Optimal. Leaf size=66 \[ -\frac {(a C+b B) \cot (c+d x)}{d}-\frac {(a B-b C) \log (\sin (c+d x))}{d}-x (a C+b B)-\frac {a B \cot ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.16, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {3632, 3591, 3529, 3531, 3475} \[ -\frac {(a C+b B) \cot (c+d x)}{d}-\frac {(a B-b C) \log (\sin (c+d x))}{d}-x (a C+b B)-\frac {a B \cot ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3529
Rule 3531
Rule 3591
Rule 3632
Rubi steps
\begin {align*} \int \cot ^4(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 ^3(c+d x) (a+b \tan (c+d x)) (B+C \tan (c+d x)) \, dx\\ &=-\frac {a B \cot ^2(c+d x)}{2 d}+\int \cot ^2(c+d x) (b B+a C-(a B-b C) \tan (c+d x)) \, dx\\ &=-\frac {(b B+a C) \cot (c+d x)}{d}-\frac {a B \cot ^2(c+d x)}{2 d}+\int \cot (c+d x) (-a B+b C-(b B+a C) \tan (c+d x)) \, dx\\ &=-(b B+a C) x-\frac {(b B+a C) \cot (c+d x)}{d}-\frac {a B \cot ^2(c+d x)}{2 d}+(-a B+b C) \int \cot (c+d x) \, dx\\ &=-(b B+a C) x-\frac {(b B+a C) \cot (c+d x)}{d}-\frac {a B \cot ^2(c+d x)}{2 d}-\frac {(a B-b C) \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [C] time = 0.47, size = 77, normalized size = 1.17 \[ -\frac {2 (a C+b B) \cot (c+d x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2(c+d x)\right )+2 (a B-b C) (\log (\tan (c+d x))+\log (\cos (c+d x)))+a B \cot ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 95, normalized size = 1.44 \[ -\frac {{\left (B a - C b\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{2} + {\left (2 \, {\left (C a + B b\right )} d x + B a\right )} \tan \left (d x + c\right )^{2} + B a + 2 \, {\left (C a + B b\right )} \tan \left (d x + c\right )}{2 \, d \tan \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 5.64, size = 179, normalized size = 2.71 \[ -\frac {B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, {\left (C a + B b\right )} {\left (d x + c\right )} - 8 \, {\left (B a - C b\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) + 8 \, {\left (B a - C b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {12 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 96, normalized size = 1.45 \[ -\frac {a B \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a B \ln \left (\sin \left (d x +c \right )\right )}{d}-a C x -\frac {C \cot \left (d x +c \right ) a}{d}-\frac {C a c}{d}-B x b -\frac {B \cot \left (d x +c \right ) b}{d}-\frac {B b c}{d}+\frac {C b \ln \left (\sin \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 86, normalized size = 1.30 \[ -\frac {2 \, {\left (C a + B b\right )} {\left (d x + c\right )} - {\left (B a - C b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \, {\left (B a - C b\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {B a + 2 \, {\left (C a + B b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.94, size = 108, normalized size = 1.64 \[ -\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (B\,a-C\,b\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^2\,\left (\frac {B\,a}{2}+\mathrm {tan}\left (c+d\,x\right )\,\left (B\,b+C\,a\right )\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (B+C\,1{}\mathrm {i}\right )\,\left (a+b\,1{}\mathrm {i}\right )}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )\,\left (b+a\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.33, size = 150, normalized size = 2.27 \[ \begin {cases} \text {NaN} & \text {for}\: \left (c = 0 \vee c = - d x\right ) \wedge \left (c = - d x \vee d = 0\right ) \\x \left (a + b \tan {\relax (c )}\right ) \left (B \tan {\relax (c )} + C \tan ^{2}{\relax (c )}\right ) \cot ^{4}{\relax (c )} & \text {for}\: d = 0 \\\frac {B a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {B a \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {B a}{2 d \tan ^{2}{\left (c + d x \right )}} - B b x - \frac {B b}{d \tan {\left (c + d x \right )}} - C a x - \frac {C a}{d \tan {\left (c + d x \right )}} - \frac {C b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {C b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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