Optimal. Leaf size=60 \[ -\frac {(B-C) x}{a}+\frac {(2 B-C) \sin (c+d x)}{a d}-\frac {(B-C) \sin (c+d x)}{d (a+a \sec (c+d x))} \]
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Rubi [A]
time = 0.14, 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 = {4157, 4105,
3872, 2717, 8} \begin {gather*} \frac {(2 B-C) \sin (c+d x)}{a d}-\frac {(B-C) \sin (c+d x)}{d (a \sec (c+d x)+a)}-\frac {x (B-C)}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2717
Rule 3872
Rule 4105
Rule 4157
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx &=\int \frac {\cos (c+d x) (B+C \sec (c+d x))}{a+a \sec (c+d x)} \, dx\\ &=-\frac {(B-C) \sin (c+d x)}{d (a+a \sec (c+d x))}+\frac {\int \cos (c+d x) (a (2 B-C)-a (B-C) \sec (c+d x)) \, dx}{a^2}\\ &=-\frac {(B-C) \sin (c+d x)}{d (a+a \sec (c+d x))}-\frac {(B-C) \int 1 \, dx}{a}+\frac {(2 B-C) \int \cos (c+d x) \, dx}{a}\\ &=-\frac {(B-C) x}{a}+\frac {(2 B-C) \sin (c+d x)}{a d}-\frac {(B-C) \sin (c+d x)}{d (a+a \sec (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 76, normalized size = 1.27 \begin {gather*} \frac {2 \cos \left (\frac {1}{2} (c+d x)\right ) \left ((B-C) \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )+\cos \left (\frac {1}{2} (c+d x)\right ) ((-B+C) d x+B \sin (c+d x))\right )}{a d (1+\cos (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.87, size = 76, normalized size = 1.27
method | result | size |
derivativedivides | \(\frac {B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}-2 \left (B -C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\) | \(76\) |
default | \(\frac {B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}-2 \left (B -C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\) | \(76\) |
risch | \(-\frac {B x}{a}+\frac {x C}{a}-\frac {i B \,{\mathrm e}^{i \left (d x +c \right )}}{2 a d}+\frac {i B \,{\mathrm e}^{-i \left (d x +c \right )}}{2 a d}+\frac {2 i B}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}-\frac {2 i C}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}\) | \(99\) |
norman | \(\frac {\frac {\left (B -C \right ) x}{a}+\frac {\left (B -C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}+\frac {\left (B -C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {\left (3 B -C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {\left (B -C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {\left (B -C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {\left (B -C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {\left (3 B -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(204\) |
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. 143 vs.
\(2 (60) = 120\).
time = 0.50, size = 143, normalized size = 2.38 \begin {gather*} -\frac {B {\left (\frac {2 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 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 )} - C {\left (\frac {2 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.43, size = 63, normalized size = 1.05 \begin {gather*} -\frac {{\left (B - C\right )} d x \cos \left (d x + c\right ) + {\left (B - C\right )} d x - {\left (B \cos \left (d x + c\right ) + 2 \, B - C\right )} \sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + a d} \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*} \frac {\int \frac {B \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 79, normalized size = 1.32 \begin {gather*} -\frac {\frac {{\left (d x + c\right )} {\left (B - C\right )}}{a} - \frac {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 \, B \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.91, size = 65, normalized size = 1.08 \begin {gather*} \frac {2\,B\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )}-\frac {x\,\left (B-C\right )}{a}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (B-C\right )}{a\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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