Optimal. Leaf size=117 \[ \left (a^4-6 a^2 b^2+b^4\right ) x+\frac {a^2 \left (3 a^2-17 b^2\right ) \cot (c+d x)}{3 d}-\frac {4 a^3 b \cot ^2(c+d x)}{3 d}-\frac {4 a b \left (a^2-b^2\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \cot ^3(c+d x) (a+b \tan (c+d x))^2}{3 d} \]
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Rubi [A]
time = 0.18, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3646, 3716,
3709, 3612, 3556} \begin {gather*} -\frac {4 a^3 b \cot ^2(c+d x)}{3 d}+\frac {a^2 \left (3 a^2-17 b^2\right ) \cot (c+d x)}{3 d}-\frac {4 a b \left (a^2-b^2\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \cot ^3(c+d x) (a+b \tan (c+d x))^2}{3 d}+x \left (a^4-6 a^2 b^2+b^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3612
Rule 3646
Rule 3709
Rule 3716
Rubi steps
\begin {align*} \int \cot ^4(c+d x) (a+b \tan (c+d x))^4 \, dx &=-\frac {a^2 \cot ^3(c+d x) (a+b \tan (c+d x))^2}{3 d}+\frac {1}{3} \int \cot ^3(c+d x) (a+b \tan (c+d x)) \left (8 a^2 b-3 a \left (a^2-3 b^2\right ) \tan (c+d x)-b \left (a^2-3 b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac {4 a^3 b \cot ^2(c+d x)}{3 d}-\frac {a^2 \cot ^3(c+d x) (a+b \tan (c+d x))^2}{3 d}+\frac {1}{3} \int \cot ^2(c+d x) \left (-a^2 \left (3 a^2-17 b^2\right )-12 a b \left (a^2-b^2\right ) \tan (c+d x)-b^2 \left (a^2-3 b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=\frac {a^2 \left (3 a^2-17 b^2\right ) \cot (c+d x)}{3 d}-\frac {4 a^3 b \cot ^2(c+d x)}{3 d}-\frac {a^2 \cot ^3(c+d x) (a+b \tan (c+d x))^2}{3 d}+\frac {1}{3} \int \cot (c+d x) \left (-12 a b \left (a^2-b^2\right )+3 \left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right ) \, dx\\ &=\left (a^4-6 a^2 b^2+b^4\right ) x+\frac {a^2 \left (3 a^2-17 b^2\right ) \cot (c+d x)}{3 d}-\frac {4 a^3 b \cot ^2(c+d x)}{3 d}-\frac {a^2 \cot ^3(c+d x) (a+b \tan (c+d x))^2}{3 d}-\left (4 a b \left (a^2-b^2\right )\right ) \int \cot (c+d x) \, dx\\ &=\left (a^4-6 a^2 b^2+b^4\right ) x+\frac {a^2 \left (3 a^2-17 b^2\right ) \cot (c+d x)}{3 d}-\frac {4 a^3 b \cot ^2(c+d x)}{3 d}-\frac {4 a b \left (a^2-b^2\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \cot ^3(c+d x) (a+b \tan (c+d x))^2}{3 d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.28, size = 125, normalized size = 1.07 \begin {gather*} -\frac {-6 a^2 \left (a^2-6 b^2\right ) \cot (c+d x)+12 a^3 b \cot ^2(c+d x)+2 a^4 \cot ^3(c+d x)+3 i (a+i b)^4 \log (i-\tan (c+d x))+24 a b \left (a^2-b^2\right ) \log (\tan (c+d x))-3 i (a-i b)^4 \log (i+\tan (c+d x))}{6 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 103, normalized size = 0.88
method | result | size |
derivativedivides | \(\frac {a^{4} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+4 a^{3} b \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+6 a^{2} b^{2} \left (-\cot \left (d x +c \right )-d x -c \right )+4 a \,b^{3} \ln \left (\sin \left (d x +c \right )\right )+b^{4} \left (d x +c \right )}{d}\) | \(103\) |
default | \(\frac {a^{4} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+4 a^{3} b \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+6 a^{2} b^{2} \left (-\cot \left (d x +c \right )-d x -c \right )+4 a \,b^{3} \ln \left (\sin \left (d x +c \right )\right )+b^{4} \left (d x +c \right )}{d}\) | \(103\) |
norman | \(\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) x \left (\tan ^{3}\left (d x +c \right )\right )+\frac {a^{2} \left (a^{2}-6 b^{2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{d}-\frac {a^{4}}{3 d}-\frac {2 a^{3} b \tan \left (d x +c \right )}{d}}{\tan \left (d x +c \right )^{3}}-\frac {4 a b \left (a^{2}-b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {2 a b \left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\) | \(134\) |
risch | \(4 i a^{3} b x -4 i a \,b^{3} x +a^{4} x -6 a^{2} b^{2} x +b^{4} x +\frac {8 i a^{3} b c}{d}-\frac {8 i a \,b^{3} c}{d}+\frac {4 i a^{2} \left (3 a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-9 b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-6 i a b \,{\mathrm e}^{4 i \left (d x +c \right )}-3 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+18 b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+6 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 a^{2}-9 b^{2}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}-\frac {4 a^{3} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {4 a \,b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(218\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 122, normalized size = 1.04 \begin {gather*} \frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )} + 6 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 12 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac {6 \, a^{3} b \tan \left (d x + c\right ) + a^{4} - 3 \, {\left (a^{4} - 6 \, a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{3}}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.25, size = 131, normalized size = 1.12 \begin {gather*} -\frac {6 \, a^{3} b \tan \left (d x + c\right ) + 6 \, {\left (a^{3} b - 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 )^{3} + a^{4} + 3 \, {\left (2 \, a^{3} b - {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} d x\right )} \tan \left (d x + c\right )^{3} - 3 \, {\left (a^{4} - 6 \, a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2}}{3 \, d \tan \left (d x + c\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.19, size = 187, normalized size = 1.60 \begin {gather*} \begin {cases} \tilde {\infty } a^{4} 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 {\left (c \right )}\right )^{4} \cot ^{4}{\left (c \right )} & \text {for}\: d = 0 \\a^{4} x + \frac {a^{4}}{d \tan {\left (c + d x \right )}} - \frac {a^{4}}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {2 a^{3} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - \frac {4 a^{3} b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {2 a^{3} b}{d \tan ^{2}{\left (c + d x \right )}} - 6 a^{2} b^{2} x - \frac {6 a^{2} b^{2}}{d \tan {\left (c + d x \right )}} - \frac {2 a b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {4 a b^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + b^{4} x & \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. 246 vs.
\(2 (111) = 222\).
time = 1.90, size = 246, normalized size = 2.10 \begin {gather*} \frac {a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 72 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )} + 96 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 96 \, {\left (a^{3} b - a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {176 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 176 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 72 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.98, size = 127, normalized size = 1.09 \begin {gather*} -\frac {{\mathrm {cot}\left (c+d\,x\right )}^3\,\left (\frac {a^4}{3}-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (a^4-6\,a^2\,b^2\right )+2\,a^3\,b\,\mathrm {tan}\left (c+d\,x\right )\right )}{d}-\frac {4\,a\,b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^2-b^2\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,{\left (a-b\,1{}\mathrm {i}\right )}^4\,1{}\mathrm {i}}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^4\,1{}\mathrm {i}}{2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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