Integrand size = 13, antiderivative size = 113 \[ \int \frac {\tan ^4(x)}{a+b \cos (x)} \, dx=\frac {2 (a-b)^{3/2} (a+b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^4}+\frac {b \left (3 a^2-2 b^2\right ) \text {arctanh}(\sin (x))}{2 a^4}-\frac {\left (4 a^2-3 b^2\right ) \tan (x)}{3 a^3}-\frac {b \sec (x) \tan (x)}{2 a^2}+\frac {\sec ^2(x) \tan (x)}{3 a} \]
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Time = 0.64 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2804, 3134, 3080, 3855, 2738, 211} \[ \int \frac {\tan ^4(x)}{a+b \cos (x)} \, dx=\frac {2 (a-b)^{3/2} (a+b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^4}-\frac {b \tan (x) \sec (x)}{2 a^2}+\frac {b \left (3 a^2-2 b^2\right ) \text {arctanh}(\sin (x))}{2 a^4}-\frac {\left (4 a^2-3 b^2\right ) \tan (x)}{3 a^3}+\frac {\tan (x) \sec ^2(x)}{3 a} \]
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Rule 211
Rule 2738
Rule 2804
Rule 3080
Rule 3134
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {b \sec (x) \tan (x)}{2 a^2}+\frac {\sec ^2(x) \tan (x)}{3 a}-\frac {\int \frac {\left (2 \left (4 a^2-3 b^2\right )-a b \cos (x)-3 \left (2 a^2-b^2\right ) \cos ^2(x)\right ) \sec ^2(x)}{a+b \cos (x)} \, dx}{6 a^2} \\ & = -\frac {\left (4 a^2-3 b^2\right ) \tan (x)}{3 a^3}-\frac {b \sec (x) \tan (x)}{2 a^2}+\frac {\sec ^2(x) \tan (x)}{3 a}-\frac {\int \frac {\left (-3 b \left (3 a^2-2 b^2\right )-3 a \left (2 a^2-b^2\right ) \cos (x)\right ) \sec (x)}{a+b \cos (x)} \, dx}{6 a^3} \\ & = -\frac {\left (4 a^2-3 b^2\right ) \tan (x)}{3 a^3}-\frac {b \sec (x) \tan (x)}{2 a^2}+\frac {\sec ^2(x) \tan (x)}{3 a}+\frac {\left (b \left (3 a^2-2 b^2\right )\right ) \int \sec (x) \, dx}{2 a^4}+\frac {\left (a^2-b^2\right )^2 \int \frac {1}{a+b \cos (x)} \, dx}{a^4} \\ & = \frac {b \left (3 a^2-2 b^2\right ) \text {arctanh}(\sin (x))}{2 a^4}-\frac {\left (4 a^2-3 b^2\right ) \tan (x)}{3 a^3}-\frac {b \sec (x) \tan (x)}{2 a^2}+\frac {\sec ^2(x) \tan (x)}{3 a}+\frac {\left (2 \left (a^2-b^2\right )^2\right ) \text {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{a^4} \\ & = \frac {2 (a-b)^{3/2} (a+b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{a^4}+\frac {b \left (3 a^2-2 b^2\right ) \text {arctanh}(\sin (x))}{2 a^4}-\frac {\left (4 a^2-3 b^2\right ) \tan (x)}{3 a^3}-\frac {b \sec (x) \tan (x)}{2 a^2}+\frac {\sec ^2(x) \tan (x)}{3 a} \\ \end{align*}
Time = 0.81 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.68 \[ \int \frac {\tan ^4(x)}{a+b \cos (x)} \, dx=-\frac {48 \left (-a^2+b^2\right )^{3/2} \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )+\sec ^3(x) \left (9 b \left (3 a^2-2 b^2\right ) \cos (x) \left (\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-\log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )\right )+3 b \left (3 a^2-2 b^2\right ) \cos (3 x) \left (\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-\log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )\right )+2 a \left (-3 b^2 \sin (x)+3 a b \sin (2 x)+\left (4 a^2-3 b^2\right ) \sin (3 x)\right )\right )}{24 a^4} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(218\) vs. \(2(95)=190\).
Time = 1.10 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.94
method | result | size |
default | \(-\frac {1}{3 a \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {-a -b}{2 a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {-2 a^{2}+a b +2 b^{2}}{2 a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )}+\frac {b \left (3 a^{2}-2 b^{2}\right ) \ln \left (\tan \left (\frac {x}{2}\right )+1\right )}{2 a^{4}}+\frac {2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {x}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{4} \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {1}{3 a \left (\tan \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {a +b}{2 a^{2} \left (\tan \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {-2 a^{2}+a b +2 b^{2}}{2 a^{3} \left (\tan \left (\frac {x}{2}\right )-1\right )}-\frac {b \left (3 a^{2}-2 b^{2}\right ) \ln \left (\tan \left (\frac {x}{2}\right )-1\right )}{2 a^{4}}\) | \(219\) |
risch | \(\frac {i \left (3 a b \,{\mathrm e}^{5 i x}-12 a^{2} {\mathrm e}^{4 i x}+6 b^{2} {\mathrm e}^{4 i x}-12 \,{\mathrm e}^{2 i x} a^{2}+12 \,{\mathrm e}^{2 i x} b^{2}-3 b \,{\mathrm e}^{i x} a -8 a^{2}+6 b^{2}\right )}{3 a^{3} \left ({\mathrm e}^{2 i x}+1\right )^{3}}-\frac {3 b \ln \left ({\mathrm e}^{i x}-i\right )}{2 a^{2}}+\frac {b^{3} \ln \left ({\mathrm e}^{i x}-i\right )}{a^{4}}+\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i x}-\frac {i \sqrt {-a^{2}+b^{2}}-a}{b}\right )}{a^{2}}-\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i x}-\frac {i \sqrt {-a^{2}+b^{2}}-a}{b}\right ) b^{2}}{a^{4}}-\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i x}+\frac {i \sqrt {-a^{2}+b^{2}}+a}{b}\right )}{a^{2}}+\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i x}+\frac {i \sqrt {-a^{2}+b^{2}}+a}{b}\right ) b^{2}}{a^{4}}+\frac {3 b \ln \left ({\mathrm e}^{i x}+i\right )}{2 a^{2}}-\frac {b^{3} \ln \left ({\mathrm e}^{i x}+i\right )}{a^{4}}\) | \(331\) |
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Time = 0.38 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.94 \[ \int \frac {\tan ^4(x)}{a+b \cos (x)} \, dx=\left [-\frac {6 \, {\left (a^{2} - b^{2}\right )} \sqrt {-a^{2} + b^{2}} \cos \left (x\right )^{3} \log \left (\frac {2 \, a b \cos \left (x\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (x\right ) + b\right )} \sin \left (x\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (x\right )^{2} + 2 \, a b \cos \left (x\right ) + a^{2}}\right ) - 3 \, {\left (3 \, a^{2} b - 2 \, b^{3}\right )} \cos \left (x\right )^{3} \log \left (\sin \left (x\right ) + 1\right ) + 3 \, {\left (3 \, a^{2} b - 2 \, b^{3}\right )} \cos \left (x\right )^{3} \log \left (-\sin \left (x\right ) + 1\right ) + 2 \, {\left (3 \, a^{2} b \cos \left (x\right ) - 2 \, a^{3} + 2 \, {\left (4 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{12 \, a^{4} \cos \left (x\right )^{3}}, \frac {12 \, {\left (a^{2} - b^{2}\right )}^{\frac {3}{2}} \arctan \left (-\frac {a \cos \left (x\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (x\right )}\right ) \cos \left (x\right )^{3} + 3 \, {\left (3 \, a^{2} b - 2 \, b^{3}\right )} \cos \left (x\right )^{3} \log \left (\sin \left (x\right ) + 1\right ) - 3 \, {\left (3 \, a^{2} b - 2 \, b^{3}\right )} \cos \left (x\right )^{3} \log \left (-\sin \left (x\right ) + 1\right ) - 2 \, {\left (3 \, a^{2} b \cos \left (x\right ) - 2 \, a^{3} + 2 \, {\left (4 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{12 \, a^{4} \cos \left (x\right )^{3}}\right ] \]
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\[ \int \frac {\tan ^4(x)}{a+b \cos (x)} \, dx=\int \frac {\tan ^{4}{\left (x \right )}}{a + b \cos {\left (x \right )}}\, dx \]
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Exception generated. \[ \int \frac {\tan ^4(x)}{a+b \cos (x)} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (95) = 190\).
Time = 0.31 (sec) , antiderivative size = 226, normalized size of antiderivative = 2.00 \[ \int \frac {\tan ^4(x)}{a+b \cos (x)} \, dx=\frac {{\left (3 \, a^{2} b - 2 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) + 1 \right |}\right )}{2 \, a^{4}} - \frac {{\left (3 \, a^{2} b - 2 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) - 1 \right |}\right )}{2 \, a^{4}} - \frac {2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, x\right ) - b \tan \left (\frac {1}{2} \, x\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{4}} + \frac {6 \, a^{2} \tan \left (\frac {1}{2} \, x\right )^{5} - 3 \, a b \tan \left (\frac {1}{2} \, x\right )^{5} - 6 \, b^{2} \tan \left (\frac {1}{2} \, x\right )^{5} - 20 \, a^{2} \tan \left (\frac {1}{2} \, x\right )^{3} + 12 \, b^{2} \tan \left (\frac {1}{2} \, x\right )^{3} + 6 \, a^{2} \tan \left (\frac {1}{2} \, x\right ) + 3 \, a b \tan \left (\frac {1}{2} \, x\right ) - 6 \, b^{2} \tan \left (\frac {1}{2} \, x\right )}{3 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} - 1\right )}^{3} a^{3}} \]
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Time = 14.85 (sec) , antiderivative size = 1666, normalized size of antiderivative = 14.74 \[ \int \frac {\tan ^4(x)}{a+b \cos (x)} \, dx=\text {Too large to display} \]
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