Optimal. Leaf size=115 \[ -\frac {2 \tan ^5(c+d x)}{5 a^2 d}-\frac {4 \tan ^3(c+d x)}{3 a^2 d}-\frac {2 \tan (c+d x)}{a^2 d}+\frac {2 \sec ^5(c+d x)}{5 a^2 d}+\frac {\sec ^3(c+d x)}{3 a^2 d}+\frac {\sec (c+d x)}{a^2 d}-\frac {\tanh ^{-1}(\cos (c+d x))}{a^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.27, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2875, 2873, 3767, 2622, 302, 207, 2606, 30} \[ -\frac {2 \tan ^5(c+d x)}{5 a^2 d}-\frac {4 \tan ^3(c+d x)}{3 a^2 d}-\frac {2 \tan (c+d x)}{a^2 d}+\frac {2 \sec ^5(c+d x)}{5 a^2 d}+\frac {\sec ^3(c+d x)}{3 a^2 d}+\frac {\sec (c+d x)}{a^2 d}-\frac {\tanh ^{-1}(\cos (c+d x))}{a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 207
Rule 302
Rule 2606
Rule 2622
Rule 2873
Rule 2875
Rule 3767
Rubi steps
\begin {align*} \int \frac {\csc (c+d x) \sec ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \csc (c+d x) \sec ^6(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac {\int \left (-2 a^2 \sec ^6(c+d x)+a^2 \csc (c+d x) \sec ^6(c+d x)+a^2 \sec ^5(c+d x) \tan (c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \csc (c+d x) \sec ^6(c+d x) \, dx}{a^2}+\frac {\int \sec ^5(c+d x) \tan (c+d x) \, dx}{a^2}-\frac {2 \int \sec ^6(c+d x) \, dx}{a^2}\\ &=\frac {\operatorname {Subst}\left (\int x^4 \, dx,x,\sec (c+d x)\right )}{a^2 d}+\frac {\operatorname {Subst}\left (\int \frac {x^6}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^2 d}+\frac {2 \operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{a^2 d}\\ &=\frac {\sec ^5(c+d x)}{5 a^2 d}-\frac {2 \tan (c+d x)}{a^2 d}-\frac {4 \tan ^3(c+d x)}{3 a^2 d}-\frac {2 \tan ^5(c+d x)}{5 a^2 d}+\frac {\operatorname {Subst}\left (\int \left (1+x^2+x^4+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac {\sec (c+d x)}{a^2 d}+\frac {\sec ^3(c+d x)}{3 a^2 d}+\frac {2 \sec ^5(c+d x)}{5 a^2 d}-\frac {2 \tan (c+d x)}{a^2 d}-\frac {4 \tan ^3(c+d x)}{3 a^2 d}-\frac {2 \tan ^5(c+d x)}{5 a^2 d}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=-\frac {\tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac {\sec (c+d x)}{a^2 d}+\frac {\sec ^3(c+d x)}{3 a^2 d}+\frac {2 \sec ^5(c+d x)}{5 a^2 d}-\frac {2 \tan (c+d x)}{a^2 d}-\frac {4 \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 = 0.52, size = 196, normalized size = 1.70 \[ \frac {\sec (c+d x) \left (160 \sin (c+d x)-316 \sin (2 (c+d x))+64 \sin (3 (c+d x))+136 \cos (2 (c+d x))+79 \cos (3 (c+d x))+240 \sin (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+60 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-5 \cos (c+d x) \left (-60 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+60 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+79\right )-60 \cos (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-240 \sin (2 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+280\right )}{240 a^2 d (\sin (c+d x)+1)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 168, normalized size = 1.46 \[ -\frac {34 \, \cos \left (d x + c\right )^{2} + 15 \, {\left (\cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 15 \, {\left (\cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 4 \, {\left (8 \, \cos \left (d x + c\right )^{2} + 3\right )} \sin \left (d x + c\right ) + 18}{30 \, {\left (a^{2} d \cos \left (d x + c\right )^{3} - 2 \, a^{2} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 109, normalized size = 0.95 \[ \frac {\frac {60 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {15}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}} + \frac {255 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 810 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 710 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 193}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.56, size = 145, normalized size = 1.26 \[ -\frac {1}{4 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}+\frac {4}{5 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {2}{a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {11}{3 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {7}{2 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {17}{4 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.33, size = 250, normalized size = 2.17 \[ \frac {\frac {4 \, {\left (\frac {37 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {35 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {30 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {15 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + 13\right )}}{a^{2} + \frac {4 \, a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {5 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {5 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {4 \, a^{2} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {15 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 10.78, size = 117, normalized size = 1.02 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d}-\frac {-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+\frac {148\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{15}+\frac {52}{15}}{a^2\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\csc {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________