Optimal. Leaf size=71 \[ \frac {(2 A-B+C) \tan (c+d x)}{a d}-\frac {(A-B+C) \tan (c+d x)}{d (a \cos (c+d x)+a)}-\frac {(A-B) \tanh ^{-1}(\sin (c+d x))}{a d} \]
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Rubi [A] time = 0.17, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {3041, 2748, 3767, 8, 3770} \[ \frac {(2 A-B+C) \tan (c+d x)}{a d}-\frac {(A-B+C) \tan (c+d x)}{d (a \cos (c+d x)+a)}-\frac {(A-B) \tanh ^{-1}(\sin (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2748
Rule 3041
Rule 3767
Rule 3770
Rubi steps
\begin {align*} \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx &=-\frac {(A-B+C) \tan (c+d x)}{d (a+a \cos (c+d x))}+\frac {\int (a (2 A-B+C)-a (A-B) \cos (c+d x)) \sec ^2(c+d x) \, dx}{a^2}\\ &=-\frac {(A-B+C) \tan (c+d x)}{d (a+a \cos (c+d x))}-\frac {(A-B) \int \sec (c+d x) \, dx}{a}+\frac {(2 A-B+C) \int \sec ^2(c+d x) \, dx}{a}\\ &=-\frac {(A-B) \tanh ^{-1}(\sin (c+d x))}{a d}-\frac {(A-B+C) \tan (c+d x)}{d (a+a \cos (c+d x))}-\frac {(2 A-B+C) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{a d}\\ &=-\frac {(A-B) \tanh ^{-1}(\sin (c+d x))}{a d}+\frac {(2 A-B+C) \tan (c+d x)}{a d}-\frac {(A-B+C) \tan (c+d x)}{d (a+a \cos (c+d x))}\\ \end {align*}
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Mathematica [B] time = 1.39, size = 256, normalized size = 3.61 \[ \frac {4 \cos \left (\frac {1}{2} (c+d x)\right ) \cos ^2(c+d x) \left (A \sec ^2(c+d x)+B \sec (c+d x)+C\right ) \left (\sec \left (\frac {c}{2}\right ) (A-B+C) \sin \left (\frac {d x}{2}\right )+\cos \left (\frac {1}{2} (c+d x)\right ) \left ((A-B) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+\frac {A \sin (d x)}{\left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}\right )\right )}{a d (\cos (c+d x)+1) (2 A+2 B \cos (c+d x)+C \cos (2 (c+d x))+C)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 128, normalized size = 1.80 \[ -\frac {{\left ({\left (A - B\right )} \cos \left (d x + c\right )^{2} + {\left (A - B\right )} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left ({\left (A - B\right )} \cos \left (d x + c\right )^{2} + {\left (A - B\right )} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left ({\left (2 \, A - B + C\right )} \cos \left (d x + c\right ) + A\right )} \sin \left (d x + c\right )}{2 \, {\left (a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 121, normalized size = 1.70 \[ -\frac {\frac {{\left (A - B\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a} - \frac {{\left (A - B\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a} - \frac {A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a} + \frac {2 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 180, normalized size = 2.54 \[ \frac {A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {A}{a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a d}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) B}{a d}-\frac {A}{a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a d}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) B}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 218, normalized size = 3.07 \[ -\frac {A {\left (\frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} - \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} - \frac {2 \, \sin \left (d x + c\right )}{{\left (a - \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} - \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - B {\left (\frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} - \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} - \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - \frac {C \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 79, normalized size = 1.11 \[ \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A-B+C\right )}{a\,d}+\frac {2\,A\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (A-B\right )}{a\,d} \]
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 \[ \frac {\int \frac {A \sec ^{2}{\left (c + d x \right )}}{\cos {\left (c + d x \right )} + 1}\, dx + \int \frac {B \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\cos {\left (c + d x \right )} + 1}\, dx + \int \frac {C \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\cos {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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