Optimal. Leaf size=310 \[ -\frac {8 a^4 (24 A+19 B+16 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {8 a^4 (187 A+132 B+113 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {4 a^4 (913 A+803 B+667 C) \sin (c+d x)}{1155 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {8 a^4 (24 A+19 B+16 C) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 a (11 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 (33 A+55 B+43 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4 (891 A+946 B+769 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)} \]
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Rubi [A]
time = 0.64, 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, 3122,
3054, 3047, 3100, 2827, 2716, 2719, 2720} \begin {gather*} \frac {8 a^4 (187 A+132 B+113 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}-\frac {8 a^4 (24 A+19 B+16 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 a^4 (913 A+803 B+667 C) \sin (c+d x)}{1155 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {4 (891 A+946 B+769 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {8 a^4 (24 A+19 B+16 C) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 (33 A+55 B+43 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a (11 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2716
Rule 2719
Rule 2720
Rule 2827
Rule 3047
Rule 3054
Rule 3100
Rule 3122
Rule 4197
Rubi steps
\begin {align*} \int \frac {(a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx &=\int \frac {(a+a \cos (c+d x))^4 \left (C+B \cos (c+d x)+A \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx\\ &=\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 \int \frac {(a+a \cos (c+d x))^4 \left (\frac {1}{2} a (11 B+8 C)+\frac {1}{2} a (11 A+C) \cos (c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx}{11 a}\\ &=\frac {2 a (11 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {4 \int \frac {(a+a \cos (c+d x))^3 \left (\frac {3}{4} a^2 (33 A+55 B+43 C)+\frac {1}{4} a^2 (99 A+11 B+17 C) \cos (c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx}{99 a}\\ &=\frac {2 a (11 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 (33 A+55 B+43 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {8 \int \frac {(a+a \cos (c+d x))^2 \left (\frac {1}{4} a^3 (891 A+946 B+769 C)+\frac {1}{4} a^3 (396 A+121 B+124 C) \cos (c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx}{693 a}\\ &=\frac {2 a (11 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 (33 A+55 B+43 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4 (891 A+946 B+769 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {16 \int \frac {(a+a \cos (c+d x)) \left (\frac {9}{8} a^4 (913 A+803 B+667 C)+\frac {3}{8} a^4 (957 A+517 B+463 C) \cos (c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{3465 a}\\ &=\frac {2 a (11 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 (33 A+55 B+43 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4 (891 A+946 B+769 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {16 \int \frac {\frac {9}{8} a^5 (913 A+803 B+667 C)+\left (\frac {3}{8} a^5 (957 A+517 B+463 C)+\frac {9}{8} a^5 (913 A+803 B+667 C)\right ) \cos (c+d x)+\frac {3}{8} a^5 (957 A+517 B+463 C) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{3465 a}\\ &=\frac {4 a^4 (913 A+803 B+667 C) \sin (c+d x)}{1155 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a (11 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 (33 A+55 B+43 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4 (891 A+946 B+769 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {32 \int \frac {\frac {693}{8} a^5 (24 A+19 B+16 C)+\frac {45}{8} a^5 (187 A+132 B+113 C) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)} \, dx}{10395 a}\\ &=\frac {4 a^4 (913 A+803 B+667 C) \sin (c+d x)}{1155 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a (11 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 (33 A+55 B+43 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4 (891 A+946 B+769 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{15} \left (4 a^4 (24 A+19 B+16 C)\right ) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)} \, dx+\frac {1}{231} \left (4 a^4 (187 A+132 B+113 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {8 a^4 (187 A+132 B+113 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {4 a^4 (913 A+803 B+667 C) \sin (c+d x)}{1155 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {8 a^4 (24 A+19 B+16 C) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 a (11 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 (33 A+55 B+43 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4 (891 A+946 B+769 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}-\frac {1}{15} \left (4 a^4 (24 A+19 B+16 C)\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {8 a^4 (24 A+19 B+16 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {8 a^4 (187 A+132 B+113 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {4 a^4 (913 A+803 B+667 C) \sin (c+d x)}{1155 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {8 a^4 (24 A+19 B+16 C) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 a (11 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 (33 A+55 B+43 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4 (891 A+946 B+769 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 7.02, size = 1795, normalized size = 5.79 \begin {gather*} \frac {\cos ^{\frac {13}{2}}(c+d x) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {(24 A+19 B+16 C) \csc (c) \sec (c)}{15 d}+\frac {C \sec (c) \sec ^6(c+d x) \sin (d x)}{44 d}+\frac {\sec (c) \sec ^5(c+d x) (9 C \sin (c)+11 B \sin (d x)+44 C \sin (d x))}{396 d}+\frac {\sec (c) \sec ^4(c+d x) (77 B \sin (c)+308 C \sin (c)+99 A \sin (d x)+396 B \sin (d x)+675 C \sin (d x))}{2772 d}+\frac {\sec (c) \sec ^3(c+d x) (495 A \sin (c)+1980 B \sin (c)+3375 C \sin (c)+2772 A \sin (d x)+4697 B \sin (d x)+4928 C \sin (d x))}{13860 d}+\frac {\sec (c) \sec (c+d x) (2585 A \sin (c)+2640 B \sin (c)+2260 C \sin (c)+7392 A \sin (d x)+5852 B \sin (d x)+4928 C \sin (d x))}{4620 d}+\frac {\sec (c) \sec ^2(c+d x) (2772 A \sin (c)+4697 B \sin (c)+4928 C \sin (c)+7755 A \sin (d x)+7920 B \sin (d x)+6780 C \sin (d x))}{13860 d}\right )}{A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)}-\frac {17 A \cos ^6(c+d x) \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 ) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\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)))}}{21 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sqrt {1+\cot ^2(c)}}-\frac {4 B \cos ^6(c+d x) \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 ) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\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 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sqrt {1+\cot ^2(c)}}-\frac {113 C \cos ^6(c+d x) \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 ) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\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 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sqrt {1+\cot ^2(c)}}+\frac {4 A \cos ^6(c+d x) \csc (c) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\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 )}{5 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {19 B \cos ^6(c+d x) \csc (c) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\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 )}{30 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {8 C \cos ^6(c+d x) \csc (c) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\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 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1477\) vs.
\(2(334)=668\).
time = 0.58, size = 1478, normalized size = 4.77
method | result | size |
default | \(\text {Expression too large to display}\) | \(1478\) |
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 = 0.92, size = 327, normalized size = 1.05 \begin {gather*} -\frac {2 \, {\left (30 i \, \sqrt {2} {\left (187 \, A + 132 \, B + 113 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 30 i \, \sqrt {2} {\left (187 \, A + 132 \, B + 113 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 462 i \, \sqrt {2} {\left (24 \, A + 19 \, B + 16 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} {\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 (24 \, A + 19 \, B + 16 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} {\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 (924 \, {\left (24 \, A + 19 \, B + 16 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 15 \, {\left (517 \, A + 528 \, B + 452 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 77 \, {\left (36 \, A + 61 \, B + 64 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 45 \, {\left (11 \, A + 44 \, B + 75 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 385 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right ) + 315 \, C a^{4}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{3465 \, d \cos \left (d x + c\right )^{6}} \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 = 10.13, size = 830, normalized size = 2.68 \begin {gather*} \frac {8\,\left (\frac {11\,B\,a^4\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {3\,B\,a^4\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{7/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{21\,d}-\frac {8\,\left (\frac {61\,B\,a^4\,\sin \left (c+d\,x\right )}{\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {5\,B\,a^4\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{5/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {7}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{135\,d}+\frac {8\,\left (\frac {13\,C\,a^4\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{5/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {5\,C\,a^4\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{9/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{2};\ -\frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{45\,d}+\frac {8\,\left (\frac {75\,C\,a^4\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {7\,C\,a^4\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{7/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {5}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{231\,d}+\frac {2\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )\,\left (\frac {289\,B\,a^4\,\sin \left (c+d\,x\right )}{\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {66\,B\,a^4\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{5/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {5\,B\,a^4\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{9/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}\right )}{45\,d}+\frac {2\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )\,\left (\frac {377\,C\,a^4\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {218\,C\,a^4\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{7/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {21\,C\,a^4\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{11/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}\right )}{231\,d}+\frac {2\,A\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {8\,A\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {4\,A\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {8\,A\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{2};\ -\frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{5\,d\,{\cos \left (c+d\,x\right )}^{5/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {2\,A\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {7}{4},\frac {1}{2};\ -\frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,{\cos \left (c+d\,x\right )}^{7/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {32\,B\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {5}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{21\,d\,{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {32\,C\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{15\,d\,{\cos \left (c+d\,x\right )}^{5/2}\,\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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