Optimal. Leaf size=124 \[ -\frac{(4 A+3 C) \sin ^3(c+d x)}{3 a d}+\frac{(4 A+3 C) \sin (c+d x)}{a d}-\frac{(3 A+2 C) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}-\frac{x (3 A+2 C)}{2 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.168757, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {4085, 3787, 2633, 2635, 8} \[ -\frac{(4 A+3 C) \sin ^3(c+d x)}{3 a d}+\frac{(4 A+3 C) \sin (c+d x)}{a d}-\frac{(3 A+2 C) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}-\frac{x (3 A+2 C)}{2 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4085
Rule 3787
Rule 2633
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx &=-\frac{(A+C) \cos ^2(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}-\frac{\int \cos ^3(c+d x) (-a (4 A+3 C)+a (3 A+2 C) \sec (c+d x)) \, dx}{a^2}\\ &=-\frac{(A+C) \cos ^2(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}-\frac{(3 A+2 C) \int \cos ^2(c+d x) \, dx}{a}+\frac{(4 A+3 C) \int \cos ^3(c+d x) \, dx}{a}\\ &=-\frac{(3 A+2 C) \cos (c+d x) \sin (c+d x)}{2 a d}-\frac{(A+C) \cos ^2(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}-\frac{(3 A+2 C) \int 1 \, dx}{2 a}-\frac{(4 A+3 C) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a d}\\ &=-\frac{(3 A+2 C) x}{2 a}+\frac{(4 A+3 C) \sin (c+d x)}{a d}-\frac{(3 A+2 C) \cos (c+d x) \sin (c+d x)}{2 a d}-\frac{(A+C) \cos ^2(c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}-\frac{(4 A+3 C) \sin ^3(c+d x)}{3 a d}\\ \end{align*}
Mathematica [A] time = 0.794644, size = 225, normalized size = 1.81 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (-12 d x (3 A+2 C) \cos \left (c+\frac{d x}{2}\right )+21 A \sin \left (c+\frac{d x}{2}\right )+18 A \sin \left (c+\frac{3 d x}{2}\right )+18 A \sin \left (2 c+\frac{3 d x}{2}\right )-2 A \sin \left (2 c+\frac{5 d x}{2}\right )-2 A \sin \left (3 c+\frac{5 d x}{2}\right )+A \sin \left (3 c+\frac{7 d x}{2}\right )+A \sin \left (4 c+\frac{7 d x}{2}\right )-12 d x (3 A+2 C) \cos \left (\frac{d x}{2}\right )+69 A \sin \left (\frac{d x}{2}\right )+12 C \sin \left (c+\frac{d x}{2}\right )+12 C \sin \left (c+\frac{3 d x}{2}\right )+12 C \sin \left (2 c+\frac{3 d x}{2}\right )+60 C \sin \left (\frac{d x}{2}\right )\right )}{24 a d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.092, size = 280, normalized size = 2.3 \begin{align*}{\frac{A}{ad}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{C}{ad}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+5\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}A}{ad \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}+2\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}C}{ad \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}+{\frac{16\,A}{3\,ad} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-3}}+4\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}C}{ad \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}+3\,{\frac{A\tan \left ( 1/2\,dx+c/2 \right ) }{ad \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}+2\,{\frac{C\tan \left ( 1/2\,dx+c/2 \right ) }{ad \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}-3\,{\frac{A\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{ad}}-2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) C}{ad}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.42638, size = 363, normalized size = 2.93 \begin{align*} \frac{A{\left (\frac{\frac{9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{16 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{15 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a + \frac{3 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{3 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac{9 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac{3 \, \sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - 3 \, C{\left (\frac{2 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac{2 \, \sin \left (d x + c\right )}{{\left (a + \frac{a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}} - \frac{\sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.48975, size = 243, normalized size = 1.96 \begin{align*} -\frac{3 \,{\left (3 \, A + 2 \, C\right )} d x \cos \left (d x + c\right ) + 3 \,{\left (3 \, A + 2 \, C\right )} d x -{\left (2 \, A \cos \left (d x + c\right )^{3} - A \cos \left (d x + c\right )^{2} +{\left (7 \, A + 6 \, C\right )} \cos \left (d x + c\right ) + 16 \, A + 12 \, C\right )} \sin \left (d x + c\right )}{6 \,{\left (a d \cos \left (d x + c\right ) + a d\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 = 1.17271, size = 205, normalized size = 1.65 \begin{align*} -\frac{\frac{3 \,{\left (d x + c\right )}{\left (3 \, A + 2 \, C\right )}}{a} - \frac{6 \,{\left (A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{a} - \frac{2 \,{\left (15 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 16 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 12 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 9 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{3} a}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]