Optimal. Leaf size=93 \[ -a x-\frac {a \cot (c+d x)}{d}+\frac {b \cot ^2(c+d x)}{2 d}+\frac {a \cot ^3(c+d x)}{3 d}-\frac {b \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 d}+\frac {b \log (\sin (c+d x))}{d} \]
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Rubi [A]
time = 0.10, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 ^5(c+d x)}{5 d}+\frac {a \cot ^3(c+d x)}{3 d}-\frac {a \cot (c+d x)}{d}-a x-\frac {b \cot ^4(c+d x)}{4 d}+\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 ^6(c+d x) (a+b \tan (c+d x)) \, dx &=-\frac {a \cot ^5(c+d x)}{5 d}+\int \cot ^5(c+d x) (b-a \tan (c+d x)) \, dx\\ &=-\frac {b \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 d}+\int \cot ^4(c+d x) (-a-b \tan (c+d x)) \, dx\\ &=\frac {a \cot ^3(c+d x)}{3 d}-\frac {b \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 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}-\frac {b \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 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}-\frac {b \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 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}-\frac {b \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 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 \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 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 = 0.38, size = 82, normalized size = 0.88 \begin {gather*} -\frac {a \cot ^5(c+d x) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};-\tan ^2(c+d x)\right )}{5 d}+\frac {b \left (2 \cot ^2(c+d x)-\cot ^4(c+d x)+4 \log (\cos (c+d x))+4 \log (\tan (c+d x))\right )}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 74, normalized size = 0.80
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+b \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(74\) |
default | \(\frac {a \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+b \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(74\) |
norman | \(\frac {-\frac {a}{5 d}-a x \left (\tan ^{5}\left (d x +c \right )\right )+\frac {a \left (\tan ^{2}\left (d x +c \right )\right )}{3 d}-\frac {a \left (\tan ^{4}\left (d x +c \right )\right )}{d}-\frac {b \tan \left (d x +c \right )}{4 d}+\frac {b \left (\tan ^{3}\left (d x +c \right )\right )}{2 d}}{\tan \left (d x +c \right )^{5}}+\frac {b \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(113\) |
risch | \(-i b x -a x -\frac {2 i b c}{d}-\frac {2 \left (45 i a \,{\mathrm e}^{8 i \left (d x +c \right )}+30 b \,{\mathrm e}^{8 i \left (d x +c \right )}-90 i a \,{\mathrm e}^{6 i \left (d x +c \right )}-60 b \,{\mathrm e}^{6 i \left (d x +c \right )}+140 i a \,{\mathrm e}^{4 i \left (d x +c \right )}+60 b \,{\mathrm e}^{4 i \left (d x +c \right )}-70 i a \,{\mathrm e}^{2 i \left (d x +c \right )}-30 b \,{\mathrm e}^{2 i \left (d x +c \right )}+23 i a \right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(159\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 93, normalized size = 1.00 \begin {gather*} -\frac {60 \, {\left (d x + c\right )} a + 30 \, b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 \, b \log \left (\tan \left (d x + c\right )\right ) + \frac {60 \, a \tan \left (d x + c\right )^{4} - 30 \, b \tan \left (d x + c\right )^{3} - 20 \, a \tan \left (d x + c\right )^{2} + 15 \, b \tan \left (d x + c\right ) + 12 \, a}{\tan \left (d x + c\right )^{5}}}{60 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.92, size = 111, normalized size = 1.19 \begin {gather*} \frac {30 \, 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 )^{5} - 15 \, {\left (4 \, a d x - 3 \, b\right )} \tan \left (d x + c\right )^{5} - 60 \, a \tan \left (d x + c\right )^{4} + 30 \, b \tan \left (d x + c\right )^{3} + 20 \, a \tan \left (d x + c\right )^{2} - 15 \, b \tan \left (d x + c\right ) - 12 \, a}{60 \, d \tan \left (d x + c\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.73, size = 124, normalized size = 1.33 \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 ^{6}{\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 {a}{5 d \tan ^{5}{\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 )}} - \frac {b}{4 d \tan ^{4}{\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. 198 vs.
\(2 (85) = 170\).
time = 0.79, size = 198, normalized size = 2.13 \begin {gather*} \frac {6 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 70 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 180 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 960 \, {\left (d x + c\right )} a - 960 \, b \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) + 960 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 660 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {2192 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 660 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 180 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 70 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.14, size = 116, normalized size = 1.25 \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 )}^5\,\left (a\,{\mathrm {tan}\left (c+d\,x\right )}^4-\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^3}{2}-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{3}+\frac {b\,\mathrm {tan}\left (c+d\,x\right )}{4}+\frac {a}{5}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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