Optimal. Leaf size=108 \[ (b B+a C) x+\frac {(b B+a C) \cot (c+d x)}{d}+\frac {(a B-b C) \cot ^2(c+d x)}{2 d}-\frac {(b B+a C) \cot ^3(c+d x)}{3 d}-\frac {a B \cot ^4(c+d x)}{4 d}+\frac {(a B-b C) \log (\sin (c+d x))}{d} \]
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Rubi [A]
time = 0.17, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {3713, 3672,
3610, 3612, 3556} \begin {gather*} -\frac {(a C+b B) \cot ^3(c+d x)}{3 d}+\frac {(a B-b C) \cot ^2(c+d x)}{2 d}+\frac {(a C+b B) \cot (c+d x)}{d}+\frac {(a B-b C) \log (\sin (c+d x))}{d}+x (a C+b B)-\frac {a B \cot ^4(c+d x)}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3610
Rule 3612
Rule 3672
Rule 3713
Rubi steps
\begin {align*} \int \cot ^6(c+d x) (a+b \tan (c+d x)) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx &=\int \cot ^5(c+d x) (a+b \tan (c+d x)) (B+C \tan (c+d x)) \, dx\\ &=-\frac {a B \cot ^4(c+d x)}{4 d}+\int \cot ^4(c+d x) (b B+a C-(a B-b C) \tan (c+d x)) \, dx\\ &=-\frac {(b B+a C) \cot ^3(c+d x)}{3 d}-\frac {a B \cot ^4(c+d x)}{4 d}+\int \cot ^3(c+d x) (-a B+b C-(b B+a C) \tan (c+d x)) \, dx\\ &=\frac {(a B-b C) \cot ^2(c+d x)}{2 d}-\frac {(b B+a C) \cot ^3(c+d x)}{3 d}-\frac {a B \cot ^4(c+d x)}{4 d}+\int \cot ^2(c+d x) (-b B-a C+(a B-b C) \tan (c+d x)) \, dx\\ &=\frac {(b B+a C) \cot (c+d x)}{d}+\frac {(a B-b C) \cot ^2(c+d x)}{2 d}-\frac {(b B+a C) \cot ^3(c+d x)}{3 d}-\frac {a B \cot ^4(c+d x)}{4 d}+\int \cot (c+d x) (a B-b C+(b B+a C) \tan (c+d x)) \, dx\\ &=(b B+a C) x+\frac {(b B+a C) \cot (c+d x)}{d}+\frac {(a B-b C) \cot ^2(c+d x)}{2 d}-\frac {(b B+a C) \cot ^3(c+d x)}{3 d}-\frac {a B \cot ^4(c+d x)}{4 d}+(a B-b C) \int \cot (c+d x) \, dx\\ &=(b B+a C) x+\frac {(b B+a C) \cot (c+d x)}{d}+\frac {(a B-b C) \cot ^2(c+d x)}{2 d}-\frac {(b B+a C) \cot ^3(c+d x)}{3 d}-\frac {a B \cot ^4(c+d x)}{4 d}+\frac {(a B-b C) \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.78, size = 100, normalized size = 0.93 \begin {gather*} -\frac {4 (b B+a C) \cot ^3(c+d x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\tan ^2(c+d x)\right )+3 \left ((-2 a B+2 b C) \cot ^2(c+d x)+a B \cot ^4(c+d x)-4 (a B-b C) (\log (\cos (c+d x))+\log (\tan (c+d x)))\right )}{12 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 108, normalized size = 1.00
method | result | size |
derivativedivides | \(\frac {B b \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+C b \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+a B \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+C a \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(108\) |
default | \(\frac {B b \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+C b \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+a B \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+C a \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(108\) |
norman | \(\frac {\frac {\left (B b +C a \right ) \left (\tan ^{4}\left (d x +c \right )\right )}{d}+\left (B b +C a \right ) x \left (\tan ^{5}\left (d x +c \right )\right )-\frac {\left (B b +C a \right ) \left (\tan ^{2}\left (d x +c \right )\right )}{3 d}+\frac {\left (a B -C b \right ) \left (\tan ^{3}\left (d x +c \right )\right )}{2 d}-\frac {a B \tan \left (d x +c \right )}{4 d}}{\tan \left (d x +c \right )^{5}}+\frac {\left (a B -C b \right ) \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {\left (a B -C b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(145\) |
risch | \(B b x +C a x -i B a x +i C b x -\frac {2 i a B c}{d}+\frac {2 i C b c}{d}-\frac {2 \left (-6 i B b \,{\mathrm e}^{6 i \left (d x +c \right )}-6 i C a \,{\mathrm e}^{6 i \left (d x +c \right )}+6 B a \,{\mathrm e}^{6 i \left (d x +c \right )}-3 C b \,{\mathrm e}^{6 i \left (d x +c \right )}+12 i B b \,{\mathrm e}^{4 i \left (d x +c \right )}+12 i C a \,{\mathrm e}^{4 i \left (d x +c \right )}-6 a B \,{\mathrm e}^{4 i \left (d x +c \right )}+6 C b \,{\mathrm e}^{4 i \left (d x +c \right )}-10 i B b \,{\mathrm e}^{2 i \left (d x +c \right )}-10 i C a \,{\mathrm e}^{2 i \left (d x +c \right )}+6 a B \,{\mathrm e}^{2 i \left (d x +c \right )}-3 C b \,{\mathrm e}^{2 i \left (d x +c \right )}+4 i B b +4 i C a \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B}{d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) C b}{d}\) | \(268\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 122, normalized size = 1.13 \begin {gather*} \frac {12 \, {\left (C a + B b\right )} {\left (d x + c\right )} - 6 \, {\left (B a - C b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 12 \, {\left (B a - C b\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {12 \, {\left (C a + B b\right )} \tan \left (d x + c\right )^{3} + 6 \, {\left (B a - C b\right )} \tan \left (d x + c\right )^{2} - 3 \, B a - 4 \, {\left (C a + B b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4}}}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.60, size = 138, normalized size = 1.28 \begin {gather*} \frac {6 \, {\left (B a - C b\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 )^{4} + 3 \, {\left (4 \, {\left (C a + B b\right )} d x + 3 \, B a - 2 \, C b\right )} \tan \left (d x + c\right )^{4} + 12 \, {\left (C a + B b\right )} \tan \left (d x + c\right )^{3} + 6 \, {\left (B a - C b\right )} \tan \left (d x + c\right )^{2} - 3 \, B a - 4 \, {\left (C a + B b\right )} \tan \left (d x + c\right )}{12 \, d \tan \left (d x + c\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 211 vs.
\(2 (95) = 190\).
time = 3.03, size = 211, normalized size = 1.95 \begin {gather*} \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 {\left (c \right )}\right ) \left (B \tan {\left (c \right )} + C \tan ^{2}{\left (c \right )}\right ) \cot ^{6}{\left (c \right )} & \text {for}\: d = 0 \\- \frac {B a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {B a \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {B a}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac {B a}{4 d \tan ^{4}{\left (c + d x \right )}} + B b x + \frac {B b}{d \tan {\left (c + d x \right )}} - \frac {B b}{3 d \tan ^{3}{\left (c + d x \right )}} + C a x + \frac {C a}{d \tan {\left (c + d x \right )}} - \frac {C a}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {C b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {C b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {C b}{2 d \tan ^{2}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 299 vs.
\(2 (102) = 204\).
time = 1.44, size = 299, normalized size = 2.77 \begin {gather*} -\frac {3 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 120 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 192 \, {\left (C a + B b\right )} {\left (d x + c\right )} + 192 \, {\left (B a - C b\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 192 \, {\left (B a - C b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {400 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 400 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.82, size = 145, normalized size = 1.34 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (B\,a-C\,b\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^4\,\left (\left (-B\,b-C\,a\right )\,{\mathrm {tan}\left (c+d\,x\right )}^3+\left (\frac {C\,b}{2}-\frac {B\,a}{2}\right )\,{\mathrm {tan}\left (c+d\,x\right )}^2+\left (\frac {B\,b}{3}+\frac {C\,a}{3}\right )\,\mathrm {tan}\left (c+d\,x\right )+\frac {B\,a}{4}\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 )}{2\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )\,\left (b+a\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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