Optimal. Leaf size=120 \[ -\left (\left (a^2-b^2\right ) x\right )-\frac {\left (a^2-b^2\right ) \cot (c+d x)}{d}+\frac {a b \cot ^2(c+d x)}{d}+\frac {\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {a b \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\frac {2 a b \log (\sin (c+d x))}{d} \]
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Rubi [A]
time = 0.14, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3623, 3610,
3612, 3556} \begin {gather*} \frac {\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {\left (a^2-b^2\right ) \cot (c+d x)}{d}-x \left (a^2-b^2\right )-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a b \cot ^4(c+d x)}{2 d}+\frac {a b \cot ^2(c+d x)}{d}+\frac {2 a b \log (\sin (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3610
Rule 3612
Rule 3623
Rubi steps
\begin {align*} \int \cot ^6(c+d x) (a+b \tan (c+d x))^2 \, dx &=-\frac {a^2 \cot ^5(c+d x)}{5 d}+\int \cot ^5(c+d x) \left (2 a b-\left (a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=-\frac {a b \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\int \cot ^4(c+d x) \left (-a^2+b^2-2 a b \tan (c+d x)\right ) \, dx\\ &=\frac {\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {a b \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\int \cot ^3(c+d x) \left (-2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=\frac {a b \cot ^2(c+d x)}{d}+\frac {\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {a b \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\int \cot ^2(c+d x) \left (a^2-b^2+2 a b \tan (c+d x)\right ) \, dx\\ &=-\frac {\left (a^2-b^2\right ) \cot (c+d x)}{d}+\frac {a b \cot ^2(c+d x)}{d}+\frac {\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {a b \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\int \cot (c+d x) \left (2 a b-\left (a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=-\left (a^2-b^2\right ) x-\frac {\left (a^2-b^2\right ) \cot (c+d x)}{d}+\frac {a b \cot ^2(c+d x)}{d}+\frac {\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {a b \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+(2 a b) \int \cot (c+d x) \, dx\\ &=-\left (a^2-b^2\right ) x-\frac {\left (a^2-b^2\right ) \cot (c+d x)}{d}+\frac {a b \cot ^2(c+d x)}{d}+\frac {\left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {a b \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\frac {2 a 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 = 1.01, size = 121, normalized size = 1.01 \begin {gather*} -\frac {a^2 \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^2 \cot ^3(c+d x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\tan ^2(c+d x)\right )}{3 d}+\frac {a 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 )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 103, normalized size = 0.86
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+2 a b \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+b^{2} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(103\) |
default | \(\frac {a^{2} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+2 a b \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+b^{2} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(103\) |
norman | \(\frac {\left (-a^{2}+b^{2}\right ) x \left (\tan ^{5}\left (d x +c \right )\right )+\frac {a b \left (\tan ^{3}\left (d x +c \right )\right )}{d}-\frac {a^{2}}{5 d}-\frac {\left (a^{2}-b^{2}\right ) \left (\tan ^{4}\left (d x +c \right )\right )}{d}+\frac {\left (a^{2}-b^{2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{3 d}-\frac {a b \tan \left (d x +c \right )}{2 d}}{\tan \left (d x +c \right )^{5}}+\frac {2 a b \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {a b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\) | \(142\) |
risch | \(-2 i a b x -a^{2} x +b^{2} x -\frac {4 i a b c}{d}-\frac {2 i \left (-60 i a b \,{\mathrm e}^{8 i \left (d x +c \right )}+45 a^{2} {\mathrm e}^{8 i \left (d x +c \right )}-30 b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+120 i a b \,{\mathrm e}^{6 i \left (d x +c \right )}-90 a^{2} {\mathrm e}^{6 i \left (d x +c \right )}+90 b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-120 i a b \,{\mathrm e}^{4 i \left (d x +c \right )}+140 a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-110 b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+60 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}-70 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+70 b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+23 a^{2}-20 b^{2}\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {2 a b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(245\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 124, normalized size = 1.03 \begin {gather*} -\frac {30 \, a b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 \, a b \log \left (\tan \left (d x + c\right )\right ) + 30 \, {\left (a^{2} - b^{2}\right )} {\left (d x + c\right )} - \frac {30 \, a b \tan \left (d x + c\right )^{3} - 30 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{4} - 15 \, a b \tan \left (d x + c\right ) + 10 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{2} - 6 \, a^{2}}{\tan \left (d x + c\right )^{5}}}{30 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.09, size = 141, normalized size = 1.18 \begin {gather*} \frac {30 \, a 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 (2 \, {\left (a^{2} - b^{2}\right )} d x - 3 \, a b\right )} \tan \left (d x + c\right )^{5} + 30 \, a b \tan \left (d x + c\right )^{3} - 30 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{4} - 15 \, a b \tan \left (d x + c\right ) + 10 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{2} - 6 \, a^{2}}{30 \, 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 = 2.31, size = 172, normalized size = 1.43 \begin {gather*} \begin {cases} \tilde {\infty } a^{2} 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 )^{2} \cot ^{6}{\left (c \right )} & \text {for}\: d = 0 \\- a^{2} x - \frac {a^{2}}{d \tan {\left (c + d x \right )}} + \frac {a^{2}}{3 d \tan ^{3}{\left (c + d x \right )}} - \frac {a^{2}}{5 d \tan ^{5}{\left (c + d x \right )}} - \frac {a b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {2 a b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {a b}{d \tan ^{2}{\left (c + d x \right )}} - \frac {a b}{2 d \tan ^{4}{\left (c + d x \right )}} + b^{2} x + \frac {b^{2}}{d \tan {\left (c + d x \right )}} - \frac {b^{2}}{3 d \tan ^{3}{\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. 287 vs.
\(2 (114) = 228\).
time = 1.15, size = 287, normalized size = 2.39 \begin {gather*} \frac {3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 35 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 20 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 180 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 960 \, a b \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) + 960 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 330 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 300 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 480 \, {\left (a^{2} - b^{2}\right )} {\left (d x + c\right )} - \frac {2192 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 330 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 300 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 180 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 35 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 20 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.97, size = 145, normalized size = 1.21 \begin {gather*} \frac {2\,a\,b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^5\,\left ({\mathrm {tan}\left (c+d\,x\right )}^4\,\left (a^2-b^2\right )+\frac {a^2}{5}-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {a^2}{3}-\frac {b^2}{3}\right )+\frac {a\,b\,\mathrm {tan}\left (c+d\,x\right )}{2}-a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,{\left (a-b\,1{}\mathrm {i}\right )}^2\,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 )}^2\,1{}\mathrm {i}}{2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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