Optimal. Leaf size=117 \[ -\frac {\cos (c+d x)}{a d}+\frac {\tan ^5(c+d x)}{5 a d}-\frac {\tan ^3(c+d x)}{3 a d}+\frac {\tan (c+d x)}{a d}-\frac {\sec ^5(c+d x)}{5 a d}+\frac {\sec ^3(c+d x)}{a d}-\frac {3 \sec (c+d x)}{a d}-\frac {x}{a} \]
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Rubi [A] time = 0.16, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ -\frac {\cos (c+d x)}{a d}+\frac {\tan ^5(c+d x)}{5 a d}-\frac {\tan ^3(c+d x)}{3 a d}+\frac {\tan (c+d x)}{a d}-\frac {\sec ^5(c+d x)}{5 a d}+\frac {\sec ^3(c+d x)}{a d}-\frac {3 \sec (c+d x)}{a d}-\frac {x}{a} \]
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 ^4(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int \tan ^6(c+d x) \, dx}{a}-\frac {\int \sin (c+d x) \tan ^6(c+d x) \, dx}{a}\\ &=\frac {\tan ^5(c+d x)}{5 a d}-\frac {\int \tan ^4(c+d x) \, dx}{a}+\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^3}{x^6} \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac {\tan ^3(c+d x)}{3 a d}+\frac {\tan ^5(c+d x)}{5 a d}+\frac {\int \tan ^2(c+d x) \, dx}{a}+\frac {\operatorname {Subst}\left (\int \left (-1+\frac {1}{x^6}-\frac {3}{x^4}+\frac {3}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac {\cos (c+d x)}{a d}-\frac {3 \sec (c+d x)}{a d}+\frac {\sec ^3(c+d x)}{a d}-\frac {\sec ^5(c+d x)}{5 a d}+\frac {\tan (c+d x)}{a d}-\frac {\tan ^3(c+d x)}{3 a d}+\frac {\tan ^5(c+d x)}{5 a d}-\frac {\int 1 \, dx}{a}\\ &=-\frac {x}{a}-\frac {\cos (c+d x)}{a d}-\frac {3 \sec (c+d x)}{a d}+\frac {\sec ^3(c+d x)}{a d}-\frac {\sec ^5(c+d x)}{5 a d}+\frac {\tan (c+d x)}{a d}-\frac {\tan ^3(c+d x)}{3 a d}+\frac {\tan ^5(c+d x)}{5 a d}\\ \end {align*}
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Mathematica [A] time = 0.68, size = 224, normalized size = 1.91 \[ -\frac {216 \sin (c+d x)+240 c \sin (2 (c+d x))+240 d x \sin (2 (c+d x))-618 \sin (2 (c+d x))+532 \sin (3 (c+d x))+120 c \sin (4 (c+d x))+120 d x \sin (4 (c+d x))-309 \sin (4 (c+d x))+60 \sin (5 (c+d x))+18 (40 c+40 d x-103) \cos (c+d x)+1568 \cos (2 (c+d x))+240 c \cos (3 (c+d x))+240 d x \cos (3 (c+d x))-618 \cos (3 (c+d x))+304 \cos (4 (c+d x))+1200}{960 a d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 108, normalized size = 0.92 \[ -\frac {15 \, d x \cos \left (d x + c\right )^{3} + 38 \, \cos \left (d x + c\right )^{4} + 11 \, \cos \left (d x + c\right )^{2} + {\left (15 \, d x \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )^{4} + 22 \, \cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) - 1}{15 \, {\left (a d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 149, normalized size = 1.27 \[ -\frac {\frac {120 \, {\left (d x + c\right )}}{a} + \frac {240}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a} - \frac {5 \, {\left (21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 23\right )}}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {345 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1560 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2570 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 413}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.44, size = 210, normalized size = 1.79 \[ -\frac {1}{6 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{4 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {7}{8 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {2}{5 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {1}{a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{3 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {3}{2 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {23}{8 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 400, normalized size = 3.42 \[ -\frac {2 \, {\left (\frac {\frac {81 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {78 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {172 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {26 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {22 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {70 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {20 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {30 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {15 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + 48}{a + \frac {2 \, a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {4 \, a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {2 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {2 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {4 \, a \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {2 \, a \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} + \frac {15 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}\right )}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 18.67, size = 172, normalized size = 1.47 \[ \frac {-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}+\frac {28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {44\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{15}-\frac {52\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{15}-\frac {344\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{15}-\frac {52\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {54\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}+\frac {32}{5}}{a\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^5\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {x}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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