Optimal. Leaf size=123 \[ \frac {a^3 b \tanh ^{-1}\left (\frac {a \cos (x)-b \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac {a^2 b \cos (x)}{\left (a^2+b^2\right )^2}-\frac {b \cos ^3(x)}{3 \left (a^2+b^2\right )}-\frac {a b^2 \sin (x)}{\left (a^2+b^2\right )^2}+\frac {a \sin (x)}{a^2+b^2}-\frac {a \sin ^3(x)}{3 \left (a^2+b^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.16, antiderivative size = 123, normalized size of antiderivative = 1.00, 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 = {3599, 3188,
2713, 2645, 30, 3179, 2717, 3153, 212} \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 212
Rule 2645
Rule 2713
Rule 2717
Rule 3153
Rule 3179
Rule 3188
Rule 3599
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 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (x)\right )}{a^2+b^2}-\frac {b \text {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 ) \text {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 = 1.20, size = 112, normalized size = 0.91 \begin {gather*} -\frac {2 a^3 b \tanh ^{-1}\left (\frac {-a+b \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac {3 b \left (5 a^2+b^2\right ) \cos (x)+b \left (a^2+b^2\right ) \cos (3 x)-2 a \left (5 a^2-b^2+\left (a^2+b^2\right ) \cos (2 x)\right ) \sin (x)}{12 \left (a^2+b^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.39, size = 170, normalized size = 1.38
method | result | size |
default | \(-\frac {2 \left (-a^{3} \left (\tan ^{5}\left (\frac {x}{2}\right )\right )+\left (2 a^{2} b +b^{3}\right ) \left (\tan ^{4}\left (\frac {x}{2}\right )\right )+\left (-\frac {2}{3} a^{3}+\frac {4}{3} a \,b^{2}\right ) \left (\tan ^{3}\left (\frac {x}{2}\right )\right )+2 a^{2} b \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-a^{3} \tan \left (\frac {x}{2}\right )+\frac {4 a^{2} b}{3}+\frac {b^{3}}{3}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{3}}+\frac {4 a^{3} b \arctanh \left (\frac {-2 b \tan \left (\frac {x}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (2 a^{4}+4 a^{2} b^{2}+2 b^{4}\right ) \sqrt {a^{2}+b^{2}}}\) | \(170\) |
risch | \(\frac {{\mathrm e}^{i x} b}{16 i a b +8 a^{2}-8 b^{2}}-\frac {3 i {\mathrm e}^{i x} a}{8 \left (2 i a b +a^{2}-b^{2}\right )}+\frac {{\mathrm e}^{-i x} b}{8 \left (-i b +a \right )^{2}}+\frac {3 i {\mathrm e}^{-i x} a}{8 \left (-i b +a \right )^{2}}+\frac {b \,a^{3} \ln \left ({\mathrm e}^{i x}-\frac {i a^{4} b +2 i a^{2} b^{3}+i b^{5}-a^{5}-2 a^{3} b^{2}-a \,b^{4}}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}-\frac {b \,a^{3} \ln \left ({\mathrm e}^{i x}+\frac {i a^{4} b +2 i a^{2} b^{3}+i b^{5}-a^{5}-2 a^{3} b^{2}-a \,b^{4}}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}+\frac {b \cos \left (3 x \right )}{-12 a^{2}-12 b^{2}}-\frac {a \sin \left (3 x \right )}{12 \left (-a^{2}-b^{2}\right )}\) | \(272\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 279 vs.
\(2 (115) = 230\).
time = 0.53, size = 279, normalized size = 2.27 \begin {gather*} \frac {a^{3} b \log \left (\frac {a - \frac {b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \sqrt {a^{2} + b^{2}}}{a - \frac {b \sin \left (x\right )}{\cos \left (x\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (4 \, a^{2} b + b^{3} - \frac {3 \, a^{3} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {6 \, a^{2} b \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {3 \, a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} - \frac {2 \, {\left (a^{3} - 2 \, a b^{2}\right )} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {3 \, {\left (2 \, a^{2} b + b^{3}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}}\right )}}{3 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4} + \frac {3 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {3 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.90, size = 214, normalized size = 1.74 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos ^{3}{\left (x \right )}}{a + b \cot {\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.45, size = 201, normalized size = 1.63 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.67, size = 291, normalized size = 2.37 \begin {gather*} \frac {2\,a^3\,b\,\mathrm {atanh}\left (\frac {2\,a\,b^4+2\,a^5+4\,a^3\,b^2-2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}{2\,{\left (a^2+b^2\right )}^{5/2}}\right )}{{\left (a^2+b^2\right )}^{5/2}}-\frac {\frac {2\,\left (4\,a^2\,b+b^3\right )}{3\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3\,\left (2\,a\,b^2-a^3\right )}{3\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {2\,a^3\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^4+2\,a^2\,b^2+b^4}+\frac {2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4\,\left (2\,a^2\,b+b^3\right )}{a^4+2\,a^2\,b^2+b^4}-\frac {2\,a^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5}{a^4+2\,a^2\,b^2+b^4}+\frac {4\,a^2\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{a^4+2\,a^2\,b^2+b^4}}{{\mathrm {tan}\left (\frac {x}{2}\right )}^6+3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________