Integrand size = 29, antiderivative size = 73 \[ \int \frac {\sec ^2(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\sec ^3(c+d x)}{3 a d}-\frac {\sec ^5(c+d x)}{5 a d}+\frac {\tan ^3(c+d x)}{3 a d}+\frac {\tan ^5(c+d x)}{5 a d} \]
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Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2918, 2687, 14, 2686} \[ \int \frac {\sec ^2(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\tan ^5(c+d x)}{5 a d}+\frac {\tan ^3(c+d x)}{3 a d}-\frac {\sec ^5(c+d x)}{5 a d}+\frac {\sec ^3(c+d x)}{3 a d} \]
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Rule 14
Rule 2686
Rule 2687
Rule 2918
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^4(c+d x) \tan ^2(c+d x) \, dx}{a}-\frac {\int \sec ^3(c+d x) \tan ^3(c+d x) \, dx}{a} \\ & = -\frac {\text {Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{a d}+\frac {\text {Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,\tan (c+d x)\right )}{a d} \\ & = -\frac {\text {Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\sec (c+d x)\right )}{a d}+\frac {\text {Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,\tan (c+d x)\right )}{a d} \\ & = \frac {\sec ^3(c+d x)}{3 a d}-\frac {\sec ^5(c+d x)}{5 a d}+\frac {\tan ^3(c+d x)}{3 a d}+\frac {\tan ^5(c+d x)}{5 a d} \\ \end{align*}
Time = 0.74 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.45 \[ \int \frac {\sec ^2(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\sec ^3(c+d x) (-80+66 \cos (c+d x)-32 \cos (2 (c+d x))+22 \cos (3 (c+d x))-16 \cos (4 (c+d x))-224 \sin (c+d x)+22 \sin (2 (c+d x))+32 \sin (3 (c+d x))+11 \sin (4 (c+d x)))}{960 a d (1+\sin (c+d x))} \]
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.33
method | result | size |
risch | \(\frac {-\frac {16 \,{\mathrm e}^{3 i \left (d x +c \right )}}{15}+\frac {4 i {\mathrm e}^{4 i \left (d x +c \right )}}{3}+\frac {8 i {\mathrm e}^{2 i \left (d x +c \right )}}{15}+\frac {8 \,{\mathrm e}^{5 i \left (d x +c \right )}}{3}+\frac {8 \,{\mathrm e}^{i \left (d x +c \right )}}{15}+\frac {4 i}{15}}{\left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{5} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d a}\) | \(97\) |
parallelrisch | \(\frac {-\frac {4}{15}-\frac {8 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {16 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15}+\frac {8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15}-\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{15}}{d a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\) | \(100\) |
derivativedivides | \(\frac {-\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {8}{64 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+64}}{d a}\) | \(130\) |
default | \(\frac {-\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {8}{64 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+64}}{d a}\) | \(130\) |
norman | \(\frac {-\frac {4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {4}{15 a d}-\frac {8 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{15 d a}+\frac {8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d a}-\frac {16 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d a}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}\) | \(130\) |
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Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00 \[ \int \frac {\sec ^2(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2 \, \cos \left (d x + c\right )^{4} - \cos \left (d x + c\right )^{2} - 2 \, {\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) + 1}{15 \, {\left (a d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{3}\right )}} \]
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\[ \int \frac {\sec ^2(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {\sin ^{2}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (65) = 130\).
Time = 0.24 (sec) , antiderivative size = 254, normalized size of antiderivative = 3.48 \[ \int \frac {\sec ^2(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {4 \, {\left (\frac {2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {4 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {5 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {10 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + 1\right )}}{15 \, {\left (a + \frac {2 \, a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {6 \, a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {6 \, a \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {2 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {2 \, a \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d} \]
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Time = 0.31 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.49 \[ \int \frac {\sec ^2(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {5 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} - \frac {3 \, {\left (5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 40 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 50 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 40 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9\right )}}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}}{120 \, d} \]
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Time = 11.31 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.36 \[ \int \frac {\sec ^2(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {4\,\left (10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{15\,a\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^5} \]
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