Optimal. Leaf size=233 \[ \frac {a \left (5 a^2 B-15 a b C-12 b^2 B\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (5 a C+7 b B) \cot ^4(c+d x)}{20 d}+\frac {\left (a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right ) \cot ^2(c+d x)}{2 d}-\frac {\left (a^3 B-3 a^2 b C-3 a b^2 B+b^3 C\right ) \cot (c+d x)}{d}+\frac {\left (a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right ) \log (\sin (c+d x))}{d}-x \left (a^3 B-3 a^2 b C-3 a b^2 B+b^3 C\right )-\frac {a B \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d} \]
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Rubi [A] time = 0.56, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {3632, 3605, 3635, 3628, 3529, 3531, 3475} \[ \frac {a \left (5 a^2 B-15 a b C-12 b^2 B\right ) \cot ^3(c+d x)}{15 d}+\frac {\left (3 a^2 b B+a^3 C-3 a b^2 C-b^3 B\right ) \cot ^2(c+d x)}{2 d}-\frac {\left (-3 a^2 b C+a^3 B-3 a b^2 B+b^3 C\right ) \cot (c+d x)}{d}+\frac {\left (3 a^2 b B+a^3 C-3 a b^2 C-b^3 B\right ) \log (\sin (c+d x))}{d}-x \left (-3 a^2 b C+a^3 B-3 a b^2 B+b^3 C\right )-\frac {a^2 (5 a C+7 b B) \cot ^4(c+d x)}{20 d}-\frac {a B \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3529
Rule 3531
Rule 3605
Rule 3628
Rule 3632
Rule 3635
Rubi steps
\begin {align*} \int \cot ^7(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx &=\int \cot ^6(c+d x) (a+b \tan (c+d x))^3 (B+C \tan (c+d x)) \, dx\\ &=-\frac {a B \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 b B+5 a C)-5 \left (a^2 B-b^2 B-2 a b C\right ) \tan (c+d x)-b (3 a B-5 b C) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac {a^2 (7 b B+5 a C) \cot ^4(c+d x)}{20 d}-\frac {a B \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 B-12 b^2 B-15 a b C\right )-5 \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \tan (c+d x)-b^2 (3 a B-5 b C) \tan ^2(c+d x)\right ) \, dx\\ &=\frac {a \left (5 a^2 B-12 b^2 B-15 a b C\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (7 b B+5 a C) \cot ^4(c+d x)}{20 d}-\frac {a B \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 b B-b^3 B+a^3 C-3 a b^2 C\right )+5 \left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) \tan (c+d x)\right ) \, dx\\ &=\frac {\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (5 a^2 B-12 b^2 B-15 a b C\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (7 b B+5 a C) \cot ^4(c+d x)}{20 d}-\frac {a B \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 B-3 a b^2 B-3 a^2 b C+b^3 C\right )+5 \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \tan (c+d x)\right ) \, dx\\ &=-\frac {\left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) \cot (c+d x)}{d}+\frac {\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (5 a^2 B-12 b^2 B-15 a b C\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (7 b B+5 a C) \cot ^4(c+d x)}{20 d}-\frac {a B \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 b B-b^3 B+a^3 C-3 a b^2 C\right )-5 \left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) \tan (c+d x)\right ) \, dx\\ &=-\left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) x-\frac {\left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) \cot (c+d x)}{d}+\frac {\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (5 a^2 B-12 b^2 B-15 a b C\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (7 b B+5 a C) \cot ^4(c+d x)}{20 d}-\frac {a B \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \int \cot (c+d x) \, dx\\ &=-\left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) x-\frac {\left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) \cot (c+d x)}{d}+\frac {\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (5 a^2 B-12 b^2 B-15 a b C\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (7 b B+5 a C) \cot ^4(c+d x)}{20 d}+\frac {\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \log (\sin (c+d x))}{d}-\frac {a B \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\\ \end {align*}
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Mathematica [C] time = 1.21, size = 237, normalized size = 1.02 \[ \frac {-12 a^3 B \cot ^5(c+d x)+20 a \left (a^2 B-3 a b C-3 b^2 B\right ) \cot ^3(c+d x)-15 a^2 (a C+3 b B) \cot ^4(c+d x)+30 \left (a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right ) \cot ^2(c+d x)-60 \left (a^3 B-3 a^2 b C-3 a b^2 B+b^3 C\right ) \cot (c+d x)+60 \left (a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right ) \log (\tan (c+d x))+30 i (a+i b)^3 (B+i C) \log (-\tan (c+d x)+i)+30 (b+i a)^3 (B-i C) \log (\tan (c+d x)+i)}{60 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.41, size = 266, normalized size = 1.14 \[ \frac {30 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{5} + 15 \, {\left (3 \, C a^{3} + 9 \, B a^{2} b - 6 \, C a b^{2} - 2 \, B b^{3} - 4 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{5} - 60 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} \tan \left (d x + c\right )^{4} - 12 \, B a^{3} + 30 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \tan \left (d x + c\right )^{3} + 20 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2}\right )} \tan \left (d x + c\right )^{2} - 15 \, {\left (C a^{3} + 3 \, B a^{2} b\right )} \tan \left (d x + c\right )}{60 \, d \tan \left (d x + c\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 61.54, size = 670, normalized size = 2.88 \[ \frac {6 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 45 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 70 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 180 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 540 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 360 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 120 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 660 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1800 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1800 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 480 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 960 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} {\left (d x + c\right )} - 960 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) + 960 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2192 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6576 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6576 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2192 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 660 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1800 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1800 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 480 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 180 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 540 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 360 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 70 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 45 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, B 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.
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maple [A] time = 0.54, size = 376, normalized size = 1.61 \[ -a^{3} B x -\frac {3 C a \,b^{2} \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}-\frac {C \,a^{2} b \left (\cot ^{3}\left (d x +c \right )\right )}{d}-\frac {3 a^{2} b B \left (\cot ^{4}\left (d x +c \right )\right )}{4 d}-\frac {B a \,b^{2} \left (\cot ^{3}\left (d x +c \right )\right )}{d}-\frac {a^{3} B c}{d}+\frac {C \,a^{3} \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}+\frac {C \,a^{3} \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {b^{3} C c}{d}+3 C x \,a^{2} b +3 B x a \,b^{2}-\frac {B \cot \left (d x +c \right ) a^{3}}{d}+\frac {3 a^{2} b B \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}-\frac {b^{3} B \ln \left (\sin \left (d x +c \right )\right )}{d}+\frac {a^{3} B \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}-\frac {C \,a^{3} \left (\cot ^{4}\left (d x +c \right )\right )}{4 d}-\frac {a^{3} B \left (\cot ^{5}\left (d x +c \right )\right )}{5 d}-\frac {b^{3} B \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}-\frac {C \cot \left (d x +c \right ) b^{3}}{d}+\frac {3 C \cot \left (d x +c \right ) a^{2} b}{d}+\frac {3 B \cot \left (d x +c \right ) a \,b^{2}}{d}-\frac {3 C a \,b^{2} \ln \left (\sin \left (d x +c \right )\right )}{d}+\frac {3 C \,a^{2} b c}{d}+\frac {3 B a \,b^{2} c}{d}-b^{3} C x +\frac {3 a^{2} b B \ln \left (\sin \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.76, size = 250, normalized size = 1.07 \[ -\frac {60 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} {\left (d x + c\right )} + 30 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {60 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} \tan \left (d x + c\right )^{4} + 12 \, B a^{3} - 30 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \tan \left (d x + c\right )^{3} - 20 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2}\right )} \tan \left (d x + c\right )^{2} + 15 \, {\left (C a^{3} + 3 \, B a^{2} b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{5}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.12, size = 238, normalized size = 1.02 \[ -\frac {{\mathrm {cot}\left (c+d\,x\right )}^5\,\left (\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {C\,a^3}{4}+\frac {3\,B\,b\,a^2}{4}\right )+\frac {B\,a^3}{5}+{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-\frac {B\,a^3}{3}+C\,a^2\,b+B\,a\,b^2\right )+{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (B\,a^3-3\,C\,a^2\,b-3\,B\,a\,b^2+C\,b^3\right )+{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (-\frac {C\,a^3}{2}-\frac {3\,B\,a^2\,b}{2}+\frac {3\,C\,a\,b^2}{2}+\frac {B\,b^3}{2}\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-C\,a^3-3\,B\,a^2\,b+3\,C\,a\,b^2+B\,b^3\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (B+C\,1{}\mathrm {i}\right )\,{\left (a+b\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )\,{\left (a-b\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 27.65, size = 469, normalized size = 2.01 \[ \begin {cases} \text {NaN} & \text {for}\: \left (c = 0 \vee c = - d x\right ) \wedge \left (c = - d x \vee d = 0\right ) \\x \left (a + b \tan {\relax (c )}\right )^{3} \left (B \tan {\relax (c )} + C \tan ^{2}{\relax (c )}\right ) \cot ^{7}{\relax (c )} & \text {for}\: d = 0 \\- B a^{3} x - \frac {B a^{3}}{d \tan {\left (c + d x \right )}} + \frac {B a^{3}}{3 d \tan ^{3}{\left (c + d x \right )}} - \frac {B a^{3}}{5 d \tan ^{5}{\left (c + d x \right )}} - \frac {3 B a^{2} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {3 B a^{2} b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {3 B a^{2} b}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac {3 B a^{2} b}{4 d \tan ^{4}{\left (c + d x \right )}} + 3 B a b^{2} x + \frac {3 B a b^{2}}{d \tan {\left (c + d x \right )}} - \frac {B a b^{2}}{d \tan ^{3}{\left (c + d x \right )}} + \frac {B b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {B b^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {B b^{3}}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac {C a^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {C a^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {C a^{3}}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac {C a^{3}}{4 d \tan ^{4}{\left (c + d x \right )}} + 3 C a^{2} b x + \frac {3 C a^{2} b}{d \tan {\left (c + d x \right )}} - \frac {C a^{2} b}{d \tan ^{3}{\left (c + d x \right )}} + \frac {3 C a b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {3 C a b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {3 C a b^{2}}{2 d \tan ^{2}{\left (c + d x \right )}} - C b^{3} x - \frac {C b^{3}}{d \tan {\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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