Integrand size = 29, antiderivative size = 250 \[ \int \frac {\csc ^2(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\csc (c+d x)}{a d}-\frac {\left (15 a^2+37 a b+24 b^2\right ) \log (1-\sin (c+d x))}{16 (a+b)^3 d}-\frac {b \log (\sin (c+d x))}{a^2 d}+\frac {\left (15 a^2-37 a b+24 b^2\right ) \log (1+\sin (c+d x))}{16 (a-b)^3 d}-\frac {b^7 \log (a+b \sin (c+d x))}{a^2 \left (a^2-b^2\right )^3 d}+\frac {1}{16 (a+b) d (1-\sin (c+d x))^2}+\frac {7 a+9 b}{16 (a+b)^2 d (1-\sin (c+d x))}-\frac {1}{16 (a-b) d (1+\sin (c+d x))^2}-\frac {7 a-9 b}{16 (a-b)^2 d (1+\sin (c+d x))} \]
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Time = 0.30 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2916, 12, 908} \[ \int \frac {\csc ^2(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\left (15 a^2+37 a b+24 b^2\right ) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac {\left (15 a^2-37 a b+24 b^2\right ) \log (\sin (c+d x)+1)}{16 d (a-b)^3}-\frac {b^7 \log (a+b \sin (c+d x))}{a^2 d \left (a^2-b^2\right )^3}-\frac {b \log (\sin (c+d x))}{a^2 d}+\frac {7 a+9 b}{16 d (a+b)^2 (1-\sin (c+d x))}-\frac {7 a-9 b}{16 d (a-b)^2 (\sin (c+d x)+1)}+\frac {1}{16 d (a+b) (1-\sin (c+d x))^2}-\frac {1}{16 d (a-b) (\sin (c+d x)+1)^2}-\frac {\csc (c+d x)}{a d} \]
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Rule 12
Rule 908
Rule 2916
Rubi steps \begin{align*} \text {integral}& = \frac {b^5 \text {Subst}\left (\int \frac {b^2}{x^2 (a+x) \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d} \\ & = \frac {b^7 \text {Subst}\left (\int \frac {1}{x^2 (a+x) \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d} \\ & = \frac {b^7 \text {Subst}\left (\int \left (\frac {1}{8 b^5 (a+b) (b-x)^3}+\frac {7 a+9 b}{16 b^6 (a+b)^2 (b-x)^2}+\frac {15 a^2+37 a b+24 b^2}{16 b^7 (a+b)^3 (b-x)}+\frac {1}{a b^6 x^2}-\frac {1}{a^2 b^6 x}-\frac {1}{a^2 (a-b)^3 (a+b)^3 (a+x)}-\frac {1}{8 b^5 (-a+b) (b+x)^3}+\frac {7 a-9 b}{16 (a-b)^2 b^6 (b+x)^2}+\frac {15 a^2-37 a b+24 b^2}{16 (a-b)^3 b^7 (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{d} \\ & = -\frac {\csc (c+d x)}{a d}-\frac {\left (15 a^2+37 a b+24 b^2\right ) \log (1-\sin (c+d x))}{16 (a+b)^3 d}-\frac {b \log (\sin (c+d x))}{a^2 d}+\frac {\left (15 a^2-37 a b+24 b^2\right ) \log (1+\sin (c+d x))}{16 (a-b)^3 d}-\frac {b^7 \log (a+b \sin (c+d x))}{a^2 \left (a^2-b^2\right )^3 d}+\frac {1}{16 (a+b) d (1-\sin (c+d x))^2}+\frac {7 a+9 b}{16 (a+b)^2 d (1-\sin (c+d x))}-\frac {1}{16 (a-b) d (1+\sin (c+d x))^2}-\frac {7 a-9 b}{16 (a-b)^2 d (1+\sin (c+d x))} \\ \end{align*}
Time = 4.24 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.94 \[ \int \frac {\csc ^2(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {b^7 \left (-\frac {16 \csc (c+d x)}{a b^7}-\frac {\left (15 a^2+37 a b+24 b^2\right ) \log (1-\sin (c+d x))}{b^7 (a+b)^3}-\frac {16 \log (\sin (c+d x))}{a^2 b^6}+\frac {\left (15 a^2-37 a b+24 b^2\right ) \log (1+\sin (c+d x))}{(a-b)^3 b^7}-\frac {16 \log (a+b \sin (c+d x))}{a^2 (a-b)^3 (a+b)^3}+\frac {1}{b^7 (a+b) (-1+\sin (c+d x))^2}+\frac {-7 a-9 b}{b^7 (a+b)^2 (-1+\sin (c+d x))}-\frac {1}{(a-b) b^7 (1+\sin (c+d x))^2}+\frac {-7 a+9 b}{(a-b)^2 b^7 (1+\sin (c+d x))}\right )}{16 d} \]
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Time = 1.31 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {-\frac {1}{2 \left (8 a -8 b \right ) \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {7 a -9 b}{16 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (15 a^{2}-37 a b +24 b^{2}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{3}}-\frac {1}{a \sin \left (d x +c \right )}-\frac {b \ln \left (\sin \left (d x +c \right )\right )}{a^{2}}+\frac {1}{2 \left (8 a +8 b \right ) \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {7 a +9 b}{16 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-15 a^{2}-37 a b -24 b^{2}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{3}}-\frac {b^{7} \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} a^{2}}}{d}\) | \(219\) |
default | \(\frac {-\frac {1}{2 \left (8 a -8 b \right ) \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {7 a -9 b}{16 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (15 a^{2}-37 a b +24 b^{2}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{3}}-\frac {1}{a \sin \left (d x +c \right )}-\frac {b \ln \left (\sin \left (d x +c \right )\right )}{a^{2}}+\frac {1}{2 \left (8 a +8 b \right ) \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {7 a +9 b}{16 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-15 a^{2}-37 a b -24 b^{2}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{3}}-\frac {b^{7} \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} a^{2}}}{d}\) | \(219\) |
parallelrisch | \(\frac {-8 b^{7} \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )-15 \left (a^{2}+\frac {37}{15} a b +\frac {8}{5} b^{2}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a -b \right )^{3} a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+15 \left (a +b \right ) \left (\left (a^{2}-\frac {37}{15} a b +\frac {8}{5} b^{2}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a +b \right )^{2} a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {\left (\frac {8 b \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (a +b \right )^{2} \left (a -b \right )^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\left (\left (\left (a^{4}-\frac {9}{5} a^{2} b^{2}+\frac {4}{5} b^{4}\right ) \cos \left (2 d x +2 c \right )+\left (\frac {3}{8} a^{4}-\frac {27}{40} a^{2} b^{2}+\frac {1}{5} b^{4}\right ) \cos \left (4 d x +4 c \right )+\frac {9 a^{4}}{40}-\frac {29 a^{2} b^{2}}{40}+\frac {3 b^{4}}{5}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {2 b \left (\left (a^{2}-b^{2}\right ) \cos \left (2 d x +2 c \right )+\frac {\left (3 a^{2}-5 b^{2}\right ) \cos \left (4 d x +4 c \right )}{4}-\frac {7 a^{2}}{4}+\frac {9 b^{2}}{4}\right ) a}{5}\right ) a \right ) \left (a -b \right )}{3}\right )}{2 \left (a -b \right )^{3} \left (a +b \right )^{3} a^{2} d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(429\) |
norman | \(\frac {-\frac {1}{2 a d}-\frac {\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}-\frac {2 \left (2 a^{2} b -3 b^{3}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {2 \left (2 a^{2} b -3 b^{3}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {2 \left (2 a^{2} b -4 b^{3}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {\left (15 a^{4}-25 a^{2} b^{2}+6 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {\left (15 a^{4}-25 a^{2} b^{2}+6 b^{4}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (5 a^{4}-13 a^{2} b^{2}+4 b^{4}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (5 a^{4}-13 a^{2} b^{2}+4 b^{4}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d}+\frac {\left (15 a^{2}-37 a b +24 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}-\frac {\left (15 a^{2}+37 a b +24 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}-\frac {b^{7} \ln \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}{a^{2} d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}\) | \(605\) |
risch | \(\text {Expression too large to display}\) | \(1209\) |
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Time = 2.42 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.70 \[ \int \frac {\csc ^2(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {16 \, b^{7} \cos \left (d x + c\right )^{4} \log \left (b \sin \left (d x + c\right ) + a\right ) \sin \left (d x + c\right ) - 4 \, a^{7} + 8 \, a^{5} b^{2} - 4 \, a^{3} b^{4} + 16 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{4} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - {\left (15 \, a^{7} + 8 \, a^{6} b - 42 \, a^{5} b^{2} - 24 \, a^{4} b^{3} + 35 \, a^{3} b^{4} + 24 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + {\left (15 \, a^{7} - 8 \, a^{6} b - 42 \, a^{5} b^{2} + 24 \, a^{4} b^{3} + 35 \, a^{3} b^{4} - 24 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 2 \, {\left (15 \, a^{7} - 42 \, a^{5} b^{2} + 35 \, a^{3} b^{4} - 8 \, a b^{6}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (5 \, a^{7} - 14 \, a^{5} b^{2} + 9 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5} + 2 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 2 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, {\left (a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}\right )} d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \frac {\csc ^2(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.44 \[ \int \frac {\csc ^2(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\frac {16 \, b^{7} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}} - \frac {{\left (15 \, a^{2} - 37 \, a b + 24 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac {{\left (15 \, a^{2} + 37 \, a b + 24 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {2 \, {\left ({\left (15 \, a^{4} - 27 \, a^{2} b^{2} + 8 \, b^{4}\right )} \sin \left (d x + c\right )^{4} + 8 \, a^{4} - 16 \, a^{2} b^{2} + 8 \, b^{4} - 4 \, {\left (a^{3} b - 2 \, a b^{3}\right )} \sin \left (d x + c\right )^{3} - {\left (25 \, a^{4} - 45 \, a^{2} b^{2} + 16 \, b^{4}\right )} \sin \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{3} b - 5 \, a b^{3}\right )} \sin \left (d x + c\right )\right )}}{{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (d x + c\right )^{5} - 2 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (d x + c\right )^{3} + {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (d x + c\right )} + \frac {16 \, b \log \left (\sin \left (d x + c\right )\right )}{a^{2}}}{16 \, d} \]
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Time = 0.62 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.67 \[ \int \frac {\csc ^2(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\frac {16 \, b^{8} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}} - \frac {{\left (15 \, a^{2} - 37 \, a b + 24 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac {{\left (15 \, a^{2} + 37 \, a b + 24 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {16 \, b \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{2}} + \frac {2 \, {\left (6 \, a^{4} b \sin \left (d x + c\right )^{4} - 18 \, a^{2} b^{3} \sin \left (d x + c\right )^{4} + 18 \, b^{5} \sin \left (d x + c\right )^{4} + 7 \, a^{5} \sin \left (d x + c\right )^{3} - 18 \, a^{3} b^{2} \sin \left (d x + c\right )^{3} + 11 \, a b^{4} \sin \left (d x + c\right )^{3} - 16 \, a^{4} b \sin \left (d x + c\right )^{2} + 48 \, a^{2} b^{3} \sin \left (d x + c\right )^{2} - 44 \, b^{5} \sin \left (d x + c\right )^{2} - 9 \, a^{5} \sin \left (d x + c\right ) + 22 \, a^{3} b^{2} \sin \left (d x + c\right ) - 13 \, a b^{4} \sin \left (d x + c\right ) + 12 \, a^{4} b - 34 \, a^{2} b^{3} + 28 \, b^{5}\right )}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}} - \frac {16 \, {\left (b \sin \left (d x + c\right ) - a\right )}}{a^{2} \sin \left (d x + c\right )}}{16 \, d} \]
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Time = 12.31 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.49 \[ \int \frac {\csc ^2(c+d x) \sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (\frac {b^2}{8\,{\left (a-b\right )}^3}-\frac {7\,b}{16\,{\left (a-b\right )}^2}+\frac {15}{16\,\left (a-b\right )}\right )}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (\frac {7\,b}{16\,{\left (a+b\right )}^2}+\frac {15}{16\,\left (a+b\right )}+\frac {b^2}{8\,{\left (a+b\right )}^3}\right )}{d}-\frac {\frac {1}{a}-\frac {{\sin \left (c+d\,x\right )}^3\,\left (a^2\,b-2\,b^3\right )}{2\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {\sin \left (c+d\,x\right )\,\left (3\,a^2\,b-5\,b^3\right )}{4\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {{\sin \left (c+d\,x\right )}^4\,\left (15\,a^4-27\,a^2\,b^2+8\,b^4\right )}{8\,a\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (25\,a^4-45\,a^2\,b^2+16\,b^4\right )}{8\,a\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left ({\sin \left (c+d\,x\right )}^5-2\,{\sin \left (c+d\,x\right )}^3+\sin \left (c+d\,x\right )\right )}-\frac {b\,\ln \left (\sin \left (c+d\,x\right )\right )}{a^2\,d}-\frac {b^7\,\ln \left (a+b\,\sin \left (c+d\,x\right )\right )}{d\,\left (a^8-3\,a^6\,b^2+3\,a^4\,b^4-a^2\,b^6\right )} \]
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