Optimal. Leaf size=161 \[ -\frac {\left (a^2 (4 A+5 C)+2 A b^2\right ) \sin ^3(c+d x)}{15 d}+\frac {\left (a^2+b^2\right ) (4 A+5 C) \sin (c+d x)}{5 d}+\frac {a b (3 A+4 C) \sin (c+d x) \cos (c+d x)}{4 d}+\frac {a A b \sin (c+d x) \cos ^3(c+d x)}{10 d}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}+\frac {1}{4} a b x (3 A+4 C) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.40, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4095, 4074, 4047, 2635, 8, 4044, 3013} \[ -\frac {\left (a^2 (4 A+5 C)+2 A b^2\right ) \sin ^3(c+d x)}{15 d}+\frac {\left (a^2+b^2\right ) (4 A+5 C) \sin (c+d x)}{5 d}+\frac {a b (3 A+4 C) \sin (c+d x) \cos (c+d x)}{4 d}+\frac {a A b \sin (c+d x) \cos ^3(c+d x)}{10 d}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}+\frac {1}{4} a b x (3 A+4 C) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2635
Rule 3013
Rule 4044
Rule 4047
Rule 4074
Rule 4095
Rubi steps
\begin {align*} \int \cos ^5(c+d x) (a+b \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {1}{5} \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (2 A b+a (4 A+5 C) \sec (c+d x)+b (2 A+5 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a A b \cos ^3(c+d x) \sin (c+d x)}{10 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}-\frac {1}{20} \int \cos ^3(c+d x) \left (-4 \left (2 A b^2+a^2 (4 A+5 C)\right )-10 a b (3 A+4 C) \sec (c+d x)-4 b^2 (2 A+5 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a A b \cos ^3(c+d x) \sin (c+d x)}{10 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}-\frac {1}{20} \int \cos ^3(c+d x) \left (-4 \left (2 A b^2+a^2 (4 A+5 C)\right )-4 b^2 (2 A+5 C) \sec ^2(c+d x)\right ) \, dx+\frac {1}{2} (a b (3 A+4 C)) \int \cos ^2(c+d x) \, dx\\ &=\frac {a b (3 A+4 C) \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a A b \cos ^3(c+d x) \sin (c+d x)}{10 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}-\frac {1}{20} \int \cos (c+d x) \left (-4 b^2 (2 A+5 C)-4 \left (2 A b^2+a^2 (4 A+5 C)\right ) \cos ^2(c+d x)\right ) \, dx+\frac {1}{4} (a b (3 A+4 C)) \int 1 \, dx\\ &=\frac {1}{4} a b (3 A+4 C) x+\frac {a b (3 A+4 C) \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a A b \cos ^3(c+d x) \sin (c+d x)}{10 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {\operatorname {Subst}\left (\int \left (-4 b^2 (2 A+5 C)-4 \left (2 A b^2+a^2 (4 A+5 C)\right )+4 \left (2 A b^2+a^2 (4 A+5 C)\right ) x^2\right ) \, dx,x,-\sin (c+d x)\right )}{20 d}\\ &=\frac {1}{4} a b (3 A+4 C) x+\frac {\left (a^2+b^2\right ) (4 A+5 C) \sin (c+d x)}{5 d}+\frac {a b (3 A+4 C) \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a A b \cos ^3(c+d x) \sin (c+d x)}{10 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}-\frac {\left (2 A b^2+a^2 (4 A+5 C)\right ) \sin ^3(c+d x)}{15 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.45, size = 126, normalized size = 0.78 \[ \frac {30 \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \sin (c+d x)+5 \left (a^2 (5 A+4 C)+4 A b^2\right ) \sin (3 (c+d x))+3 a^2 A \sin (5 (c+d x))+60 a b (3 A+4 C) (c+d x)+120 a b (A+C) \sin (2 (c+d x))+15 a A b \sin (4 (c+d x))}{240 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 123, normalized size = 0.76 \[ \frac {15 \, {\left (3 \, A + 4 \, C\right )} a b d x + {\left (12 \, A a^{2} \cos \left (d x + c\right )^{4} + 30 \, A a b \cos \left (d x + c\right )^{3} + 15 \, {\left (3 \, A + 4 \, C\right )} a b \cos \left (d x + c\right ) + 8 \, {\left (4 \, A + 5 \, C\right )} a^{2} + 20 \, {\left (2 \, A + 3 \, C\right )} b^{2} + 4 \, {\left ({\left (4 \, A + 5 \, C\right )} a^{2} + 5 \, A b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.24, size = 498, normalized size = 3.09 \[ \frac {15 \, {\left (3 \, A a b + 4 \, C a b\right )} {\left (d x + c\right )} + \frac {2 \, {\left (60 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 60 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 75 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 60 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 60 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 60 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 80 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 160 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 30 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 120 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 160 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 240 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 232 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 200 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 200 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 360 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 80 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 160 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 30 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 160 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 240 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 60 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 60 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 75 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 60 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 60 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 60 \, C 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 )}^{5}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.58, size = 158, normalized size = 0.98 \[ \frac {\frac {a^{2} A \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {a^{2} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+2 A a b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 C a b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {A \,b^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+b^{2} C \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.36, size = 154, normalized size = 0.96 \[ \frac {16 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{2} - 80 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a b + 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a b - 80 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b^{2} + 240 \, C b^{2} \sin \left (d x + c\right )}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 6.34, size = 342, normalized size = 2.12 \[ \frac {\left (2\,A\,a^2+2\,A\,b^2+2\,C\,a^2+2\,C\,b^2-\frac {5\,A\,a\,b}{2}-2\,C\,a\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {8\,A\,a^2}{3}+\frac {16\,A\,b^2}{3}+\frac {16\,C\,a^2}{3}+8\,C\,b^2-A\,a\,b-4\,C\,a\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {116\,A\,a^2}{15}+\frac {20\,A\,b^2}{3}+\frac {20\,C\,a^2}{3}+12\,C\,b^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {8\,A\,a^2}{3}+\frac {16\,A\,b^2}{3}+\frac {16\,C\,a^2}{3}+8\,C\,b^2+A\,a\,b+4\,C\,a\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,a^2+2\,A\,b^2+2\,C\,a^2+2\,C\,b^2+\frac {5\,A\,a\,b}{2}+2\,C\,a\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a\,b\,\mathrm {atan}\left (\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,A+4\,C\right )}{2\,\left (\frac {3\,A\,a\,b}{2}+2\,C\,a\,b\right )}\right )\,\left (3\,A+4\,C\right )}{2\,d} \]
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]
________________________________________________________________________________________