Integrand size = 22, antiderivative size = 80 \[ \int (e x)^{-1+n} \left (a+b \csc \left (c+d x^n\right )\right )^2 \, dx=\frac {a^2 (e x)^n}{e n}-\frac {2 a b x^{-n} (e x)^n \text {arctanh}\left (\cos \left (c+d x^n\right )\right )}{d e n}-\frac {b^2 x^{-n} (e x)^n \cot \left (c+d x^n\right )}{d e n} \]
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Time = 0.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4294, 4290, 3858, 3855, 3852, 8} \[ \int (e x)^{-1+n} \left (a+b \csc \left (c+d x^n\right )\right )^2 \, dx=\frac {a^2 (e x)^n}{e n}-\frac {2 a b x^{-n} (e x)^n \text {arctanh}\left (\cos \left (c+d x^n\right )\right )}{d e n}-\frac {b^2 x^{-n} (e x)^n \cot \left (c+d x^n\right )}{d e n} \]
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Rule 8
Rule 3852
Rule 3855
Rule 3858
Rule 4290
Rule 4294
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{-n} (e x)^n\right ) \int x^{-1+n} \left (a+b \csc \left (c+d x^n\right )\right )^2 \, dx}{e} \\ & = \frac {\left (x^{-n} (e x)^n\right ) \text {Subst}\left (\int (a+b \csc (c+d x))^2 \, dx,x,x^n\right )}{e n} \\ & = \frac {a^2 (e x)^n}{e n}+\frac {\left (2 a b x^{-n} (e x)^n\right ) \text {Subst}\left (\int \csc (c+d x) \, dx,x,x^n\right )}{e n}+\frac {\left (b^2 x^{-n} (e x)^n\right ) \text {Subst}\left (\int \csc ^2(c+d x) \, dx,x,x^n\right )}{e n} \\ & = \frac {a^2 (e x)^n}{e n}-\frac {2 a b x^{-n} (e x)^n \text {arctanh}\left (\cos \left (c+d x^n\right )\right )}{d e n}-\frac {\left (b^2 x^{-n} (e x)^n\right ) \text {Subst}\left (\int 1 \, dx,x,\cot \left (c+d x^n\right )\right )}{d e n} \\ & = \frac {a^2 (e x)^n}{e n}-\frac {2 a b x^{-n} (e x)^n \text {arctanh}\left (\cos \left (c+d x^n\right )\right )}{d e n}-\frac {b^2 x^{-n} (e x)^n \cot \left (c+d x^n\right )}{d e n} \\ \end{align*}
Time = 2.10 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.28 \[ \int (e x)^{-1+n} \left (a+b \csc \left (c+d x^n\right )\right )^2 \, dx=\frac {x^{-n} (e x)^n \left (-b^2 \cot \left (\frac {1}{2} \left (c+d x^n\right )\right )+2 a \left (a c+a d x^n-2 b \log \left (\cos \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )+2 b \log \left (\sin \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )\right )+b^2 \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}{2 d e n} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.42 (sec) , antiderivative size = 275, normalized size of antiderivative = 3.44
method | result | size |
risch | \(\frac {a^{2} x \,{\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right ) \pi +i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi +i \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi -i \operatorname {csgn}\left (i e x \right )^{3} \pi +2 \ln \left (e \right )+2 \ln \left (x \right )\right )}{2}}}{n}-\frac {2 i x \,b^{2} {\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right ) \pi +i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi +i \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi -i \operatorname {csgn}\left (i e x \right )^{3} \pi +2 \ln \left (e \right )+2 \ln \left (x \right )\right )}{2}} x^{-n}}{d n \left ({\mathrm e}^{2 i \left (c +d \,x^{n}\right )}-1\right )}-\frac {4 \,\operatorname {arctanh}\left ({\mathrm e}^{i \left (c +d \,x^{n}\right )}\right ) e^{n} a b \,{\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i e x \right ) \left (-1+n \right ) \left (\operatorname {csgn}\left (i e x \right )-\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i e x \right )+\operatorname {csgn}\left (i e \right )\right )}{2}}}{d e n}\) | \(275\) |
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Time = 0.26 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.45 \[ \int (e x)^{-1+n} \left (a+b \csc \left (c+d x^n\right )\right )^2 \, dx=\frac {a^{2} d e^{n - 1} x^{n} \sin \left (d x^{n} + c\right ) - a b e^{n - 1} \log \left (\frac {1}{2} \, \cos \left (d x^{n} + c\right ) + \frac {1}{2}\right ) \sin \left (d x^{n} + c\right ) + a b e^{n - 1} \log \left (-\frac {1}{2} \, \cos \left (d x^{n} + c\right ) + \frac {1}{2}\right ) \sin \left (d x^{n} + c\right ) - b^{2} e^{n - 1} \cos \left (d x^{n} + c\right )}{d n \sin \left (d x^{n} + c\right )} \]
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\[ \int (e x)^{-1+n} \left (a+b \csc \left (c+d x^n\right )\right )^2 \, dx=\int \left (e x\right )^{n - 1} \left (a + b \csc {\left (c + d x^{n} \right )}\right )^{2}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (80) = 160\).
Time = 0.26 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.59 \[ \int (e x)^{-1+n} \left (a+b \csc \left (c+d x^n\right )\right )^2 \, dx=-\frac {2 \, b^{2} e^{n} \sin \left (2 \, d x^{n} + 2 \, c\right )}{d e n \cos \left (2 \, d x^{n} + 2 \, c\right )^{2} + d e n \sin \left (2 \, d x^{n} + 2 \, c\right )^{2} - 2 \, d e n \cos \left (2 \, d x^{n} + 2 \, c\right ) + d e n} + \frac {\left (e x\right )^{n} a^{2}}{e n} - \frac {{\left (e^{n} \log \left (\cos \left (d x^{n}\right )^{2} + 2 \, \cos \left (d x^{n}\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (d x^{n}\right )^{2} - 2 \, \sin \left (d x^{n}\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) - e^{n} \log \left (\cos \left (d x^{n}\right )^{2} - 2 \, \cos \left (d x^{n}\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (d x^{n}\right )^{2} + 2 \, \sin \left (d x^{n}\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right )\right )} a b}{d e n} \]
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\[ \int (e x)^{-1+n} \left (a+b \csc \left (c+d x^n\right )\right )^2 \, dx=\int { {\left (b \csc \left (d x^{n} + c\right ) + a\right )}^{2} \left (e x\right )^{n - 1} \,d x } \]
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Time = 20.61 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.28 \[ \int (e x)^{-1+n} \left (a+b \csc \left (c+d x^n\right )\right )^2 \, dx=\frac {a^2\,x\,{\left (e\,x\right )}^{n-1}}{n}-\frac {b^2\,x\,{\left (e\,x\right )}^{n-1}\,2{}\mathrm {i}}{d\,n\,x^n\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x^n\,2{}\mathrm {i}}-1\right )}-\frac {2\,a\,b\,x\,\ln \left (-a\,b\,{\left (e\,x\right )}^{n-1}\,4{}\mathrm {i}-a\,b\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{d\,x^n\,1{}\mathrm {i}}\,{\left (e\,x\right )}^{n-1}\,4{}\mathrm {i}\right )\,{\left (e\,x\right )}^{n-1}}{d\,n\,x^n}+\frac {2\,a\,b\,x\,\ln \left (a\,b\,{\left (e\,x\right )}^{n-1}\,4{}\mathrm {i}-a\,b\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{d\,x^n\,1{}\mathrm {i}}\,{\left (e\,x\right )}^{n-1}\,4{}\mathrm {i}\right )\,{\left (e\,x\right )}^{n-1}}{d\,n\,x^n} \]
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