Optimal. Leaf size=117 \[ -\frac{b \left (3 a^2 B+3 a A b-b^2 B\right ) \log (\cos (c+d x))}{d}+x \left (3 a^2 A b+a^3 B-3 a b^2 B-A b^3\right )+\frac{a^3 A \log (\sin (c+d x))}{d}+\frac{b^2 (2 a B+A b) \tan (c+d x)}{d}+\frac{b B (a+b \tan (c+d x))^2}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.27013, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {3607, 3637, 3624, 3475} \[ -\frac{b \left (3 a^2 B+3 a A b-b^2 B\right ) \log (\cos (c+d x))}{d}+x \left (3 a^2 A b+a^3 B-3 a b^2 B-A b^3\right )+\frac{a^3 A \log (\sin (c+d x))}{d}+\frac{b^2 (2 a B+A b) \tan (c+d x)}{d}+\frac{b B (a+b \tan (c+d x))^2}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3607
Rule 3637
Rule 3624
Rule 3475
Rubi steps
\begin{align*} \int \cot (c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx &=\frac{b B (a+b \tan (c+d x))^2}{2 d}+\frac{1}{2} \int \cot (c+d x) (a+b \tan (c+d x)) \left (2 a^2 A+2 \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)+2 b (A b+2 a B) \tan ^2(c+d x)\right ) \, dx\\ &=\frac{b^2 (A b+2 a B) \tan (c+d x)}{d}+\frac{b B (a+b \tan (c+d x))^2}{2 d}-\frac{1}{2} \int \cot (c+d x) \left (-2 a^3 A-2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)-2 b \left (3 a A b+3 a^2 B-b^2 B\right ) \tan ^2(c+d x)\right ) \, dx\\ &=\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) x+\frac{b^2 (A b+2 a B) \tan (c+d x)}{d}+\frac{b B (a+b \tan (c+d x))^2}{2 d}+\left (a^3 A\right ) \int \cot (c+d x) \, dx+\left (b \left (3 a A b+3 a^2 B-b^2 B\right )\right ) \int \tan (c+d x) \, dx\\ &=\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) x-\frac{b \left (3 a A b+3 a^2 B-b^2 B\right ) \log (\cos (c+d x))}{d}+\frac{a^3 A \log (\sin (c+d x))}{d}+\frac{b^2 (A b+2 a B) \tan (c+d x)}{d}+\frac{b B (a+b \tan (c+d x))^2}{2 d}\\ \end{align*}
Mathematica [C] time = 0.573259, size = 115, normalized size = 0.98 \[ \frac{2 a^3 A \log (\tan (c+d x))+2 b^2 (2 a B+A b) \tan (c+d x)-(a+i b)^3 (A+i B) \log (-\tan (c+d x)+i)-(a-i b)^3 (A-i B) \log (\tan (c+d x)+i)+b B (a+b \tan (c+d x))^2}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.08, size = 183, normalized size = 1.6 \begin{align*} -A{b}^{3}x+{\frac{A{b}^{3}\tan \left ( dx+c \right ) }{d}}-{\frac{A{b}^{3}c}{d}}+{\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}}-3\,{\frac{Aa{b}^{2}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}-3\,Ba{b}^{2}x+3\,{\frac{Ba{b}^{2}\tan \left ( dx+c \right ) }{d}}-3\,{\frac{Ba{b}^{2}c}{d}}+3\,Ax{a}^{2}b+3\,{\frac{A{a}^{2}bc}{d}}-3\,{\frac{B{a}^{2}b\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{A{a}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+B{a}^{3}x+{\frac{B{a}^{3}c}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.51568, size = 167, normalized size = 1.43 \begin{align*} \frac{B b^{3} \tan \left (d x + c\right )^{2} + 2 \, A a^{3} \log \left (\tan \left (d x + c\right )\right ) + 2 \,{\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )}{\left (d x + c\right )} -{\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \,{\left (3 \, B a b^{2} + A b^{3}\right )} \tan \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.14395, size = 305, normalized size = 2.61 \begin{align*} \frac{B b^{3} \tan \left (d x + c\right )^{2} + A a^{3} \log \left (\frac{\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + 2 \,{\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} d x -{\left (3 \, B a^{2} b + 3 \, A a b^{2} - B b^{3}\right )} \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 2 \,{\left (3 \, B a b^{2} + A b^{3}\right )} \tan \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.698, size = 204, normalized size = 1.74 \begin{align*} \begin{cases} - \frac{A a^{3} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{A a^{3} \log{\left (\tan{\left (c + d x \right )} \right )}}{d} + 3 A a^{2} b x + \frac{3 A a b^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - A b^{3} x + \frac{A b^{3} \tan{\left (c + d x \right )}}{d} + 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} & \text{for}\: d \neq 0 \\x \left (A + B \tan{\left (c \right )}\right ) \left (a + b \tan{\left (c \right )}\right )^{3} \cot{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.8813, size = 174, normalized size = 1.49 \begin{align*} \frac{B b^{3} \tan \left (d x + c\right )^{2} + 2 \, A a^{3} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) + 6 \, B a b^{2} \tan \left (d x + c\right ) + 2 \, A b^{3} \tan \left (d x + c\right ) + 2 \,{\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )}{\left (d x + c\right )} -{\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]