Optimal. Leaf size=118 \[ \frac{\left (a^2 A-2 a b B-A b^2\right ) \cot (c+d x)}{d}+x \left (a^2 A-2 a b B-A b^2\right )-\frac{a^2 A \cot ^3(c+d x)}{3 d}+\frac{\left (b^2 B-a (a B+2 A b)\right ) \log (\sin (c+d x))}{d}-\frac{a (a B+2 A b) \cot ^2(c+d x)}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.242897, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {3604, 3628, 3529, 3531, 3475} \[ \frac{\left (a^2 A-2 a b B-A b^2\right ) \cot (c+d x)}{d}+x \left (a^2 A-2 a b B-A b^2\right )-\frac{a^2 A \cot ^3(c+d x)}{3 d}+\frac{\left (b^2 B-a (a B+2 A b)\right ) \log (\sin (c+d x))}{d}-\frac{a (a B+2 A b) \cot ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3604
Rule 3628
Rule 3529
Rule 3531
Rule 3475
Rubi steps
\begin{align*} \int \cot ^4(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx &=-\frac{a^2 A \cot ^3(c+d x)}{3 d}+\int \cot ^3(c+d x) \left (a (2 A b+a B)-\left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)+b^2 B \tan ^2(c+d x)\right ) \, dx\\ &=-\frac{a (2 A b+a B) \cot ^2(c+d x)}{2 d}-\frac{a^2 A \cot ^3(c+d x)}{3 d}+\int \cot ^2(c+d x) \left (-a^2 A+A b^2+2 a b B+\left (b^2 B-a (2 A b+a B)\right ) \tan (c+d x)\right ) \, dx\\ &=\frac{\left (a^2 A-A b^2-2 a b B\right ) \cot (c+d x)}{d}-\frac{a (2 A b+a B) \cot ^2(c+d x)}{2 d}-\frac{a^2 A \cot ^3(c+d x)}{3 d}+\int \cot (c+d x) \left (b^2 B-a (2 A b+a B)+\left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)\right ) \, dx\\ &=\left (a^2 A-A b^2-2 a b B\right ) x+\frac{\left (a^2 A-A b^2-2 a b B\right ) \cot (c+d x)}{d}-\frac{a (2 A b+a B) \cot ^2(c+d x)}{2 d}-\frac{a^2 A \cot ^3(c+d x)}{3 d}+\left (b^2 B-a (2 A b+a B)\right ) \int \cot (c+d x) \, dx\\ &=\left (a^2 A-A b^2-2 a b B\right ) x+\frac{\left (a^2 A-A b^2-2 a b B\right ) \cot (c+d x)}{d}-\frac{a (2 A b+a B) \cot ^2(c+d x)}{2 d}-\frac{a^2 A \cot ^3(c+d x)}{3 d}+\frac{\left (b^2 B-a (2 A b+a B)\right ) \log (\sin (c+d x))}{d}\\ \end{align*}
Mathematica [C] time = 1.38635, size = 152, normalized size = 1.29 \[ \frac{6 \left (a^2 A-2 a b B-A b^2\right ) \cot (c+d x)-6 \left (a^2 B+2 a A b-b^2 B\right ) \log (\tan (c+d x))-2 a^2 A \cot ^3(c+d x)-3 a (a B+2 A b) \cot ^2(c+d x)+3 (a+i b)^2 (B-i A) \log (-\tan (c+d x)+i)+3 (a-i b)^2 (B+i A) \log (\tan (c+d x)+i)}{6 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.072, size = 188, normalized size = 1.6 \begin{align*} -A{b}^{2}x-{\frac{A\cot \left ( dx+c \right ){b}^{2}}{d}}-{\frac{A{b}^{2}c}{d}}+{\frac{{b}^{2}B\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{Aab \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}-2\,{\frac{Aab\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-2\,Babx-2\,{\frac{B\cot \left ( dx+c \right ) ab}{d}}-2\,{\frac{Babc}{d}}-{\frac{{a}^{2}A \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{{a}^{2}A\cot \left ( dx+c \right ) }{d}}+{a}^{2}Ax+{\frac{{a}^{2}Ac}{d}}-{\frac{{a}^{2}B \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{{a}^{2}B\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.46717, size = 201, normalized size = 1.7 \begin{align*} \frac{6 \,{\left (A a^{2} - 2 \, B a b - A b^{2}\right )}{\left (d x + c\right )} + 3 \,{\left (B a^{2} + 2 \, A a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 \,{\left (B a^{2} + 2 \, A a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac{2 \, A a^{2} - 6 \,{\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.03331, size = 367, normalized size = 3.11 \begin{align*} -\frac{3 \,{\left (B a^{2} + 2 \, A a b - B b^{2}\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 )^{3} + 3 \,{\left (B a^{2} + 2 \, A a b - 2 \,{\left (A a^{2} - 2 \, B a b - A b^{2}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 2 \, A a^{2} - 6 \,{\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} \tan \left (d x + c\right )}{6 \, d \tan \left (d x + c\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 7.87257, size = 260, normalized size = 2.2 \begin{align*} \begin{cases} \tilde{\infty } A a^{2} x & \text{for}\: \left (c = 0 \vee c = - d x\right ) \wedge \left (c = - d x \vee d = 0\right ) \\x \left (A + B \tan{\left (c \right )}\right ) \left (a + b \tan{\left (c \right )}\right )^{2} \cot ^{4}{\left (c \right )} & \text{for}\: d = 0 \\A a^{2} x + \frac{A a^{2}}{d \tan{\left (c + d x \right )}} - \frac{A a^{2}}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac{A a b \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - \frac{2 A a b \log{\left (\tan{\left (c + d x \right )} \right )}}{d} - \frac{A a b}{d \tan ^{2}{\left (c + d x \right )}} - A b^{2} x - \frac{A b^{2}}{d \tan{\left (c + d x \right )}} + \frac{B a^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac{B a^{2} \log{\left (\tan{\left (c + d x \right )} \right )}}{d} - \frac{B a^{2}}{2 d \tan ^{2}{\left (c + d x \right )}} - 2 B a b x - \frac{2 B a b}{d \tan{\left (c + d x \right )}} - \frac{B b^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{B b^{2} \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.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.49365, size = 451, normalized size = 3.82 \begin{align*} \frac{A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 6 \, A a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 24 \, B a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 12 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 24 \,{\left (A a^{2} - 2 \, B a b - A b^{2}\right )}{\left (d x + c\right )} + 24 \,{\left (B a^{2} + 2 \, A a b - B b^{2}\right )} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) - 24 \,{\left (B a^{2} + 2 \, A a b - B b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + \frac{44 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 88 \, A a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 44 \, B b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 15 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 24 \, B a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 12 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 6 \, A a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - A a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]