Optimal. Leaf size=310 \[ \frac {8 a^4 (185 A+208 B+247 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {8 a^4 (100 A+113 B+132 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {8 a^4 (100 A+113 B+132 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {4 a^4 (5255 A+6019 B+6721 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15015 d}+\frac {2 a (8 A+13 B) \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {2 (13 A+17 B+11 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 d}+\frac {4 (1355 A+1612 B+1573 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{9009 d} \]
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Rubi [A]
time = 0.62, antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 9, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.209, Rules used = {4197, 3124,
3055, 3047, 3102, 2827, 2719, 2715, 2720} \begin {gather*} \frac {8 a^4 (100 A+113 B+132 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {8 a^4 (185 A+208 B+247 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {4 a^4 (5255 A+6019 B+6721 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{15015 d}+\frac {4 (1355 A+1612 B+1573 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9009 d}+\frac {8 a^4 (100 A+113 B+132 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{231 d}+\frac {2 (13 A+17 B+11 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{99 d}+\frac {2 a (8 A+13 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{143 d}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^4}{13 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2715
Rule 2719
Rule 2720
Rule 2827
Rule 3047
Rule 3055
Rule 3102
Rule 3124
Rule 4197
Rubi steps
\begin {align*} \int \cos ^{\frac {13}{2}}(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 &=\int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^4 \left (C+B \cos (c+d x)+A \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {2 \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^4 \left (\frac {1}{2} a (3 A+13 C)+\frac {1}{2} a (8 A+13 B) \cos (c+d x)\right ) \, dx}{13 a}\\ &=\frac {2 a (8 A+13 B) \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {4 \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \left (\frac {1}{4} a^2 (57 A+39 B+143 C)+\frac {13}{4} a^2 (13 A+17 B+11 C) \cos (c+d x)\right ) \, dx}{143 a}\\ &=\frac {2 a (8 A+13 B) \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {2 (13 A+17 B+11 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 d}+\frac {8 \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^2 \left (\frac {3}{4} a^3 (170 A+169 B+286 C)+\frac {1}{4} a^3 (1355 A+1612 B+1573 C) \cos (c+d x)\right ) \, dx}{1287 a}\\ &=\frac {2 a (8 A+13 B) \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {2 (13 A+17 B+11 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 d}+\frac {4 (1355 A+1612 B+1573 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{9009 d}+\frac {16 \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x)) \left (\frac {15}{8} a^4 (509 A+559 B+715 C)+\frac {3}{8} a^4 (5255 A+6019 B+6721 C) \cos (c+d x)\right ) \, dx}{9009 a}\\ &=\frac {2 a (8 A+13 B) \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {2 (13 A+17 B+11 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 d}+\frac {4 (1355 A+1612 B+1573 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{9009 d}+\frac {16 \int \sqrt {\cos (c+d x)} \left (\frac {15}{8} a^5 (509 A+559 B+715 C)+\left (\frac {15}{8} a^5 (509 A+559 B+715 C)+\frac {3}{8} a^5 (5255 A+6019 B+6721 C)\right ) \cos (c+d x)+\frac {3}{8} a^5 (5255 A+6019 B+6721 C) \cos ^2(c+d x)\right ) \, dx}{9009 a}\\ &=\frac {4 a^4 (5255 A+6019 B+6721 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15015 d}+\frac {2 a (8 A+13 B) \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {2 (13 A+17 B+11 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 d}+\frac {4 (1355 A+1612 B+1573 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{9009 d}+\frac {32 \int \sqrt {\cos (c+d x)} \left (\frac {231}{8} a^5 (185 A+208 B+247 C)+\frac {585}{8} a^5 (100 A+113 B+132 C) \cos (c+d x)\right ) \, dx}{45045 a}\\ &=\frac {4 a^4 (5255 A+6019 B+6721 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15015 d}+\frac {2 a (8 A+13 B) \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {2 (13 A+17 B+11 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 d}+\frac {4 (1355 A+1612 B+1573 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{9009 d}+\frac {1}{77} \left (4 a^4 (100 A+113 B+132 C)\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{195} \left (4 a^4 (185 A+208 B+247 C)\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {8 a^4 (185 A+208 B+247 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {8 a^4 (100 A+113 B+132 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {4 a^4 (5255 A+6019 B+6721 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15015 d}+\frac {2 a (8 A+13 B) \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {2 (13 A+17 B+11 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 d}+\frac {4 (1355 A+1612 B+1573 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{9009 d}+\frac {1}{231} \left (4 a^4 (100 A+113 B+132 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {8 a^4 (185 A+208 B+247 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {8 a^4 (100 A+113 B+132 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {8 a^4 (100 A+113 B+132 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {4 a^4 (5255 A+6019 B+6721 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15015 d}+\frac {2 a (8 A+13 B) \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {2 (13 A+17 B+11 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 d}+\frac {4 (1355 A+1612 B+1573 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{9009 d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 6.34, size = 1416, normalized size = 4.57 \begin {gather*} a^4 \left (\sqrt {\cos (c+d x)} (1+\cos (c+d x))^4 \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (-\frac {(185 A+208 B+247 C) \cot (c)}{390 d}+\frac {(3764 A+4087 B+4488 C) \cos (d x) \sin (c)}{14784 d}+\frac {(15625 A+15392 B+13208 C) \cos (2 d x) \sin (2 c)}{149760 d}+\frac {(404 A+321 B+176 C) \cos (3 d x) \sin (3 c)}{9856 d}+\frac {(98 A+52 B+13 C) \cos (4 d x) \sin (4 c)}{7488 d}+\frac {(4 A+B) \cos (5 d x) \sin (5 c)}{1408 d}+\frac {A \cos (6 d x) \sin (6 c)}{3328 d}+\frac {(3764 A+4087 B+4488 C) \cos (c) \sin (d x)}{14784 d}+\frac {(15625 A+15392 B+13208 C) \cos (2 c) \sin (2 d x)}{149760 d}+\frac {(404 A+321 B+176 C) \cos (3 c) \sin (3 d x)}{9856 d}+\frac {(98 A+52 B+13 C) \cos (4 c) \sin (4 d x)}{7488 d}+\frac {(4 A+B) \cos (5 c) \sin (5 d x)}{1408 d}+\frac {A \cos (6 c) \sin (6 d x)}{3328 d}\right )-\frac {50 A (1+\cos (c+d x))^4 \csc (c) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\text {ArcTan}(\cot (c)))\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (d x-\text {ArcTan}(\cot (c))) \sqrt {1-\sin (d x-\text {ArcTan}(\cot (c)))} \sqrt {-\sqrt {1+\cot ^2(c)} \sin (c) \sin (d x-\text {ArcTan}(\cot (c)))} \sqrt {1+\sin (d x-\text {ArcTan}(\cot (c)))}}{231 d \sqrt {1+\cot ^2(c)}}-\frac {113 B (1+\cos (c+d x))^4 \csc (c) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\text {ArcTan}(\cot (c)))\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (d x-\text {ArcTan}(\cot (c))) \sqrt {1-\sin (d x-\text {ArcTan}(\cot (c)))} \sqrt {-\sqrt {1+\cot ^2(c)} \sin (c) \sin (d x-\text {ArcTan}(\cot (c)))} \sqrt {1+\sin (d x-\text {ArcTan}(\cot (c)))}}{462 d \sqrt {1+\cot ^2(c)}}-\frac {2 C (1+\cos (c+d x))^4 \csc (c) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\text {ArcTan}(\cot (c)))\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (d x-\text {ArcTan}(\cot (c))) \sqrt {1-\sin (d x-\text {ArcTan}(\cot (c)))} \sqrt {-\sqrt {1+\cot ^2(c)} \sin (c) \sin (d x-\text {ArcTan}(\cot (c)))} \sqrt {1+\sin (d x-\text {ArcTan}(\cot (c)))}}{7 d \sqrt {1+\cot ^2(c)}}-\frac {37 A (1+\cos (c+d x))^4 \csc (c) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\text {ArcTan}(\tan (c)))\right ) \sin (d x+\text {ArcTan}(\tan (c))) \tan (c)}{\sqrt {1-\cos (d x+\text {ArcTan}(\tan (c)))} \sqrt {1+\cos (d x+\text {ArcTan}(\tan (c)))} \sqrt {\cos (c) \cos (d x+\text {ArcTan}(\tan (c))) \sqrt {1+\tan ^2(c)}} \sqrt {1+\tan ^2(c)}}-\frac {\frac {\sin (d x+\text {ArcTan}(\tan (c))) \tan (c)}{\sqrt {1+\tan ^2(c)}}+\frac {2 \cos ^2(c) \cos (d x+\text {ArcTan}(\tan (c))) \sqrt {1+\tan ^2(c)}}{\cos ^2(c)+\sin ^2(c)}}{\sqrt {\cos (c) \cos (d x+\text {ArcTan}(\tan (c))) \sqrt {1+\tan ^2(c)}}}\right )}{156 d}-\frac {4 B (1+\cos (c+d x))^4 \csc (c) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\text {ArcTan}(\tan (c)))\right ) \sin (d x+\text {ArcTan}(\tan (c))) \tan (c)}{\sqrt {1-\cos (d x+\text {ArcTan}(\tan (c)))} \sqrt {1+\cos (d x+\text {ArcTan}(\tan (c)))} \sqrt {\cos (c) \cos (d x+\text {ArcTan}(\tan (c))) \sqrt {1+\tan ^2(c)}} \sqrt {1+\tan ^2(c)}}-\frac {\frac {\sin (d x+\text {ArcTan}(\tan (c))) \tan (c)}{\sqrt {1+\tan ^2(c)}}+\frac {2 \cos ^2(c) \cos (d x+\text {ArcTan}(\tan (c))) \sqrt {1+\tan ^2(c)}}{\cos ^2(c)+\sin ^2(c)}}{\sqrt {\cos (c) \cos (d x+\text {ArcTan}(\tan (c))) \sqrt {1+\tan ^2(c)}}}\right )}{15 d}-\frac {19 C (1+\cos (c+d x))^4 \csc (c) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\text {ArcTan}(\tan (c)))\right ) \sin (d x+\text {ArcTan}(\tan (c))) \tan (c)}{\sqrt {1-\cos (d x+\text {ArcTan}(\tan (c)))} \sqrt {1+\cos (d x+\text {ArcTan}(\tan (c)))} \sqrt {\cos (c) \cos (d x+\text {ArcTan}(\tan (c))) \sqrt {1+\tan ^2(c)}} \sqrt {1+\tan ^2(c)}}-\frac {\frac {\sin (d x+\text {ArcTan}(\tan (c))) \tan (c)}{\sqrt {1+\tan ^2(c)}}+\frac {2 \cos ^2(c) \cos (d x+\text {ArcTan}(\tan (c))) \sqrt {1+\tan ^2(c)}}{\cos ^2(c)+\sin ^2(c)}}{\sqrt {\cos (c) \cos (d x+\text {ArcTan}(\tan (c))) \sqrt {1+\tan ^2(c)}}}\right )}{60 d}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.16, size = 576, normalized size = 1.86
method | result | size |
default | \(-\frac {8 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, a^{4} \left (-110880 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (594720 A +65520 B \right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-1345120 A -323960 B -40040 C \right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (1667840 A +659620 B +183040 C \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-1237490 A -713518 B -336622 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (572110 A +448448 B +322322 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-117945 A -110097 B -97383 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+19500 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-42735 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+22035 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-48048 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+25740 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-57057 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{45045 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\) | \(576\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.89, size = 287, normalized size = 0.93 \begin {gather*} -\frac {2 \, {\left (390 i \, \sqrt {2} {\left (100 \, A + 113 \, B + 132 \, C\right )} a^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 390 i \, \sqrt {2} {\left (100 \, A + 113 \, B + 132 \, C\right )} a^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 462 i \, \sqrt {2} {\left (185 \, A + 208 \, B + 247 \, C\right )} a^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 462 i \, \sqrt {2} {\left (185 \, A + 208 \, B + 247 \, C\right )} a^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (3465 \, A a^{4} \cos \left (d x + c\right )^{5} + 4095 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{4} + 385 \, {\left (89 \, A + 52 \, B + 13 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 585 \, {\left (80 \, A + 75 \, B + 44 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 77 \, {\left (740 \, A + 832 \, B + 793 \, C\right )} a^{4} \cos \left (d x + c\right ) + 780 \, {\left (100 \, A + 113 \, B + 132 \, C\right )} a^{4}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{45045 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.26, size = 764, normalized size = 2.46 \begin {gather*} \frac {2\,\left (3\,C\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+4\,C\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+4\,C\,a^4\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{3\,d}-\frac {8\,\left (\frac {13\,A\,a^4\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {5\,A\,a^4\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {13}{4};\ \frac {17}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{117\,d}-\frac {136\,\left (\frac {11\,A\,a^4\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {51\,A\,a^4\,{\cos \left (c+d\,x\right )}^{15/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {15}{4};\ \frac {23}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{21945\,d}-\frac {2\,\left (\frac {66\,C\,a^4\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {17\,C\,a^4\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{77\,d}-\frac {2\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {15}{4};\ \frac {19}{4};\ {\cos \left (c+d\,x\right )}^2\right )\,\left (\frac {165\,A\,a^4\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {578\,A\,a^4\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {127\,A\,a^4\,{\cos \left (c+d\,x\right )}^{15/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}\right )}{1155\,d}+\frac {B\,a^4\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}-\frac {160\,A\,a^4\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {13}{4};\ \frac {21}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{663\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {8\,B\,a^4\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,B\,a^4\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {8\,B\,a^4\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,a^4\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {13}{4};\ \frac {17}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{13\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {8\,C\,a^4\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {208\,C\,a^4\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {19}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{385\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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