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 = 0.159387, antiderivative size = 66, normalized size of antiderivative = 1., 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 3632
Rule 3591
Rule 3529
Rule 3531
Rule 3475
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*}
Mathematica [C] time = 0.435101, size = 77, normalized size = 1.17 \[ -\frac{2 (a C+b B) \cot (c+d x) \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-\tan ^2(c+d x)\right )+2 (a B-b C) (\log (\tan (c+d x))+\log (\cos (c+d x)))+a B \cot ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.076, size = 96, normalized size = 1.5 \begin{align*} -Bxb-{\frac{B\cot \left ( dx+c \right ) b}{d}}-{\frac{Bbc}{d}}+{\frac{Cb\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{aB \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{aB\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-Cxa-{\frac{C\cot \left ( dx+c \right ) a}{d}}-{\frac{Cac}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.70778, size = 116, normalized size = 1.76 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39201, size = 234, normalized size = 3.55 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 14.0008, size = 143, normalized size = 2.17 \begin{align*} \begin{cases} \text{NaN} & \text{for}\: c = 0 \wedge d = 0 \\x \left (a + b \tan{\left (c \right )}\right ) \left (B \tan{\left (c \right )} + C \tan ^{2}{\left (c \right )}\right ) \cot ^{4}{\left (c \right )} & \text{for}\: d = 0 \\\text{NaN} & \text{for}\: c = - d x \\\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.55907, size = 242, normalized size = 3.67 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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