Optimal. Leaf size=32 \[ a B x+\frac {a (A+B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a A \tan (c+d x)}{d} \]
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Rubi [A]
time = 0.07, antiderivative size = 32, 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 = {3047, 3100,
2814, 3855} \begin {gather*} \frac {a (A+B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a A \tan (c+d x)}{d}+a B x \end {gather*}
Antiderivative was successfully verified.
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Rule 2814
Rule 3047
Rule 3100
Rule 3855
Rubi steps
\begin {align*} \int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx &=\int \left (a A+(a A+a B) \cos (c+d x)+a B \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac {a A \tan (c+d x)}{d}+\int (a (A+B)+a B \cos (c+d x)) \sec (c+d x) \, dx\\ &=a B x+\frac {a A \tan (c+d x)}{d}+(a (A+B)) \int \sec (c+d x) \, dx\\ &=a B x+\frac {a (A+B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a A \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 43, normalized size = 1.34 \begin {gather*} a B x+\frac {a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a B \tanh ^{-1}(\sin (c+d x))}{d}+\frac {a A \tan (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 57, normalized size = 1.78
method | result | size |
derivativedivides | \(\frac {a A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a B \left (d x +c \right )+a A \tan \left (d x +c \right )+a B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(57\) |
default | \(\frac {a A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a B \left (d x +c \right )+a A \tan \left (d x +c \right )+a B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(57\) |
risch | \(a B x +\frac {2 i a A}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {a A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{d}-\frac {a A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{d}\) | \(105\) |
norman | \(\frac {a B x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a B x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a B x -\frac {2 a A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {4 a A \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 a A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-a B x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (A +B \right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {\left (A +B \right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(178\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 73 vs.
\(2 (32) = 64\).
time = 0.27, size = 73, normalized size = 2.28 \begin {gather*} \frac {2 \, {\left (d x + c\right )} B a + A a {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + B a {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, A a \tan \left (d x + c\right )}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 79 vs.
\(2 (32) = 64\).
time = 0.36, size = 79, normalized size = 2.47 \begin {gather*} \frac {2 \, B a d x \cos \left (d x + c\right ) + {\left (A + B\right )} a \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (A + B\right )} a \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, A a \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \end {gather*}
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 \begin {gather*} a \left (\int A \sec ^{2}{\left (c + d x \right )}\, dx + \int A \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int B \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int B \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 84 vs.
\(2 (32) = 64\).
time = 0.48, size = 84, normalized size = 2.62 \begin {gather*} \frac {{\left (d x + c\right )} B a + {\left (A a + B a\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (A a + B a\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, A a \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}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.31, size = 100, normalized size = 3.12 \begin {gather*} \frac {A\,a\,\mathrm {tan}\left (c+d\,x\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 {2\,B\,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\,B\,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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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