Optimal. Leaf size=66 \[ -\frac{2 \tan ^5(c+d x)}{5 a^2 d}+\frac{2 \sec ^5(c+d x)}{5 a^2 d}-\frac{\sec ^3(c+d x)}{a^2 d}+\frac{\sec (c+d x)}{a^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.258677, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {2875, 2873, 2606, 14, 2607, 30, 194} \[ -\frac{2 \tan ^5(c+d x)}{5 a^2 d}+\frac{2 \sec ^5(c+d x)}{5 a^2 d}-\frac{\sec ^3(c+d x)}{a^2 d}+\frac{\sec (c+d x)}{a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2875
Rule 2873
Rule 2606
Rule 14
Rule 2607
Rule 30
Rule 194
Rubi steps
\begin{align*} \int \frac{\sin (c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\int \sec ^3(c+d x) (a-a \sin (c+d x))^2 \tan ^3(c+d x) \, dx}{a^4}\\ &=\frac{\int \left (a^2 \sec ^3(c+d x) \tan ^3(c+d x)-2 a^2 \sec ^2(c+d x) \tan ^4(c+d x)+a^2 \sec (c+d x) \tan ^5(c+d x)\right ) \, dx}{a^4}\\ &=\frac{\int \sec ^3(c+d x) \tan ^3(c+d x) \, dx}{a^2}+\frac{\int \sec (c+d x) \tan ^5(c+d x) \, dx}{a^2}-\frac{2 \int \sec ^2(c+d x) \tan ^4(c+d x) \, dx}{a^2}\\ &=\frac{\operatorname{Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d}+\frac{\operatorname{Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{a^2 d}-\frac{2 \operatorname{Subst}\left (\int x^4 \, dx,x,\tan (c+d x)\right )}{a^2 d}\\ &=-\frac{2 \tan ^5(c+d x)}{5 a^2 d}+\frac{\operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d}+\frac{\operatorname{Subst}\left (\int \left (-x^2+x^4\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)}{a^2 d}+\frac{2 \sec ^5(c+d x)}{5 a^2 d}-\frac{2 \tan ^5(c+d x)}{5 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.27068, size = 84, normalized size = 1.27 \[ \frac{\sec (c+d x) (40 \sin (c+d x)-52 \sin (2 (c+d x))+8 \sin (3 (c+d x))-65 \cos (c+d x)-8 \cos (2 (c+d x))+13 \cos (3 (c+d x))+40)}{80 a^2 d (\sin (c+d x)+1)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.088, size = 100, normalized size = 1.5 \begin{align*} 16\,{\frac{1}{d{a}^{2}} \left ( -{\frac{1}{64\,\tan \left ( 1/2\,dx+c/2 \right ) -64}}+1/20\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-5}-1/8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-4}+1/16\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-3}+1/32\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-2}+{\frac{1}{64\,\tan \left ( 1/2\,dx+c/2 \right ) +64}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.03021, size = 221, normalized size = 3.35 \begin{align*} \frac{4 \,{\left (\frac{4 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{5 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}}{5 \,{\left (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}}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.12167, size = 197, normalized size = 2.98 \begin{align*} \frac{\cos \left (d x + c\right )^{2} - 2 \,{\left (\cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) - 3}{5 \,{\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.19185, size = 127, normalized size = 1.92 \begin{align*} -\frac{\frac{5}{a^{2}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}} - \frac{5 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 30 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 80 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 50 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 11}{a^{2}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{5}}}{20 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]