Optimal. Leaf size=127 \[ -\left (\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) x\right )-\frac {a^2 (2 b B+a C) \cot (c+d x)}{d}-\frac {b^3 C \log (\cos (c+d x))}{d}-\frac {a \left (a^2 B-3 b^2 B-3 a b C\right ) \log (\sin (c+d x))}{d}-\frac {a B \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d} \]
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Rubi [A]
time = 0.25, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3713, 3686,
3716, 3705, 3556} \begin {gather*} -\frac {a \left (a^2 B-3 a b C-3 b^2 B\right ) \log (\sin (c+d x))}{d}-\frac {a^2 (a C+2 b B) \cot (c+d x)}{d}-x \left (a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right )-\frac {a B \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac {b^3 C \log (\cos (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3686
Rule 3705
Rule 3713
Rule 3716
Rubi steps
\begin {align*} \int \cot ^4(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 ^3(c+d x) (a+b \tan (c+d x))^3 (B+C \tan (c+d x)) \, dx\\ &=-\frac {a B \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}+\frac {1}{2} \int \cot ^2(c+d x) (a+b \tan (c+d x)) \left (2 a (2 b B+a C)-2 \left (a^2 B-b^2 B-2 a b C\right ) \tan (c+d x)+2 b^2 C \tan ^2(c+d x)\right ) \, dx\\ &=-\frac {a^2 (2 b B+a C) \cot (c+d x)}{d}-\frac {a B \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}+\frac {1}{2} \int \cot (c+d x) \left (-2 a \left (a^2 B-3 b^2 B-3 a b C\right )-2 \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \tan (c+d x)+2 b^3 C \tan ^2(c+d x)\right ) \, dx\\ &=-\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) x-\frac {a^2 (2 b B+a C) \cot (c+d x)}{d}-\frac {a B \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}+\left (b^3 C\right ) \int \tan (c+d x) \, dx-\left (a \left (a^2 B-3 b^2 B-3 a b C\right )\right ) \int \cot (c+d x) \, dx\\ &=-\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) x-\frac {a^2 (2 b B+a C) \cot (c+d x)}{d}-\frac {b^3 C \log (\cos (c+d x))}{d}-\frac {a \left (a^2 B-3 b^2 B-3 a b C\right ) \log (\sin (c+d x))}{d}-\frac {a B \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.31, size = 126, normalized size = 0.99 \begin {gather*} \frac {-2 a^2 (3 b B+a C) \cot (c+d x)-a^3 B \cot ^2(c+d x)+(a+i b)^3 (B+i C) \log (i-\tan (c+d x))-2 a \left (a^2 B-3 b^2 B-3 a b C\right ) \log (\tan (c+d x))+(a-i b)^3 (B-i C) \log (i+\tan (c+d x))}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 138, normalized size = 1.09
method | result | size |
derivativedivides | \(\frac {B \,b^{3} \left (d x +c \right )-C \,b^{3} \ln \left (\cos \left (d x +c \right )\right )+3 B a \,b^{2} \ln \left (\sin \left (d x +c \right )\right )+3 C a \,b^{2} \left (d x +c \right )+3 B \,a^{2} b \left (-\cot \left (d x +c \right )-d x -c \right )+3 C \,a^{2} b \ln \left (\sin \left (d x +c \right )\right )+B \,a^{3} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+C \,a^{3} \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(138\) |
default | \(\frac {B \,b^{3} \left (d x +c \right )-C \,b^{3} \ln \left (\cos \left (d x +c \right )\right )+3 B a \,b^{2} \ln \left (\sin \left (d x +c \right )\right )+3 C a \,b^{2} \left (d x +c \right )+3 B \,a^{2} b \left (-\cot \left (d x +c \right )-d x -c \right )+3 C \,a^{2} b \ln \left (\sin \left (d x +c \right )\right )+B \,a^{3} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+C \,a^{3} \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(138\) |
norman | \(\frac {\left (-3 B \,a^{2} b +B \,b^{3}-C \,a^{3}+3 C a \,b^{2}\right ) x \left (\tan ^{3}\left (d x +c \right )\right )-\frac {B \,a^{3} \tan \left (d x +c \right )}{2 d}-\frac {a^{2} \left (3 B b +C a \right ) \left (\tan ^{2}\left (d x +c \right )\right )}{d}}{\tan \left (d x +c \right )^{3}}+\frac {\left (B \,a^{3}-3 B a \,b^{2}-3 C \,a^{2} b +C \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a \left (a^{2} B -3 b^{2} B -3 C a b \right ) \ln \left (\tan \left (d x +c \right )\right )}{d}\) | \(158\) |
risch | \(\frac {2 i B \,a^{3} c}{d}-3 i B a \,b^{2} x +i B \,a^{3} x -3 i C \,a^{2} b x -3 B \,a^{2} b x +B \,b^{3} x -C \,a^{3} x +3 C a \,b^{2} x -\frac {2 i a^{2} \left (3 B b \,{\mathrm e}^{2 i \left (d x +c \right )}+C a \,{\mathrm e}^{2 i \left (d x +c \right )}+i B a \,{\mathrm e}^{2 i \left (d x +c \right )}-3 B b -C a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {6 i b C \,a^{2} c}{d}+i C \,b^{3} x -\frac {6 i B a \,b^{2} c}{d}+\frac {2 i C \,b^{3} c}{d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) C \,b^{3}}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B}{d}+\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B a \,b^{2}}{d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) C b}{d}\) | \(267\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 142, normalized size = 1.12 \begin {gather*} -\frac {2 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} {\left (d x + c\right )} - {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2}\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {B a^{3} + 2 \, {\left (C a^{3} + 3 \, B a^{2} b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.15, size = 162, normalized size = 1.28 \begin {gather*} -\frac {C b^{3} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{2} + B a^{3} + {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2}\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 )^{2} + {\left (B a^{3} + 2 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (C a^{3} + 3 \, B a^{2} b\right )} \tan \left (d x + c\right )}{2 \, d \tan \left (d x + c\right )^{2}} \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. 260 vs.
\(2 (121) = 242\).
time = 2.98, size = 260, normalized size = 2.05 \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 )^{3} \left (B \tan {\left (c \right )} + C \tan ^{2}{\left (c \right )}\right ) \cot ^{4}{\left (c \right )} & \text {for}\: d = 0 \\\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 )}} - 3 B a^{2} b x - \frac {3 B a^{2} b}{d \tan {\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} + B b^{3} x - C a^{3} x - \frac {C a^{3}}{d \tan {\left (c + d x \right )}} - \frac {3 C a^{2} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {3 C a^{2} b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + 3 C a b^{2} x + \frac {C b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.42, size = 193, normalized size = 1.52 \begin {gather*} -\frac {2 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} {\left (d x + c\right )} - {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) - \frac {3 \, B a^{3} \tan \left (d x + c\right )^{2} - 9 \, C a^{2} b \tan \left (d x + c\right )^{2} - 9 \, B a b^{2} \tan \left (d x + c\right )^{2} - 2 \, C a^{3} \tan \left (d x + c\right ) - 6 \, B a^{2} b \tan \left (d x + c\right ) - B a^{3}}{\tan \left (d x + c\right )^{2}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.97, size = 135, normalized size = 1.06 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-B\,a^3+3\,C\,a^2\,b+3\,B\,a\,b^2\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^2\,\left (\mathrm {tan}\left (c+d\,x\right )\,\left (C\,a^3+3\,B\,b\,a^2\right )+\frac {B\,a^3}{2}\right )}{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 )}^3\,1{}\mathrm {i}}{2\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (B+C\,1{}\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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