Integrand size = 8, antiderivative size = 27 \[ \int \csc ^4(a+b x) \, dx=-\frac {\cot (a+b x)}{b}-\frac {\cot ^3(a+b x)}{3 b} \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3852} \[ \int \csc ^4(a+b x) \, dx=-\frac {\cot ^3(a+b x)}{3 b}-\frac {\cot (a+b x)}{b} \]
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Rule 3852
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (a+b x)\right )}{b} \\ & = -\frac {\cot (a+b x)}{b}-\frac {\cot ^3(a+b x)}{3 b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \csc ^4(a+b x) \, dx=-\frac {2 \cot (a+b x)}{3 b}-\frac {\cot (a+b x) \csc ^2(a+b x)}{3 b} \]
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Time = 0.44 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {\left (-\frac {2}{3}-\frac {\csc \left (x b +a \right )^{2}}{3}\right ) \cot \left (x b +a \right )}{b}\) | \(23\) |
default | \(\frac {\left (-\frac {2}{3}-\frac {\csc \left (x b +a \right )^{2}}{3}\right ) \cot \left (x b +a \right )}{b}\) | \(23\) |
risch | \(\frac {4 i \left (3 \,{\mathrm e}^{2 i \left (x b +a \right )}-1\right )}{3 b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{3}}\) | \(33\) |
parallelrisch | \(\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}-\cot \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}+9 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )-9 \cot \left (\frac {a}{2}+\frac {x b}{2}\right )}{24 b}\) | \(53\) |
norman | \(\frac {-\frac {1}{24 b}-\frac {3 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{8 b}+\frac {3 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{8 b}+\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{24 b}}{\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}\) | \(67\) |
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none
Time = 0.23 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67 \[ \int \csc ^4(a+b x) \, dx=-\frac {2 \, \cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )}{3 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \]
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\[ \int \csc ^4(a+b x) \, dx=\int \csc ^{4}{\left (a + b x \right )}\, dx \]
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none
Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \csc ^4(a+b x) \, dx=-\frac {3 \, \tan \left (b x + a\right )^{2} + 1}{3 \, b \tan \left (b x + a\right )^{3}} \]
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none
Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \csc ^4(a+b x) \, dx=-\frac {3 \, \tan \left (b x + a\right )^{2} + 1}{3 \, b \tan \left (b x + a\right )^{3}} \]
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Time = 21.59 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \csc ^4(a+b x) \, dx=-\frac {\mathrm {cot}\left (a+b\,x\right )\,\left ({\mathrm {cot}\left (a+b\,x\right )}^2+3\right )}{3\,b} \]
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