )\right )}{5}+\frac {b^{2} \left (\tan ^{5}\left (x \right )\right )}{5}+\frac {2 a^{2} \left (\tan ^{3}\left (x \right )\right )}{3}+\frac {5 a b \left (\tan ^{3}\left (x \right )\right )}{3}+b^{2} \left (\tan ^{3}\left (x \right )\right )+a^{2} \tan \left (x \right )+3 a b \tan \left (x \right )+3 b^{2} \tan \left (x \right )}{\left (a +b \right )^{3}}+\frac {b^{3} \arctan \left (\frac {\left (a +b \right ) \tan \left (x \right )}{\sqrt {a \left (a +b \right )}}\right )}{\left (a +b \right )^{3} \sqrt {a \left (a +b \right )}}\) | \(109\) |
risch | \(\frac {2 i \left (15 b^{2} {\mathrm e}^{8 i x}+30 a b \,{\mathrm e}^{6 i x}+90 b^{2} {\mathrm e}^{6 i x}+80 a^{2} {\mathrm e}^{4 i x}+230 a b \,{\mathrm e}^{4 i x}+240 b^{2} {\mathrm e}^{4 i x}+40 \,{\mathrm e}^{2 i x} a^{2}+130 b \,{\mathrm e}^{2 i x} a +150 b^{2} {\mathrm e}^{2 i x}+8 a^{2}+26 a b +33 b^{2}\right )}{15 \left (a +b \right )^{3} \left ({\mathrm e}^{2 i x}+1\right )^{5}}-\frac {b^{3} \ln \left ({\mathrm e}^{2 i x}-\frac {2 i a^{2}+2 i a b +2 a \sqrt {-a^{2}-a b}+b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3}}+\frac {b^{3} \ln \left ({\mathrm e}^{2 i x}-\frac {-2 i a^{2}-2 i a b +2 a \sqrt {-a^{2}-a b}+b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3}}\) | \(295\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.49, size = 126, normalized size = 1.45 \begin {gather*} \frac {b^{3} \arctan \left (\frac {{\left (a + b\right )} \tan \left (x\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {{\left (a + b\right )} a}} + \frac {3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \tan \left (x\right )^{5} + 5 \, {\left (2 \, a^{2} + 5 \, a b + 3 \, b^{2}\right )} \tan \left (x\right )^{3} + 15 \, {\left (a^{2} + 3 \, a b + 3 \, b^{2}\right )} \tan \left (x\right )}{15 \, {\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. 187 vs.
\(2 (75) = 150\).
time = 0.47, size = 459, normalized size = 5.28 \begin {gather*} \left [-\frac {15 \, \sqrt {-a^{2} - a b} b^{3} \cos \left (x\right )^{5} \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 ({\left (8 \, a^{4} + 34 \, a^{3} b + 59 \, a^{2} b^{2} + 33 \, a b^{3}\right )} \cos \left (x\right )^{4} + 3 \, a^{4} + 9 \, a^{3} b + 9 \, a^{2} b^{2} + 3 \, a b^{3} + {\left (4 \, a^{4} + 17 \, a^{3} b + 22 \, a^{2} b^{2} + 9 \, a b^{3}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{60 \, {\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} \cos \left (x\right )^{5}}, -\frac {15 \, \sqrt {a^{2} + a b} b^{3} \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 ) \cos \left (x\right )^{5} - 2 \, {\left ({\left (8 \, a^{4} + 34 \, a^{3} b + 59 \, a^{2} b^{2} + 33 \, a b^{3}\right )} \cos \left (x\right )^{4} + 3 \, a^{4} + 9 \, a^{3} b + 9 \, a^{2} b^{2} + 3 \, a b^{3} + {\left (4 \, a^{4} + 17 \, a^{3} b + 22 \, a^{2} b^{2} + 9 \, a b^{3}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{30 \, {\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} \cos \left (x\right )^{5}}\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 ^{6}{\left (x \right )}}{a + b \sin ^{2}{\left (x \right )}}\, 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. 254 vs.
\(2 (75) = 150\).
time = 0.44, size = 254, normalized size = 2.92 \begin {gather*} \frac {{\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 )} b^{3}}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {a^{2} + a b}} + \frac {3 \, a^{4} \tan \left (x\right )^{5} + 12 \, a^{3} b \tan \left (x\right )^{5} + 18 \, a^{2} b^{2} \tan \left (x\right )^{5} + 12 \, a b^{3} \tan \left (x\right )^{5} + 3 \, b^{4} \tan \left (x\right )^{5} + 10 \, a^{4} \tan \left (x\right )^{3} + 45 \, a^{3} b \tan \left (x\right )^{3} + 75 \, a^{2} b^{2} \tan \left (x\right )^{3} + 55 \, a b^{3} \tan \left (x\right )^{3} + 15 \, b^{4} \tan \left (x\right )^{3} + 15 \, a^{4} \tan \left (x\right ) + 75 \, a^{3} b \tan \left (x\right ) + 150 \, a^{2} b^{2} \tan \left (x\right ) + 135 \, a b^{3} \tan \left (x\right ) + 45 \, b^{4} \tan \left (x\right )}{15 \, {\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 13.99, size = 121, normalized size = 1.39 \begin {gather*} \frac {{\mathrm {tan}\left (x\right )}^5}{5\,\left (a+b\right )}-{\mathrm {tan}\left (x\right )}^3\,\left (\frac {a}{3\,{\left (a+b\right )}^2}-\frac {1}{a+b}\right )+\mathrm {tan}\left (x\right )\,\left (\frac {3}{a+b}+\frac {a\,\left (\frac {a}{{\left (a+b\right )}^2}-\frac {3}{a+b}\right )}{a+b}\right )+\frac {b^3\,\mathrm {atan}\left (\frac {\mathrm {tan}\left (x\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}}\right )}{\sqrt {a}\,{\left (a+b\right )}^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________