Integrand size = 21, antiderivative size = 114 \[ \int \csc ^3(c+d x) (a+b \sec (c+d x))^2 \, dx=-\frac {\left (2 a b+\left (a^2+b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 d}+\frac {(a+b) (a+3 b) \log (1-\cos (c+d x))}{4 d}-\frac {2 a b \log (\cos (c+d x))}{d}-\frac {(a-3 b) (a-b) \log (1+\cos (c+d x))}{4 d}+\frac {b^2 \sec (c+d x)}{d} \]
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Time = 0.37 (sec) , antiderivative size = 114, 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, 1819, 1816} \[ \int \csc ^3(c+d x) (a+b \sec (c+d x))^2 \, dx=-\frac {\csc ^2(c+d x) \left (\left (a^2+b^2\right ) \cos (c+d x)+2 a b\right )}{2 d}-\frac {2 a b \log (\cos (c+d x))}{d}+\frac {(a+b) (a+3 b) \log (1-\cos (c+d x))}{4 d}-\frac {(a-3 b) (a-b) \log (\cos (c+d x)+1)}{4 d}+\frac {b^2 \sec (c+d x)}{d} \]
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Rule 12
Rule 1816
Rule 1819
Rule 2916
Rule 3957
Rubi steps \begin{align*} \text {integral}& = \int (-b-a \cos (c+d x))^2 \csc ^3(c+d x) \sec ^2(c+d x) \, dx \\ & = \frac {a^3 \text {Subst}\left (\int \frac {a^2 (-b+x)^2}{x^2 \left (a^2-x^2\right )^2} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^5 \text {Subst}\left (\int \frac {(-b+x)^2}{x^2 \left (a^2-x^2\right )^2} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = -\frac {a \left (2 b+\frac {\left (a^2+b^2\right ) \cos (c+d x)}{a}\right ) \csc ^2(c+d x)}{2 d}-\frac {a^3 \text {Subst}\left (\int \frac {-2 b^2+4 b x-\frac {\left (a^2+b^2\right ) x^2}{a^2}}{x^2 \left (a^2-x^2\right )} \, dx,x,-a \cos (c+d x)\right )}{2 d} \\ & = -\frac {a \left (2 b+\frac {\left (a^2+b^2\right ) \cos (c+d x)}{a}\right ) \csc ^2(c+d x)}{2 d}-\frac {a^3 \text {Subst}\left (\int \left (\frac {(a-3 b) (-a+b)}{2 a^3 (a-x)}-\frac {2 b^2}{a^2 x^2}+\frac {4 b}{a^2 x}+\frac {(-a-3 b) (a+b)}{2 a^3 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{2 d} \\ & = -\frac {a \left (2 b+\frac {\left (a^2+b^2\right ) \cos (c+d x)}{a}\right ) \csc ^2(c+d x)}{2 d}+\frac {(a+b) (a+3 b) \log (1-\cos (c+d x))}{4 d}-\frac {2 a b \log (\cos (c+d x))}{d}-\frac {(a-3 b) (a-b) \log (1+\cos (c+d x))}{4 d}+\frac {b^2 \sec (c+d x)}{d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(329\) vs. \(2(114)=228\).
Time = 1.38 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.89 \[ \int \csc ^3(c+d x) (a+b \sec (c+d x))^2 \, dx=-\frac {\csc ^4(c+d x) \left (2 a^2-2 b^2+2 \left (a^2+3 b^2\right ) \cos (2 (c+d x))-a^2 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+4 a b \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-3 b^2 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-4 a b \cos (3 (c+d x)) \log (\cos (c+d x))+a^2 \cos (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 a b \cos (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+3 b^2 \cos (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\cos (c+d x) \left (8 a b+\left (a^2-4 a b+3 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+4 a b \log (\cos (c+d x))-a^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-4 a b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-3 b^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )}{2 d \left (\csc ^2\left (\frac {1}{2} (c+d x)\right )-\sec ^2\left (\frac {1}{2} (c+d x)\right )\right )} \]
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Time = 0.94 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.02
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\frac {\cot \left (d x +c \right ) \csc \left (d x +c \right )}{2}+\frac {\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )+2 a b \left (-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+b^{2} \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )}{d}\) | \(116\) |
default | \(\frac {a^{2} \left (-\frac {\cot \left (d x +c \right ) \csc \left (d x +c \right )}{2}+\frac {\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )+2 a b \left (-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+b^{2} \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )}{d}\) | \(116\) |
norman | \(\frac {\frac {a^{2}+2 a b +b^{2}}{8 d}+\frac {\left (a^{2}-2 a b +b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{8 d}-\frac {\left (a^{2}+9 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{4 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}+\frac {\left (a^{2}+4 a b +3 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {2 a b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {2 a b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(164\) |
parallelrisch | \(\frac {16 b a \left (1-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+16 b a \left (1-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \left (a +3 b \right ) \left (a +b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a -b \right )^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (a +b \right )^{2} \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 a^{2}-18 b^{2}}{8 d \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}\) | \(172\) |
risch | \(\frac {a^{2} {\mathrm e}^{5 i \left (d x +c \right )}+3 b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+4 a b \,{\mathrm e}^{4 i \left (d x +c \right )}+2 a^{2} {\mathrm e}^{3 i \left (d x +c \right )}-2 b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+4 a b \,{\mathrm e}^{2 i \left (d x +c \right )}+a^{2} {\mathrm e}^{i \left (d x +c \right )}+3 b^{2} {\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) a b}{d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{2 d}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) a b}{d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{2 d}-\frac {2 a b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(278\) |
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Time = 0.30 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.80 \[ \int \csc ^3(c+d x) (a+b \sec (c+d x))^2 \, dx=\frac {4 \, a b \cos \left (d x + c\right ) + 2 \, {\left (a^{2} + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 4 \, b^{2} - 8 \, {\left (a b \cos \left (d x + c\right )^{3} - a b \cos \left (d x + c\right )\right )} \log \left (-\cos \left (d x + c\right )\right ) - {\left ({\left (a^{2} - 4 \, a b + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{2} - 4 \, a b + 3 \, b^{2}\right )} \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left ({\left (a^{2} + 4 \, a b + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{2} + 4 \, a b + 3 \, b^{2}\right )} \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{4 \, {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )\right )}} \]
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\[ \int \csc ^3(c+d x) (a+b \sec (c+d x))^2 \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{2} \csc ^{3}{\left (c + d x \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.04 \[ \int \csc ^3(c+d x) (a+b \sec (c+d x))^2 \, dx=-\frac {8 \, a b \log \left (\cos \left (d x + c\right )\right ) + {\left (a^{2} - 4 \, a b + 3 \, b^{2}\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) - {\left (a^{2} + 4 \, a b + 3 \, b^{2}\right )} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (2 \, a b \cos \left (d x + c\right ) + {\left (a^{2} + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, b^{2}\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )}}{4 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (108) = 216\).
Time = 0.33 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.75 \[ \int \csc ^3(c+d x) (a+b \sec (c+d x))^2 \, dx=-\frac {16 \, a b \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {2 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - 2 \, {\left (a^{2} + 4 \, a b + 3 \, b^{2}\right )} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - \frac {a^{2} + 2 \, a b + b^{2} + \frac {6 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {14 \, b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {4 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {3 \, b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + \frac {{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}}{8 \, d} \]
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Time = 0.13 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.05 \[ \int \csc ^3(c+d x) (a+b \sec (c+d x))^2 \, dx=\frac {\ln \left (\cos \left (c+d\,x\right )-1\right )\,\left (a+b\right )\,\left (a+3\,b\right )}{4\,d}-\frac {\ln \left (\cos \left (c+d\,x\right )+1\right )\,\left (a-b\right )\,\left (a-3\,b\right )}{4\,d}-\frac {2\,a\,b\,\ln \left (\cos \left (c+d\,x\right )\right )}{d}-\frac {{\cos \left (c+d\,x\right )}^2\,\left (\frac {a^2}{2}+\frac {3\,b^2}{2}\right )-b^2+a\,b\,\cos \left (c+d\,x\right )}{d\,\left (\cos \left (c+d\,x\right )-{\cos \left (c+d\,x\right )}^3\right )} \]
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