Integrand size = 21, antiderivative size = 20 \[ \int \frac {\csc ^2(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=-\frac {2 d}{3 b (d \tan (a+b x))^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2671, 30} \[ \int \frac {\csc ^2(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=-\frac {2 d}{3 b (d \tan (a+b x))^{3/2}} \]
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Rule 30
Rule 2671
Rubi steps \begin{align*} \text {integral}& = \frac {d \text {Subst}\left (\int \frac {1}{x^{5/2}} \, dx,x,d \tan (a+b x)\right )}{b} \\ & = -\frac {2 d}{3 b (d \tan (a+b x))^{3/2}} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\csc ^2(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=-\frac {2 d}{3 b (d \tan (a+b x))^{3/2}} \]
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Time = 0.16 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(-\frac {2 d}{3 b \left (d \tan \left (b x +a \right )\right )^{\frac {3}{2}}}\) | \(17\) |
default | \(-\frac {2 d}{3 b \left (d \tan \left (b x +a \right )\right )^{\frac {3}{2}}}\) | \(17\) |
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (16) = 32\).
Time = 0.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.30 \[ \int \frac {\csc ^2(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=\frac {2 \, \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \cos \left (b x + a\right )^{2}}{3 \, {\left (b d \cos \left (b x + a\right )^{2} - b d\right )}} \]
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\[ \int \frac {\csc ^2(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=\int \frac {\csc ^{2}{\left (a + b x \right )}}{\sqrt {d \tan {\left (a + b x \right )}}}\, dx \]
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none
Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {\csc ^2(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=-\frac {2}{3 \, \sqrt {d \tan \left (b x + a\right )} b \tan \left (b x + a\right )} \]
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none
Time = 0.32 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {\csc ^2(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=-\frac {2}{3 \, \sqrt {d \tan \left (b x + a\right )} b \tan \left (b x + a\right )} \]
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Time = 3.43 (sec) , antiderivative size = 102, normalized size of antiderivative = 5.10 \[ \int \frac {\csc ^2(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=-\frac {2\,\sqrt {\frac {d\,\sin \left (2\,a+2\,b\,x\right )}{\cos \left (2\,a+2\,b\,x\right )+1}}\,\left (\cos \left (2\,a+2\,b\,x\right )+2\,\cos \left (4\,a+4\,b\,x\right )-\cos \left (6\,a+6\,b\,x\right )-2\right )}{3\,b\,d\,\left (15\,\cos \left (2\,a+2\,b\,x\right )-6\,\cos \left (4\,a+4\,b\,x\right )+\cos \left (6\,a+6\,b\,x\right )-10\right )} \]
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