Optimal. Leaf size=78 \[ \left (a^2-b^2\right ) x+\frac {\left (a^2-b^2\right ) \cot (c+d x)}{d}-\frac {a b \cot ^2(c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {2 a b \log (\sin (c+d x))}{d} \]
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Rubi [A]
time = 0.10, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 5, 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 (c+d x)}{d}+x \left (a^2-b^2\right )-\frac {a^2 \cot ^3(c+d x)}{3 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 ^4(c+d x) (a+b \tan (c+d x))^2 \, dx &=-\frac {a^2 \cot ^3(c+d x)}{3 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 {a^2 \cot ^3(c+d x)}{3 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 {a^2 \cot ^3(c+d x)}{3 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 {a^2 \cot ^3(c+d x)}{3 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 {a^2 \cot ^3(c+d x)}{3 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 = 0.74, size = 103, normalized size = 1.32 \begin {gather*} -\frac {a^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 {b^2 \cot (c+d x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2(c+d x)\right )}{d}-\frac {a b \left (\cot ^2(c+d x)+2 \log (\cos (c+d x))+2 \log (\tan (c+d x))\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 75, normalized size = 0.96
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+2 a b \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+b^{2} \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(75\) |
default | \(\frac {a^{2} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+2 a b \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+b^{2} \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(75\) |
norman | \(\frac {\left (a^{2}-b^{2}\right ) x \left (\tan ^{3}\left (d x +c \right )\right )+\frac {\left (a^{2}-b^{2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{d}-\frac {a^{2}}{3 d}-\frac {a b \tan \left (d x +c \right )}{d}}{\tan \left (d x +c \right )^{3}}+\frac {a b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}-\frac {2 a b \ln \left (\tan \left (d x +c \right )\right )}{d}\) | \(104\) |
risch | \(2 i a b x +a^{2} x -b^{2} x +\frac {4 i a b c}{d}-\frac {2 i \left (6 i a b \,{\mathrm e}^{4 i \left (d x +c \right )}-6 a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+3 b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-6 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}+6 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-6 b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-4 a^{2}+3 b^{2}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}-\frac {2 a b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(161\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 91, normalized size = 1.17 \begin {gather*} \frac {3 \, a b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 \, a b \log \left (\tan \left (d x + c\right )\right ) + 3 \, {\left (a^{2} - b^{2}\right )} {\left (d x + c\right )} - \frac {3 \, a b \tan \left (d x + c\right ) - 3 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{2} + a^{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 = 1.01, size = 107, normalized size = 1.37 \begin {gather*} -\frac {3 \, 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 )^{3} - 3 \, {\left ({\left (a^{2} - b^{2}\right )} d x - a b\right )} \tan \left (d x + c\right )^{3} + 3 \, a b \tan \left (d x + c\right ) - 3 \, {\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{2} + a^{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 = 1.01, size = 126, normalized size = 1.62 \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 ^{4}{\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 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 )}} - b^{2} x - \frac {b^{2}}{d \tan {\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. 191 vs.
\(2 (76) = 152\).
time = 0.90, size = 191, normalized size = 2.45 \begin {gather*} \frac {a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 48 \, a b \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 48 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, {\left (a^{2} - b^{2}\right )} {\left (d x + c\right )} + \frac {88 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2}}{\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 = 4.02, size = 112, normalized size = 1.44 \begin {gather*} -\frac {{\mathrm {cot}\left (c+d\,x\right )}^3\,\left (\frac {a^2}{3}-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (a^2-b^2\right )+a\,b\,\mathrm {tan}\left (c+d\,x\right )\right )}{d}-\frac {2\,a\,b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\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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