Optimal. Leaf size=265 \[ -\frac {a^3 (108 A+119 B+133 C) \sin ^3(c+d x)}{105 d}+\frac {a^3 (108 A+119 B+133 C) \sin (c+d x)}{35 d}+\frac {a^3 (129 A+147 B+154 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac {a^3 (21 A+23 B+26 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {(3 A+4 B+3 C) \sin (c+d x) \cos ^4(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{15 d}+\frac {1}{16} a^3 x (21 A+23 B+26 C)+\frac {(3 A+7 B) \sin (c+d x) \cos ^5(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{42 a d}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.61, antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {4086, 4017, 3996, 3787, 2633, 2635, 8} \[ -\frac {a^3 (108 A+119 B+133 C) \sin ^3(c+d x)}{105 d}+\frac {a^3 (108 A+119 B+133 C) \sin (c+d x)}{35 d}+\frac {a^3 (129 A+147 B+154 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac {a^3 (21 A+23 B+26 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {(3 A+4 B+3 C) \sin (c+d x) \cos ^4(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{15 d}+\frac {(3 A+7 B) \sin (c+d x) \cos ^5(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{42 a d}+\frac {1}{16} a^3 x (21 A+23 B+26 C)+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2633
Rule 2635
Rule 3787
Rule 3996
Rule 4017
Rule 4086
Rubi steps
\begin {align*} \int \cos ^7(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {\int \cos ^6(c+d x) (a+a \sec (c+d x))^3 (a (3 A+7 B)+a (3 A+7 C) \sec (c+d x)) \, dx}{7 a}\\ &=\frac {A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(3 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac {\int \cos ^5(c+d x) (a+a \sec (c+d x))^2 \left (14 a^2 (3 A+4 B+3 C)+3 a^2 (9 A+7 B+14 C) \sec (c+d x)\right ) \, dx}{42 a}\\ &=\frac {A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(3 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac {(3 A+4 B+3 C) \cos ^4(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {\int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (3 a^3 (129 A+147 B+154 C)+3 a^3 (87 A+91 B+112 C) \sec (c+d x)\right ) \, dx}{210 a}\\ &=\frac {a^3 (129 A+147 B+154 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(3 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac {(3 A+4 B+3 C) \cos ^4(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {\int \cos ^3(c+d x) \left (-24 a^4 (108 A+119 B+133 C)-105 a^4 (21 A+23 B+26 C) \sec (c+d x)\right ) \, dx}{840 a}\\ &=\frac {a^3 (129 A+147 B+154 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(3 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac {(3 A+4 B+3 C) \cos ^4(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{8} \left (a^3 (21 A+23 B+26 C)\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{35} \left (a^3 (108 A+119 B+133 C)\right ) \int \cos ^3(c+d x) \, dx\\ &=\frac {a^3 (21 A+23 B+26 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (129 A+147 B+154 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(3 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac {(3 A+4 B+3 C) \cos ^4(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{16} \left (a^3 (21 A+23 B+26 C)\right ) \int 1 \, dx-\frac {\left (a^3 (108 A+119 B+133 C)\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 d}\\ &=\frac {1}{16} a^3 (21 A+23 B+26 C) x+\frac {a^3 (108 A+119 B+133 C) \sin (c+d x)}{35 d}+\frac {a^3 (21 A+23 B+26 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (129 A+147 B+154 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(3 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac {(3 A+4 B+3 C) \cos ^4(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {a^3 (108 A+119 B+133 C) \sin ^3(c+d x)}{105 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.31, size = 204, normalized size = 0.77 \[ \frac {a^3 (105 (155 A+168 B+184 C) \sin (c+d x)+105 (61 A+63 B+64 C) \sin (2 (c+d x))+2835 A \sin (3 (c+d x))+1155 A \sin (4 (c+d x))+399 A \sin (5 (c+d x))+105 A \sin (6 (c+d x))+15 A \sin (7 (c+d x))+3360 A c+8820 A d x+2660 B \sin (3 (c+d x))+945 B \sin (4 (c+d x))+252 B \sin (5 (c+d x))+35 B \sin (6 (c+d x))+9660 B c+9660 B d x+2380 C \sin (3 (c+d x))+630 C \sin (4 (c+d x))+84 C \sin (5 (c+d x))+10920 C d x)}{6720 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 168, normalized size = 0.63 \[ \frac {105 \, {\left (21 \, A + 23 \, B + 26 \, C\right )} a^{3} d x + {\left (240 \, A a^{3} \cos \left (d x + c\right )^{6} + 280 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{5} + 48 \, {\left (27 \, A + 21 \, B + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 70 \, {\left (21 \, A + 23 \, B + 18 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 16 \, {\left (108 \, A + 119 \, B + 133 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 105 \, {\left (21 \, A + 23 \, B + 26 \, C\right )} a^{3} \cos \left (d x + c\right ) + 32 \, {\left (108 \, A + 119 \, B + 133 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{1680 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.34, size = 401, normalized size = 1.51 \[ \frac {105 \, {\left (21 \, A a^{3} + 23 \, B a^{3} + 26 \, C a^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (2205 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 2415 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 2730 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 14700 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 16100 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 18200 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 41601 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 45563 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 51506 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 62592 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 72576 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 77952 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 63231 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 62853 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 71246 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 25620 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 33180 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40040 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 11235 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 11025 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 10710 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{7}}}{1680 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 2.63, size = 427, normalized size = 1.61 \[ \frac {\frac {A \,a^{3} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+a^{3} B \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {C \,a^{3} \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}+3 A \,a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {3 a^{3} B \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}+3 C \,a^{3} \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 )+\frac {3 A \,a^{3} \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}+3 a^{3} 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 )+C \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+A \,a^{3} \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 )+\frac {a^{3} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+C \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.36, size = 425, normalized size = 1.60 \[ -\frac {192 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} A a^{3} - 1344 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{3} + 105 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 1344 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{3} + 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} + 2240 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} - 630 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 448 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{3} + 6720 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} - 630 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 1680 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3}}{6720 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 6.07, size = 377, normalized size = 1.42 \[ \frac {\left (\frac {21\,A\,a^3}{8}+\frac {23\,B\,a^3}{8}+\frac {13\,C\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {35\,A\,a^3}{2}+\frac {115\,B\,a^3}{6}+\frac {65\,C\,a^3}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {1981\,A\,a^3}{40}+\frac {6509\,B\,a^3}{120}+\frac {3679\,C\,a^3}{60}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {2608\,A\,a^3}{35}+\frac {432\,B\,a^3}{5}+\frac {464\,C\,a^3}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {3011\,A\,a^3}{40}+\frac {2993\,B\,a^3}{40}+\frac {5089\,C\,a^3}{60}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {61\,A\,a^3}{2}+\frac {79\,B\,a^3}{2}+\frac {143\,C\,a^3}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {107\,A\,a^3}{8}+\frac {105\,B\,a^3}{8}+\frac {51\,C\,a^3}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (21\,A+23\,B+26\,C\right )}{8\,\left (\frac {21\,A\,a^3}{8}+\frac {23\,B\,a^3}{8}+\frac {13\,C\,a^3}{4}\right )}\right )\,\left (21\,A+23\,B+26\,C\right )}{8\,d} \]
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]
________________________________________________________________________________________