Optimal. Leaf size=279 \[ -\frac {4 a^4 (63 A+52 C) \sin ^3(c+d x)}{105 d}+\frac {4 a^4 (63 A+52 C) \sin (c+d x)}{35 d}+\frac {a^4 (2408 A+2007 C) \sin (c+d x) \cos ^3(c+d x)}{2240 d}+\frac {7 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{120 d}+\frac {a^4 (392 A+323 C) \sin (c+d x) \cos (c+d x)}{128 d}+\frac {1}{128} a^4 x (392 A+323 C)+\frac {(56 A+61 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{336 d}+\frac {a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{14 d}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.79, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 11, 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} \[ -\frac {4 a^4 (63 A+52 C) \sin ^3(c+d x)}{105 d}+\frac {4 a^4 (63 A+52 C) \sin (c+d x)}{35 d}+\frac {a^4 (2408 A+2007 C) \sin (c+d x) \cos ^3(c+d x)}{2240 d}+\frac {(56 A+61 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{336 d}+\frac {7 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{120 d}+\frac {a^4 (392 A+323 C) \sin (c+d x) \cos (c+d x)}{128 d}+\frac {1}{128} a^4 x (392 A+323 C)+\frac {a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{14 d}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d} \]
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))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {\int \cos ^2(c+d x) (a+a \cos (c+d x))^4 (a (8 A+3 C)+4 a C \cos (c+d x)) \, dx}{8 a}\\ &=\frac {a C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {\int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \left (a^2 (56 A+33 C)+a^2 (56 A+61 C) \cos (c+d x)\right ) \, dx}{56 a}\\ &=\frac {a C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {\int \cos ^2(c+d x) (a+a \cos (c+d x))^2 \left (3 a^3 (168 A+127 C)+98 a^3 (8 A+7 C) \cos (c+d x)\right ) \, dx}{336 a}\\ &=\frac {a C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {7 (8 A+7 C) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{120 d}+\frac {\int \cos ^2(c+d x) (a+a \cos (c+d x)) \left (3 a^4 (1624 A+1321 C)+3 a^4 (2408 A+2007 C) \cos (c+d x)\right ) \, dx}{1680 a}\\ &=\frac {a C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {7 (8 A+7 C) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{120 d}+\frac {\int \cos ^2(c+d x) \left (3 a^5 (1624 A+1321 C)+\left (3 a^5 (1624 A+1321 C)+3 a^5 (2408 A+2007 C)\right ) \cos (c+d x)+3 a^5 (2408 A+2007 C) \cos ^2(c+d x)\right ) \, dx}{1680 a}\\ &=\frac {a^4 (2408 A+2007 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac {a C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {7 (8 A+7 C) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{120 d}+\frac {\int \cos ^2(c+d x) \left (105 a^5 (392 A+323 C)+768 a^5 (63 A+52 C) \cos (c+d x)\right ) \, dx}{6720 a}\\ &=\frac {a^4 (2408 A+2007 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac {a C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {7 (8 A+7 C) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{120 d}+\frac {1}{35} \left (4 a^4 (63 A+52 C)\right ) \int \cos ^3(c+d x) \, dx+\frac {1}{64} \left (a^4 (392 A+323 C)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac {a^4 (392 A+323 C) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {a^4 (2408 A+2007 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac {a C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {7 (8 A+7 C) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{120 d}+\frac {1}{128} \left (a^4 (392 A+323 C)\right ) \int 1 \, dx-\frac {\left (4 a^4 (63 A+52 C)\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 d}\\ &=\frac {1}{128} a^4 (392 A+323 C) x+\frac {4 a^4 (63 A+52 C) \sin (c+d x)}{35 d}+\frac {a^4 (392 A+323 C) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {a^4 (2408 A+2007 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac {a C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {7 (8 A+7 C) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{120 d}-\frac {4 a^4 (63 A+52 C) \sin ^3(c+d x)}{105 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.93, size = 167, normalized size = 0.60 \[ \frac {a^4 (6720 (88 A+75 C) \sin (c+d x)+1680 (127 A+120 C) \sin (2 (c+d x))+80640 A \sin (3 (c+d x))+25200 A \sin (4 (c+d x))+5376 A \sin (5 (c+d x))+560 A \sin (6 (c+d x))+329280 A d x+91840 C \sin (3 (c+d x))+39480 C \sin (4 (c+d x))+14784 C \sin (5 (c+d x))+4480 C \sin (6 (c+d x))+960 C \sin (7 (c+d x))+105 C \sin (8 (c+d x))+164640 c C+271320 C d x)}{107520 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.61, size = 166, normalized size = 0.59 \[ \frac {105 \, {\left (392 \, A + 323 \, C\right )} a^{4} d x + {\left (1680 \, C a^{4} \cos \left (d x + c\right )^{7} + 7680 \, C a^{4} \cos \left (d x + c\right )^{6} + 280 \, {\left (8 \, A + 55 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 1536 \, {\left (7 \, A + 13 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (328 \, A + 323 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 512 \, {\left (63 \, A + 52 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \, {\left (392 \, A + 323 \, C\right )} a^{4} \cos \left (d x + c\right ) + 1024 \, {\left (63 \, A + 52 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{13440 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.51, size = 211, normalized size = 0.76 \[ \frac {C a^{4} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {C a^{4} \sin \left (7 \, d x + 7 \, c\right )}{112 \, d} + \frac {1}{128} \, {\left (392 \, A a^{4} + 323 \, C a^{4}\right )} x + \frac {{\left (A a^{4} + 8 \, C a^{4}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (4 \, A a^{4} + 11 \, C a^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {{\left (30 \, A a^{4} + 47 \, C a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {{\left (36 \, A a^{4} + 41 \, C a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (127 \, A a^{4} + 120 \, C a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (88 \, A a^{4} + 75 \, C a^{4}\right )} \sin \left (d x + c\right )}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.38, size = 393, normalized size = 1.41 \[ \frac {A \,a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+a^{4} C \left (\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )+\frac {4 A \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {4 a^{4} C \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+6 A \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+6 a^{4} C \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {4 A \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {4 a^{4} C \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+A \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 393, normalized size = 1.41 \[ \frac {28672 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{4} - 560 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 143360 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} + 20160 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 26880 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 12288 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} C a^{4} + 28672 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{4} - 35 \, {\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 840 \, d x - 840 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 3360 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 3360 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4}}{107520 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.42, size = 391, normalized size = 1.40 \[ \frac {\left (\frac {49\,A\,a^4}{8}+\frac {323\,C\,a^4}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+\left (\frac {1127\,A\,a^4}{24}+\frac {7429\,C\,a^4}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {18767\,A\,a^4}{120}+\frac {123709\,C\,a^4}{960}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {35371\,A\,a^4}{120}+\frac {1632119\,C\,a^4}{6720}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {40661\,A\,a^4}{120}+\frac {624003\,C\,a^4}{2240}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {29617\,A\,a^4}{120}+\frac {68673\,C\,a^4}{320}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {2713\,A\,a^4}{24}+\frac {5033\,C\,a^4}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {207\,A\,a^4}{8}+\frac {1725\,C\,a^4}{64}\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 )}^{16}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a^4\,\left (392\,A+323\,C\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{64\,d}+\frac {a^4\,\mathrm {atan}\left (\frac {a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (392\,A+323\,C\right )}{64\,\left (\frac {49\,A\,a^4}{8}+\frac {323\,C\,a^4}{64}\right )}\right )\,\left (392\,A+323\,C\right )}{64\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 14.52, size = 1149, normalized size = 4.12 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________