Optimal. Leaf size=75 \[ -\frac {a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^2(c+d x)}{2 d}+\frac {a \log (\sin (c+d x))}{d}-\frac {b \cot ^3(c+d x)}{3 d}+\frac {b \cot (c+d x)}{d}+b x \]
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Rubi [A] time = 0.11, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3529, 3531, 3475} \[ -\frac {a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^2(c+d x)}{2 d}+\frac {a \log (\sin (c+d x))}{d}-\frac {b \cot ^3(c+d x)}{3 d}+\frac {b \cot (c+d x)}{d}+b x \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3529
Rule 3531
Rubi steps
\begin {align*} \int \cot ^5(c+d x) (a+b \tan (c+d x)) \, dx &=-\frac {a \cot ^4(c+d x)}{4 d}+\int \cot ^4(c+d x) (b-a \tan (c+d x)) \, dx\\ &=-\frac {b \cot ^3(c+d x)}{3 d}-\frac {a \cot ^4(c+d x)}{4 d}+\int \cot ^3(c+d x) (-a-b \tan (c+d x)) \, dx\\ &=\frac {a \cot ^2(c+d x)}{2 d}-\frac {b \cot ^3(c+d x)}{3 d}-\frac {a \cot ^4(c+d x)}{4 d}+\int \cot ^2(c+d x) (-b+a \tan (c+d x)) \, dx\\ &=\frac {b \cot (c+d x)}{d}+\frac {a \cot ^2(c+d x)}{2 d}-\frac {b \cot ^3(c+d x)}{3 d}-\frac {a \cot ^4(c+d x)}{4 d}+\int \cot (c+d x) (a+b \tan (c+d x)) \, dx\\ &=b x+\frac {b \cot (c+d x)}{d}+\frac {a \cot ^2(c+d x)}{2 d}-\frac {b \cot ^3(c+d x)}{3 d}-\frac {a \cot ^4(c+d x)}{4 d}+a \int \cot (c+d x) \, dx\\ &=b x+\frac {b \cot (c+d x)}{d}+\frac {a \cot ^2(c+d x)}{2 d}-\frac {b \cot ^3(c+d x)}{3 d}-\frac {a \cot ^4(c+d x)}{4 d}+\frac {a \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [C] time = 0.40, size = 82, normalized size = 1.09 \[ \frac {a \left (-\cot ^4(c+d x)+2 \cot ^2(c+d x)+4 \log (\tan (c+d x))+4 \log (\cos (c+d x))\right )}{4 d}-\frac {b \cot ^3(c+d x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\tan ^2(c+d x)\right )}{3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 100, normalized size = 1.33 \[ \frac {6 \, a \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{4} + 3 \, {\left (4 \, b d x + 3 \, a\right )} \tan \left (d x + c\right )^{4} + 12 \, b \tan \left (d x + c\right )^{3} + 6 \, a \tan \left (d x + c\right )^{2} - 4 \, b \tan \left (d x + c\right ) - 3 \, a}{12 \, d \tan \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.77, size = 169, normalized size = 2.25 \[ -\frac {3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 192 \, {\left (d x + c\right )} b + 192 \, a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 192 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 120 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {400 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 76, normalized size = 1.01 \[ -\frac {a \left (\cot ^{4}\left (d x +c \right )\right )}{4 d}+\frac {a \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {b \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}+\frac {b \cot \left (d x +c \right )}{d}+b x +\frac {b c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 82, normalized size = 1.09 \[ \frac {12 \, {\left (d x + c\right )} b - 6 \, a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 12 \, a \log \left (\tan \left (d x + c\right )\right ) + \frac {12 \, b \tan \left (d x + c\right )^{3} + 6 \, a \tan \left (d x + c\right )^{2} - 4 \, b \tan \left (d x + c\right ) - 3 \, a}{\tan \left (d x + c\right )^{4}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.11, size = 107, normalized size = 1.43 \[ \frac {a\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (\frac {a}{2}-\frac {b\,1{}\mathrm {i}}{2}\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^4\,\left (-b\,{\mathrm {tan}\left (c+d\,x\right )}^3-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+\frac {b\,\mathrm {tan}\left (c+d\,x\right )}{3}+\frac {a}{4}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (\frac {a}{2}+\frac {b\,1{}\mathrm {i}}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.10, size = 110, normalized size = 1.47 \[ \begin {cases} \tilde {\infty } a x & \text {for}\: \left (c = 0 \vee c = - d x\right ) \wedge \left (c = - d x \vee d = 0\right ) \\x \left (a + b \tan {\relax (c )}\right ) \cot ^{5}{\relax (c )} & \text {for}\: d = 0 \\- \frac {a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {a}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac {a}{4 d \tan ^{4}{\left (c + d x \right )}} + b x + \frac {b}{d \tan {\left (c + d x \right )}} - \frac {b}{3 d \tan ^{3}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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