Integrand size = 29, antiderivative size = 152 \[ \int \frac {\sec ^2(c+d x) \tan ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {5 \text {arctanh}(\sin (c+d x))}{128 a d}+\frac {5 \sec (c+d x) \tan (c+d x)}{128 a d}-\frac {5 \sec ^3(c+d x) \tan (c+d x)}{64 a d}+\frac {5 \sec ^3(c+d x) \tan ^3(c+d x)}{48 a d}-\frac {\sec ^3(c+d x) \tan ^5(c+d x)}{8 a d}+\frac {\tan ^6(c+d x)}{6 a d}+\frac {\tan ^8(c+d x)}{8 a d} \]
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Time = 0.19 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2914, 2687, 14, 2691, 3853, 3855} \[ \int \frac {\sec ^2(c+d x) \tan ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {5 \text {arctanh}(\sin (c+d x))}{128 a d}+\frac {\tan ^8(c+d x)}{8 a d}+\frac {\tan ^6(c+d x)}{6 a d}-\frac {\tan ^5(c+d x) \sec ^3(c+d x)}{8 a d}+\frac {5 \tan ^3(c+d x) \sec ^3(c+d x)}{48 a d}-\frac {5 \tan (c+d x) \sec ^3(c+d x)}{64 a d}+\frac {5 \tan (c+d x) \sec (c+d x)}{128 a d} \]
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Rule 14
Rule 2687
Rule 2691
Rule 2914
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^4(c+d x) \tan ^5(c+d x) \, dx}{a}-\frac {\int \sec ^3(c+d x) \tan ^6(c+d x) \, dx}{a} \\ & = -\frac {\sec ^3(c+d x) \tan ^5(c+d x)}{8 a d}+\frac {5 \int \sec ^3(c+d x) \tan ^4(c+d x) \, dx}{8 a}+\frac {\text {Subst}\left (\int x^5 \left (1+x^2\right ) \, dx,x,\tan (c+d x)\right )}{a d} \\ & = \frac {5 \sec ^3(c+d x) \tan ^3(c+d x)}{48 a d}-\frac {\sec ^3(c+d x) \tan ^5(c+d x)}{8 a d}-\frac {5 \int \sec ^3(c+d x) \tan ^2(c+d x) \, dx}{16 a}+\frac {\text {Subst}\left (\int \left (x^5+x^7\right ) \, dx,x,\tan (c+d x)\right )}{a d} \\ & = -\frac {5 \sec ^3(c+d x) \tan (c+d x)}{64 a d}+\frac {5 \sec ^3(c+d x) \tan ^3(c+d x)}{48 a d}-\frac {\sec ^3(c+d x) \tan ^5(c+d x)}{8 a d}+\frac {\tan ^6(c+d x)}{6 a d}+\frac {\tan ^8(c+d x)}{8 a d}+\frac {5 \int \sec ^3(c+d x) \, dx}{64 a} \\ & = \frac {5 \sec (c+d x) \tan (c+d x)}{128 a d}-\frac {5 \sec ^3(c+d x) \tan (c+d x)}{64 a d}+\frac {5 \sec ^3(c+d x) \tan ^3(c+d x)}{48 a d}-\frac {\sec ^3(c+d x) \tan ^5(c+d x)}{8 a d}+\frac {\tan ^6(c+d x)}{6 a d}+\frac {\tan ^8(c+d x)}{8 a d}+\frac {5 \int \sec (c+d x) \, dx}{128 a} \\ & = \frac {5 \text {arctanh}(\sin (c+d x))}{128 a d}+\frac {5 \sec (c+d x) \tan (c+d x)}{128 a d}-\frac {5 \sec ^3(c+d x) \tan (c+d x)}{64 a d}+\frac {5 \sec ^3(c+d x) \tan ^3(c+d x)}{48 a d}-\frac {\sec ^3(c+d x) \tan ^5(c+d x)}{8 a d}+\frac {\tan ^6(c+d x)}{6 a d}+\frac {\tan ^8(c+d x)}{8 a d} \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.61 \[ \int \frac {\sec ^2(c+d x) \tan ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {15 \text {arctanh}(\sin (c+d x))-\frac {4}{(-1+\sin (c+d x))^3}-\frac {15}{(-1+\sin (c+d x))^2}-\frac {15}{-1+\sin (c+d x)}+\frac {6}{(1+\sin (c+d x))^4}-\frac {24}{(1+\sin (c+d x))^3}+\frac {30}{(1+\sin (c+d x))^2}}{384 a d} \]
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Time = 0.92 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.68
method | result | size |
derivativedivides | \(\frac {-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {5}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {5}{128 \left (\sin \left (d x +c \right )-1\right )}-\frac {5 \ln \left (\sin \left (d x +c \right )-1\right )}{256}+\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {1}{16 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {5}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\) | \(103\) |
default | \(\frac {-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {5}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {5}{128 \left (\sin \left (d x +c \right )-1\right )}-\frac {5 \ln \left (\sin \left (d x +c \right )-1\right )}{256}+\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {1}{16 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {5}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\) | \(103\) |
risch | \(-\frac {i \left (625 \,{\mathrm e}^{5 i \left (d x +c \right )}+15 \,{\mathrm e}^{13 i \left (d x +c \right )}-442 \,{\mathrm e}^{11 i \left (d x +c \right )}+86 i {\mathrm e}^{4 i \left (d x +c \right )}-140 i {\mathrm e}^{6 i \left (d x +c \right )}-30 i {\mathrm e}^{2 i \left (d x +c \right )}+15 \,{\mathrm e}^{i \left (d x +c \right )}+625 \,{\mathrm e}^{9 i \left (d x +c \right )}+30 i {\mathrm e}^{12 i \left (d x +c \right )}-442 \,{\mathrm e}^{3 i \left (d x +c \right )}+140 i {\mathrm e}^{8 i \left (d x +c \right )}-86 i {\mathrm e}^{10 i \left (d x +c \right )}-1420 \,{\mathrm e}^{7 i \left (d x +c \right )}\right )}{192 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{6} d a}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d a}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{128 a d}\) | \(231\) |
parallelrisch | \(\frac {\left (-840 \cos \left (2 d x +2 c \right )-420 \cos \left (4 d x +4 c \right )-120 \cos \left (6 d x +6 c \right )-15 \cos \left (8 d x +8 c \right )-525\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (840 \cos \left (2 d x +2 c \right )+420 \cos \left (4 d x +4 c \right )+120 \cos \left (6 d x +6 c \right )+15 \cos \left (8 d x +8 c \right )+525\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+1790 \sin \left (3 d x +3 c \right )-794 \sin \left (5 d x +5 c \right )+30 \sin \left (7 d x +7 c \right )-2944 \cos \left (2 d x +2 c \right )+1088 \cos \left (4 d x +4 c \right )-128 \cos \left (6 d x +6 c \right )-16 \cos \left (8 d x +8 c \right )-3530 \sin \left (d x +c \right )+2000}{384 a d \left (\cos \left (8 d x +8 c \right )+8 \cos \left (6 d x +6 c \right )+28 \cos \left (4 d x +4 c \right )+56 \cos \left (2 d x +2 c \right )+35\right )}\) | \(260\) |
norman | \(\frac {-\frac {5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}-\frac {5 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d a}-\frac {5 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}+\frac {413 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d a}+\frac {413 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d a}+\frac {85 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}+\frac {85 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}+\frac {35 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}+\frac {35 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}+\frac {157 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d a}-\frac {113 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}-\frac {113 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{6} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{128 a d}+\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{128 a d}\) | \(315\) |
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Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.10 \[ \int \frac {\sec ^2(c+d x) \tan ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {30 \, \cos \left (d x + c\right )^{6} - 266 \, \cos \left (d x + c\right )^{4} + 316 \, \cos \left (d x + c\right )^{2} - 15 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (15 \, \cos \left (d x + c\right )^{4} - 22 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) - 112}{768 \, {\left (a d \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{6}\right )}} \]
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Timed out. \[ \int \frac {\sec ^2(c+d x) \tan ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.14 \[ \int \frac {\sec ^2(c+d x) \tan ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{6} + 15 \, \sin \left (d x + c\right )^{5} + 88 \, \sin \left (d x + c\right )^{4} - 8 \, \sin \left (d x + c\right )^{3} - 63 \, \sin \left (d x + c\right )^{2} + \sin \left (d x + c\right ) + 16\right )}}{a \sin \left (d x + c\right )^{7} + a \sin \left (d x + c\right )^{6} - 3 \, a \sin \left (d x + c\right )^{5} - 3 \, a \sin \left (d x + c\right )^{4} + 3 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - a} - \frac {15 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {15 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{768 \, d} \]
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Time = 0.41 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.89 \[ \int \frac {\sec ^2(c+d x) \tan ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {2 \, {\left (55 \, \sin \left (d x + c\right )^{3} - 225 \, \sin \left (d x + c\right )^{2} + 225 \, \sin \left (d x + c\right ) - 71\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac {125 \, \sin \left (d x + c\right )^{4} + 500 \, \sin \left (d x + c\right )^{3} + 510 \, \sin \left (d x + c\right )^{2} + 212 \, \sin \left (d x + c\right ) + 29}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \]
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Time = 20.71 (sec) , antiderivative size = 388, normalized size of antiderivative = 2.55 \[ \int \frac {\sec ^2(c+d x) \tan ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{64\,a\,d}+\frac {-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{64}-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{32}+\frac {35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{96}+\frac {85\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{96}-\frac {113\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{192}+\frac {413\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{48}+\frac {157\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{48}+\frac {413\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{48}-\frac {113\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{192}+\frac {85\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{96}+\frac {35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96}-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{32}-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-12\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+9\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+30\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-40\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+30\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+9\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-12\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-5\,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 )+a\right )} \]
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