Optimal. Leaf size=237 \[ -\frac {a^3 (133 A+108 C) \sin ^3(c+d x)}{105 d}+\frac {a^3 (133 A+108 C) \sin (c+d x)}{35 d}+\frac {a^3 (154 A+129 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac {(A+C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{5 d}+\frac {a^3 (26 A+21 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} a^3 x (26 A+21 C)+\frac {C \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{14 a d}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.61, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3046, 2976, 2968, 3023, 2748, 2635, 8, 2633} \[ -\frac {a^3 (133 A+108 C) \sin ^3(c+d x)}{105 d}+\frac {a^3 (133 A+108 C) \sin (c+d x)}{35 d}+\frac {a^3 (154 A+129 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac {(A+C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{5 d}+\frac {a^3 (26 A+21 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} a^3 x (26 A+21 C)+\frac {C \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{14 a d}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2633
Rule 2635
Rule 2748
Rule 2968
Rule 2976
Rule 3023
Rule 3046
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {\int \cos ^2(c+d x) (a+a \cos (c+d x))^3 (a (7 A+3 C)+3 a C \cos (c+d x)) \, dx}{7 a}\\ &=\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{14 a d}+\frac {\int \cos ^2(c+d x) (a+a \cos (c+d x))^2 \left (3 a^2 (14 A+9 C)+42 a^2 (A+C) \cos (c+d x)\right ) \, dx}{42 a}\\ &=\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{14 a d}+\frac {(A+C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac {\int \cos ^2(c+d x) (a+a \cos (c+d x)) \left (3 a^3 (112 A+87 C)+3 a^3 (154 A+129 C) \cos (c+d x)\right ) \, dx}{210 a}\\ &=\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{14 a d}+\frac {(A+C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac {\int \cos ^2(c+d x) \left (3 a^4 (112 A+87 C)+\left (3 a^4 (112 A+87 C)+3 a^4 (154 A+129 C)\right ) \cos (c+d x)+3 a^4 (154 A+129 C) \cos ^2(c+d x)\right ) \, dx}{210 a}\\ &=\frac {a^3 (154 A+129 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{14 a d}+\frac {(A+C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac {\int \cos ^2(c+d x) \left (105 a^4 (26 A+21 C)+24 a^4 (133 A+108 C) \cos (c+d x)\right ) \, dx}{840 a}\\ &=\frac {a^3 (154 A+129 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{14 a d}+\frac {(A+C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac {1}{8} \left (a^3 (26 A+21 C)\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{35} \left (a^3 (133 A+108 C)\right ) \int \cos ^3(c+d x) \, dx\\ &=\frac {a^3 (26 A+21 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (154 A+129 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{14 a d}+\frac {(A+C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac {1}{16} \left (a^3 (26 A+21 C)\right ) \int 1 \, dx-\frac {\left (a^3 (133 A+108 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 (26 A+21 C) x+\frac {a^3 (133 A+108 C) \sin (c+d x)}{35 d}+\frac {a^3 (26 A+21 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (154 A+129 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{14 a d}+\frac {(A+C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{5 d}-\frac {a^3 (133 A+108 C) \sin ^3(c+d x)}{105 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.69, size = 145, normalized size = 0.61 \[ \frac {a^3 (105 (184 A+155 C) \sin (c+d x)+105 (64 A+61 C) \sin (2 (c+d x))+2380 A \sin (3 (c+d x))+630 A \sin (4 (c+d x))+84 A \sin (5 (c+d x))+10920 A d x+2835 C \sin (3 (c+d x))+1155 C \sin (4 (c+d x))+399 C \sin (5 (c+d x))+105 C \sin (6 (c+d x))+15 C \sin (7 (c+d x))+5460 c C+8820 C d x)}{6720 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.62, size = 146, normalized size = 0.62 \[ \frac {105 \, {\left (26 \, A + 21 \, C\right )} a^{3} d x + {\left (240 \, C a^{3} \cos \left (d x + c\right )^{6} + 840 \, C a^{3} \cos \left (d x + c\right )^{5} + 48 \, {\left (7 \, A + 27 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 210 \, {\left (6 \, A + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 16 \, {\left (133 \, A + 108 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 105 \, {\left (26 \, A + 21 \, C\right )} a^{3} \cos \left (d x + c\right ) + 32 \, {\left (133 \, A + 108 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{1680 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.55, size = 185, normalized size = 0.78 \[ \frac {C a^{3} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {C a^{3} \sin \left (6 \, d x + 6 \, c\right )}{64 \, d} + \frac {1}{16} \, {\left (26 \, A a^{3} + 21 \, C a^{3}\right )} x + \frac {{\left (4 \, A a^{3} + 19 \, C a^{3}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (6 \, A a^{3} + 11 \, C a^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (68 \, A a^{3} + 81 \, C a^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {{\left (64 \, A a^{3} + 61 \, C a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (184 \, A a^{3} + 155 \, C a^{3}\right )} \sin \left (d x + c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.33, size = 286, normalized size = 1.21 \[ \frac {\frac {A \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {C \,a^{3} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+3 A \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 C \,a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+A \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\frac {3 C \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+A \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.41, size = 284, normalized size = 1.20 \[ \frac {448 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{3} - 6720 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} + 630 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} + 1680 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 192 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} C a^{3} + 1344 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{3} - 105 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3}}{6720 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.28, size = 353, normalized size = 1.49 \[ \frac {\left (\frac {13\,A\,a^3}{4}+\frac {21\,C\,a^3}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {65\,A\,a^3}{3}+\frac {35\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {3679\,A\,a^3}{60}+\frac {1981\,C\,a^3}{40}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {464\,A\,a^3}{5}+\frac {2608\,C\,a^3}{35}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {5089\,A\,a^3}{60}+\frac {3011\,C\,a^3}{40}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {143\,A\,a^3}{3}+\frac {61\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {51\,A\,a^3}{4}+\frac {107\,C\,a^3}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a^3\,\left (26\,A+21\,C\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{8\,d}+\frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (26\,A+21\,C\right )}{8\,\left (\frac {13\,A\,a^3}{4}+\frac {21\,C\,a^3}{8}\right )}\right )\,\left (26\,A+21\,C\right )}{8\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 7.77, size = 750, normalized size = 3.16 \[ \begin {cases} \frac {9 A a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {9 A a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {A a^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {9 A a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {A a^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {8 A a^{3} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 A a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {9 A a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {2 A a^{3} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {A a^{3} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {15 A a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {3 A a^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {A a^{3} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {15 C a^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {45 C a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 C a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {45 C a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 C a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {15 C a^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 C a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {16 C a^{3} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 C a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {15 C a^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {8 C a^{3} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {2 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac {4 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {C a^{3} \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} + \frac {33 C a^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {3 C a^{3} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 C a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\relax (c )}\right ) \left (a \cos {\relax (c )} + a\right )^{3} \cos ^{2}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________