Optimal. Leaf size=215 \[ \frac{b (A b-a B)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{b \left (3 a^2 A b-2 a^3 B+A b^3\right )}{a^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac{b \left (3 a^2 A b^3+6 a^4 A b+a^3 b^2 B-3 a^5 B+A b^5\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left (a^2+b^2\right )^3}-\frac{x \left (3 a^2 A b+a^3 (-B)+3 a b^2 B-A b^3\right )}{\left (a^2+b^2\right )^3}+\frac{A \log (\sin (c+d x))}{a^3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.62136, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {3609, 3649, 3651, 3530, 3475} \[ \frac{b (A b-a B)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{b \left (3 a^2 A b-2 a^3 B+A b^3\right )}{a^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac{b \left (3 a^2 A b^3+6 a^4 A b+a^3 b^2 B-3 a^5 B+A b^5\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left (a^2+b^2\right )^3}-\frac{x \left (3 a^2 A b+a^3 (-B)+3 a b^2 B-A b^3\right )}{\left (a^2+b^2\right )^3}+\frac{A \log (\sin (c+d x))}{a^3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3609
Rule 3649
Rule 3651
Rule 3530
Rule 3475
Rubi steps
\begin{align*} \int \frac{\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx &=\frac{b (A b-a B)}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{\int \frac{\cot (c+d x) \left (2 A \left (a^2+b^2\right )-2 a (A b-a B) \tan (c+d x)+2 b (A b-a B) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 a \left (a^2+b^2\right )}\\ &=\frac{b (A b-a B)}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{b \left (3 a^2 A b+A b^3-2 a^3 B\right )}{a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac{\int \frac{\cot (c+d x) \left (2 A \left (a^2+b^2\right )^2-2 a^2 \left (2 a A b-a^2 B+b^2 B\right ) \tan (c+d x)+2 b \left (3 a^2 A b+A b^3-2 a^3 B\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 a^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{b (A b-a B)}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{b \left (3 a^2 A b+A b^3-2 a^3 B\right )}{a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac{A \int \cot (c+d x) \, dx}{a^3}-\frac{\left (b \left (6 a^4 A b+3 a^2 A b^3+A b^5-3 a^5 B+a^3 b^2 B\right )\right ) \int \frac{b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^3 \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{A \log (\sin (c+d x))}{a^3 d}-\frac{b \left (6 a^4 A b+3 a^2 A b^3+A b^5-3 a^5 B+a^3 b^2 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 \left (a^2+b^2\right )^3 d}+\frac{b (A b-a B)}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{b \left (3 a^2 A b+A b^3-2 a^3 B\right )}{a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end{align*}
Mathematica [C] time = 3.34972, size = 254, normalized size = 1.18 \[ \frac{\frac{4 a b (A b-a B)}{\left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac{2 b \left (3 a^2 A b^3+6 a^4 A b+a^3 b^2 B-3 a^5 B+A b^5\right ) \log (a+b \tan (c+d x))}{a^2 \left (a^2+b^2\right )^2}+\frac{2 A b^2}{a^2+a b \tan (c+d x)}+\frac{2 A \left (a^2+b^2\right ) \log (\tan (c+d x))}{a^2}+\frac{b (A b-a B)}{(a+b \tan (c+d x))^2}-\frac{a (a-i b) (A+i B) \log (-\tan (c+d x)+i)}{(a+i b)^2}-\frac{a (a+i b) (A-i B) \log (\tan (c+d x)+i)}{(a-i b)^2}}{2 a d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.181, size = 540, normalized size = 2.5 \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 = 1.59958, size = 502, normalized size = 2.33 \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 (3 \, B a^{5} b - 6 \, A a^{4} b^{2} - B a^{3} b^{3} - 3 \, A a^{2} b^{4} - A b^{6}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} 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{5 \, B a^{4} b - 7 \, A a^{3} b^{2} + B a^{2} b^{3} - 3 \, A a b^{4} + 2 \,{\left (2 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3} - A b^{5}\right )} \tan \left (d x + c\right )}{a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4} +{\left (a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} \tan \left (d x + c\right )} + \frac{2 \, A \log \left (\tan \left (d x + c\right )\right )}{a^{3}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.56288, size = 1451, normalized size = 6.75 \begin{align*} -\frac{7 \, B a^{5} b^{3} - 9 \, A a^{4} b^{4} + B a^{3} b^{5} - 3 \, A a^{2} b^{6} - 2 \,{\left (B a^{8} - 3 \, A a^{7} b - 3 \, B a^{6} b^{2} + A a^{5} b^{3}\right )} d x -{\left (5 \, B a^{5} b^{3} - 7 \, A a^{4} b^{4} - B a^{3} b^{5} - A a^{2} b^{6} + 2 \,{\left (B a^{6} b^{2} - 3 \, A a^{5} b^{3} - 3 \, B a^{4} b^{4} + A a^{3} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} -{\left (A a^{8} + 3 \, A a^{6} b^{2} + 3 \, A a^{4} b^{4} + A a^{2} b^{6} +{\left (A a^{6} b^{2} + 3 \, A a^{4} b^{4} + 3 \, A a^{2} b^{6} + A b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (A a^{7} b + 3 \, A a^{5} b^{3} + 3 \, A a^{3} b^{5} + A a b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac{\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) -{\left (3 \, B a^{7} b - 6 \, A a^{6} b^{2} - B a^{5} b^{3} - 3 \, A a^{4} b^{4} - A a^{2} b^{6} +{\left (3 \, B a^{5} b^{3} - 6 \, A a^{4} b^{4} - B a^{3} b^{5} - 3 \, A a^{2} b^{6} - A b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (3 \, B a^{6} b^{2} - 6 \, A a^{5} b^{3} - B a^{4} b^{4} - 3 \, A a^{3} b^{5} - A a b^{7}\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 ) - 2 \,{\left (3 \, B a^{6} b^{2} - 4 \, A a^{5} b^{3} - 3 \, B a^{4} b^{4} + 3 \, A a^{3} b^{5} + A a b^{7} + 2 \,{\left (B a^{7} b - 3 \, A a^{6} b^{2} - 3 \, B a^{5} b^{3} + A a^{4} b^{4}\right )} d x\right )} \tan \left (d x + c\right )}{2 \,{\left ({\left (a^{9} b^{2} + 3 \, a^{7} b^{4} + 3 \, a^{5} b^{6} + a^{3} b^{8}\right )} d \tan \left (d x + c\right )^{2} + 2 \,{\left (a^{10} b + 3 \, a^{8} b^{3} + 3 \, a^{6} b^{5} + a^{4} b^{7}\right )} d \tan \left (d x + c\right ) +{\left (a^{11} + 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} + a^{5} b^{6}\right )} d\right )}} \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 = 1.3471, size = 647, normalized size = 3.01 \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 (3 \, B a^{5} b^{2} - 6 \, A a^{4} b^{3} - B a^{3} b^{4} - 3 \, A a^{2} b^{5} - A b^{7}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{9} b + 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} + a^{3} b^{7}} + \frac{2 \, A \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{3}} - \frac{9 \, B a^{5} b^{3} \tan \left (d x + c\right )^{2} - 18 \, A a^{4} b^{4} \tan \left (d x + c\right )^{2} - 3 \, B a^{3} b^{5} \tan \left (d x + c\right )^{2} - 9 \, A a^{2} b^{6} \tan \left (d x + c\right )^{2} - 3 \, A b^{8} \tan \left (d x + c\right )^{2} + 22 \, B a^{6} b^{2} \tan \left (d x + c\right ) - 42 \, A a^{5} b^{3} \tan \left (d x + c\right ) - 2 \, B a^{4} b^{4} \tan \left (d x + c\right ) - 26 \, A a^{3} b^{5} \tan \left (d x + c\right ) - 8 \, A a b^{7} \tan \left (d x + c\right ) + 14 \, B a^{7} b - 25 \, A a^{6} b^{2} + 3 \, B a^{5} b^{3} - 19 \, A a^{4} b^{4} + B a^{3} b^{5} - 6 \, A a^{2} b^{6}}{{\left (a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6}\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]