Optimal. Leaf size=58 \[ \frac {a (A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a A \tan (c+d x)}{d}+\frac {a A \tan (c+d x) \sec (c+d x)}{2 d}+a C x \]
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Rubi [A] time = 0.12, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {3032, 3021, 2735, 3770} \[ \frac {a (A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a A \tan (c+d x)}{d}+\frac {a A \tan (c+d x) \sec (c+d x)}{2 d}+a C x \]
Antiderivative was successfully verified.
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Rule 2735
Rule 3021
Rule 3032
Rule 3770
Rubi steps
\begin {align*} \int (a+a \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx &=\frac {a A \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int \left (2 a A+a (A+2 C) \cos (c+d x)+2 a C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac {a A \tan (c+d x)}{d}+\frac {a A \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int (a (A+2 C)+2 a C \cos (c+d x)) \sec (c+d x) \, dx\\ &=a C x+\frac {a A \tan (c+d x)}{d}+\frac {a A \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} (a (A+2 C)) \int \sec (c+d x) \, dx\\ &=a C x+\frac {a (A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a A \tan (c+d x)}{d}+\frac {a A \sec (c+d x) \tan (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 67, normalized size = 1.16 \[ \frac {a A \tan (c+d x)}{d}+\frac {a A \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a A \tan (c+d x) \sec (c+d x)}{2 d}+\frac {a C \tanh ^{-1}(\sin (c+d x))}{d}+a C x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 101, normalized size = 1.74 \[ \frac {4 \, C a d x \cos \left (d x + c\right )^{2} + {\left (A + 2 \, C\right )} a \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (A + 2 \, C\right )} a \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, A a \cos \left (d x + c\right ) + A a\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 105, normalized size = 1.81 \[ \frac {2 \, {\left (d x + c\right )} C a + {\left (A a + 2 \, C a\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (A a + 2 \, C a\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 85, normalized size = 1.47 \[ \frac {a A \tan \left (d x +c \right )}{d}+a C x +\frac {a C c}{d}+\frac {a A \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {a A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {a C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 95, normalized size = 1.64 \[ \frac {4 \, {\left (d x + c\right )} C a - A a {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, C a {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, A a \tan \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.90, size = 128, normalized size = 2.21 \[ \frac {A\,a\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,C\,a\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {A\,a\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {A\,a\,\sin \left (c+d\,x\right )}{2\,d\,{\cos \left (c+d\,x\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a \left (\int A \sec ^{3}{\left (c + d x \right )}\, dx + \int A \cos {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int C \cos ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int C \cos ^{3}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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