Integrand size = 21, antiderivative size = 162 \[ \int (a+b \sin (c+d x))^2 \tan ^5(c+d x) \, dx=-\frac {\left (4 a^2+15 a b+12 b^2\right ) \log (1-\sin (c+d x))}{8 d}+\frac {\left (15 a b-4 \left (a^2+3 b^2\right )\right ) \log (1+\sin (c+d x))}{8 d}-\frac {2 a b \sin (c+d x)}{d}-\frac {b^2 \sin ^2(c+d x)}{2 d}+\frac {\sec ^4(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac {\sec ^2(c+d x) (a+b \sin (c+d x)) (4 a+5 b \sin (c+d x))}{4 d} \]
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Time = 0.20 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2800, 1659, 1824, 647, 31} \[ \int (a+b \sin (c+d x))^2 \tan ^5(c+d x) \, dx=-\frac {\left (4 a^2+15 a b+12 b^2\right ) \log (1-\sin (c+d x))}{8 d}+\frac {\left (15 a b-4 \left (a^2+3 b^2\right )\right ) \log (\sin (c+d x)+1)}{8 d}-\frac {2 a b \sin (c+d x)}{d}+\frac {\sec ^4(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac {\sec ^2(c+d x) (a+b \sin (c+d x)) (4 a+5 b \sin (c+d x))}{4 d}-\frac {b^2 \sin ^2(c+d x)}{2 d} \]
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Rule 31
Rule 647
Rule 1659
Rule 1824
Rule 2800
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^5 (a+x)^2}{\left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d} \\ & = \frac {\sec ^4(c+d x) (a+b \sin (c+d x))^2}{4 d}+\frac {\text {Subst}\left (\int \frac {(a+x) \left (-2 b^6-4 a b^4 x-4 b^4 x^2-4 a b^2 x^3-4 b^2 x^4\right )}{\left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 b^2 d} \\ & = \frac {\sec ^4(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac {\sec ^2(c+d x) (a+b \sin (c+d x)) (4 a+5 b \sin (c+d x))}{4 d}+\frac {\text {Subst}\left (\int \frac {14 a b^6+8 b^4 \left (a^2+2 b^2\right ) x+16 a b^4 x^2+8 b^4 x^3}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{8 b^4 d} \\ & = \frac {\sec ^4(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac {\sec ^2(c+d x) (a+b \sin (c+d x)) (4 a+5 b \sin (c+d x))}{4 d}+\frac {\text {Subst}\left (\int \left (-16 a b^4-8 b^4 x+\frac {2 \left (15 a b^6+4 b^4 \left (a^2+3 b^2\right ) x\right )}{b^2-x^2}\right ) \, dx,x,b \sin (c+d x)\right )}{8 b^4 d} \\ & = -\frac {2 a b \sin (c+d x)}{d}-\frac {b^2 \sin ^2(c+d x)}{2 d}+\frac {\sec ^4(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac {\sec ^2(c+d x) (a+b \sin (c+d x)) (4 a+5 b \sin (c+d x))}{4 d}+\frac {\text {Subst}\left (\int \frac {15 a b^6+4 b^4 \left (a^2+3 b^2\right ) x}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{4 b^4 d} \\ & = -\frac {2 a b \sin (c+d x)}{d}-\frac {b^2 \sin ^2(c+d x)}{2 d}+\frac {\sec ^4(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac {\sec ^2(c+d x) (a+b \sin (c+d x)) (4 a+5 b \sin (c+d x))}{4 d}+\frac {\left (4 a^2-15 a b+12 b^2\right ) \text {Subst}\left (\int \frac {1}{-b-x} \, dx,x,b \sin (c+d x)\right )}{8 d}+\frac {\left (4 a^2+15 a b+12 b^2\right ) \text {Subst}\left (\int \frac {1}{b-x} \, dx,x,b \sin (c+d x)\right )}{8 d} \\ & = -\frac {\left (4 a^2+15 a b+12 b^2\right ) \log (1-\sin (c+d x))}{8 d}-\frac {\left (4 a^2-15 a b+12 b^2\right ) \log (1+\sin (c+d x))}{8 d}-\frac {2 a b \sin (c+d x)}{d}-\frac {b^2 \sin ^2(c+d x)}{2 d}+\frac {\sec ^4(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac {\sec ^2(c+d x) (a+b \sin (c+d x)) (4 a+5 b \sin (c+d x))}{4 d} \\ \end{align*}
Time = 1.59 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.01 \[ \int (a+b \sin (c+d x))^2 \tan ^5(c+d x) \, dx=\frac {-2 \left (4 a^2+15 a b+12 b^2\right ) \log (1-\sin (c+d x))-2 \left (4 a^2-15 a b+12 b^2\right ) \log (1+\sin (c+d x))+\frac {(a+b)^2}{(-1+\sin (c+d x))^2}+\frac {(a+b) (7 a+11 b)}{-1+\sin (c+d x)}-32 a b \sin (c+d x)-8 b^2 \sin ^2(c+d x)+\frac {(a-b)^2}{(1+\sin (c+d x))^2}-\frac {(7 a-11 b) (a-b)}{1+\sin (c+d x)}}{16 d} \]
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Time = 0.95 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.27
method | result | size |
derivativedivides | \(\frac {a^{2} \left (\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+2 a b \left (\frac {\sin ^{7}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {3 \left (\sin ^{7}\left (d x +c \right )\right )}{8 \cos \left (d x +c \right )^{2}}-\frac {3 \left (\sin ^{5}\left (d x +c \right )\right )}{8}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{8}-\frac {15 \sin \left (d x +c \right )}{8}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+b^{2} \left (\frac {\sin ^{8}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin ^{8}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{2}-\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {3 \left (\sin ^{2}\left (d x +c \right )\right )}{2}-3 \ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(205\) |
default | \(\frac {a^{2} \left (\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+2 a b \left (\frac {\sin ^{7}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {3 \left (\sin ^{7}\left (d x +c \right )\right )}{8 \cos \left (d x +c \right )^{2}}-\frac {3 \left (\sin ^{5}\left (d x +c \right )\right )}{8}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{8}-\frac {15 \sin \left (d x +c \right )}{8}+\frac {15 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+b^{2} \left (\frac {\sin ^{8}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}-\frac {\sin ^{8}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{2}-\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {3 \left (\sin ^{2}\left (d x +c \right )\right )}{2}-3 \ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(205\) |
parallelrisch | \(\frac {32 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a^{2}+3 b^{2}\right ) \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-32 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a^{2}+\frac {15}{4} a b +3 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-32 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a^{2}-\frac {15}{4} a b +3 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (-8 a^{2}-9 b^{2}\right ) \cos \left (2 d x +2 c \right )+6 \left (a^{2}+2 b^{2}\right ) \cos \left (4 d x +4 c \right )-60 a b \sin \left (3 d x +3 c \right )-8 a b \sin \left (5 d x +5 c \right )-20 a b \sin \left (d x +c \right )+b^{2} \cos \left (6 d x +6 c \right )+2 a^{2}-4 b^{2}}{8 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(274\) |
norman | \(\frac {\frac {\left (4 a^{2}+12 b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (4 a^{2}+12 b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 \left (a^{2}+3 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 \left (a^{2}+3 b^{2}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 \left (3 a^{2}+b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {15 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}+\frac {25 a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {11 a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {11 a b \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {25 a b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {15 a b \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (a^{2}+3 b^{2}\right ) \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (4 a^{2}-15 a b +12 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{4 d}-\frac {\left (4 a^{2}+15 a b +12 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{4 d}\) | \(358\) |
risch | \(3 i b^{2} x +\frac {i a b \,{\mathrm e}^{i \left (d x +c \right )}}{d}+\frac {b^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {i a b \,{\mathrm e}^{-i \left (d x +c \right )}}{d}+\frac {i \left (8 i a^{2} {\mathrm e}^{6 i \left (d x +c \right )}+12 i b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+9 a b \,{\mathrm e}^{7 i \left (d x +c \right )}+8 i a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+16 i b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+a b \,{\mathrm e}^{5 i \left (d x +c \right )}+8 i a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+12 i b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-a b \,{\mathrm e}^{3 i \left (d x +c \right )}-9 a b \,{\mathrm e}^{i \left (d x +c \right )}\right )}{2 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}+\frac {b^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {2 i a^{2} c}{d}+\frac {6 i b^{2} c}{d}+i a^{2} x -\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}+\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a b}{4 d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b^{2}}{d}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a b}{4 d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b^{2}}{d}\) | \(389\) |
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Time = 0.29 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.10 \[ \int (a+b \sin (c+d x))^2 \tan ^5(c+d x) \, dx=\frac {4 \, b^{2} \cos \left (d x + c\right )^{6} - 2 \, b^{2} \cos \left (d x + c\right )^{4} - {\left (4 \, a^{2} - 15 \, a b + 12 \, b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (4 \, a^{2} + 15 \, a b + 12 \, b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 4 \, {\left (2 \, a^{2} + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, a^{2} + 2 \, b^{2} - 2 \, {\left (8 \, a b \cos \left (d x + c\right )^{4} + 9 \, a b \cos \left (d x + c\right )^{2} - 2 \, a b\right )} \sin \left (d x + c\right )}{8 \, d \cos \left (d x + c\right )^{4}} \]
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Timed out. \[ \int (a+b \sin (c+d x))^2 \tan ^5(c+d x) \, dx=\text {Timed out} \]
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Time = 0.32 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.97 \[ \int (a+b \sin (c+d x))^2 \tan ^5(c+d x) \, dx=-\frac {4 \, b^{2} \sin \left (d x + c\right )^{2} + 16 \, a b \sin \left (d x + c\right ) + {\left (4 \, a^{2} - 15 \, a b + 12 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (4 \, a^{2} + 15 \, a b + 12 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (9 \, a b \sin \left (d x + c\right )^{3} - 7 \, a b \sin \left (d x + c\right ) + 2 \, {\left (2 \, a^{2} + 3 \, b^{2}\right )} \sin \left (d x + c\right )^{2} - 3 \, a^{2} - 5 \, b^{2}\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1}}{8 \, d} \]
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Time = 0.50 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.08 \[ \int (a+b \sin (c+d x))^2 \tan ^5(c+d x) \, dx=-\frac {4 \, b^{2} \sin \left (d x + c\right )^{2} + 16 \, a b \sin \left (d x + c\right ) + {\left (4 \, a^{2} - 15 \, a b + 12 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + {\left (4 \, a^{2} + 15 \, a b + 12 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (3 \, a^{2} \sin \left (d x + c\right )^{4} + 9 \, b^{2} \sin \left (d x + c\right )^{4} + 9 \, a b \sin \left (d x + c\right )^{3} - 2 \, a^{2} \sin \left (d x + c\right )^{2} - 12 \, b^{2} \sin \left (d x + c\right )^{2} - 7 \, a b \sin \left (d x + c\right ) + 4 \, b^{2}\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{8 \, d} \]
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Time = 8.18 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.33 \[ \int (a+b \sin (c+d x))^2 \tan ^5(c+d x) \, dx=\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )\,\left (a^2+3\,b^2\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )\,\left (a^2-\frac {15\,a\,b}{4}+3\,b^2\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,\left (a^2+\frac {15\,a\,b}{4}+3\,b^2\right )}{d}-\frac {-\frac {15\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{2}+\left (-2\,a^2-6\,b^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {25\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{2}+\left (4\,a^2+12\,b^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+11\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (12\,a^2+4\,b^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+11\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (4\,a^2+12\,b^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {25\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}+\left (-2\,a^2-6\,b^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {15\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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