Optimal. Leaf size=57 \[ -\frac {5 \cot ^3(a+b x)}{6 b}+\frac {5 \cot (a+b x)}{2 b}+\frac {\cos ^2(a+b x) \cot ^3(a+b x)}{2 b}+\frac {5 x}{2} \]
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Rubi [A] time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2591, 288, 302, 203} \[ -\frac {5 \cot ^3(a+b x)}{6 b}+\frac {5 \cot (a+b x)}{2 b}+\frac {\cos ^2(a+b x) \cot ^3(a+b x)}{2 b}+\frac {5 x}{2} \]
Antiderivative was successfully verified.
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Rule 203
Rule 288
Rule 302
Rule 2591
Rubi steps
\begin {align*} \int \cos ^2(a+b x) \cot ^4(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^2} \, dx,x,\cot (a+b x)\right )}{b}\\ &=\frac {\cos ^2(a+b x) \cot ^3(a+b x)}{2 b}-\frac {5 \operatorname {Subst}\left (\int \frac {x^4}{1+x^2} \, dx,x,\cot (a+b x)\right )}{2 b}\\ &=\frac {\cos ^2(a+b x) \cot ^3(a+b x)}{2 b}-\frac {5 \operatorname {Subst}\left (\int \left (-1+x^2+\frac {1}{1+x^2}\right ) \, dx,x,\cot (a+b x)\right )}{2 b}\\ &=\frac {5 \cot (a+b x)}{2 b}-\frac {5 \cot ^3(a+b x)}{6 b}+\frac {\cos ^2(a+b x) \cot ^3(a+b x)}{2 b}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cot (a+b x)\right )}{2 b}\\ &=\frac {5 x}{2}+\frac {5 \cot (a+b x)}{2 b}-\frac {5 \cot ^3(a+b x)}{6 b}+\frac {\cos ^2(a+b x) \cot ^3(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 43, normalized size = 0.75 \[ \frac {30 (a+b x)+3 \sin (2 (a+b x))-4 \cot (a+b x) \left (\csc ^2(a+b x)-7\right )}{12 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 79, normalized size = 1.39 \[ -\frac {3 \, \cos \left (b x + a\right )^{5} - 20 \, \cos \left (b x + a\right )^{3} - 15 \, {\left (b x \cos \left (b x + a\right )^{2} - b x\right )} \sin \left (b x + a\right ) + 15 \, \cos \left (b x + a\right )}{6 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.74, size = 55, normalized size = 0.96 \[ \frac {15 \, b x + 15 \, a + \frac {3 \, \tan \left (b x + a\right )}{\tan \left (b x + a\right )^{2} + 1} + \frac {2 \, {\left (6 \, \tan \left (b x + a\right )^{2} - 1\right )}}{\tan \left (b x + a\right )^{3}}}{6 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 84, normalized size = 1.47 \[ \frac {-\frac {\cos ^{7}\left (b x +a \right )}{3 \sin \left (b x +a \right )^{3}}+\frac {4 \left (\cos ^{7}\left (b x +a \right )\right )}{3 \sin \left (b x +a \right )}+\frac {4 \left (\cos ^{5}\left (b x +a \right )+\frac {5 \left (\cos ^{3}\left (b x +a \right )\right )}{4}+\frac {15 \cos \left (b x +a \right )}{8}\right ) \sin \left (b x +a \right )}{3}+\frac {5 b x}{2}+\frac {5 a}{2}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 55, normalized size = 0.96 \[ \frac {15 \, b x + 15 \, a + \frac {15 \, \tan \left (b x + a\right )^{4} + 10 \, \tan \left (b x + a\right )^{2} - 2}{\tan \left (b x + a\right )^{5} + \tan \left (b x + a\right )^{3}}}{6 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.72, size = 46, normalized size = 0.81 \[ \frac {5\,x}{2}+\frac {{\cos \left (a+b\,x\right )}^2\,\left (\frac {5\,{\mathrm {tan}\left (a+b\,x\right )}^4}{2}+\frac {5\,{\mathrm {tan}\left (a+b\,x\right )}^2}{3}-\frac {1}{3}\right )}{b\,{\mathrm {tan}\left (a+b\,x\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.72, size = 97, normalized size = 1.70 \[ \begin {cases} \frac {5 x \sin ^{2}{\left (a + b x \right )}}{2} + \frac {5 x \cos ^{2}{\left (a + b x \right )}}{2} + \frac {5 \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{2 b} + \frac {5 \cos ^{3}{\left (a + b x \right )}}{3 b \sin {\left (a + b x \right )}} - \frac {\cos ^{5}{\left (a + b x \right )}}{3 b \sin ^{3}{\left (a + b x \right )}} & \text {for}\: b \neq 0 \\\frac {x \cos ^{6}{\relax (a )}}{\sin ^{4}{\relax (a )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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