Optimal. Leaf size=123 \[ -\frac{a \sin ^3(x)}{3 \left (a^2+b^2\right )}+\frac{a \sin (x)}{a^2+b^2}-\frac{a b^2 \sin (x)}{\left (a^2+b^2\right )^2}-\frac{b \cos ^3(x)}{3 \left (a^2+b^2\right )}-\frac{a^2 b \cos (x)}{\left (a^2+b^2\right )^2}+\frac{a^3 b \tanh ^{-1}\left (\frac{a \cos (x)-b \sin (x)}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}} \]
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Rubi [A] time = 0.208198, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.692, Rules used = {3518, 3109, 2633, 2565, 30, 3100, 2637, 3074, 206} \[ -\frac{a \sin ^3(x)}{3 \left (a^2+b^2\right )}+\frac{a \sin (x)}{a^2+b^2}-\frac{a b^2 \sin (x)}{\left (a^2+b^2\right )^2}-\frac{b \cos ^3(x)}{3 \left (a^2+b^2\right )}-\frac{a^2 b \cos (x)}{\left (a^2+b^2\right )^2}+\frac{a^3 b \tanh ^{-1}\left (\frac{a \cos (x)-b \sin (x)}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3518
Rule 3109
Rule 2633
Rule 2565
Rule 30
Rule 3100
Rule 2637
Rule 3074
Rule 206
Rubi steps
\begin{align*} \int \frac{\cos ^3(x)}{a+b \cot (x)} \, dx &=-\int \frac{\cos ^3(x) \sin (x)}{-b \cos (x)-a \sin (x)} \, dx\\ &=\frac{a \int \cos ^3(x) \, dx}{a^2+b^2}+\frac{b \int \cos ^2(x) \sin (x) \, dx}{a^2+b^2}+\frac{(a b) \int \frac{\cos ^2(x)}{-b \cos (x)-a \sin (x)} \, dx}{a^2+b^2}\\ &=-\frac{a^2 b \cos (x)}{\left (a^2+b^2\right )^2}+\frac{\left (a^3 b\right ) \int \frac{1}{-b \cos (x)-a \sin (x)} \, dx}{\left (a^2+b^2\right )^2}-\frac{\left (a b^2\right ) \int \cos (x) \, dx}{\left (a^2+b^2\right )^2}-\frac{a \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (x)\right )}{a^2+b^2}-\frac{b \operatorname{Subst}\left (\int x^2 \, dx,x,\cos (x)\right )}{a^2+b^2}\\ &=-\frac{a^2 b \cos (x)}{\left (a^2+b^2\right )^2}-\frac{b \cos ^3(x)}{3 \left (a^2+b^2\right )}-\frac{a b^2 \sin (x)}{\left (a^2+b^2\right )^2}+\frac{a \sin (x)}{a^2+b^2}-\frac{a \sin ^3(x)}{3 \left (a^2+b^2\right )}-\frac{\left (a^3 b\right ) \operatorname{Subst}\left (\int \frac{1}{a^2+b^2-x^2} \, dx,x,-a \cos (x)+b \sin (x)\right )}{\left (a^2+b^2\right )^2}\\ &=\frac{a^3 b \tanh ^{-1}\left (\frac{a \cos (x)-b \sin (x)}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac{a^2 b \cos (x)}{\left (a^2+b^2\right )^2}-\frac{b \cos ^3(x)}{3 \left (a^2+b^2\right )}-\frac{a b^2 \sin (x)}{\left (a^2+b^2\right )^2}+\frac{a \sin (x)}{a^2+b^2}-\frac{a \sin ^3(x)}{3 \left (a^2+b^2\right )}\\ \end{align*}
Mathematica [A] time = 1.05929, size = 112, normalized size = 0.91 \[ -\frac{3 b \left (5 a^2+b^2\right ) \cos (x)+b \left (a^2+b^2\right ) \cos (3 x)-2 a \sin (x) \left (\left (a^2+b^2\right ) \cos (2 x)+5 a^2-b^2\right )}{12 \left (a^2+b^2\right )^2}-\frac{2 a^3 b \tanh ^{-1}\left (\frac{b \tan \left (\frac{x}{2}\right )-a}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 170, normalized size = 1.4 \begin{align*} -2\,{\frac{-{a}^{3} \left ( \tan \left ( x/2 \right ) \right ) ^{5}+ \left ( 2\,{a}^{2}b+{b}^{3} \right ) \left ( \tan \left ( x/2 \right ) \right ) ^{4}+ \left ( -2/3\,{a}^{3}+4/3\,a{b}^{2} \right ) \left ( \tan \left ( x/2 \right ) \right ) ^{3}+2\,{a}^{2}b \left ( \tan \left ( x/2 \right ) \right ) ^{2}-{a}^{3}\tan \left ( x/2 \right ) +4/3\,{a}^{2}b+1/3\,{b}^{3}}{ \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}-4\,{\frac{{a}^{3}b}{ \left ( 2\,{a}^{4}+4\,{a}^{2}{b}^{2}+2\,{b}^{4} \right ) \sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,b\tan \left ( x/2 \right ) -2\,a}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17203, size = 494, normalized size = 4.02 \begin{align*} \frac{3 \, \sqrt{a^{2} + b^{2}} a^{3} b \log \left (\frac{2 \, a b \cos \left (x\right ) \sin \left (x\right ) -{\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - a^{2} - 2 \, b^{2} - 2 \, \sqrt{a^{2} + b^{2}}{\left (a \cos \left (x\right ) - b \sin \left (x\right )\right )}}{2 \, a b \cos \left (x\right ) \sin \left (x\right ) -{\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + a^{2}}\right ) - 2 \,{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (x\right )^{3} - 6 \,{\left (a^{4} b + a^{2} b^{3}\right )} \cos \left (x\right ) + 2 \,{\left (2 \, a^{5} + a^{3} b^{2} - a b^{4} +{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{6 \,{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos ^{3}{\left (x \right )}}{a + b \cot{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32361, size = 271, normalized size = 2.2 \begin{align*} \frac{a^{3} b \log \left (\frac{{\left | 2 \, b \tan \left (\frac{1}{2} \, x\right ) - 2 \, a - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, b \tan \left (\frac{1}{2} \, x\right ) - 2 \, a + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt{a^{2} + b^{2}}} + \frac{2 \,{\left (3 \, a^{3} \tan \left (\frac{1}{2} \, x\right )^{5} - 6 \, a^{2} b \tan \left (\frac{1}{2} \, x\right )^{4} - 3 \, b^{3} \tan \left (\frac{1}{2} \, x\right )^{4} + 2 \, a^{3} \tan \left (\frac{1}{2} \, x\right )^{3} - 4 \, a b^{2} \tan \left (\frac{1}{2} \, x\right )^{3} - 6 \, a^{2} b \tan \left (\frac{1}{2} \, x\right )^{2} + 3 \, a^{3} \tan \left (\frac{1}{2} \, x\right ) - 4 \, a^{2} b - b^{3}\right )}}{3 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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