Optimal. Leaf size=55 \[ \frac{x}{a}-\frac{\tan ^5(x) (1-\csc (x))}{5 a}+\frac{\tan ^3(x) (5-4 \csc (x))}{15 a}-\frac{\tan (x) (15-8 \csc (x))}{15 a} \]
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Rubi [A] time = 0.102983, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3888, 3882, 8} \[ \frac{x}{a}-\frac{\tan ^5(x) (1-\csc (x))}{5 a}+\frac{\tan ^3(x) (5-4 \csc (x))}{15 a}-\frac{\tan (x) (15-8 \csc (x))}{15 a} \]
Antiderivative was successfully verified.
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Rule 3888
Rule 3882
Rule 8
Rubi steps
\begin{align*} \int \frac{\tan ^4(x)}{a+a \csc (x)} \, dx &=\frac{\int (-a+a \csc (x)) \tan ^6(x) \, dx}{a^2}\\ &=-\frac{(1-\csc (x)) \tan ^5(x)}{5 a}+\frac{\int (5 a-4 a \csc (x)) \tan ^4(x) \, dx}{5 a^2}\\ &=\frac{(5-4 \csc (x)) \tan ^3(x)}{15 a}-\frac{(1-\csc (x)) \tan ^5(x)}{5 a}+\frac{\int (-15 a+8 a \csc (x)) \tan ^2(x) \, dx}{15 a^2}\\ &=-\frac{(15-8 \csc (x)) \tan (x)}{15 a}+\frac{(5-4 \csc (x)) \tan ^3(x)}{15 a}-\frac{(1-\csc (x)) \tan ^5(x)}{5 a}+\frac{\int 15 a \, dx}{15 a^2}\\ &=\frac{x}{a}-\frac{(15-8 \csc (x)) \tan (x)}{15 a}+\frac{(5-4 \csc (x)) \tan ^3(x)}{15 a}-\frac{(1-\csc (x)) \tan ^5(x)}{5 a}\\ \end{align*}
Mathematica [B] time = 0.148421, size = 111, normalized size = 2.02 \[ \frac{-64 \sin (x)+240 x \sin (2 x)-178 \sin (2 x)-128 \sin (3 x)+120 x \sin (4 x)-89 \sin (4 x)+6 (120 x-89) \cos (x)+128 \cos (2 x)+240 x \cos (3 x)-178 \cos (3 x)+184 \cos (4 x)+200}{960 a \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )^3 \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.053, size = 102, normalized size = 1.9 \begin{align*} 2\,{\frac{\arctan \left ( \tan \left ( x/2 \right ) \right ) }{a}}+{\frac{2}{5\,a} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-5}}-{\frac{1}{a} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-4}}+{\frac{1}{a} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}+{\frac{11}{8\,a} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{1}{6\,a} \left ( \tan \left ({\frac{x}{2}} \right ) -1 \right ) ^{-3}}-{\frac{1}{4\,a} \left ( \tan \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}+{\frac{5}{8\,a} \left ( \tan \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.48135, size = 262, normalized size = 4.76 \begin{align*} \frac{2 \,{\left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac{46 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac{13 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{100 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{35 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} - \frac{30 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac{15 \, \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} + 8\right )}}{15 \,{\left (a + \frac{2 \, a \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac{2 \, a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac{6 \, a \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{6 \, a \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac{2 \, a \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac{2 \, a \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} - \frac{a \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}}\right )}} + \frac{2 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.488463, size = 174, normalized size = 3.16 \begin{align*} \frac{15 \, x \cos \left (x\right )^{3} + 23 \, \cos \left (x\right )^{4} - 19 \, \cos \left (x\right )^{2} +{\left (15 \, x \cos \left (x\right )^{3} - 8 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) + 4}{15 \,{\left (a \cos \left (x\right )^{3} \sin \left (x\right ) + a \cos \left (x\right )^{3}\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*} \frac{\int \frac{\tan ^{4}{\left (x \right )}}{\csc{\left (x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.40345, size = 108, normalized size = 1.96 \begin{align*} \frac{x}{a} + \frac{15 \, \tan \left (\frac{1}{2} \, x\right )^{2} - 36 \, \tan \left (\frac{1}{2} \, x\right ) + 17}{24 \, a{\left (\tan \left (\frac{1}{2} \, x\right ) - 1\right )}^{3}} + \frac{55 \, \tan \left (\frac{1}{2} \, x\right )^{4} + 260 \, \tan \left (\frac{1}{2} \, x\right )^{3} + 450 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 300 \, \tan \left (\frac{1}{2} \, x\right ) + 71}{40 \, a{\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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