Optimal. Leaf size=191 \[ \frac{a \left (2 a^2 B-6 a b C-5 b^2 B\right ) \cot ^2(c+d x)}{4 d}+\frac{\left (3 a^2 b B+a^3 C-3 a b^2 C-b^3 B\right ) \cot (c+d x)}{d}+\frac{\left (-3 a^2 b C+a^3 B-3 a b^2 B+b^3 C\right ) \log (\sin (c+d x))}{d}+x \left (3 a^2 b B+a^3 C-3 a b^2 C-b^3 B\right )-\frac{a^2 (2 a C+3 b B) \cot ^3(c+d x)}{6 d}-\frac{a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.513793, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.175, Rules used = {3632, 3605, 3635, 3628, 3529, 3531, 3475} \[ \frac{a \left (2 a^2 B-6 a b C-5 b^2 B\right ) \cot ^2(c+d x)}{4 d}+\frac{\left (3 a^2 b B+a^3 C-3 a b^2 C-b^3 B\right ) \cot (c+d x)}{d}+\frac{\left (-3 a^2 b C+a^3 B-3 a b^2 B+b^3 C\right ) \log (\sin (c+d x))}{d}+x \left (3 a^2 b B+a^3 C-3 a b^2 C-b^3 B\right )-\frac{a^2 (2 a C+3 b B) \cot ^3(c+d x)}{6 d}-\frac{a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3632
Rule 3605
Rule 3635
Rule 3628
Rule 3529
Rule 3531
Rule 3475
Rubi steps
\begin{align*} \int \cot ^6(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 \cot ^5(c+d x) (a+b \tan (c+d x))^3 (B+C \tan (c+d x)) \, dx\\ &=-\frac{a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac{1}{4} \int \cot ^4(c+d x) (a+b \tan (c+d x)) \left (2 a (3 b B+2 a C)-4 \left (a^2 B-b^2 B-2 a b C\right ) \tan (c+d x)-2 b (a B-2 b C) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac{a^2 (3 b B+2 a C) \cot ^3(c+d x)}{6 d}-\frac{a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac{1}{4} \int \cot ^3(c+d x) \left (-2 a \left (2 a^2 B-5 b^2 B-6 a b C\right )-4 \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \tan (c+d x)-2 b^2 (a B-2 b C) \tan ^2(c+d x)\right ) \, dx\\ &=\frac{a \left (2 a^2 B-5 b^2 B-6 a b C\right ) \cot ^2(c+d x)}{4 d}-\frac{a^2 (3 b B+2 a C) \cot ^3(c+d x)}{6 d}-\frac{a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac{1}{4} \int \cot ^2(c+d x) \left (-4 \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right )+4 \left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) \tan (c+d x)\right ) \, dx\\ &=\frac{\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \cot (c+d x)}{d}+\frac{a \left (2 a^2 B-5 b^2 B-6 a b C\right ) \cot ^2(c+d x)}{4 d}-\frac{a^2 (3 b B+2 a C) \cot ^3(c+d x)}{6 d}-\frac{a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac{1}{4} \int \cot (c+d x) \left (4 \left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right )+4 \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \tan (c+d x)\right ) \, dx\\ &=\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) x+\frac{\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \cot (c+d x)}{d}+\frac{a \left (2 a^2 B-5 b^2 B-6 a b C\right ) \cot ^2(c+d x)}{4 d}-\frac{a^2 (3 b B+2 a C) \cot ^3(c+d x)}{6 d}-\frac{a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) \int \cot (c+d x) \, dx\\ &=\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) x+\frac{\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \cot (c+d x)}{d}+\frac{a \left (2 a^2 B-5 b^2 B-6 a b C\right ) \cot ^2(c+d x)}{4 d}-\frac{a^2 (3 b B+2 a C) \cot ^3(c+d x)}{6 d}+\frac{\left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) \log (\sin (c+d x))}{d}-\frac{a B \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}\\ \end{align*}
Mathematica [C] time = 0.74581, size = 199, normalized size = 1.04 \[ \frac{6 a \left (a^2 B-3 a b C-3 b^2 B\right ) \cot ^2(c+d x)+12 \left (3 a^2 b B+a^3 C-3 a b^2 C-b^3 B\right ) \cot (c+d x)+12 \left (-3 a^2 b C+a^3 B-3 a b^2 B+b^3 C\right ) \log (\tan (c+d x))-4 a^2 (a C+3 b B) \cot ^3(c+d x)-3 a^3 B \cot ^4(c+d x)-6 (a+i b)^3 (B+i C) \log (-\tan (c+d x)+i)-6 (a-i b)^3 (B-i C) \log (\tan (c+d x)+i)}{12 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.101, size = 302, normalized size = 1.6 \begin{align*} -Bx{b}^{3}-{\frac{B\cot \left ( dx+c \right ){b}^{3}}{d}}-{\frac{B{b}^{3}c}{d}}+{\frac{C{b}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{3\,Ba{b}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-3\,{\frac{Ba{b}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-3\,Ca{b}^{2}x-3\,{\frac{C\cot \left ( dx+c \right ) a{b}^{2}}{d}}-3\,{\frac{Ca{b}^{2}c}{d}}-{\frac{B{a}^{2}b \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{d}}+3\,B{a}^{2}bx+3\,{\frac{B\cot \left ( dx+c \right ){a}^{2}b}{d}}+3\,{\frac{B{a}^{2}bc}{d}}-{\frac{3\,C{a}^{2}b \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-3\,{\frac{C{a}^{2}b\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{B{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{B{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{B{a}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{C{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{C\cot \left ( dx+c \right ){a}^{3}}{d}}+Cx{a}^{3}+{\frac{C{a}^{3}c}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.71969, size = 290, normalized size = 1.52 \begin{align*} \frac{12 \,{\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )}{\left (d x + c\right )} - 6 \,{\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 12 \,{\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac{3 \, B a^{3} - 12 \,{\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \tan \left (d x + c\right )^{3} - 6 \,{\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2}\right )} \tan \left (d x + c\right )^{2} + 4 \,{\left (C a^{3} + 3 \, B a^{2} b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.14495, size = 518, normalized size = 2.71 \begin{align*} \frac{6 \,{\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\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 )^{4} + 3 \,{\left (3 \, B a^{3} - 6 \, C a^{2} b - 6 \, B a b^{2} + 4 \,{\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{4} - 3 \, B a^{3} + 12 \,{\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \tan \left (d x + c\right )^{3} + 6 \,{\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2}\right )} \tan \left (d x + c\right )^{2} - 4 \,{\left (C a^{3} + 3 \, B a^{2} b\right )} \tan \left (d x + c\right )}{12 \, d \tan \left (d x + c\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 2.83771, size = 713, normalized size = 3.73 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]