Integrand size = 19, antiderivative size = 64 \[ \int \csc ^3(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {a \text {arctanh}(\cos (c+d x))}{2 d}-\frac {b \cot ^2(c+d x)}{2 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d}+\frac {b \log (\tan (c+d x))}{d} \]
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Time = 0.13 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3957, 2913, 2700, 14, 3853, 3855} \[ \int \csc ^3(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {a \text {arctanh}(\cos (c+d x))}{2 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d}-\frac {b \cot ^2(c+d x)}{2 d}+\frac {b \log (\tan (c+d x))}{d} \]
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Rule 14
Rule 2700
Rule 2913
Rule 3853
Rule 3855
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int (-b-a \cos (c+d x)) \csc ^3(c+d x) \sec (c+d x) \, dx \\ & = a \int \csc ^3(c+d x) \, dx+b \int \csc ^3(c+d x) \sec (c+d x) \, dx \\ & = -\frac {a \cot (c+d x) \csc (c+d x)}{2 d}+\frac {1}{2} a \int \csc (c+d x) \, dx+\frac {b \text {Subst}\left (\int \frac {1+x^2}{x^3} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {a \text {arctanh}(\cos (c+d x))}{2 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d}+\frac {b \text {Subst}\left (\int \left (\frac {1}{x^3}+\frac {1}{x}\right ) \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {a \text {arctanh}(\cos (c+d x))}{2 d}-\frac {b \cot ^2(c+d x)}{2 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d}+\frac {b \log (\tan (c+d x))}{d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.88 \[ \int \csc ^3(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {b \csc ^2(c+d x)}{2 d}-\frac {a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {b \log (\cos (c+d x))}{d}+\frac {a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {b \log (\sin (c+d x))}{d}+\frac {a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d} \]
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Time = 1.16 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {\cot \left (d x +c \right ) \csc \left (d x +c \right )}{2}+\frac {\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )+b \left (-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(61\) |
default | \(\frac {a \left (-\frac {\cot \left (d x +c \right ) \csc \left (d x +c \right )}{2}+\frac {\ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{2}\right )+b \left (-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(61\) |
parallelrisch | \(\frac {-8 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-8 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (4 a +8 b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a -b \right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (a -b \right )}{8 d}\) | \(91\) |
norman | \(\frac {-\frac {a +b}{8 d}+\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}+\frac {\left (a +2 b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\) | \(100\) |
risch | \(\frac {a \,{\mathrm e}^{3 i \left (d x +c \right )}+2 b \,{\mathrm e}^{2 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )} a}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b}{d}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b}{d}-\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(142\) |
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Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (58) = 116\).
Time = 0.29 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.92 \[ \int \csc ^3(c+d x) (a+b \sec (c+d x)) \, dx=\frac {2 \, a \cos \left (d x + c\right ) - 4 \, {\left (b \cos \left (d x + c\right )^{2} - b\right )} \log \left (-\cos \left (d x + c\right )\right ) - {\left ({\left (a - 2 \, b\right )} \cos \left (d x + c\right )^{2} - a + 2 \, b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left ({\left (a + 2 \, b\right )} \cos \left (d x + c\right )^{2} - a - 2 \, b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, b}{4 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
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\[ \int \csc ^3(c+d x) (a+b \sec (c+d x)) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right ) \csc ^{3}{\left (c + d x \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.11 \[ \int \csc ^3(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {{\left (a - 2 \, b\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) - {\left (a + 2 \, b\right )} \log \left (\cos \left (d x + c\right ) - 1\right ) + 4 \, b \log \left (\cos \left (d x + c\right )\right ) - \frac {2 \, {\left (a \cos \left (d x + c\right ) + b\right )}}{\cos \left (d x + c\right )^{2} - 1}}{4 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (58) = 116\).
Time = 0.30 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.64 \[ \int \csc ^3(c+d x) (a+b \sec (c+d x)) \, dx=\frac {2 \, {\left (a + 2 \, b\right )} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 8 \, b \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {{\left (a + b - \frac {2 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {4 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}{\cos \left (d x + c\right ) - 1} - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{8 \, d} \]
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Time = 0.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.19 \[ \int \csc ^3(c+d x) (a+b \sec (c+d x)) \, dx=\frac {\frac {\frac {b}{2}+\frac {a\,\cos \left (c+d\,x\right )}{2}}{{\cos \left (c+d\,x\right )}^2-1}+\ln \left (\cos \left (c+d\,x\right )-1\right )\,\left (\frac {a}{4}+\frac {b}{2}\right )-\ln \left (\cos \left (c+d\,x\right )+1\right )\,\left (\frac {a}{4}-\frac {b}{2}\right )-b\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
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