Optimal. Leaf size=233 \[ \frac {a \left (5 a^2 A-15 a b B-12 A b^2\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (5 a B+7 A b) \cot ^4(c+d x)}{20 d}+\frac {\left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \cot ^2(c+d x)}{2 d}-\frac {\left (a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right ) \cot (c+d x)}{d}+\frac {\left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \log (\sin (c+d x))}{d}-x \left (a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right )-\frac {a A \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.50, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3605, 3635, 3628, 3529, 3531, 3475} \[ \frac {a \left (5 a^2 A-15 a b B-12 A b^2\right ) \cot ^3(c+d x)}{15 d}+\frac {\left (3 a^2 A b+a^3 B-3 a b^2 B-A b^3\right ) \cot ^2(c+d x)}{2 d}-\frac {\left (a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right ) \cot (c+d x)}{d}+\frac {\left (3 a^2 A b+a^3 B-3 a b^2 B-A b^3\right ) \log (\sin (c+d x))}{d}-x \left (a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right )-\frac {a^2 (5 a B+7 A b) \cot ^4(c+d x)}{20 d}-\frac {a A \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3475
Rule 3529
Rule 3531
Rule 3605
Rule 3628
Rule 3635
Rubi steps
\begin {align*} \int \cot ^6(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx &=-\frac {a A \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {1}{5} \int \cot ^5(c+d x) (a+b \tan (c+d x)) \left (a (7 A b+5 a B)-5 \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)-b (3 a A-5 b B) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac {a^2 (7 A b+5 a B) \cot ^4(c+d x)}{20 d}-\frac {a A \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {1}{5} \int \cot ^4(c+d x) \left (-a \left (5 a^2 A-12 A b^2-15 a b B\right )-5 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)-b^2 (3 a A-5 b B) \tan ^2(c+d x)\right ) \, dx\\ &=\frac {a \left (5 a^2 A-12 A b^2-15 a b B\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (7 A b+5 a B) \cot ^4(c+d x)}{20 d}-\frac {a A \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {1}{5} \int \cot ^3(c+d x) \left (-5 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )+5 \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \tan (c+d x)\right ) \, dx\\ &=\frac {\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (5 a^2 A-12 A b^2-15 a b B\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (7 A b+5 a B) \cot ^4(c+d x)}{20 d}-\frac {a A \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {1}{5} \int \cot ^2(c+d x) \left (5 \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )+5 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)\right ) \, dx\\ &=-\frac {\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \cot (c+d x)}{d}+\frac {\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (5 a^2 A-12 A b^2-15 a b B\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (7 A b+5 a B) \cot ^4(c+d x)}{20 d}-\frac {a A \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {1}{5} \int \cot (c+d x) \left (5 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )-5 \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \tan (c+d x)\right ) \, dx\\ &=-\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) x-\frac {\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \cot (c+d x)}{d}+\frac {\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (5 a^2 A-12 A b^2-15 a b B\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (7 A b+5 a B) \cot ^4(c+d x)}{20 d}-\frac {a A \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \int \cot (c+d x) \, dx\\ &=-\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) x-\frac {\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \cot (c+d x)}{d}+\frac {\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (5 a^2 A-12 A b^2-15 a b B\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (7 A b+5 a B) \cot ^4(c+d x)}{20 d}+\frac {\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \log (\sin (c+d x))}{d}-\frac {a A \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 1.21, size = 237, normalized size = 1.02 \[ \frac {-12 a^3 A \cot ^5(c+d x)+20 a \left (a^2 A-3 a b B-3 A b^2\right ) \cot ^3(c+d x)-15 a^2 (a B+3 A b) \cot ^4(c+d x)+30 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \cot ^2(c+d x)-60 \left (a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right ) \cot (c+d x)+60 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \log (\tan (c+d x))+30 i (a+i b)^3 (A+i B) \log (-\tan (c+d x)+i)+30 (b+i a)^3 (A-i B) \log (\tan (c+d x)+i)}{60 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.55, size = 266, normalized size = 1.14 \[ \frac {30 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{5} + 15 \, {\left (3 \, B a^{3} + 9 \, A a^{2} b - 6 \, B a b^{2} - 2 \, A b^{3} - 4 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{5} - 60 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )} \tan \left (d x + c\right )^{4} - 12 \, A a^{3} + 30 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} \tan \left (d x + c\right )^{3} + 20 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2}\right )} \tan \left (d x + c\right )^{2} - 15 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \tan \left (d x + c\right )}{60 \, d \tan \left (d x + c\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 6.11, size = 670, normalized size = 2.88 \[ \frac {6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 45 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 70 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 180 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 540 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 360 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 120 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 660 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1800 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1800 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 480 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 960 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )} {\left (d x + c\right )} - 960 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) + 960 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2192 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6576 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6576 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2192 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 660 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1800 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1800 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 480 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 180 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 540 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 360 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 70 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 45 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, A a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.44, size = 376, normalized size = 1.61 \[ -\frac {a^{2} b B \left (\cot ^{3}\left (d x +c \right )\right )}{d}-\frac {3 B a \,b^{2} \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}-\frac {A \,b^{3} \ln \left (\sin \left (d x +c \right )\right )}{d}+\frac {A \,a^{3} \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}+\frac {a^{3} B \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}-\frac {A \,a^{3} c}{d}-\frac {B \,b^{3} c}{d}+3 A x a \,b^{2}+3 B x \,a^{2} b -\frac {3 A \,a^{2} b \left (\cot ^{4}\left (d x +c \right )\right )}{4 d}-\frac {A a \,b^{2} \left (\cot ^{3}\left (d x +c \right )\right )}{d}-\frac {A \,b^{3} \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}-\frac {A \,a^{3} \left (\cot ^{5}\left (d x +c \right )\right )}{5 d}-\frac {a^{3} B \left (\cot ^{4}\left (d x +c \right )\right )}{4 d}-\frac {B \cot \left (d x +c \right ) b^{3}}{d}-\frac {3 B a \,b^{2} \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {A \cot \left (d x +c \right ) a^{3}}{d}+\frac {a^{3} B \ln \left (\sin \left (d x +c \right )\right )}{d}-A \,a^{3} x -B x \,b^{3}+\frac {3 A \,a^{2} b \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}+\frac {3 A \,a^{2} b \ln \left (\sin \left (d x +c \right )\right )}{d}+\frac {3 A a \,b^{2} c}{d}+\frac {3 B \,a^{2} b c}{d}+\frac {3 B \cot \left (d x +c \right ) a^{2} b}{d}+\frac {3 A \cot \left (d x +c \right ) a \,b^{2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.78, size = 250, normalized size = 1.07 \[ -\frac {60 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )} {\left (d x + c\right )} + 30 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {60 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )} \tan \left (d x + c\right )^{4} + 12 \, A a^{3} - 30 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} \tan \left (d x + c\right )^{3} - 20 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2}\right )} \tan \left (d x + c\right )^{2} + 15 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{5}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 6.57, size = 238, normalized size = 1.02 \[ -\frac {{\mathrm {cot}\left (c+d\,x\right )}^5\,\left (\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {B\,a^3}{4}+\frac {3\,A\,b\,a^2}{4}\right )+\frac {A\,a^3}{5}+{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-\frac {A\,a^3}{3}+B\,a^2\,b+A\,a\,b^2\right )+{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (A\,a^3-3\,B\,a^2\,b-3\,A\,a\,b^2+B\,b^3\right )+{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (-\frac {B\,a^3}{2}-\frac {3\,A\,a^2\,b}{2}+\frac {3\,B\,a\,b^2}{2}+\frac {A\,b^3}{2}\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-B\,a^3-3\,A\,a^2\,b+3\,B\,a\,b^2+A\,b^3\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,1{}\mathrm {i}\right )\,{\left (a+b\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )\,{\left (a-b\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 9.15, size = 471, normalized size = 2.02 \[ \begin {cases} \tilde {\infty } A a^{3} x & \text {for}\: \left (c = 0 \vee c = - d x\right ) \wedge \left (c = - d x \vee d = 0\right ) \\x \left (A + B \tan {\relax (c )}\right ) \left (a + b \tan {\relax (c )}\right )^{3} \cot ^{6}{\relax (c )} & \text {for}\: d = 0 \\- A a^{3} x - \frac {A a^{3}}{d \tan {\left (c + d x \right )}} + \frac {A a^{3}}{3 d \tan ^{3}{\left (c + d x \right )}} - \frac {A a^{3}}{5 d \tan ^{5}{\left (c + d x \right )}} - \frac {3 A a^{2} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {3 A a^{2} b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {3 A a^{2} b}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac {3 A a^{2} b}{4 d \tan ^{4}{\left (c + d x \right )}} + 3 A a b^{2} x + \frac {3 A a b^{2}}{d \tan {\left (c + d x \right )}} - \frac {A a b^{2}}{d \tan ^{3}{\left (c + d x \right )}} + \frac {A b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {A b^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {A b^{3}}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac {B a^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {B a^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {B a^{3}}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac {B a^{3}}{4 d \tan ^{4}{\left (c + d x \right )}} + 3 B a^{2} b x + \frac {3 B a^{2} b}{d \tan {\left (c + d x \right )}} - \frac {B a^{2} b}{d \tan ^{3}{\left (c + d x \right )}} + \frac {3 B a b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {3 B a b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {3 B a b^{2}}{2 d \tan ^{2}{\left (c + d x \right )}} - B b^{3} x - \frac {B b^{3}}{d \tan {\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________