Optimal. Leaf size=60 \[ a x+\frac {a \cot (c+d x)}{d}-\frac {b \cot ^2(c+d x)}{2 d}-\frac {a \cot ^3(c+d x)}{3 d}-\frac {b \log (\sin (c+d x))}{d} \]
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Rubi [A]
time = 0.06, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3610, 3612,
3556} \begin {gather*} -\frac {a \cot ^3(c+d x)}{3 d}+\frac {a \cot (c+d x)}{d}+a x-\frac {b \cot ^2(c+d x)}{2 d}-\frac {b \log (\sin (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3610
Rule 3612
Rubi steps
\begin {align*} \int \cot ^4(c+d x) (a+b \tan (c+d x)) \, dx &=-\frac {a \cot ^3(c+d x)}{3 d}+\int \cot ^3(c+d x) (b-a \tan (c+d x)) \, dx\\ &=-\frac {b \cot ^2(c+d x)}{2 d}-\frac {a \cot ^3(c+d x)}{3 d}+\int \cot ^2(c+d x) (-a-b \tan (c+d x)) \, dx\\ &=\frac {a \cot (c+d x)}{d}-\frac {b \cot ^2(c+d x)}{2 d}-\frac {a \cot ^3(c+d x)}{3 d}+\int \cot (c+d x) (-b+a \tan (c+d x)) \, dx\\ &=a x+\frac {a \cot (c+d x)}{d}-\frac {b \cot ^2(c+d x)}{2 d}-\frac {a \cot ^3(c+d x)}{3 d}-b \int \cot (c+d x) \, dx\\ &=a x+\frac {a \cot (c+d x)}{d}-\frac {b \cot ^2(c+d x)}{2 d}-\frac {a \cot ^3(c+d x)}{3 d}-\frac {b \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 = 0.30, size = 70, normalized size = 1.17 \begin {gather*} -\frac {a \cot ^3(c+d x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\tan ^2(c+d x)\right )}{3 d}-\frac {b \left (\cot ^2(c+d x)+2 \log (\cos (c+d x))+2 \log (\tan (c+d x))\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 51, normalized size = 0.85
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+b \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(51\) |
default | \(\frac {a \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+b \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(51\) |
norman | \(\frac {a x \left (\tan ^{3}\left (d x +c \right )\right )+\frac {a \left (\tan ^{2}\left (d x +c \right )\right )}{d}-\frac {a}{3 d}-\frac {b \tan \left (d x +c \right )}{2 d}}{\tan \left (d x +c \right )^{3}}-\frac {b \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(84\) |
risch | \(i b x +a x +\frac {2 i b c}{d}+\frac {4 i a \,{\mathrm e}^{4 i \left (d x +c \right )}+2 b \,{\mathrm e}^{4 i \left (d x +c \right )}-4 i a \,{\mathrm e}^{2 i \left (d x +c \right )}-2 b \,{\mathrm e}^{2 i \left (d x +c \right )}+\frac {8 i a}{3}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}-\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(109\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 71, normalized size = 1.18 \begin {gather*} \frac {6 \, {\left (d x + c\right )} a + 3 \, b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 \, b \log \left (\tan \left (d x + c\right )\right ) + \frac {6 \, a \tan \left (d x + c\right )^{2} - 3 \, b \tan \left (d x + c\right ) - 2 \, a}{\tan \left (d x + c\right )^{3}}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.77, size = 89, normalized size = 1.48 \begin {gather*} -\frac {3 \, b \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} - 3 \, {\left (2 \, a d x - b\right )} \tan \left (d x + c\right )^{3} - 6 \, a \tan \left (d x + c\right )^{2} + 3 \, b \tan \left (d x + c\right ) + 2 \, a}{6 \, 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 = 0.80, size = 97, normalized size = 1.62 \begin {gather*} \begin {cases} \tilde {\infty } a 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 ) \cot ^{4}{\left (c \right )} & \text {for}\: d = 0 \\a x + \frac {a}{d \tan {\left (c + d x \right )}} - \frac {a}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {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. 140 vs.
\(2 (56) = 112\).
time = 0.62, size = 140, normalized size = 2.33 \begin {gather*} \frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, {\left (d x + c\right )} a + 24 \, b \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 24 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {44 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a}{\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 = 96, normalized size = 1.60 \begin {gather*} -\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-\frac {b}{2}+\frac {a\,1{}\mathrm {i}}{2}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (\frac {b}{2}+\frac {a\,1{}\mathrm {i}}{2}\right )}{d}-\frac {b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^3\,\left (-a\,{\mathrm {tan}\left (c+d\,x\right )}^2+\frac {b\,\mathrm {tan}\left (c+d\,x\right )}{2}+\frac {a}{3}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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