Optimal. Leaf size=278 \[ \frac {1}{16} a^4 (44 A+49 B+56 C) x+\frac {a^4 (454 A+504 B+581 C) \sin (c+d x)}{105 d}+\frac {a^4 (44 A+49 B+56 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^4 (988 A+1113 B+1232 C) \cos ^2(c+d x) \sin (c+d x)}{840 d}+\frac {a (4 A+7 B) \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac {A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac {(16 A+21 B+14 C) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{70 d}+\frac {(436 A+511 B+504 C) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{840 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.53, antiderivative size = 278, 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 = {4171, 4102,
4081, 3872, 2715, 8, 2717} \begin {gather*} \frac {a^4 (454 A+504 B+581 C) \sin (c+d x)}{105 d}+\frac {a^4 (988 A+1113 B+1232 C) \sin (c+d x) \cos ^2(c+d x)}{840 d}+\frac {a^4 (44 A+49 B+56 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {(436 A+511 B+504 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{840 d}+\frac {1}{16} a^4 x (44 A+49 B+56 C)+\frac {(16 A+21 B+14 C) \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{70 d}+\frac {a (4 A+7 B) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{42 d}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2715
Rule 2717
Rule 3872
Rule 4081
Rule 4102
Rule 4171
Rubi steps
\begin {align*} \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \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))^4 \sin (c+d x)}{7 d}+\frac {\int \cos ^6(c+d x) (a+a \sec (c+d x))^4 (a (4 A+7 B)+a (2 A+7 C) \sec (c+d x)) \, dx}{7 a}\\ &=\frac {a (4 A+7 B) \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac {A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac {\int \cos ^5(c+d x) (a+a \sec (c+d x))^3 \left (3 a^2 (16 A+21 B+14 C)+2 a^2 (10 A+7 B+21 C) \sec (c+d x)\right ) \, dx}{42 a}\\ &=\frac {a (4 A+7 B) \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac {A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac {(16 A+21 B+14 C) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{70 d}+\frac {\int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left (a^3 (436 A+511 B+504 C)+98 a^3 (2 A+2 B+3 C) \sec (c+d x)\right ) \, dx}{210 a}\\ &=\frac {a (4 A+7 B) \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac {A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac {(16 A+21 B+14 C) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{70 d}+\frac {(436 A+511 B+504 C) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{840 d}+\frac {\int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (3 a^4 (988 A+1113 B+1232 C)+6 a^4 (276 A+301 B+364 C) \sec (c+d x)\right ) \, dx}{840 a}\\ &=\frac {a^4 (988 A+1113 B+1232 C) \cos ^2(c+d x) \sin (c+d x)}{840 d}+\frac {a (4 A+7 B) \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac {A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac {(16 A+21 B+14 C) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{70 d}+\frac {(436 A+511 B+504 C) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{840 d}-\frac {\int \cos ^2(c+d x) \left (-315 a^5 (44 A+49 B+56 C)-24 a^5 (454 A+504 B+581 C) \sec (c+d x)\right ) \, dx}{2520 a}\\ &=\frac {a^4 (988 A+1113 B+1232 C) \cos ^2(c+d x) \sin (c+d x)}{840 d}+\frac {a (4 A+7 B) \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac {A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac {(16 A+21 B+14 C) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{70 d}+\frac {(436 A+511 B+504 C) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{840 d}+\frac {1}{8} \left (a^4 (44 A+49 B+56 C)\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{105} \left (a^4 (454 A+504 B+581 C)\right ) \int \cos (c+d x) \, dx\\ &=\frac {a^4 (454 A+504 B+581 C) \sin (c+d x)}{105 d}+\frac {a^4 (44 A+49 B+56 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^4 (988 A+1113 B+1232 C) \cos ^2(c+d x) \sin (c+d x)}{840 d}+\frac {a (4 A+7 B) \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac {A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac {(16 A+21 B+14 C) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{70 d}+\frac {(436 A+511 B+504 C) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{840 d}+\frac {1}{16} \left (a^4 (44 A+49 B+56 C)\right ) \int 1 \, dx\\ &=\frac {1}{16} a^4 (44 A+49 B+56 C) x+\frac {a^4 (454 A+504 B+581 C) \sin (c+d x)}{105 d}+\frac {a^4 (44 A+49 B+56 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^4 (988 A+1113 B+1232 C) \cos ^2(c+d x) \sin (c+d x)}{840 d}+\frac {a (4 A+7 B) \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac {A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac {(16 A+21 B+14 C) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{70 d}+\frac {(436 A+511 B+504 C) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{840 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.11, size = 204, normalized size = 0.73 \begin {gather*} \frac {a^4 (11760 A c+20580 B c+18480 A d x+20580 B d x+23520 C d x+105 (323 A+352 B+392 C) \sin (c+d x)+105 (124 A+127 B+128 C) \sin (2 (c+d x))+5495 A \sin (3 (c+d x))+5040 B \sin (3 (c+d x))+4060 C \sin (3 (c+d x))+2100 A \sin (4 (c+d x))+1575 B \sin (4 (c+d x))+840 C \sin (4 (c+d x))+651 A \sin (5 (c+d x))+336 B \sin (5 (c+d x))+84 C \sin (5 (c+d x))+140 A \sin (6 (c+d x))+35 B \sin (6 (c+d x))+15 A \sin (7 (c+d x)))}{6720 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.30, size = 490, normalized size = 1.76 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 483, normalized size = 1.74 \begin {gather*} -\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^{4} - 2688 \, {\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} + 140 \, {\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} + 2240 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 840 \, {\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} - 1792 \, {\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^{4} + 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^{4} + 8960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} - 1260 \, {\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^{4} - 1680 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 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^{4} + 13440 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} - 840 \, {\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} - 6720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 6720 \, C a^{4} \sin \left (d x + c\right )}{6720 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.52, size = 168, normalized size = 0.60 \begin {gather*} \frac {105 \, {\left (44 \, A + 49 \, B + 56 \, C\right )} a^{4} d x + {\left (240 \, A a^{4} \cos \left (d x + c\right )^{6} + 280 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{5} + 48 \, {\left (48 \, A + 28 \, B + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (44 \, A + 41 \, B + 24 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 16 \, {\left (227 \, A + 252 \, B + 238 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \, {\left (44 \, A + 49 \, B + 56 \, C\right )} a^{4} \cos \left (d x + c\right ) + 16 \, {\left (454 \, A + 504 \, B + 581 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{1680 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.54, size = 401, normalized size = 1.44 \begin {gather*} \frac {105 \, {\left (44 \, A a^{4} + 49 \, B a^{4} + 56 \, C a^{4}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (4620 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 5145 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 5880 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 30800 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 34300 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 39200 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 87164 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 97069 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 110936 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 135168 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 150528 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 172032 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 126084 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 134099 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 159656 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 58800 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 73220 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 86240 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 22260 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 21735 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 21000 \, C a^{4} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 5.88, size = 377, normalized size = 1.36 \begin {gather*} \frac {\left (\frac {11\,A\,a^4}{2}+\frac {49\,B\,a^4}{8}+7\,C\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {110\,A\,a^4}{3}+\frac {245\,B\,a^4}{6}+\frac {140\,C\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {3113\,A\,a^4}{30}+\frac {13867\,B\,a^4}{120}+\frac {1981\,C\,a^4}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {5632\,A\,a^4}{35}+\frac {896\,B\,a^4}{5}+\frac {1024\,C\,a^4}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {1501\,A\,a^4}{10}+\frac {19157\,B\,a^4}{120}+\frac {2851\,C\,a^4}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (70\,A\,a^4+\frac {523\,B\,a^4}{6}+\frac {308\,C\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {53\,A\,a^4}{2}+\frac {207\,B\,a^4}{8}+25\,C\,a^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^4\,\mathrm {atan}\left (\frac {a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (44\,A+49\,B+56\,C\right )}{8\,\left (\frac {11\,A\,a^4}{2}+\frac {49\,B\,a^4}{8}+7\,C\,a^4\right )}\right )\,\left (44\,A+49\,B+56\,C\right )}{8\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________