Optimal. Leaf size=250 \[ \frac{a (A b-a B) \tan ^2(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac{a^2 \left (2 A b^3-a B \left (a^2+3 b^2\right )\right )}{b^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac{a \left (a^2 A b^3+3 a^3 b^2 B+a^5 B+6 a b^4 B-3 A b^5\right ) \log (a+b \tan (c+d x))}{b^3 d \left (a^2+b^2\right )^3}+\frac{\left (a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^3}-\frac{x \left (3 a^2 A b+a^3 (-B)+3 a b^2 B-A b^3\right )}{\left (a^2+b^2\right )^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.492649, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {3605, 3635, 3626, 3617, 31, 3475} \[ \frac{a (A b-a B) \tan ^2(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac{a^2 \left (2 A b^3-a B \left (a^2+3 b^2\right )\right )}{b^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac{a \left (a^2 A b^3+3 a^3 b^2 B+a^5 B+6 a b^4 B-3 A b^5\right ) \log (a+b \tan (c+d x))}{b^3 d \left (a^2+b^2\right )^3}+\frac{\left (a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^3}-\frac{x \left (3 a^2 A b+a^3 (-B)+3 a b^2 B-A b^3\right )}{\left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3605
Rule 3635
Rule 3626
Rule 3617
Rule 31
Rule 3475
Rubi steps
\begin{align*} \int \frac{\tan ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx &=\frac{a (A b-a B) \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{\int \frac{\tan (c+d x) \left (-2 a (A b-a B)+2 b (A b-a B) \tan (c+d x)+2 \left (a^2+b^2\right ) B \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 b \left (a^2+b^2\right )}\\ &=\frac{a (A b-a B) \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac{a^2 \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right )}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac{\int \frac{-2 a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right )-2 b^2 \left (a^2 A-A b^2+2 a b B\right ) \tan (c+d x)+2 \left (a^2+b^2\right )^2 B \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{2 b^2 \left (a^2+b^2\right )^2}\\ &=-\frac{\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) x}{\left (a^2+b^2\right )^3}+\frac{a (A b-a B) \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac{a^2 \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right )}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac{\left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right ) \int \tan (c+d x) \, dx}{\left (a^2+b^2\right )^3}+\frac{\left (a \left (a^2 A b^3-3 A b^5+a^5 B+3 a^3 b^2 B+6 a b^4 B\right )\right ) \int \frac{1+\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b^2 \left (a^2+b^2\right )^3}\\ &=-\frac{\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) x}{\left (a^2+b^2\right )^3}+\frac{\left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac{a (A b-a B) \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac{a^2 \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right )}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac{\left (a \left (a^2 A b^3-3 A b^5+a^5 B+3 a^3 b^2 B+6 a b^4 B\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^3 \left (a^2+b^2\right )^3 d}\\ &=-\frac{\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) x}{\left (a^2+b^2\right )^3}+\frac{\left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac{a \left (a^2 A b^3-3 A b^5+a^5 B+3 a^3 b^2 B+6 a b^4 B\right ) \log (a+b \tan (c+d x))}{b^3 \left (a^2+b^2\right )^3 d}+\frac{a (A b-a B) \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac{a^2 \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right )}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end{align*}
Mathematica [C] time = 4.49186, size = 462, normalized size = 1.85 \[ \frac{\sec ^2(c+d x) (A+B \tan (c+d x)) (a \cos (c+d x)+b \sin (c+d x)) \left (2 i a (c+d x) \left (a^2 A b^3+3 a^3 b^2 B+a^5 B+6 a b^4 B-3 A b^5\right ) (a \cos (c+d x)+b \sin (c+d x))^2-2 a b \left (a^2+b^2\right ) \left (a B \left (a^2+4 b^2\right )-3 A b^3\right ) \sin (c+d x) (a \cos (c+d x)+b \sin (c+d x))+2 b^3 (c+d x) \left (-3 a^2 A b+a^3 B-3 a b^2 B+A b^3\right ) (a \cos (c+d x)+b \sin (c+d x))^2+a \left (a^2 A b^3+3 a^3 b^2 B+a^5 B+6 a b^4 B-3 A b^5\right ) (a \cos (c+d x)+b \sin (c+d x))^2 \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right )-2 i a \left (a^2 A b^3+3 a^3 b^2 B+a^5 B+6 a b^4 B-3 A b^5\right ) \tan ^{-1}(\tan (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^2+a^3 b^2 \left (a^2+b^2\right ) (A b-a B)-2 B \left (a^2+b^2\right )^3 \log (\cos (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^2\right )}{2 b^3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))^3 (A \cos (c+d x)+B \sin (c+d x))} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.054, size = 566, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.62286, size = 494, normalized size = 1.98 \begin{align*} \frac{\frac{2 \,{\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )}{\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{2 \,{\left (B a^{6} + 3 \, B a^{4} b^{2} + A a^{3} b^{3} + 6 \, B a^{2} b^{4} - 3 \, A a b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}} - \frac{{\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 )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{3 \, B a^{6} - A a^{5} b + 7 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3} + 2 \,{\left (2 \, B a^{5} b - A a^{4} b^{2} + 4 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} \tan \left (d x + c\right )}{a^{6} b^{3} + 2 \, a^{4} b^{5} + a^{2} b^{7} +{\left (a^{4} b^{5} + 2 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (a^{5} b^{4} + 2 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.54733, size = 1432, normalized size = 5.73 \begin{align*} \frac{B a^{6} b^{2} + A a^{5} b^{3} + 7 \, B a^{4} b^{4} - 5 \, A a^{3} b^{5} + 2 \,{\left (B a^{5} b^{3} - 3 \, A a^{4} b^{4} - 3 \, B a^{3} b^{5} + A a^{2} b^{6}\right )} d x -{\left (3 \, B a^{6} b^{2} - A a^{5} b^{3} + 9 \, B a^{4} b^{4} - 7 \, A a^{3} b^{5} - 2 \,{\left (B a^{3} b^{5} - 3 \, A a^{2} b^{6} - 3 \, B a b^{7} + A b^{8}\right )} d x\right )} \tan \left (d x + c\right )^{2} +{\left (B a^{8} + 3 \, B a^{6} b^{2} + A a^{5} b^{3} + 6 \, B a^{4} b^{4} - 3 \, A a^{3} b^{5} +{\left (B a^{6} b^{2} + 3 \, B a^{4} b^{4} + A a^{3} b^{5} + 6 \, B a^{2} b^{6} - 3 \, A a b^{7}\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (B a^{7} b + 3 \, B a^{5} b^{3} + A a^{4} b^{4} + 6 \, B a^{3} b^{5} - 3 \, A a^{2} b^{6}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac{b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) -{\left (B a^{8} + 3 \, B a^{6} b^{2} + 3 \, B a^{4} b^{4} + B a^{2} b^{6} +{\left (B a^{6} b^{2} + 3 \, B a^{4} b^{4} + 3 \, B a^{2} b^{6} + B b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (B a^{7} b + 3 \, B a^{5} b^{3} + 3 \, B a^{3} b^{5} + B a b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \,{\left (B a^{7} b + 3 \, B a^{5} b^{3} - 3 \, A a^{4} b^{4} - 4 \, B a^{3} b^{5} + 3 \, A a^{2} b^{6} - 2 \,{\left (B a^{4} b^{4} - 3 \, A a^{3} b^{5} - 3 \, B a^{2} b^{6} + A a b^{7}\right )} d x\right )} \tan \left (d x + c\right )}{2 \,{\left ({\left (a^{6} b^{5} + 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} + b^{11}\right )} d \tan \left (d x + c\right )^{2} + 2 \,{\left (a^{7} b^{4} + 3 \, a^{5} b^{6} + 3 \, a^{3} b^{8} + a b^{10}\right )} d \tan \left (d x + c\right ) +{\left (a^{8} b^{3} + 3 \, a^{6} b^{5} + 3 \, a^{4} b^{7} + a^{2} b^{9}\right )} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.74616, size = 618, normalized size = 2.47 \begin{align*} \frac{\frac{2 \,{\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )}{\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{{\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 )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{2 \,{\left (B a^{6} + 3 \, B a^{4} b^{2} + A a^{3} b^{3} + 6 \, B a^{2} b^{4} - 3 \, A a b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}} - \frac{3 \, B a^{6} b \tan \left (d x + c\right )^{2} + 9 \, B a^{4} b^{3} \tan \left (d x + c\right )^{2} + 3 \, A a^{3} b^{4} \tan \left (d x + c\right )^{2} + 18 \, B a^{2} b^{5} \tan \left (d x + c\right )^{2} - 9 \, A a b^{6} \tan \left (d x + c\right )^{2} + 2 \, B a^{7} \tan \left (d x + c\right ) + 2 \, A a^{6} b \tan \left (d x + c\right ) + 6 \, B a^{5} b^{2} \tan \left (d x + c\right ) + 14 \, A a^{4} b^{3} \tan \left (d x + c\right ) + 28 \, B a^{3} b^{4} \tan \left (d x + c\right ) - 12 \, A a^{2} b^{5} \tan \left (d x + c\right ) + A a^{7} - B a^{6} b + 9 \, A a^{5} b^{2} + 11 \, B a^{4} b^{3} - 4 \, A a^{3} b^{4}}{{\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )}{\left (b \tan \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]