Optimal. Leaf size=140 \[ \frac{b \left (a^2 C+2 a b B-b^2 C\right ) \tan (c+d x)}{d}-\frac{\left (3 a^2 b B+a^3 C-3 a b^2 C-b^3 B\right ) \log (\cos (c+d x))}{d}+x \left (-3 a^2 b C+a^3 B-3 a b^2 B+b^3 C\right )+\frac{(a C+b B) (a+b \tan (c+d x))^2}{2 d}+\frac{C (a+b \tan (c+d x))^3}{3 d} \]
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Rubi [A] time = 0.207881, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3632, 3528, 3525, 3475} \[ \frac{b \left (a^2 C+2 a b B-b^2 C\right ) \tan (c+d x)}{d}-\frac{\left (3 a^2 b B+a^3 C-3 a b^2 C-b^3 B\right ) \log (\cos (c+d x))}{d}+x \left (-3 a^2 b C+a^3 B-3 a b^2 B+b^3 C\right )+\frac{(a C+b B) (a+b \tan (c+d x))^2}{2 d}+\frac{C (a+b \tan (c+d x))^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 3632
Rule 3528
Rule 3525
Rule 3475
Rubi steps
\begin{align*} \int \cot (c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx &=\int (a+b \tan (c+d x))^3 (B+C \tan (c+d x)) \, dx\\ &=\frac{C (a+b \tan (c+d x))^3}{3 d}+\int (a+b \tan (c+d x))^2 (a B-b C+(b B+a C) \tan (c+d x)) \, dx\\ &=\frac{(b B+a C) (a+b \tan (c+d x))^2}{2 d}+\frac{C (a+b \tan (c+d x))^3}{3 d}+\int (a+b \tan (c+d x)) \left (a^2 B-b^2 B-2 a b C+\left (2 a b B+a^2 C-b^2 C\right ) \tan (c+d x)\right ) \, dx\\ &=\left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) x+\frac{b \left (2 a b B+a^2 C-b^2 C\right ) \tan (c+d x)}{d}+\frac{(b B+a C) (a+b \tan (c+d x))^2}{2 d}+\frac{C (a+b \tan (c+d x))^3}{3 d}+\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \int \tan (c+d x) \, dx\\ &=\left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) x-\frac{\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \log (\cos (c+d x))}{d}+\frac{b \left (2 a b B+a^2 C-b^2 C\right ) \tan (c+d x)}{d}+\frac{(b B+a C) (a+b \tan (c+d x))^2}{2 d}+\frac{C (a+b \tan (c+d x))^3}{3 d}\\ \end{align*}
Mathematica [C] time = 1.05835, size = 130, normalized size = 0.93 \[ \frac{6 b \left (3 a^2 C+3 a b B-b^2 C\right ) \tan (c+d x)+3 b^2 (3 a C+b B) \tan ^2(c+d x)+3 (a-i b)^3 (C+i B) \log (\tan (c+d x)+i)+3 (a+i b)^3 (C-i B) \log (-\tan (c+d x)+i)+2 b^3 C \tan ^3(c+d x)}{6 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.089, size = 234, normalized size = 1.7 \begin{align*}{\frac{B{b}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{B{b}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{C{b}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{C{b}^{3}\tan \left ( dx+c \right ) }{d}}+C{b}^{3}x+{\frac{C{b}^{3}c}{d}}-3\,Ba{b}^{2}x+3\,{\frac{B\tan \left ( dx+c \right ) a{b}^{2}}{d}}-3\,{\frac{Ba{b}^{2}c}{d}}+{\frac{3\,Ca{b}^{2} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+3\,{\frac{Ca{b}^{2}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}-3\,{\frac{B{a}^{2}b\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}-3\,Cx{a}^{2}b+3\,{\frac{C\tan \left ( dx+c \right ){a}^{2}b}{d}}-3\,{\frac{C{a}^{2}bc}{d}}+B{a}^{3}x+{\frac{B{a}^{3}c}{d}}-{\frac{C{a}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.76893, size = 193, normalized size = 1.38 \begin{align*} \frac{2 \, C b^{3} \tan \left (d x + c\right )^{3} + 3 \,{\left (3 \, C a b^{2} + B b^{3}\right )} \tan \left (d x + c\right )^{2} + 6 \,{\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )}{\left (d x + c\right )} + 3 \,{\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 6 \,{\left (3 \, C a^{2} b + 3 \, B a b^{2} - C b^{3}\right )} \tan \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64078, size = 324, normalized size = 2.31 \begin{align*} \frac{2 \, C b^{3} \tan \left (d x + c\right )^{3} + 6 \,{\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} d x + 3 \,{\left (3 \, C a b^{2} + B b^{3}\right )} \tan \left (d x + c\right )^{2} - 3 \,{\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 6 \,{\left (3 \, C a^{2} b + 3 \, B a b^{2} - C b^{3}\right )} \tan \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.4206, size = 248, normalized size = 1.77 \begin{align*} \begin{cases} B a^{3} x + \frac{3 B a^{2} b \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - 3 B a b^{2} x + \frac{3 B a b^{2} \tan{\left (c + d x \right )}}{d} - \frac{B b^{3} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{B b^{3} \tan ^{2}{\left (c + d x \right )}}{2 d} + \frac{C a^{3} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - 3 C a^{2} b x + \frac{3 C a^{2} b \tan{\left (c + d x \right )}}{d} - \frac{3 C a b^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{3 C a b^{2} \tan ^{2}{\left (c + d x \right )}}{2 d} + C b^{3} x + \frac{C b^{3} \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac{C b^{3} \tan{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \tan{\left (c \right )}\right )^{3} \left (B \tan{\left (c \right )} + C \tan ^{2}{\left (c \right )}\right ) \cot{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.37083, size = 213, normalized size = 1.52 \begin{align*} \frac{2 \, C b^{3} \tan \left (d x + c\right )^{3} + 9 \, C a b^{2} \tan \left (d x + c\right )^{2} + 3 \, B b^{3} \tan \left (d x + c\right )^{2} + 18 \, C a^{2} b \tan \left (d x + c\right ) + 18 \, B a b^{2} \tan \left (d x + c\right ) - 6 \, C b^{3} \tan \left (d x + c\right ) + 6 \,{\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )}{\left (d x + c\right )} + 3 \,{\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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