Optimal. Leaf size=83 \[ \frac {\cos (c+d x)}{a d}+\frac {\tan ^3(c+d x)}{3 a d}-\frac {\tan (c+d x)}{a d}-\frac {\sec ^3(c+d x)}{3 a d}+\frac {2 \sec (c+d x)}{a d}+\frac {x}{a} \]
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Rubi [A] time = 0.14, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2839, 3473, 8, 2590, 270} \[ \frac {\cos (c+d x)}{a d}+\frac {\tan ^3(c+d x)}{3 a d}-\frac {\tan (c+d x)}{a d}-\frac {\sec ^3(c+d x)}{3 a d}+\frac {2 \sec (c+d x)}{a d}+\frac {x}{a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 270
Rule 2590
Rule 2839
Rule 3473
Rubi steps
\begin {align*} \int \frac {\sin ^2(c+d x) \tan ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int \tan ^4(c+d x) \, dx}{a}-\frac {\int \sin (c+d x) \tan ^4(c+d x) \, dx}{a}\\ &=\frac {\tan ^3(c+d x)}{3 a d}-\frac {\int \tan ^2(c+d x) \, dx}{a}+\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^4} \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac {\tan (c+d x)}{a d}+\frac {\tan ^3(c+d x)}{3 a d}+\frac {\int 1 \, dx}{a}+\frac {\operatorname {Subst}\left (\int \left (1+\frac {1}{x^4}-\frac {2}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac {x}{a}+\frac {\cos (c+d x)}{a d}+\frac {2 \sec (c+d x)}{a d}-\frac {\sec ^3(c+d x)}{3 a d}-\frac {\tan (c+d x)}{a d}+\frac {\tan ^3(c+d x)}{3 a d}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 148, normalized size = 1.78 \[ \frac {11 \sin (c+d x)+6 c \sin (2 (c+d x))+6 d x \sin (2 (c+d x))-11 \sin (2 (c+d x))+3 \sin (3 (c+d x))+2 (6 c+6 d x-11) \cos (c+d x)+14 \cos (2 (c+d x))+18}{12 a d (\sin (c+d x)+1) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 80, normalized size = 0.96 \[ \frac {3 \, d x \cos \left (d x + c\right ) + 7 \, \cos \left (d x + c\right )^{2} + {\left (3 \, d x \cos \left (d x + c\right ) + 3 \, \cos \left (d x + c\right )^{2} + 2\right )} \sin \left (d x + c\right ) + 1}{3 \, {\left (a d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 125, normalized size = 1.51 \[ \frac {\frac {6 \, {\left (d x + c\right )}}{a} - \frac {3 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )} a} + \frac {15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 17}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 126, normalized size = 1.52 \[ -\frac {1}{2 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {2}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {2}{3 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {5}{2 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 236, normalized size = 2.84 \[ \frac {2 \, {\left (\frac {\frac {13 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {2 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {6 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + 8}{a + \frac {2 \, a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {2 \, a \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.52, size = 129, normalized size = 1.55 \[ \frac {x}{a}-\frac {-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+\frac {26\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+\frac {16}{3}}{a\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^3\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )} \]
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 \[ \frac {\int \frac {\sin ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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