Integrand size = 16, antiderivative size = 29 \[ \int \frac {\sec ^4(x)}{a-a \sin ^2(x)} \, dx=\frac {\tan (x)}{a}+\frac {2 \tan ^3(x)}{3 a}+\frac {\tan ^5(x)}{5 a} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3254, 3852} \[ \int \frac {\sec ^4(x)}{a-a \sin ^2(x)} \, dx=\frac {\tan ^5(x)}{5 a}+\frac {2 \tan ^3(x)}{3 a}+\frac {\tan (x)}{a} \]
[In]
[Out]
Rule 3254
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^6(x) \, dx}{a} \\ & = -\frac {\text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (x)\right )}{a} \\ & = \frac {\tan (x)}{a}+\frac {2 \tan ^3(x)}{3 a}+\frac {\tan ^5(x)}{5 a} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {\sec ^4(x)}{a-a \sin ^2(x)} \, dx=\frac {\tan (x)+\frac {2 \tan ^3(x)}{3}+\frac {\tan ^5(x)}{5}}{a} \]
[In]
[Out]
Time = 0.65 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69
method | result | size |
default | \(\frac {\frac {\left (\tan ^{5}\left (x \right )\right )}{5}+\frac {2 \left (\tan ^{3}\left (x \right )\right )}{3}+\tan \left (x \right )}{a}\) | \(20\) |
parallelrisch | \(\frac {\tan \left (x \right ) \left (3 \left (\sec ^{4}\left (x \right )\right )+4 \left (\sec ^{2}\left (x \right )\right )+8\right )}{15 a}\) | \(22\) |
risch | \(\frac {16 i \left (10 \,{\mathrm e}^{4 i x}+5 \,{\mathrm e}^{2 i x}+1\right )}{15 \left ({\mathrm e}^{2 i x}+1\right )^{5} a}\) | \(32\) |
norman | \(\frac {-\frac {2 \tan \left (\frac {x}{2}\right )}{a}+\frac {8 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {116 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{15 a}+\frac {8 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {2 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{a}}{\left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )^{5}}\) | \(66\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {\sec ^4(x)}{a-a \sin ^2(x)} \, dx=\frac {{\left (8 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{2} + 3\right )} \sin \left (x\right )}{15 \, a \cos \left (x\right )^{5}} \]
[In]
[Out]
\[ \int \frac {\sec ^4(x)}{a-a \sin ^2(x)} \, dx=- \frac {\int \frac {\sec ^{4}{\left (x \right )}}{\sin ^{2}{\left (x \right )} - 1}\, dx}{a} \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {\sec ^4(x)}{a-a \sin ^2(x)} \, dx=\frac {3 \, \tan \left (x\right )^{5} + 10 \, \tan \left (x\right )^{3} + 15 \, \tan \left (x\right )}{15 \, a} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {\sec ^4(x)}{a-a \sin ^2(x)} \, dx=\frac {3 \, \tan \left (x\right )^{5} + 10 \, \tan \left (x\right )^{3} + 15 \, \tan \left (x\right )}{15 \, a} \]
[In]
[Out]
Time = 13.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {\sec ^4(x)}{a-a \sin ^2(x)} \, dx=\frac {\mathrm {tan}\left (x\right )\,\left (3\,{\mathrm {tan}\left (x\right )}^4+10\,{\mathrm {tan}\left (x\right )}^2+15\right )}{15\,a} \]
[In]
[Out]