Optimal. Leaf size=60 \[ \frac {a^2 B \tan (c+d x)}{d}+\frac {a (a C+2 b B) \tanh ^{-1}(\sin (c+d x))}{d}+b x (2 a C+b B)+\frac {b^2 C \sin (c+d x)}{d} \]
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Rubi [A] time = 0.24, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3029, 2988, 3023, 2735, 3770} \[ \frac {a^2 B \tan (c+d x)}{d}+\frac {a (a C+2 b B) \tanh ^{-1}(\sin (c+d x))}{d}+b x (2 a C+b B)+\frac {b^2 C \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2988
Rule 3023
Rule 3029
Rule 3770
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx &=\int (a+b \cos (c+d x))^2 (B+C \cos (c+d x)) \sec ^2(c+d x) \, dx\\ &=\frac {a^2 B \tan (c+d x)}{d}-\int \left (-a (2 b B+a C)-b (b B+2 a C) \cos (c+d x)-b^2 C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {b^2 C \sin (c+d x)}{d}+\frac {a^2 B \tan (c+d x)}{d}-\int (-a (2 b B+a C)-b (b B+2 a C) \cos (c+d x)) \sec (c+d x) \, dx\\ &=b (b B+2 a C) x+\frac {b^2 C \sin (c+d x)}{d}+\frac {a^2 B \tan (c+d x)}{d}+(a (2 b B+a C)) \int \sec (c+d x) \, dx\\ &=b (b B+2 a C) x+\frac {a (2 b B+a C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b^2 C \sin (c+d x)}{d}+\frac {a^2 B \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.48, size = 109, normalized size = 1.82 \[ \frac {a^2 B \tan (c+d x)+b (c+d x) (2 a C+b B)-a (a C+2 b B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+a (a C+2 b B) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+b^2 C \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 117, normalized size = 1.95 \[ \frac {2 \, {\left (2 \, C a b + B b^{2}\right )} d x \cos \left (d x + c\right ) + {\left (C a^{2} + 2 \, B a b\right )} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (C a^{2} + 2 \, B a b\right )} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (C b^{2} \cos \left (d x + c\right ) + B a^{2}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.38, size = 152, normalized size = 2.53 \[ \frac {{\left (2 \, C a b + B b^{2}\right )} {\left (d x + c\right )} + {\left (C a^{2} + 2 \, B a b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (C a^{2} + 2 \, B a b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 104, normalized size = 1.73 \[ b^{2} B x +2 a b C x +\frac {2 B a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a^{2} B \tan \left (d x +c \right )}{d}+\frac {B \,b^{2} c}{d}+\frac {a^{2} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {b^{2} C \sin \left (d x +c \right )}{d}+\frac {2 C a b c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 103, normalized size = 1.72 \[ \frac {4 \, {\left (d x + c\right )} C a b + 2 \, {\left (d x + c\right )} B b^{2} + C a^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, B a b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, C b^{2} \sin \left (d x + c\right ) + 2 \, B a^{2} \tan \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.25, size = 169, normalized size = 2.82 \[ \frac {B\,a^2\,\mathrm {tan}\left (c+d\,x\right )}{d}+\frac {2\,B\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {C\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d\,\cos \left (c+d\,x\right )}+\frac {4\,C\,a\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {C\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,2{}\mathrm {i}}{d}-\frac {B\,a\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,4{}\mathrm {i}}{d} \]
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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