Optimal. Leaf size=66 \[ -\frac {5 \cos ^3(a+b x)}{6 b}-\frac {5 \cos (a+b x)}{2 b}-\frac {\cos ^3(a+b x) \cot ^2(a+b x)}{2 b}+\frac {5 \tanh ^{-1}(\cos (a+b x))}{2 b} \]
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Rubi [A] time = 0.04, antiderivative size = 66, 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 = {2592, 288, 302, 206} \[ -\frac {5 \cos ^3(a+b x)}{6 b}-\frac {5 \cos (a+b x)}{2 b}-\frac {\cos ^3(a+b x) \cot ^2(a+b x)}{2 b}+\frac {5 \tanh ^{-1}(\cos (a+b x))}{2 b} \]
Antiderivative was successfully verified.
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Rule 206
Rule 288
Rule 302
Rule 2592
Rubi steps
\begin {align*} \int \cos ^3(a+b x) \cot ^3(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^2} \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {\cos ^3(a+b x) \cot ^2(a+b x)}{2 b}+\frac {5 \operatorname {Subst}\left (\int \frac {x^4}{1-x^2} \, dx,x,\cos (a+b x)\right )}{2 b}\\ &=-\frac {\cos ^3(a+b x) \cot ^2(a+b x)}{2 b}+\frac {5 \operatorname {Subst}\left (\int \left (-1-x^2+\frac {1}{1-x^2}\right ) \, dx,x,\cos (a+b x)\right )}{2 b}\\ &=-\frac {5 \cos (a+b x)}{2 b}-\frac {5 \cos ^3(a+b x)}{6 b}-\frac {\cos ^3(a+b x) \cot ^2(a+b x)}{2 b}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (a+b x)\right )}{2 b}\\ &=\frac {5 \tanh ^{-1}(\cos (a+b x))}{2 b}-\frac {5 \cos (a+b x)}{2 b}-\frac {5 \cos ^3(a+b x)}{6 b}-\frac {\cos ^3(a+b x) \cot ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 103, normalized size = 1.56 \[ -\frac {9 \cos (a+b x)}{4 b}-\frac {\cos (3 (a+b x))}{12 b}-\frac {\csc ^2\left (\frac {1}{2} (a+b x)\right )}{8 b}+\frac {\sec ^2\left (\frac {1}{2} (a+b x)\right )}{8 b}-\frac {5 \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{2 b}+\frac {5 \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 93, normalized size = 1.41 \[ -\frac {4 \, \cos \left (b x + a\right )^{5} + 20 \, \cos \left (b x + a\right )^{3} - 15 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 30 \, \cos \left (b x + a\right )}{12 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.60, size = 163, normalized size = 2.47 \[ \frac {\frac {3 \, {\left (\frac {10 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}}{\cos \left (b x + a\right ) - 1} - \frac {3 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {16 \, {\left (\frac {12 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {9 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - 7\right )}}{{\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{3}} - 30 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{24 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 81, normalized size = 1.23 \[ -\frac {\cos ^{7}\left (b x +a \right )}{2 b \sin \left (b x +a \right )^{2}}-\frac {\cos ^{5}\left (b x +a \right )}{2 b}-\frac {5 \left (\cos ^{3}\left (b x +a \right )\right )}{6 b}-\frac {5 \cos \left (b x +a \right )}{2 b}-\frac {5 \ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 66, normalized size = 1.00 \[ -\frac {4 \, \cos \left (b x + a\right )^{3} - \frac {6 \, \cos \left (b x + a\right )}{\cos \left (b x + a\right )^{2} - 1} + 24 \, \cos \left (b x + a\right ) - 15 \, \log \left (\cos \left (b x + a\right ) + 1\right ) + 15 \, \log \left (\cos \left (b x + a\right ) - 1\right )}{12 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.35, size = 129, normalized size = 1.95 \[ \frac {{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^2}{8\,b}-\frac {5\,\ln \left (\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )\right )}{2\,b}-\frac {\frac {49\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^6}{8}+\frac {67\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^4}{8}+\frac {121\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^2}{24}+\frac {1}{8}}{b\,\left ({\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^8+3\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^6+3\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^4+{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.08, size = 719, normalized size = 10.89 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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