Optimal. Leaf size=163 \[ \frac {a \left (a^2 (2 A+3 C)+3 A b^2\right ) \sin (c+d x)}{3 d}+\frac {1}{2} b x \left (3 a^2 (A+2 C)+2 A b^2\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}+\frac {A b \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}+\frac {3 a b^2 C \tanh ^{-1}(\sin (c+d x))}{d}-\frac {b^3 (5 A-6 C) \tan (c+d x)}{6 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.52, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4095, 4094, 4076, 4047, 8, 4045, 3770} \[ \frac {a \left (a^2 (2 A+3 C)+3 A b^2\right ) \sin (c+d x)}{3 d}+\frac {1}{2} b x \left (3 a^2 (A+2 C)+2 A b^2\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}+\frac {A b \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}+\frac {3 a b^2 C \tanh ^{-1}(\sin (c+d x))}{d}-\frac {b^3 (5 A-6 C) \tan (c+d x)}{6 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3770
Rule 4045
Rule 4047
Rule 4076
Rule 4094
Rule 4095
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {1}{3} \int \cos ^2(c+d x) (a+b \sec (c+d x))^2 \left (3 A b+a (2 A+3 C) \sec (c+d x)-b (A-3 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {A b \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {1}{6} \int \cos (c+d x) (a+b \sec (c+d x)) \left (2 \left (3 A b^2+\frac {1}{2} a^2 (4 A+6 C)\right )+a b (5 A+12 C) \sec (c+d x)-b^2 (5 A-6 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {A b \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}-\frac {b^3 (5 A-6 C) \tan (c+d x)}{6 d}+\frac {1}{6} \int \cos (c+d x) \left (2 a \left (3 A b^2+\frac {1}{2} a^2 (4 A+6 C)\right )+3 b \left (2 A b^2+3 a^2 (A+2 C)\right ) \sec (c+d x)+18 a b^2 C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {A b \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}-\frac {b^3 (5 A-6 C) \tan (c+d x)}{6 d}+\frac {1}{6} \int \cos (c+d x) \left (2 a \left (3 A b^2+\frac {1}{2} a^2 (4 A+6 C)\right )+18 a b^2 C \sec ^2(c+d x)\right ) \, dx+\frac {1}{2} \left (b \left (2 A b^2+3 a^2 (A+2 C)\right )\right ) \int 1 \, dx\\ &=\frac {1}{2} b \left (2 A b^2+3 a^2 (A+2 C)\right ) x+\frac {a \left (3 A b^2+a^2 (2 A+3 C)\right ) \sin (c+d x)}{3 d}+\frac {A b \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}-\frac {b^3 (5 A-6 C) \tan (c+d x)}{6 d}+\left (3 a b^2 C\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{2} b \left (2 A b^2+3 a^2 (A+2 C)\right ) x+\frac {3 a b^2 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a \left (3 A b^2+a^2 (2 A+3 C)\right ) \sin (c+d x)}{3 d}+\frac {A b \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}-\frac {b^3 (5 A-6 C) \tan (c+d x)}{6 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.01, size = 184, normalized size = 1.13 \[ \frac {a^3 A \sin (3 (c+d x))+3 a \left (a^2 (3 A+4 C)+12 A b^2\right ) \sin (c+d x)+9 a^2 A b \sin (2 (c+d x))+18 a^2 A b c+18 a^2 A b d x+36 a^2 b c C+36 a^2 b C d x-36 a b^2 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+36 a b^2 C \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+12 A b^3 c+12 A b^3 d x+12 b^3 C \tan (c+d x)}{12 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 158, normalized size = 0.97 \[ \frac {9 \, C a b^{2} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 9 \, C a b^{2} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (3 \, {\left (A + 2 \, C\right )} a^{2} b + 2 \, A b^{3}\right )} d x \cos \left (d x + c\right ) + {\left (2 \, A a^{3} \cos \left (d x + c\right )^{3} + 9 \, A a^{2} b \cos \left (d x + c\right )^{2} + 6 \, C b^{3} + 2 \, {\left ({\left (2 \, A + 3 \, C\right )} a^{3} + 9 \, A a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.29, size = 306, normalized size = 1.88 \[ \frac {18 \, C a b^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 18 \, C a b^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {12 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + 3 \, {\left (3 \, A a^{2} b + 6 \, C a^{2} b + 2 \, A b^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.10, size = 183, normalized size = 1.12 \[ \frac {A \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a^{3}}{3 d}+\frac {2 a^{3} A \sin \left (d x +c \right )}{3 d}+\frac {a^{3} C \sin \left (d x +c \right )}{d}+\frac {3 A \,a^{2} b \sin \left (d x +c \right ) \cos \left (d x +c \right )}{2 d}+\frac {3 a^{2} A b x}{2}+\frac {3 A \,a^{2} b c}{2 d}+3 C x \,a^{2} b +\frac {3 C \,a^{2} b c}{d}+\frac {3 A a \,b^{2} \sin \left (d x +c \right )}{d}+\frac {3 C a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+A x \,b^{3}+\frac {A \,b^{3} c}{d}+\frac {b^{3} C \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.37, size = 141, normalized size = 0.87 \[ -\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} - 9 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} b - 36 \, {\left (d x + c\right )} C a^{2} b - 12 \, {\left (d x + c\right )} A b^{3} - 18 \, C a b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, C a^{3} \sin \left (d x + c\right ) - 36 \, A a b^{2} \sin \left (d x + c\right ) - 12 \, C b^{3} \tan \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.07, size = 238, normalized size = 1.46 \[ \frac {2\,A\,b^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+3\,A\,a^2\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+6\,C\,a^2\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )-C\,a\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,6{}\mathrm {i}}{d}+\frac {\frac {5\,A\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{12}+\frac {A\,a^3\,\sin \left (4\,c+4\,d\,x\right )}{24}+\frac {C\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{2}+C\,b^3\,\sin \left (c+d\,x\right )+\frac {3\,A\,a^2\,b\,\sin \left (c+d\,x\right )}{8}+\frac {3\,A\,a\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{2}+\frac {3\,A\,a^2\,b\,\sin \left (3\,c+3\,d\,x\right )}{8}}{d\,\cos \left (c+d\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________