Optimal. Leaf size=157 \[ \frac{a \left (a^2-3 b^2\right ) \cot ^3(c+d x)}{3 d}+\frac{b \left (3 a^2-b^2\right ) \cot ^2(c+d x)}{2 d}-\frac{a \left (a^2-3 b^2\right ) \cot (c+d x)}{d}+\frac{b \left (3 a^2-b^2\right ) \log (\sin (c+d x))}{d}-a x \left (a^2-3 b^2\right )-\frac{11 a^2 b \cot ^4(c+d x)}{20 d}-\frac{a^2 \cot ^5(c+d x) (a+b \tan (c+d x))}{5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.277536, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3565, 3628, 3529, 3531, 3475} \[ \frac{a \left (a^2-3 b^2\right ) \cot ^3(c+d x)}{3 d}+\frac{b \left (3 a^2-b^2\right ) \cot ^2(c+d x)}{2 d}-\frac{a \left (a^2-3 b^2\right ) \cot (c+d x)}{d}+\frac{b \left (3 a^2-b^2\right ) \log (\sin (c+d x))}{d}-a x \left (a^2-3 b^2\right )-\frac{11 a^2 b \cot ^4(c+d x)}{20 d}-\frac{a^2 \cot ^5(c+d x) (a+b \tan (c+d x))}{5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3565
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 \, dx &=-\frac{a^2 \cot ^5(c+d x) (a+b \tan (c+d x))}{5 d}+\frac{1}{5} \int \cot ^5(c+d x) \left (11 a^2 b-5 a \left (a^2-3 b^2\right ) \tan (c+d x)-b \left (4 a^2-5 b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac{11 a^2 b \cot ^4(c+d x)}{20 d}-\frac{a^2 \cot ^5(c+d x) (a+b \tan (c+d x))}{5 d}+\frac{1}{5} \int \cot ^4(c+d x) \left (-5 a \left (a^2-3 b^2\right )-5 b \left (3 a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=\frac{a \left (a^2-3 b^2\right ) \cot ^3(c+d x)}{3 d}-\frac{11 a^2 b \cot ^4(c+d x)}{20 d}-\frac{a^2 \cot ^5(c+d x) (a+b \tan (c+d x))}{5 d}+\frac{1}{5} \int \cot ^3(c+d x) \left (-5 b \left (3 a^2-b^2\right )+5 a \left (a^2-3 b^2\right ) \tan (c+d x)\right ) \, dx\\ &=\frac{b \left (3 a^2-b^2\right ) \cot ^2(c+d x)}{2 d}+\frac{a \left (a^2-3 b^2\right ) \cot ^3(c+d x)}{3 d}-\frac{11 a^2 b \cot ^4(c+d x)}{20 d}-\frac{a^2 \cot ^5(c+d x) (a+b \tan (c+d x))}{5 d}+\frac{1}{5} \int \cot ^2(c+d x) \left (5 a \left (a^2-3 b^2\right )+5 b \left (3 a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=-\frac{a \left (a^2-3 b^2\right ) \cot (c+d x)}{d}+\frac{b \left (3 a^2-b^2\right ) \cot ^2(c+d x)}{2 d}+\frac{a \left (a^2-3 b^2\right ) \cot ^3(c+d x)}{3 d}-\frac{11 a^2 b \cot ^4(c+d x)}{20 d}-\frac{a^2 \cot ^5(c+d x) (a+b \tan (c+d x))}{5 d}+\frac{1}{5} \int \cot (c+d x) \left (5 b \left (3 a^2-b^2\right )-5 a \left (a^2-3 b^2\right ) \tan (c+d x)\right ) \, dx\\ &=-a \left (a^2-3 b^2\right ) x-\frac{a \left (a^2-3 b^2\right ) \cot (c+d x)}{d}+\frac{b \left (3 a^2-b^2\right ) \cot ^2(c+d x)}{2 d}+\frac{a \left (a^2-3 b^2\right ) \cot ^3(c+d x)}{3 d}-\frac{11 a^2 b \cot ^4(c+d x)}{20 d}-\frac{a^2 \cot ^5(c+d x) (a+b \tan (c+d x))}{5 d}+\left (b \left (3 a^2-b^2\right )\right ) \int \cot (c+d x) \, dx\\ &=-a \left (a^2-3 b^2\right ) x-\frac{a \left (a^2-3 b^2\right ) \cot (c+d x)}{d}+\frac{b \left (3 a^2-b^2\right ) \cot ^2(c+d x)}{2 d}+\frac{a \left (a^2-3 b^2\right ) \cot ^3(c+d x)}{3 d}-\frac{11 a^2 b \cot ^4(c+d x)}{20 d}+\frac{b \left (3 a^2-b^2\right ) \log (\sin (c+d x))}{d}-\frac{a^2 \cot ^5(c+d x) (a+b \tan (c+d x))}{5 d}\\ \end{align*}
Mathematica [C] time = 0.942461, size = 152, normalized size = 0.97 \[ -\frac{-\frac{1}{3} a \left (a^2-3 b^2\right ) \cot ^3(c+d x)-\frac{1}{2} b \left (3 a^2-b^2\right ) \cot ^2(c+d x)+a \left (a^2-3 b^2\right ) \cot (c+d x)+\frac{3}{4} a^2 b \cot ^4(c+d x)+\frac{1}{5} a^3 \cot ^5(c+d x)-\frac{1}{2} (b+i a)^3 \log (-\cot (c+d x)+i)+\frac{1}{2} (-b+i a)^3 \log (\cot (c+d x)+i)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.056, size = 193, normalized size = 1.2 \begin{align*} -{\frac{{b}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{{b}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{a{b}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{d}}+3\,a{b}^{2}x+3\,{\frac{a{b}^{2}\cot \left ( dx+c \right ) }{d}}+3\,{\frac{a{b}^{2}c}{d}}-{\frac{3\,b{a}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{3\,b{a}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+3\,{\frac{b{a}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{{a}^{3}\cot \left ( dx+c \right ) }{d}}-{a}^{3}x-{\frac{{a}^{3}c}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.61354, size = 213, normalized size = 1.36 \begin{align*} -\frac{60 \,{\left (a^{3} - 3 \, a b^{2}\right )}{\left (d x + c\right )} + 30 \,{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 \,{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac{60 \,{\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{4} + 45 \, a^{2} b \tan \left (d x + c\right ) - 30 \,{\left (3 \, a^{2} b - b^{3}\right )} \tan \left (d x + c\right )^{3} + 12 \, a^{3} - 20 \,{\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{5}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.82719, size = 412, normalized size = 2.62 \begin{align*} \frac{30 \,{\left (3 \, a^{2} b - 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 )^{5} + 15 \,{\left (9 \, a^{2} b - 2 \, b^{3} - 4 \,{\left (a^{3} - 3 \, a b^{2}\right )} d x\right )} \tan \left (d x + c\right )^{5} - 60 \,{\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{4} - 45 \, a^{2} b \tan \left (d x + c\right ) + 30 \,{\left (3 \, a^{2} b - b^{3}\right )} \tan \left (d x + c\right )^{3} - 12 \, a^{3} + 20 \,{\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{2}}{60 \, d \tan \left (d x + c\right )^{5}} \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 = 1.96325, size = 500, normalized size = 3.18 \begin{align*} \frac{6 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 45 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 70 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 120 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 540 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 120 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 660 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1800 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 960 \,{\left (a^{3} - 3 \, a b^{2}\right )}{\left (d x + c\right )} - 960 \,{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) + 960 \,{\left (3 \, a^{2} b - b^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - \frac{6576 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 2192 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 660 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 1800 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 540 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 120 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 70 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 120 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 45 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}}}{960 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]