Optimal. Leaf size=123 \[ -\frac {a^2 (3 A+2 B-2 C) \sin (c+d x)}{2 d}+\frac {a^2 (3 A+4 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {(A+B) \tan (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{d}+a^2 x (B+2 C)+\frac {A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d} \]
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Rubi [A] time = 0.39, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {3043, 2975, 2968, 3023, 2735, 3770} \[ -\frac {a^2 (3 A+2 B-2 C) \sin (c+d x)}{2 d}+\frac {a^2 (3 A+4 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {(A+B) \tan (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{d}+a^2 x (B+2 C)+\frac {A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2968
Rule 2975
Rule 3023
Rule 3043
Rule 3770
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx &=\frac {A (a+a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {\int (a+a \cos (c+d x))^2 (2 a (A+B)-a (A-2 C) \cos (c+d x)) \sec ^2(c+d x) \, dx}{2 a}\\ &=\frac {(A+B) \left (a^2+a^2 \cos (c+d x)\right ) \tan (c+d x)}{d}+\frac {A (a+a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {\int (a+a \cos (c+d x)) \left (a^2 (3 A+4 B+2 C)-a^2 (3 A+2 B-2 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{2 a}\\ &=\frac {(A+B) \left (a^2+a^2 \cos (c+d x)\right ) \tan (c+d x)}{d}+\frac {A (a+a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {\int \left (a^3 (3 A+4 B+2 C)+\left (-a^3 (3 A+2 B-2 C)+a^3 (3 A+4 B+2 C)\right ) \cos (c+d x)-a^3 (3 A+2 B-2 C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{2 a}\\ &=-\frac {a^2 (3 A+2 B-2 C) \sin (c+d x)}{2 d}+\frac {(A+B) \left (a^2+a^2 \cos (c+d x)\right ) \tan (c+d x)}{d}+\frac {A (a+a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {\int \left (a^3 (3 A+4 B+2 C)+2 a^3 (B+2 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{2 a}\\ &=a^2 (B+2 C) x-\frac {a^2 (3 A+2 B-2 C) \sin (c+d x)}{2 d}+\frac {(A+B) \left (a^2+a^2 \cos (c+d x)\right ) \tan (c+d x)}{d}+\frac {A (a+a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \left (a^2 (3 A+4 B+2 C)\right ) \int \sec (c+d x) \, dx\\ &=a^2 (B+2 C) x+\frac {a^2 (3 A+4 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {a^2 (3 A+2 B-2 C) \sin (c+d x)}{2 d}+\frac {(A+B) \left (a^2+a^2 \cos (c+d x)\right ) \tan (c+d x)}{d}+\frac {A (a+a \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d}\\ \end {align*}
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Mathematica [B] time = 1.46, size = 259, normalized size = 2.11 \[ \frac {a^2 (\cos (c+d x)+1)^2 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \left (-2 (3 A+4 B+2 C) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 (3 A+4 B+2 C) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+\frac {4 (2 A+B) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {4 (2 A+B) \sin \left (\frac {1}{2} (c+d x)\right )}{\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )}+\frac {A}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {A}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}+4 (B+2 C) (c+d x)+4 C \sin (c+d x)\right )}{16 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 143, normalized size = 1.16 \[ \frac {4 \, {\left (B + 2 \, C\right )} a^{2} d x \cos \left (d x + c\right )^{2} + {\left (3 \, A + 4 \, B + 2 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (3 \, A + 4 \, B + 2 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, C a^{2} \cos \left (d x + c\right )^{2} + 2 \, {\left (2 \, A + B\right )} a^{2} \cos \left (d x + c\right ) + A a^{2}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.85, size = 204, normalized size = 1.66 \[ \frac {\frac {4 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + 2 \, {\left (B a^{2} + 2 \, C a^{2}\right )} {\left (d x + c\right )} + {\left (3 \, A a^{2} + 4 \, B a^{2} + 2 \, C a^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (3 \, A a^{2} + 4 \, B a^{2} + 2 \, C a^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (3 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 166, normalized size = 1.35 \[ \frac {3 a^{2} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+a^{2} B x +\frac {B \,a^{2} c}{d}+\frac {a^{2} C \sin \left (d x +c \right )}{d}+\frac {2 a^{2} A \tan \left (d x +c \right )}{d}+\frac {2 B \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+2 a^{2} C x +\frac {2 a^{2} C c}{d}+\frac {a^{2} A \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {a^{2} B \tan \left (d x +c \right )}{d}+\frac {a^{2} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 192, normalized size = 1.56 \[ \frac {4 \, {\left (d x + c\right )} B a^{2} + 8 \, {\left (d x + c\right )} C a^{2} - A a^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, A a^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, B a^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, C a^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, C a^{2} \sin \left (d x + c\right ) + 8 \, A a^{2} \tan \left (d x + c\right ) + 4 \, B a^{2} \tan \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.95, size = 244, normalized size = 1.98 \[ \frac {A\,a^2\,\sin \left (2\,c+2\,d\,x\right )+\frac {B\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2}+\frac {C\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{4}+\frac {A\,a^2\,\sin \left (c+d\,x\right )}{2}+\frac {C\,a^2\,\sin \left (c+d\,x\right )}{4}}{d\,\left (\frac {\cos \left (2\,c+2\,d\,x\right )}{2}+\frac {1}{2}\right )}-\frac {2\,\left (\frac {A\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,3{}\mathrm {i}}{2}-B\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+B\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,2{}\mathrm {i}-2\,C\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+C\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,1{}\mathrm {i}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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