Optimal. Leaf size=126 \[ \frac {\tan ^5(c+d x)}{5 a d}+\frac {\tan ^3(c+d x)}{a d}+\frac {3 \tan (c+d x)}{a d}-\frac {\cot (c+d x)}{a d}-\frac {\sec ^5(c+d x)}{5 a d}-\frac {\sec ^3(c+d x)}{3 a d}-\frac {\sec (c+d x)}{a d}+\frac {\tanh ^{-1}(\cos (c+d x))}{a d} \]
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Rubi [A] time = 0.17, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2839, 2620, 270, 2622, 302, 207} \[ \frac {\tan ^5(c+d x)}{5 a d}+\frac {\tan ^3(c+d x)}{a d}+\frac {3 \tan (c+d x)}{a d}-\frac {\cot (c+d x)}{a d}-\frac {\sec ^5(c+d x)}{5 a d}-\frac {\sec ^3(c+d x)}{3 a d}-\frac {\sec (c+d x)}{a d}+\frac {\tanh ^{-1}(\cos (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 207
Rule 270
Rule 302
Rule 2620
Rule 2622
Rule 2839
Rubi steps
\begin {align*} \int \frac {\csc ^2(c+d x) \sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac {\int \csc (c+d x) \sec ^6(c+d x) \, dx}{a}+\frac {\int \csc ^2(c+d x) \sec ^6(c+d x) \, dx}{a}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {x^6}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a d}+\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^2} \, dx,x,\tan (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \left (3+\frac {1}{x^2}+3 x^2+x^4\right ) \, dx,x,\tan (c+d x)\right )}{a d}-\frac {\operatorname {Subst}\left (\int \left (1+x^2+x^4+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (c+d x)\right )}{a d}\\ &=-\frac {\cot (c+d x)}{a d}-\frac {\sec (c+d x)}{a d}-\frac {\sec ^3(c+d x)}{3 a d}-\frac {\sec ^5(c+d x)}{5 a d}+\frac {3 \tan (c+d x)}{a d}+\frac {\tan ^3(c+d x)}{a d}+\frac {\tan ^5(c+d x)}{5 a d}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{a d}\\ &=\frac {\tanh ^{-1}(\cos (c+d x))}{a d}-\frac {\cot (c+d x)}{a d}-\frac {\sec (c+d x)}{a d}-\frac {\sec ^3(c+d x)}{3 a d}-\frac {\sec ^5(c+d x)}{5 a d}+\frac {3 \tan (c+d x)}{a d}+\frac {\tan ^3(c+d x)}{a d}+\frac {\tan ^5(c+d x)}{5 a d}\\ \end {align*}
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Mathematica [B] time = 0.60, size = 341, normalized size = 2.71 \[ -\frac {\csc \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (352 \sin (c+d x)-596 \sin (2 (c+d x))+864 \sin (3 (c+d x))-298 \sin (4 (c+d x))+384 \sin (5 (c+d x))+1216 \cos (2 (c+d x))+149 \cos (3 (c+d x))+528 \cos (4 (c+d x))+149 \cos (5 (c+d x))+480 \sin (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+240 \sin (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+120 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+120 \cos (5 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-120 \cos (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-120 \cos (5 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\cos (c+d x) \left (240 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-240 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-298\right )-480 \sin (2 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-240 \sin (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+176\right )}{3840 a d (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 194, normalized size = 1.54 \[ \frac {66 \, \cos \left (d x + c\right )^{4} - 28 \, \cos \left (d x + c\right )^{2} + 15 \, {\left (\cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - \cos \left (d x + c\right )^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 15 \, {\left (\cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - \cos \left (d x + c\right )^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (48 \, \cos \left (d x + c\right )^{4} - 9 \, \cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - 8}{30 \, {\left (a d \cos \left (d x + c\right )^{5} - a d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - a d \cos \left (d x + c\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 178, normalized size = 1.41 \[ -\frac {\frac {120 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} - \frac {60 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a} - \frac {60 \, {\left (2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + \frac {5 \, {\left (27 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 25\right )}}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {585 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2040 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2890 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1880 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 493}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.50, size = 223, normalized size = 1.77 \[ \frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a d}-\frac {1}{6 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{4 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {9}{8 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{2 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {2}{5 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {1}{a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {7}{3 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {5}{2 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {39}{8 a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 379, normalized size = 3.01 \[ -\frac {\frac {\frac {122 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {26 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {454 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {252 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {510 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {330 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {210 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {195 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + 15}{\frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {2 \, a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {6 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {2 \, a \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {2 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {a \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}} + \frac {30 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {15 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.82, size = 257, normalized size = 2.04 \[ \frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-22\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-34\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {84\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}+\frac {454\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{15}+\frac {26\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}-\frac {122\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{15}-1}{d\,\left (-2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+12\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-12\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a\,d} \]
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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