Optimal. Leaf size=217 \[ \frac {1}{2} a^4 (12 A+13 B+8 C) x+\frac {a^4 (2 A+8 B+13 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {5 a^4 (2 A+B-C) \sin (c+d x)}{2 d}+\frac {a (4 A+3 B) \cos (c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{3 d}-\frac {(2 A+B-C) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 d}-\frac {(8 A-3 B-18 C) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{6 d} \]
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Rubi [A]
time = 0.42, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {4171, 4102,
4103, 4081, 3855} \begin {gather*} \frac {5 a^4 (2 A+B-C) \sin (c+d x)}{2 d}+\frac {a^4 (2 A+8 B+13 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {(8 A-3 B-18 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{6 d}+\frac {1}{2} a^4 x (12 A+13 B+8 C)-\frac {(2 A+B-C) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{2 d}+\frac {a (4 A+3 B) \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^3}{6 d}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^4}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3855
Rule 4081
Rule 4102
Rule 4103
Rule 4171
Rubi steps
\begin {align*} \int \cos ^3(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 ^2(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{3 d}+\frac {\int \cos ^2(c+d x) (a+a \sec (c+d x))^4 (a (4 A+3 B)-a (2 A-3 C) \sec (c+d x)) \, dx}{3 a}\\ &=\frac {a (4 A+3 B) \cos (c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{3 d}+\frac {\int \cos (c+d x) (a+a \sec (c+d x))^3 \left (a^2 (16 A+15 B+6 C)-6 a^2 (2 A+B-C) \sec (c+d x)\right ) \, dx}{6 a}\\ &=\frac {a (4 A+3 B) \cos (c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{3 d}-\frac {(2 A+B-C) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 d}+\frac {\int \cos (c+d x) (a+a \sec (c+d x))^2 \left (2 a^3 (22 A+18 B+3 C)-2 a^3 (8 A-3 B-18 C) \sec (c+d x)\right ) \, dx}{12 a}\\ &=\frac {a (4 A+3 B) \cos (c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{3 d}-\frac {(2 A+B-C) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 d}-\frac {(8 A-3 B-18 C) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {\int \cos (c+d x) (a+a \sec (c+d x)) \left (30 a^4 (2 A+B-C)+6 a^4 (2 A+8 B+13 C) \sec (c+d x)\right ) \, dx}{12 a}\\ &=\frac {5 a^4 (2 A+B-C) \sin (c+d x)}{2 d}+\frac {a (4 A+3 B) \cos (c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{3 d}-\frac {(2 A+B-C) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 d}-\frac {(8 A-3 B-18 C) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{6 d}-\frac {\int \left (-6 a^5 (12 A+13 B+8 C)-6 a^5 (2 A+8 B+13 C) \sec (c+d x)\right ) \, dx}{12 a}\\ &=\frac {1}{2} a^4 (12 A+13 B+8 C) x+\frac {5 a^4 (2 A+B-C) \sin (c+d x)}{2 d}+\frac {a (4 A+3 B) \cos (c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{3 d}-\frac {(2 A+B-C) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 d}-\frac {(8 A-3 B-18 C) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {1}{2} \left (a^4 (2 A+8 B+13 C)\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{2} a^4 (12 A+13 B+8 C) x+\frac {a^4 (2 A+8 B+13 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {5 a^4 (2 A+B-C) \sin (c+d x)}{2 d}+\frac {a (4 A+3 B) \cos (c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{3 d}-\frac {(2 A+B-C) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{2 d}-\frac {(8 A-3 B-18 C) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{6 d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1518\) vs. \(2(217)=434\).
time = 6.33, size = 1518, normalized size = 7.00 \begin {gather*} \frac {(12 A+13 B+8 C) x \cos ^6(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 )}{16 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(-2 A-8 B-13 C) \cos ^6(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\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 )}{16 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(2 A+8 B+13 C) \cos ^6(c+d x) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\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 )}{16 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(27 A+16 B+4 C) \cos (d x) \cos ^6(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 ) \sin (c)}{32 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(4 A+B) \cos (2 d x) \cos ^6(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 ) \sin (2 c)}{32 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {A \cos (3 d x) \cos ^6(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 ) \sin (3 c)}{96 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(27 A+16 B+4 C) \cos (c) \cos ^6(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 ) \sin (d x)}{32 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(4 A+B) \cos (2 c) \cos ^6(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 ) \sin (2 d x)}{32 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {A \cos (3 c) \cos ^6(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 ) \sin (3 d x)}{96 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {C \cos ^6(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 )}{32 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {\cos ^6(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 (B \sin \left (\frac {d x}{2}\right )+4 C \sin \left (\frac {d x}{2}\right )\right )}{8 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}-\frac {C \cos ^6(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 )}{32 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {\cos ^6(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 (B \sin \left (\frac {d x}{2}\right )+4 C \sin \left (\frac {d x}{2}\right )\right )}{8 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.98, size = 269, normalized size = 1.24 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 296, normalized size = 1.36 \begin {gather*} -\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 12 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 48 \, {\left (d x + c\right )} A a^{4} - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 72 \, {\left (d x + c\right )} B a^{4} - 48 \, {\left (d x + c\right )} C a^{4} + 3 \, C a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 6 \, A a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 24 \, B a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 36 \, C a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 72 \, A a^{4} \sin \left (d x + c\right ) - 48 \, B a^{4} \sin \left (d x + c\right ) - 12 \, C a^{4} \sin \left (d x + c\right ) - 12 \, B a^{4} \tan \left (d x + c\right ) - 48 \, C a^{4} \tan \left (d x + c\right )}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.01, size = 191, normalized size = 0.88 \begin {gather*} \frac {6 \, {\left (12 \, A + 13 \, B + 8 \, C\right )} a^{4} d x \cos \left (d x + c\right )^{2} + 3 \, {\left (2 \, A + 8 \, B + 13 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (2 \, A + 8 \, B + 13 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, A a^{4} \cos \left (d x + c\right )^{4} + 3 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{3} + 2 \, {\left (20 \, A + 12 \, B + 3 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 6 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right ) + 3 \, C a^{4}\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.56, size = 347, normalized size = 1.60 \begin {gather*} \frac {3 \, {\left (12 \, A a^{4} + 13 \, B a^{4} + 8 \, C a^{4}\right )} {\left (d x + c\right )} + 3 \, {\left (2 \, A a^{4} + 8 \, B a^{4} + 13 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (2 \, A a^{4} + 8 \, B a^{4} + 13 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {6 \, {\left (2 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9 \, 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 )}^{2}} + \frac {2 \, {\left (30 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 21 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 76 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 48 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 54 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 27 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, 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 )}^{3}}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.53, size = 373, normalized size = 1.72 \begin {gather*} \frac {2\,\left (6\,A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )-A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,1{}\mathrm {i}+\frac {13\,B\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2}-B\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,4{}\mathrm {i}+4\,C\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )-\frac {C\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,13{}\mathrm {i}}{2}\right )}{d}+\frac {\frac {A\,a^4\,\sin \left (2\,c+2\,d\,x\right )}{2}+\frac {83\,A\,a^4\,\sin \left (3\,c+3\,d\,x\right )}{48}+\frac {A\,a^4\,\sin \left (4\,c+4\,d\,x\right )}{4}+\frac {A\,a^4\,\sin \left (5\,c+5\,d\,x\right )}{48}+\frac {5\,B\,a^4\,\sin \left (2\,c+2\,d\,x\right )}{8}+B\,a^4\,\sin \left (3\,c+3\,d\,x\right )+\frac {B\,a^4\,\sin \left (4\,c+4\,d\,x\right )}{16}+2\,C\,a^4\,\sin \left (2\,c+2\,d\,x\right )+\frac {C\,a^4\,\sin \left (3\,c+3\,d\,x\right )}{4}+\frac {41\,A\,a^4\,\sin \left (c+d\,x\right )}{24}+B\,a^4\,\sin \left (c+d\,x\right )+\frac {3\,C\,a^4\,\sin \left (c+d\,x\right )}{4}}{d\,\left (\frac {\cos \left (2\,c+2\,d\,x\right )}{2}+\frac {1}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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