Optimal. Leaf size=106 \[ \frac {2 \tan ^5(c+d x)}{5 a^2 d}-\frac {\tan ^3(c+d x)}{3 a^2 d}+\frac {\tan (c+d x)}{a^2 d}-\frac {2 \sec ^5(c+d x)}{5 a^2 d}+\frac {4 \sec ^3(c+d x)}{3 a^2 d}-\frac {2 \sec (c+d x)}{a^2 d}-\frac {x}{a^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.28, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2875, 2873, 2607, 30, 2606, 194, 3473, 8} \[ \frac {2 \tan ^5(c+d x)}{5 a^2 d}-\frac {\tan ^3(c+d x)}{3 a^2 d}+\frac {\tan (c+d x)}{a^2 d}-\frac {2 \sec ^5(c+d x)}{5 a^2 d}+\frac {4 \sec ^3(c+d x)}{3 a^2 d}-\frac {2 \sec (c+d x)}{a^2 d}-\frac {x}{a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 30
Rule 194
Rule 2606
Rule 2607
Rule 2873
Rule 2875
Rule 3473
Rubi steps
\begin {align*} \int \frac {\sin ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \sec ^2(c+d x) (a-a \sin (c+d x))^2 \tan ^4(c+d x) \, dx}{a^4}\\ &=\frac {\int \left (a^2 \sec ^2(c+d x) \tan ^4(c+d x)-2 a^2 \sec (c+d x) \tan ^5(c+d x)+a^2 \tan ^6(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \sec ^2(c+d x) \tan ^4(c+d x) \, dx}{a^2}+\frac {\int \tan ^6(c+d x) \, dx}{a^2}-\frac {2 \int \sec (c+d x) \tan ^5(c+d x) \, dx}{a^2}\\ &=\frac {\tan ^5(c+d x)}{5 a^2 d}-\frac {\int \tan ^4(c+d x) \, dx}{a^2}+\frac {\operatorname {Subst}\left (\int x^4 \, dx,x,\tan (c+d x)\right )}{a^2 d}-\frac {2 \operatorname {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=-\frac {\tan ^3(c+d x)}{3 a^2 d}+\frac {2 \tan ^5(c+d x)}{5 a^2 d}+\frac {\int \tan ^2(c+d x) \, dx}{a^2}-\frac {2 \operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=-\frac {2 \sec (c+d x)}{a^2 d}+\frac {4 \sec ^3(c+d x)}{3 a^2 d}-\frac {2 \sec ^5(c+d x)}{5 a^2 d}+\frac {\tan (c+d x)}{a^2 d}-\frac {\tan ^3(c+d x)}{3 a^2 d}+\frac {2 \tan ^5(c+d x)}{5 a^2 d}-\frac {\int 1 \, dx}{a^2}\\ &=-\frac {x}{a^2}-\frac {2 \sec (c+d x)}{a^2 d}+\frac {4 \sec ^3(c+d x)}{3 a^2 d}-\frac {2 \sec ^5(c+d x)}{5 a^2 d}+\frac {\tan (c+d x)}{a^2 d}-\frac {\tan ^3(c+d x)}{3 a^2 d}+\frac {2 \tan ^5(c+d x)}{5 a^2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.56, size = 143, normalized size = 1.35 \[ -\frac {\sec (c+d x) \left (-10 \sin (c+d x)+60 c \sin (2 (c+d x))+60 d x \sin (2 (c+d x))-89 \sin (2 (c+d x))+26 \sin (3 (c+d x))+\frac {5}{4} (60 c+60 d x-89) \cos (c+d x)+44 \cos (2 (c+d x))-15 c \cos (3 (c+d x))-15 d x \cos (3 (c+d x))+\frac {89}{4} \cos (3 (c+d x))+20\right )}{60 a^2 d (\sin (c+d x)+1)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 112, normalized size = 1.06 \[ -\frac {15 \, d x \cos \left (d x + c\right )^{3} - 30 \, d x \cos \left (d x + c\right ) - 22 \, \cos \left (d x + c\right )^{2} - {\left (30 \, d x \cos \left (d x + c\right ) + 26 \, \cos \left (d x + c\right )^{2} - 9\right )} \sin \left (d x + c\right ) + 6}{15 \, {\left (a^{2} d \cos \left (d x + c\right )^{3} - 2 \, a^{2} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.22, size = 103, normalized size = 0.97 \[ -\frac {\frac {60 \, {\left (d x + c\right )}}{a^{2}} + \frac {15}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}} + \frac {105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 510 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 920 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 610 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 143}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.43, size = 146, normalized size = 1.38 \[ -\frac {1}{4 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}-\frac {4}{5 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {2}{a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {1}{3 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {3}{2 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {7}{4 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.43, size = 249, normalized size = 2.35 \[ -\frac {2 \, {\left (\frac {\frac {49 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {20 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {70 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {60 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {15 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + 16}{a^{2} + \frac {4 \, a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {5 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {5 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {4 \, a^{2} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {15 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 14.09, size = 105, normalized size = 0.99 \[ \frac {-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+\frac {98\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{15}+\frac {32}{15}}{a^2\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^5}-\frac {x}{a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sin ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________