Optimal. Leaf size=58 \[ \frac {a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {1}{2} a x (2 A+C)+\frac {a C \sin (c+d x)}{d}+\frac {a C \sin (c+d x) \cos (c+d x)}{2 d} \]
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Rubi [A] time = 0.11, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {3034, 3023, 2735, 3770} \[ \frac {a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {1}{2} a x (2 A+C)+\frac {a C \sin (c+d x)}{d}+\frac {a C \sin (c+d x) \cos (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 3023
Rule 3034
Rule 3770
Rubi steps
\begin {align*} \int (a+a \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx &=\frac {a C \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} \int \left (2 a A+a (2 A+C) \cos (c+d x)+2 a C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {a C \sin (c+d x)}{d}+\frac {a C \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} \int (2 a A+a (2 A+C) \cos (c+d x)) \sec (c+d x) \, dx\\ &=\frac {1}{2} a (2 A+C) x+\frac {a C \sin (c+d x)}{d}+\frac {a C \cos (c+d x) \sin (c+d x)}{2 d}+(a A) \int \sec (c+d x) \, dx\\ &=\frac {1}{2} a (2 A+C) x+\frac {a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a C \sin (c+d x)}{d}+\frac {a C \cos (c+d x) \sin (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 52, normalized size = 0.90 \[ \frac {a \left (4 A \tanh ^{-1}(\sin (c+d x))+4 A d x+4 C \sin (c+d x)+C \sin (2 (c+d x))+2 c C+2 C d x\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 63, normalized size = 1.09 \[ \frac {{\left (2 \, A + C\right )} a d x + A a \log \left (\sin \left (d x + c\right ) + 1\right ) - A a \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (C a \cos \left (d x + c\right ) + 2 \, C a\right )} \sin \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 99, normalized size = 1.71 \[ \frac {2 \, A a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 2 \, A a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + {\left (2 \, A a + C a\right )} {\left (d x + c\right )} + \frac {2 \, {\left (C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 77, normalized size = 1.33 \[ a A x +\frac {A a c}{d}+\frac {a C \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {a C x}{2}+\frac {a C c}{2 d}+\frac {a A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a C \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 63, normalized size = 1.09 \[ \frac {4 \, {\left (d x + c\right )} A a + {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a + 4 \, A a \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 4 \, C a \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.97, size = 115, normalized size = 1.98 \[ \frac {C\,a\,\sin \left (c+d\,x\right )}{d}+\frac {2\,A\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {2\,A\,a\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {C\,a\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {C\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d} \]
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 \[ a \left (\int A \sec {\left (c + d x \right )}\, dx + \int A \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int C \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int C \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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