Optimal. Leaf size=43 \[ \frac {(a C+b B) \log (\sin (c+d x))}{d}-(x (a B-b C))-\frac {a B \cot (c+d x)}{d} \]
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Rubi [A] time = 0.12, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3632, 3591, 3531, 3475} \[ \frac {(a C+b B) \log (\sin (c+d x))}{d}+x (-(a B-b C))-\frac {a B \cot (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3531
Rule 3591
Rule 3632
Rubi steps
\begin {align*} \int \cot ^3(c+d x) (a+b \tan (c+d x)) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx &=\int \cot ^2(c+d x) (a+b \tan (c+d x)) (B+C \tan (c+d x)) \, dx\\ &=-\frac {a B \cot (c+d x)}{d}+\int \cot (c+d x) (b B+a C-(a B-b C) \tan (c+d x)) \, dx\\ &=-(a B-b C) x-\frac {a B \cot (c+d x)}{d}+(b B+a C) \int \cot (c+d x) \, dx\\ &=-(a B-b C) x-\frac {a B \cot (c+d x)}{d}+\frac {(b B+a C) \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [C] time = 0.16, size = 78, normalized size = 1.81 \[ -\frac {a B \cot (c+d x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2(c+d x)\right )}{d}+\frac {a C (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+\frac {b B (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+b C x \]
Antiderivative was successfully verified.
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fricas [A] time = 1.55, size = 73, normalized size = 1.70 \[ -\frac {2 \, {\left (B a - C b\right )} d x \tan \left (d x + c\right ) - {\left (C a + B b\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right ) + 2 \, B a}{2 \, d \tan \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.31, size = 119, normalized size = 2.77 \[ \frac {B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, {\left (B a - C b\right )} {\left (d x + c\right )} - 2 \, {\left (C a + B b\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) + 2 \, {\left (C a + B b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + B a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.44, size = 65, normalized size = 1.51 \[ -a B x +b C x -\frac {a B \cot \left (d x +c \right )}{d}+\frac {B b \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {B a c}{d}+\frac {a C \ln \left (\sin \left (d x +c \right )\right )}{d}+\frac {C b c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 68, normalized size = 1.58 \[ -\frac {2 \, {\left (B a - C b\right )} {\left (d x + c\right )} + {\left (C a + B b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 \, {\left (C a + B b\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {2 \, B a}{\tan \left (d x + c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.87, size = 87, normalized size = 2.02 \[ \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (B\,b+C\,a\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )\,\left (b+a\,1{}\mathrm {i}\right )}{2\,d}-\frac {B\,a\,\mathrm {cot}\left (c+d\,x\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (B+C\,1{}\mathrm {i}\right )\,\left (a+b\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.66, size = 116, normalized size = 2.70 \[ \begin {cases} \text {NaN} & \text {for}\: c = 0 \wedge d = 0 \\x \left (a + b \tan {\relax (c )}\right ) \left (B \tan {\relax (c )} + C \tan ^{2}{\relax (c )}\right ) \cot ^{3}{\relax (c )} & \text {for}\: d = 0 \\\text {NaN} & \text {for}\: c = - d x \\- B a x - \frac {B a}{d \tan {\left (c + d x \right )}} - \frac {B b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {B b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {C a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {C a \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + C b x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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