Optimal. Leaf size=34 \[ -\frac {\tan ^5(x)}{5 a}-\frac {\tan ^3(x)}{3 a}+\frac {\sec ^5(x)}{5 a} \]
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Rubi [A] time = 0.12, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3872, 2839, 2606, 30, 2607, 14} \[ -\frac {\tan ^5(x)}{5 a}-\frac {\tan ^3(x)}{3 a}+\frac {\sec ^5(x)}{5 a} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 2606
Rule 2607
Rule 2839
Rule 3872
Rubi steps
\begin {align*} \int \frac {\sec ^4(x)}{a+a \csc (x)} \, dx &=\int \frac {\sec ^3(x) \tan (x)}{a+a \sin (x)} \, dx\\ &=\frac {\int \sec ^5(x) \tan (x) \, dx}{a}-\frac {\int \sec ^4(x) \tan ^2(x) \, dx}{a}\\ &=\frac {\operatorname {Subst}\left (\int x^4 \, dx,x,\sec (x)\right )}{a}-\frac {\operatorname {Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,\tan (x)\right )}{a}\\ &=\frac {\sec ^5(x)}{5 a}-\frac {\operatorname {Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,\tan (x)\right )}{a}\\ &=\frac {\sec ^5(x)}{5 a}-\frac {\tan ^3(x)}{3 a}-\frac {\tan ^5(x)}{5 a}\\ \end {align*}
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Mathematica [B] time = 0.15, size = 85, normalized size = 2.50 \[ -\frac {-96 \sin (x)+18 \sin (2 x)-32 \sin (3 x)+9 \sin (4 x)+54 \cos (x)+32 \cos (2 x)+18 \cos (3 x)+16 \cos (4 x)-240}{960 a \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^3 \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 45, normalized size = 1.32 \[ -\frac {2 \, \cos \relax (x)^{4} - \cos \relax (x)^{2} - {\left (2 \, \cos \relax (x)^{2} + 1\right )} \sin \relax (x) - 4}{15 \, {\left (a \cos \relax (x)^{3} \sin \relax (x) + a \cos \relax (x)^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 75, normalized size = 2.21 \[ -\frac {9 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 12 \, \tan \left (\frac {1}{2} \, x\right ) + 7}{24 \, a {\left (\tan \left (\frac {1}{2} \, x\right ) - 1\right )}^{3}} + \frac {45 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 60 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 70 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 20 \, \tan \left (\frac {1}{2} \, x\right ) + 13}{120 \, a {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.53, size = 87, normalized size = 2.56 \[ \frac {-\frac {1}{6 \left (\tan \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{4 \left (\tan \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {3}{8 \left (\tan \left (\frac {x}{2}\right )-1\right )}-\frac {1}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {2}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {4}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {1}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {3}{8 \left (\tan \left (\frac {x}{2}\right )+1\right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 167, normalized size = 4.91 \[ \frac {2 \, {\left (\frac {6 \, \sin \relax (x)}{\cos \relax (x) + 1} + \frac {9 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - \frac {8 \, \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {5 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {10 \, \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}} + \frac {15 \, \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} + 3\right )}}{15 \, {\left (a + \frac {2 \, a \sin \relax (x)}{\cos \relax (x) + 1} - \frac {2 \, a \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - \frac {6 \, a \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {6 \, a \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}} + \frac {2 \, a \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} - \frac {2 \, a \sin \relax (x)^{7}}{{\left (\cos \relax (x) + 1\right )}^{7}} - \frac {a \sin \relax (x)^{8}}{{\left (\cos \relax (x) + 1\right )}^{8}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 69, normalized size = 2.03 \[ -\frac {2\,\left (15\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+5\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4-8\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+9\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+6\,\mathrm {tan}\left (\frac {x}{2}\right )+3\right )}{15\,a\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )-1\right )}^3\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sec ^{4}{\relax (x )}}{\csc {\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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