Integrand size = 29, antiderivative size = 171 \[ \int \frac {\csc ^2(c+d x) \sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\csc (c+d x)}{a d}-\frac {(3 a+4 b) \log (1-\sin (c+d x))}{4 (a+b)^2 d}-\frac {b \log (\sin (c+d x))}{a^2 d}+\frac {(3 a-4 b) \log (1+\sin (c+d x))}{4 (a-b)^2 d}+\frac {b^5 \log (a+b \sin (c+d x))}{a^2 \left (a^2-b^2\right )^2 d}+\frac {1}{4 (a+b) d (1-\sin (c+d x))}-\frac {1}{4 (a-b) d (1+\sin (c+d x))} \]
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Time = 0.20 (sec) , antiderivative size = 171, 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} \[ \int \frac {\csc ^2(c+d x) \sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {b^5 \log (a+b \sin (c+d x))}{a^2 d \left (a^2-b^2\right )^2}-\frac {b \log (\sin (c+d x))}{a^2 d}+\frac {1}{4 d (a+b) (1-\sin (c+d x))}-\frac {1}{4 d (a-b) (\sin (c+d x)+1)}-\frac {(3 a+4 b) \log (1-\sin (c+d x))}{4 d (a+b)^2}+\frac {(3 a-4 b) \log (\sin (c+d x)+1)}{4 d (a-b)^2}-\frac {\csc (c+d x)}{a d} \]
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Rule 12
Rule 908
Rule 2916
Rubi steps \begin{align*} \text {integral}& = \frac {b^3 \text {Subst}\left (\int \frac {b^2}{x^2 (a+x) \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{d} \\ & = \frac {b^5 \text {Subst}\left (\int \frac {1}{x^2 (a+x) \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{d} \\ & = \frac {b^5 \text {Subst}\left (\int \left (\frac {1}{4 b^4 (a+b) (b-x)^2}+\frac {3 a+4 b}{4 b^5 (a+b)^2 (b-x)}+\frac {1}{a b^4 x^2}-\frac {1}{a^2 b^4 x}+\frac {1}{a^2 (a-b)^2 (a+b)^2 (a+x)}+\frac {1}{4 (a-b) b^4 (b+x)^2}+\frac {3 a-4 b}{4 (a-b)^2 b^5 (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{d} \\ & = -\frac {\csc (c+d x)}{a d}-\frac {(3 a+4 b) \log (1-\sin (c+d x))}{4 (a+b)^2 d}-\frac {b \log (\sin (c+d x))}{a^2 d}+\frac {(3 a-4 b) \log (1+\sin (c+d x))}{4 (a-b)^2 d}+\frac {b^5 \log (a+b \sin (c+d x))}{a^2 \left (a^2-b^2\right )^2 d}+\frac {1}{4 (a+b) d (1-\sin (c+d x))}-\frac {1}{4 (a-b) d (1+\sin (c+d x))} \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.02 \[ \int \frac {\csc ^2(c+d x) \sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\csc (c+d x) (a+b \sin (c+d x)) \left (\frac {4 \csc (c+d x)}{a}+\frac {(3 a+4 b) \log (1-\sin (c+d x))}{(a+b)^2}+\frac {4 b \log (\sin (c+d x))}{a^2}-\frac {(3 a-4 b) \log (1+\sin (c+d x))}{(a-b)^2}-\frac {4 b^5 \log (a+b \sin (c+d x))}{a^2 (a-b)^2 (a+b)^2}+\frac {1}{(a+b) (-1+\sin (c+d x))}+\frac {1}{(a-b) (1+\sin (c+d x))}\right )}{4 d (b+a \csc (c+d x))} \]
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Time = 0.92 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {-\frac {1}{\left (4 a -4 b \right ) \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (3 a -4 b \right ) \ln \left (1+\sin \left (d x +c \right )\right )}{4 \left (a -b \right )^{2}}+\frac {b^{5} \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{2} \left (a -b \right )^{2} a^{2}}-\frac {1}{\left (4 a +4 b \right ) \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-3 a -4 b \right ) \ln \left (\sin \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{2}}-\frac {1}{a \sin \left (d x +c \right )}-\frac {b \ln \left (\sin \left (d x +c \right )\right )}{a^{2}}}{d}\) | \(152\) |
default | \(\frac {-\frac {1}{\left (4 a -4 b \right ) \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (3 a -4 b \right ) \ln \left (1+\sin \left (d x +c \right )\right )}{4 \left (a -b \right )^{2}}+\frac {b^{5} \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{2} \left (a -b \right )^{2} a^{2}}-\frac {1}{\left (4 a +4 b \right ) \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-3 a -4 b \right ) \ln \left (\sin \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{2}}-\frac {1}{a \sin \left (d x +c \right )}-\frac {b \ln \left (\sin \left (d x +c \right )\right )}{a^{2}}}{d}\) | \(152\) |
parallelrisch | \(\frac {4 b^{5} \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )-6 \left (a +\frac {4 b}{3}\right ) \left (a -b \right )^{2} \left (1+\cos \left (2 d x +2 c \right )\right ) a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+6 \left (a +b \right ) \left (\left (a +b \right ) \left (a -\frac {4 b}{3}\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {\left (\frac {4 b \left (a -b \right ) \left (a +b \right ) \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (a^{2}-\frac {2 b^{2}}{3}\right ) \cos \left (2 d x +2 c \right )+\frac {a^{2}}{3}-\frac {2 b^{2}}{3}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {2 b a \left (-1+\cos \left (2 d x +2 c \right )\right )}{3}\right ) a \right ) \left (a -b \right )}{2}\right )}{4 \left (a -b \right )^{2} \left (a +b \right )^{2} a^{2} d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(266\) |
norman | \(\frac {-\frac {1}{2 a d}-\frac {\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}-\frac {2 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (a^{2}-b^{2}\right )}+\frac {\left (3 a^{2}-b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a \left (a^{2}-b^{2}\right )}+\frac {\left (3 a^{2}-b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a \left (a^{2}-b^{2}\right )}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {b^{5} \ln \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}{a^{2} d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d}+\frac {\left (3 a -4 b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (3 a +4 b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 \left (a^{2}+2 a b +b^{2}\right ) d}\) | \(315\) |
risch | \(\frac {2 i b x}{a^{2}-2 a b +b^{2}}-\frac {3 i a c}{2 \left (a^{2}-2 a b +b^{2}\right ) d}-\frac {2 i b^{5} x}{a^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {2 i b c}{\left (a^{2}+2 a b +b^{2}\right ) d}-\frac {i \left (3 a^{2} {\mathrm e}^{5 i \left (d x +c \right )}-2 b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+2 a^{2} {\mathrm e}^{3 i \left (d x +c \right )}-4 b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-2 i a b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 a^{2} {\mathrm e}^{i \left (d x +c \right )}-2 b^{2} {\mathrm e}^{i \left (d x +c \right )}+2 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}\right )}{d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left (a^{2}-b^{2}\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {3 i a c}{2 d \left (a^{2}+2 a b +b^{2}\right )}-\frac {2 i b^{5} c}{a^{2} d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {2 i b x}{a^{2}}+\frac {2 i b c}{a^{2} d}+\frac {2 i b c}{d \left (a^{2}-2 a b +b^{2}\right )}-\frac {3 i a x}{2 \left (a^{2}-2 a b +b^{2}\right )}+\frac {2 i b x}{a^{2}+2 a b +b^{2}}+\frac {3 i a x}{2 \left (a^{2}+2 a b +b^{2}\right )}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a}{2 \left (a^{2}+2 a b +b^{2}\right ) d}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b}{\left (a^{2}+2 a b +b^{2}\right ) d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a}{2 \left (a^{2}-2 a b +b^{2}\right ) d}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b}{\left (a^{2}-2 a b +b^{2}\right ) d}+\frac {b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{a^{2} d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{2} d}\) | \(597\) |
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Time = 0.94 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.68 \[ \int \frac {\csc ^2(c+d x) \sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {4 \, b^{5} \cos \left (d x + c\right )^{2} \log \left (b \sin \left (d x + c\right ) + a\right ) \sin \left (d x + c\right ) + 2 \, a^{5} - 2 \, a^{3} b^{2} - 4 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + {\left (3 \, a^{5} + 2 \, a^{4} b - 5 \, a^{3} b^{2} - 4 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) - {\left (3 \, a^{5} - 2 \, a^{4} b - 5 \, a^{3} b^{2} + 4 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) - 2 \, {\left (3 \, a^{5} - 5 \, a^{3} b^{2} + 2 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{4} b - a^{2} b^{3}\right )} \sin \left (d x + c\right )}{4 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right )} \]
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\[ \int \frac {\csc ^2(c+d x) \sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\csc ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.17 \[ \int \frac {\csc ^2(c+d x) \sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {4 \, b^{5} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}} + \frac {{\left (3 \, a - 4 \, b\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2} - 2 \, a b + b^{2}} - \frac {{\left (3 \, a + 4 \, b\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{2} + 2 \, a b + b^{2}} + \frac {2 \, {\left (a b \sin \left (d x + c\right ) - {\left (3 \, a^{2} - 2 \, b^{2}\right )} \sin \left (d x + c\right )^{2} + 2 \, a^{2} - 2 \, b^{2}\right )}}{{\left (a^{3} - a b^{2}\right )} \sin \left (d x + c\right )^{3} - {\left (a^{3} - a b^{2}\right )} \sin \left (d x + c\right )} - \frac {4 \, b \log \left (\sin \left (d x + c\right )\right )}{a^{2}}}{4 \, d} \]
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Time = 0.67 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.63 \[ \int \frac {\csc ^2(c+d x) \sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {12 \, b^{6} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}} + \frac {3 \, {\left (3 \, a - 4 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{2} - 2 \, a b + b^{2}} - \frac {3 \, {\left (3 \, a + 4 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{2} + 2 \, a b + b^{2}} - \frac {12 \, b \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{2}} + \frac {2 \, {\left (2 \, b^{5} \sin \left (d x + c\right )^{3} - 9 \, a^{5} \sin \left (d x + c\right )^{2} + 15 \, a^{3} b^{2} \sin \left (d x + c\right )^{2} - 6 \, a b^{4} \sin \left (d x + c\right )^{2} + 3 \, a^{4} b \sin \left (d x + c\right ) - 3 \, a^{2} b^{3} \sin \left (d x + c\right ) - 2 \, b^{5} \sin \left (d x + c\right ) + 6 \, a^{5} - 12 \, a^{3} b^{2} + 6 \, a b^{4}\right )}}{{\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} {\left (\sin \left (d x + c\right )^{3} - \sin \left (d x + c\right )\right )}}}{12 \, d} \]
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Time = 12.41 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.14 \[ \int \frac {\csc ^2(c+d x) \sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (3\,a-4\,b\right )}{4\,d\,{\left (a-b\right )}^2}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (\frac {b}{4\,{\left (a+b\right )}^2}+\frac {3}{4\,\left (a+b\right )}\right )}{d}-\frac {\frac {1}{a}+\frac {b\,\sin \left (c+d\,x\right )}{2\,\left (a^2-b^2\right )}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (3\,a^2-2\,b^2\right )}{2\,a\,\left (a^2-b^2\right )}}{d\,\left (\sin \left (c+d\,x\right )-{\sin \left (c+d\,x\right )}^3\right )}-\frac {b\,\ln \left (\sin \left (c+d\,x\right )\right )}{a^2\,d}+\frac {b^5\,\ln \left (a+b\,\sin \left (c+d\,x\right )\right )}{a^2\,d\,{\left (a^2-b^2\right )}^2} \]
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