Optimal. Leaf size=214 \[ \frac {a^2 (e x)^{2 n}}{2 e n}+\frac {2 i a b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-e^{i \left (d x^n+c\right )}\right )}{d^2 e n}-\frac {2 i a b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (e^{i \left (d x^n+c\right )}\right )}{d^2 e n}-\frac {4 a b x^{-n} (e x)^{2 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (\sin \left (c+d x^n\right )\right )}{d^2 e n}-\frac {b^2 x^{-n} (e x)^{2 n} \cot \left (c+d x^n\right )}{d e n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.20, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4209, 4205, 4190, 4183, 2279, 2391, 4184, 3475} \[ \frac {2 i a b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,-e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {2 i a b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,e^{i \left (c+d x^n\right )}\right )}{d^2 e n}+\frac {a^2 (e x)^{2 n}}{2 e n}-\frac {4 a b x^{-n} (e x)^{2 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (\sin \left (c+d x^n\right )\right )}{d^2 e n}-\frac {b^2 x^{-n} (e x)^{2 n} \cot \left (c+d x^n\right )}{d e n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2279
Rule 2391
Rule 3475
Rule 4183
Rule 4184
Rule 4190
Rule 4205
Rule 4209
Rubi steps
\begin {align*} \int (e x)^{-1+2 n} \left (a+b \csc \left (c+d x^n\right )\right )^2 \, dx &=\frac {\left (x^{-2 n} (e x)^{2 n}\right ) \int x^{-1+2 n} \left (a+b \csc \left (c+d x^n\right )\right )^2 \, dx}{e}\\ &=\frac {\left (x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int x (a+b \csc (c+d x))^2 \, dx,x,x^n\right )}{e n}\\ &=\frac {\left (x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \left (a^2 x+2 a b x \csc (c+d x)+b^2 x \csc ^2(c+d x)\right ) \, dx,x,x^n\right )}{e n}\\ &=\frac {a^2 (e x)^{2 n}}{2 e n}+\frac {\left (2 a b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int x \csc (c+d x) \, dx,x,x^n\right )}{e n}+\frac {\left (b^2 x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int x \csc ^2(c+d x) \, dx,x,x^n\right )}{e n}\\ &=\frac {a^2 (e x)^{2 n}}{2 e n}-\frac {4 a b x^{-n} (e x)^{2 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}-\frac {b^2 x^{-n} (e x)^{2 n} \cot \left (c+d x^n\right )}{d e n}-\frac {\left (2 a b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \log \left (1-e^{i (c+d x)}\right ) \, dx,x,x^n\right )}{d e n}+\frac {\left (2 a b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \log \left (1+e^{i (c+d x)}\right ) \, dx,x,x^n\right )}{d e n}+\frac {\left (b^2 x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \cot (c+d x) \, dx,x,x^n\right )}{d e n}\\ &=\frac {a^2 (e x)^{2 n}}{2 e n}-\frac {4 a b x^{-n} (e x)^{2 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}-\frac {b^2 x^{-n} (e x)^{2 n} \cot \left (c+d x^n\right )}{d e n}+\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (\sin \left (c+d x^n\right )\right )}{d^2 e n}+\frac {\left (2 i a b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {\left (2 i a b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{d^2 e n}\\ &=\frac {a^2 (e x)^{2 n}}{2 e n}-\frac {4 a b x^{-n} (e x)^{2 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}-\frac {b^2 x^{-n} (e x)^{2 n} \cot \left (c+d x^n\right )}{d e n}+\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (\sin \left (c+d x^n\right )\right )}{d^2 e n}+\frac {2 i a b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {2 i a b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (e^{i \left (c+d x^n\right )}\right )}{d^2 e n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 6.28, size = 286, normalized size = 1.34 \[ \frac {x^{-2 n} (e x)^{2 n} \left (d x^n \left (a^2 d x^n-2 b^2 \cot (c)\right )+4 a b \left (2 \tan ^{-1}(\tan (c)) \tanh ^{-1}\left (\cos (c)-\sin (c) \tan \left (\frac {d x^n}{2}\right )\right )+\frac {\sec (c) \left (i \text {Li}_2\left (-e^{i \left (d x^n+\tan ^{-1}(\tan (c))\right )}\right )-i \text {Li}_2\left (e^{i \left (d x^n+\tan ^{-1}(\tan (c))\right )}\right )+\left (\tan ^{-1}(\tan (c))+d x^n\right ) \left (\log \left (1-e^{i \left (\tan ^{-1}(\tan (c))+d x^n\right )}\right )-\log \left (1+e^{i \left (\tan ^{-1}(\tan (c))+d x^n\right )}\right )\right )\right )}{\sqrt {\sec ^2(c)}}\right )+2 b^2 d \cot (c) x^n+b^2 d \csc \left (\frac {c}{2}\right ) x^n \sin \left (\frac {d x^n}{2}\right ) \csc \left (\frac {1}{2} \left (c+d x^n\right )\right )+b^2 d \sec \left (\frac {c}{2}\right ) x^n \sin \left (\frac {d x^n}{2}\right ) \sec \left (\frac {1}{2} \left (c+d x^n\right )\right )-2 b^2 \left (d \cot (c) x^n-\log \left (\sin \left (c+d x^n\right )\right )\right )\right )}{2 d^2 e n} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.60, size = 568, normalized size = 2.65 \[ \frac {a^{2} d^{2} e^{2 \, n - 1} x^{2 \, n} \sin \left (d x^{n} + c\right ) - 2 \, b^{2} d e^{2 \, n - 1} x^{n} \cos \left (d x^{n} + c\right ) - 2 i \, a b e^{2 \, n - 1} {\rm Li}_2\left (\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right )\right ) \sin \left (d x^{n} + c\right ) + 2 i \, a b e^{2 \, n - 1} {\rm Li}_2\left (\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right )\right ) \sin \left (d x^{n} + c\right ) - 2 i \, a b e^{2 \, n - 1} {\rm Li}_2\left (-\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right )\right ) \sin \left (d x^{n} + c\right ) + 2 i \, a b e^{2 \, n - 1} {\rm Li}_2\left (-\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right )\right ) \sin \left (d x^{n} + c\right ) - {\left (2 \, a b c - b^{2}\right )} e^{2 \, n - 1} \log \left (-\frac {1}{2} \, \cos \left (d x^{n} + c\right ) + \frac {1}{2} i \, \sin \left (d x^{n} + c\right ) + \frac {1}{2}\right ) \sin \left (d x^{n} + c\right ) - {\left (2 \, a b c - b^{2}\right )} e^{2 \, n - 1} \log \left (-\frac {1}{2} \, \cos \left (d x^{n} + c\right ) - \frac {1}{2} i \, \sin \left (d x^{n} + c\right ) + \frac {1}{2}\right ) \sin \left (d x^{n} + c\right ) - {\left (2 \, a b d e^{2 \, n - 1} x^{n} - b^{2} e^{2 \, n - 1}\right )} \log \left (\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right ) + 1\right ) \sin \left (d x^{n} + c\right ) - {\left (2 \, a b d e^{2 \, n - 1} x^{n} - b^{2} e^{2 \, n - 1}\right )} \log \left (\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right ) + 1\right ) \sin \left (d x^{n} + c\right ) + 2 \, {\left (a b d e^{2 \, n - 1} x^{n} + a b c e^{2 \, n - 1}\right )} \log \left (-\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right ) + 1\right ) \sin \left (d x^{n} + c\right ) + 2 \, {\left (a b d e^{2 \, n - 1} x^{n} + a b c e^{2 \, n - 1}\right )} \log \left (-\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right ) + 1\right ) \sin \left (d x^{n} + c\right )}{2 \, d^{2} n \sin \left (d x^{n} + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \csc \left (d x^{n} + c\right ) + a\right )}^{2} \left (e x\right )^{2 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 1.95, size = 674, normalized size = 3.15 \[ \frac {a^{2} x \,{\mathrm e}^{\frac {\left (-1+2 n \right ) \left (-i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )+i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}-i \pi \mathrm {csgn}\left (i e x \right )^{3}+2 \ln \relax (x )+2 \ln \relax (e )\right )}{2}}}{2 n}-\frac {2 i x \,{\mathrm e}^{\frac {\left (-1+2 n \right ) \left (-i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )+i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}-i \pi \mathrm {csgn}\left (i e x \right )^{3}+2 \ln \relax (x )+2 \ln \relax (e )\right )}{2}} b^{2} x^{-n}}{d n \left ({\mathrm e}^{2 i \left (c +d \,x^{n}\right )}-1\right )}-\frac {2 b^{2} \ln \left ({\mathrm e}^{i x^{n} d}\right ) e^{2 n} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i e x \right ) \left (-1+2 n \right ) \left (\mathrm {csgn}\left (i e x \right )-\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i e x \right )+\mathrm {csgn}\left (i e \right )\right )}{2}}}{d^{2} n e}+\frac {b^{2} \ln \left ({\mathrm e}^{2 i \left (c +d \,x^{n}\right )}-1\right ) e^{2 n} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i e x \right ) \left (-1+2 n \right ) \left (\mathrm {csgn}\left (i e x \right )-\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i e x \right )+\mathrm {csgn}\left (i e \right )\right )}{2}}}{d^{2} n e}+\frac {2 b \ln \left (1-{\mathrm e}^{i \left (c +d \,x^{n}\right )}\right ) x^{n} a \,e^{2 n} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i e x \right ) \left (-1+2 n \right ) \left (\mathrm {csgn}\left (i e x \right )-\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i e x \right )+\mathrm {csgn}\left (i e \right )\right )}{2}}}{d n e}-\frac {2 b \ln \left ({\mathrm e}^{i \left (c +d \,x^{n}\right )}+1\right ) x^{n} a \,e^{2 n} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i e x \right ) \left (-1+2 n \right ) \left (\mathrm {csgn}\left (i e x \right )-\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i e x \right )+\mathrm {csgn}\left (i e \right )\right )}{2}}}{d n e}-\frac {2 i b \dilog \left (1-{\mathrm e}^{i \left (c +d \,x^{n}\right )}\right ) a \,e^{2 n} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i e x \right ) \left (-1+2 n \right ) \left (\mathrm {csgn}\left (i e x \right )-\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i e x \right )+\mathrm {csgn}\left (i e \right )\right )}{2}}}{d^{2} n e}+\frac {2 i b \dilog \left ({\mathrm e}^{i \left (c +d \,x^{n}\right )}+1\right ) a \,e^{2 n} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i e x \right ) \left (-1+2 n \right ) \left (\mathrm {csgn}\left (i e x \right )-\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i e x \right )+\mathrm {csgn}\left (i e \right )\right )}{2}}}{d^{2} n e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\left (e x\right )^{2 \, n} a^{2}}{2 \, e n} - \frac {2 \, b^{2} e^{2 \, n} x^{n} \sin \left (2 \, d x^{n} + 2 \, c\right ) - \frac {1}{2} \, {\left (4 \, a b d^{2} e^{2 \, n + 1} \int \frac {x^{2 \, n} \sin \left (d x^{n} + c\right )}{d^{2} e^{2} x \cos \left (d x^{n} + c\right )^{2} + d^{2} e^{2} x \sin \left (d x^{n} + c\right )^{2} + 2 \, d^{2} e^{2} x \cos \left (d x^{n} + c\right ) + d^{2} e^{2} x}\,{d x} + \frac {b^{2} e^{2 \, n - 1} \log \left (\cos \left (d x^{n}\right )^{2} + 2 \, \cos \left (d x^{n}\right ) \cos \relax (c) + \cos \relax (c)^{2} + \sin \left (d x^{n}\right )^{2} - 2 \, \sin \left (d x^{n}\right ) \sin \relax (c) + \sin \relax (c)^{2}\right )}{d^{2} n}\right )} {\left (d e n \cos \left (2 \, d x^{n} + 2 \, c\right )^{2} + d e n \sin \left (2 \, d x^{n} + 2 \, c\right )^{2} - 2 \, d e n \cos \left (2 \, d x^{n} + 2 \, c\right ) + d e n\right )} - \frac {1}{2} \, {\left (4 \, a b d^{2} e^{2 \, n + 1} \int \frac {x^{2 \, n} \sin \left (d x^{n} + c\right )}{d^{2} e^{2} x \cos \left (d x^{n} + c\right )^{2} + d^{2} e^{2} x \sin \left (d x^{n} + c\right )^{2} - 2 \, d^{2} e^{2} x \cos \left (d x^{n} + c\right ) + d^{2} e^{2} x}\,{d x} + \frac {b^{2} e^{2 \, n - 1} \log \left (\cos \left (d x^{n}\right )^{2} - 2 \, \cos \left (d x^{n}\right ) \cos \relax (c) + \cos \relax (c)^{2} + \sin \left (d x^{n}\right )^{2} + 2 \, \sin \left (d x^{n}\right ) \sin \relax (c) + \sin \relax (c)^{2}\right )}{d^{2} n}\right )} {\left (d e n \cos \left (2 \, d x^{n} + 2 \, c\right )^{2} + d e n \sin \left (2 \, d x^{n} + 2 \, c\right )^{2} - 2 \, d e n \cos \left (2 \, d x^{n} + 2 \, c\right ) + d e n\right )}}{d e n \cos \left (2 \, d x^{n} + 2 \, c\right )^{2} + d e n \sin \left (2 \, d x^{n} + 2 \, c\right )^{2} - 2 \, d e n \cos \left (2 \, d x^{n} + 2 \, c\right ) + d e n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+\frac {b}{\sin \left (c+d\,x^n\right )}\right )}^2\,{\left (e\,x\right )}^{2\,n-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{2 n - 1} \left (a + b \csc {\left (c + d x^{n} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________