Integrand size = 21, antiderivative size = 116 \[ \int \frac {\csc ^3(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {(b-a \cos (c+d x)) \csc ^2(c+d x)}{2 \left (a^2-b^2\right ) d}+\frac {a \log (1-\cos (c+d x))}{4 (a+b)^2 d}-\frac {a \log (1+\cos (c+d x))}{4 (a-b)^2 d}+\frac {a^2 b \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^2 d} \]
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Time = 0.26 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3957, 2916, 12, 837, 815} \[ \int \frac {\csc ^3(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {a^2 b \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^2}+\frac {\csc ^2(c+d x) (b-a \cos (c+d x))}{2 d \left (a^2-b^2\right )}+\frac {a \log (1-\cos (c+d x))}{4 d (a+b)^2}-\frac {a \log (\cos (c+d x)+1)}{4 d (a-b)^2} \]
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Rule 12
Rule 815
Rule 837
Rule 2916
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cot (c+d x) \csc ^2(c+d x)}{-b-a \cos (c+d x)} \, dx \\ & = \frac {a^3 \text {Subst}\left (\int \frac {x}{a (-b+x) \left (a^2-x^2\right )^2} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^2 \text {Subst}\left (\int \frac {x}{(-b+x) \left (a^2-x^2\right )^2} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {(b-a \cos (c+d x)) \csc ^2(c+d x)}{2 \left (a^2-b^2\right ) d}+\frac {\text {Subst}\left (\int \frac {a^2 b+a^2 x}{(-b+x) \left (a^2-x^2\right )} \, dx,x,-a \cos (c+d x)\right )}{2 \left (a^2-b^2\right ) d} \\ & = \frac {(b-a \cos (c+d x)) \csc ^2(c+d x)}{2 \left (a^2-b^2\right ) d}+\frac {\text {Subst}\left (\int \left (\frac {a (a+b)}{2 (a-b) (a-x)}-\frac {2 a^2 b}{(a-b) (a+b) (b-x)}+\frac {a (a-b)}{2 (a+b) (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{2 \left (a^2-b^2\right ) d} \\ & = \frac {(b-a \cos (c+d x)) \csc ^2(c+d x)}{2 \left (a^2-b^2\right ) d}+\frac {a \log (1-\cos (c+d x))}{4 (a+b)^2 d}-\frac {a \log (1+\cos (c+d x))}{4 (a-b)^2 d}+\frac {a^2 b \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^2 d} \\ \end{align*}
Time = 0.89 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.06 \[ \int \frac {\csc ^3(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {-(a-b)^2 (a+b) \csc ^2\left (\frac {1}{2} (c+d x)\right )-4 a \left ((a+b)^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-2 a b \log (b+a \cos (c+d x))-(a-b)^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+(a-b) (a+b)^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 (a-b)^2 (a+b)^2 d} \]
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Time = 0.72 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {\frac {1}{\left (4 a +4 b \right ) \left (\cos \left (d x +c \right )-1\right )}+\frac {a \ln \left (\cos \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{2}}+\frac {1}{\left (4 a -4 b \right ) \left (\cos \left (d x +c \right )+1\right )}-\frac {a \ln \left (\cos \left (d x +c \right )+1\right )}{4 \left (a -b \right )^{2}}+\frac {b \,a^{2} \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{2} \left (a -b \right )^{2}}}{d}\) | \(110\) |
default | \(\frac {\frac {1}{\left (4 a +4 b \right ) \left (\cos \left (d x +c \right )-1\right )}+\frac {a \ln \left (\cos \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{2}}+\frac {1}{\left (4 a -4 b \right ) \left (\cos \left (d x +c \right )+1\right )}-\frac {a \ln \left (\cos \left (d x +c \right )+1\right )}{4 \left (a -b \right )^{2}}+\frac {b \,a^{2} \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{2} \left (a -b \right )^{2}}}{d}\) | \(110\) |
parallelrisch | \(\frac {8 \ln \left (-2 a +\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (a -b \right )\right ) a^{2} b -\left (a -b \right ) \left (-4 a \left (a -b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a +b \right ) \left (\left (a -b \right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (a +b \right )\right )\right )}{8 d \left (a -b \right )^{2} \left (a +b \right )^{2}}\) | \(111\) |
norman | \(\frac {-\frac {1}{8 d \left (a +b \right )}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 d \left (a -b \right )}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {a^{2} b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}{d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \left (a^{2}+2 a b +b^{2}\right )}\) | \(137\) |
risch | \(-\frac {i a x}{2 \left (a^{2}+2 a b +b^{2}\right )}-\frac {i a c}{2 d \left (a^{2}+2 a b +b^{2}\right )}+\frac {i a x}{2 a^{2}-4 a b +2 b^{2}}+\frac {i a c}{2 d \left (a^{2}-2 a b +b^{2}\right )}-\frac {2 i a^{2} b x}{a^{4}-2 a^{2} b^{2}+b^{4}}-\frac {2 i a^{2} b c}{d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {a \,{\mathrm e}^{3 i \left (d x +c \right )}-2 b \,{\mathrm e}^{2 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )} a}{d \left (-a^{2}+b^{2}\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d \left (a^{2}+2 a b +b^{2}\right )}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d \left (a^{2}-2 a b +b^{2}\right )}+\frac {a^{2} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}\) | \(311\) |
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Time = 0.33 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.86 \[ \int \frac {\csc ^3(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {2 \, a^{2} b - 2 \, b^{3} - 2 \, {\left (a^{3} - a b^{2}\right )} \cos \left (d x + c\right ) - 4 \, {\left (a^{2} b \cos \left (d x + c\right )^{2} - a^{2} b\right )} \log \left (a \cos \left (d x + c\right ) + b\right ) - {\left (a^{3} + 2 \, a^{2} b + a b^{2} - {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (a^{3} - 2 \, a^{2} b + a b^{2} - {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{4 \, {\left ({\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} d\right )}} \]
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\[ \int \frac {\csc ^3(c+d x)}{a+b \sec (c+d x)} \, dx=\int \frac {\csc ^{3}{\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.14 \[ \int \frac {\csc ^3(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\frac {4 \, a^{2} b \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} - \frac {a \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2} - 2 \, a b + b^{2}} + \frac {a \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{2} + 2 \, a b + b^{2}} + \frac {2 \, {\left (a \cos \left (d x + c\right ) - b\right )}}{{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} + b^{2}}}{4 \, d} \]
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Time = 0.34 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.74 \[ \int \frac {\csc ^3(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\frac {8 \, a^{2} b \log \left ({\left | -a - b - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} \right |}\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, a \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{2} + 2 \, a b + b^{2}} + \frac {{\left (a + b - \frac {2 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}{{\left (a^{2} + 2 \, a b + b^{2}\right )} {\left (\cos \left (d x + c\right ) - 1\right )}} - \frac {\cos \left (d x + c\right ) - 1}{{\left (a - b\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}}{8 \, d} \]
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Time = 13.70 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.15 \[ \int \frac {\csc ^3(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {a\,\ln \left (\cos \left (c+d\,x\right )-1\right )}{4\,d\,{\left (a+b\right )}^2}-\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (\frac {a}{4\,{\left (a+b\right )}^2}-\frac {a}{4\,{\left (a-b\right )}^2}\right )}{d}-\frac {\frac {b}{2\,\left (a^2-b^2\right )}-\frac {a\,\cos \left (c+d\,x\right )}{2\,\left (a^2-b^2\right )}}{d\,\left ({\cos \left (c+d\,x\right )}^2-1\right )}-\frac {a\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{4\,d\,{\left (a-b\right )}^2} \]
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