3.1.74 \(\int (e x)^{-1+2 n} (a+b \csc (c+d x^n)) \, dx\) [74]

Optimal. Leaf size=141 \[ \frac {a (e x)^{2 n}}{2 e n}-\frac {2 b x^{-n} (e x)^{2 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {i b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,-e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {i b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,e^{i \left (c+d x^n\right )}\right )}{d^2 e n} \]

[Out]

1/2*a*(e*x)^(2*n)/e/n-2*b*(e*x)^(2*n)*arctanh(exp(I*(c+d*x^n)))/d/e/n/(x^n)+I*b*(e*x)^(2*n)*polylog(2,-exp(I*(
c+d*x^n)))/d^2/e/n/(x^(2*n))-I*b*(e*x)^(2*n)*polylog(2,exp(I*(c+d*x^n)))/d^2/e/n/(x^(2*n))

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Rubi [A]
time = 0.08, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {14, 4294, 4290, 4268, 2317, 2438} \begin {gather*} \frac {a (e x)^{2 n}}{2 e n}+\frac {i 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 {i 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 b x^{-n} (e x)^{2 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*x)^(-1 + 2*n)*(a + b*Csc[c + d*x^n]),x]

[Out]

(a*(e*x)^(2*n))/(2*e*n) - (2*b*(e*x)^(2*n)*ArcTanh[E^(I*(c + d*x^n))])/(d*e*n*x^n) + (I*b*(e*x)^(2*n)*PolyLog[
2, -E^(I*(c + d*x^n))])/(d^2*e*n*x^(2*n)) - (I*b*(e*x)^(2*n)*PolyLog[2, E^(I*(c + d*x^n))])/(d^2*e*n*x^(2*n))

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 4268

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*
x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[d*(m/f), Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 4290

Int[((a_.) + Csc[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Csc[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[
(m + 1)/n], 0] && IntegerQ[p]

Rule 4294

Int[((a_.) + Csc[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*((e_)*(x_))^(m_.), x_Symbol] :> Dist[e^IntPart[m]*((e*x
)^FracPart[m]/x^FracPart[m]), Int[x^m*(a + b*Csc[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]

Rubi steps

\begin {align*} \int (e x)^{-1+2 n} \left (a+b \csc \left (c+d x^n\right )\right ) \, dx &=\int \left (a (e x)^{-1+2 n}+b (e x)^{-1+2 n} \csc \left (c+d x^n\right )\right ) \, dx\\ &=\frac {a (e x)^{2 n}}{2 e n}+b \int (e x)^{-1+2 n} \csc \left (c+d x^n\right ) \, dx\\ &=\frac {a (e x)^{2 n}}{2 e n}+\frac {\left (b x^{-2 n} (e x)^{2 n}\right ) \int x^{-1+2 n} \csc \left (c+d x^n\right ) \, dx}{e}\\ &=\frac {a (e x)^{2 n}}{2 e n}+\frac {\left (b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int x \csc (c+d x) \, dx,x,x^n\right )}{e n}\\ &=\frac {a (e x)^{2 n}}{2 e n}-\frac {2 b x^{-n} (e x)^{2 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}-\frac {\left (b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \log \left (1-e^{i (c+d x)}\right ) \, dx,x,x^n\right )}{d e n}+\frac {\left (b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \log \left (1+e^{i (c+d x)}\right ) \, dx,x,x^n\right )}{d e n}\\ &=\frac {a (e x)^{2 n}}{2 e n}-\frac {2 b x^{-n} (e x)^{2 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {\left (i b x^{-2 n} (e x)^{2 n}\right ) \text {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 (i b x^{-2 n} (e x)^{2 n}\right ) \text {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 (e x)^{2 n}}{2 e n}-\frac {2 b x^{-n} (e x)^{2 n} \tanh ^{-1}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {i 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 {i 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*}

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Mathematica [A]
time = 0.24, size = 185, normalized size = 1.31 \begin {gather*} \frac {x^{-2 n} (e x)^{2 n} \left (a d^2 x^{2 n}+2 b c \log \left (1-e^{i \left (c+d x^n\right )}\right )+2 b d x^n \log \left (1-e^{i \left (c+d x^n\right )}\right )-2 b c \log \left (1+e^{i \left (c+d x^n\right )}\right )-2 b d x^n \log \left (1+e^{i \left (c+d x^n\right )}\right )-2 b c \log \left (\tan \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )+2 i b \text {PolyLog}\left (2,-e^{i \left (c+d x^n\right )}\right )-2 i b \text {PolyLog}\left (2,e^{i \left (c+d x^n\right )}\right )\right )}{2 d^2 e n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^(-1 + 2*n)*(a + b*Csc[c + d*x^n]),x]

[Out]

((e*x)^(2*n)*(a*d^2*x^(2*n) + 2*b*c*Log[1 - E^(I*(c + d*x^n))] + 2*b*d*x^n*Log[1 - E^(I*(c + d*x^n))] - 2*b*c*
Log[1 + E^(I*(c + d*x^n))] - 2*b*d*x^n*Log[1 + E^(I*(c + d*x^n))] - 2*b*c*Log[Tan[(c + d*x^n)/2]] + (2*I)*b*Po
lyLog[2, -E^(I*(c + d*x^n))] - (2*I)*b*PolyLog[2, E^(I*(c + d*x^n))]))/(2*d^2*e*n*x^(2*n))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.26, size = 699, normalized size = 4.96

method	result	size
risch	\(\frac {a x \,{\mathrm e}^{\frac {\left (-1+2 n \right ) \left (-i \mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right ) \pi +i \mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2} \pi +i \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2} \pi -i \mathrm {csgn}\left (i e x \right )^{3} \pi +2 \ln \left (e \right )+2 \ln \left (x \right )\right )}{2}}}{2 n}+\frac {e^{2 n} b \ln \left (1-{\mathrm e}^{i \left (c +d \,x^{n}\right )}\right ) x^{n} \left (-1\right )^{\frac {\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )}{2}} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i e x \right ) \left (-2 n \mathrm {csgn}\left (i e x \right )^{2}+2 n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )+2 n \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )-2 n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right )+\mathrm {csgn}\left (i e x \right )^{2}-\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )-\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )\right )}{2}}}{d n e}-\frac {e^{2 n} b \ln \left ({\mathrm e}^{i \left (c +d \,x^{n}\right )}+1\right ) x^{n} \left (-1\right )^{\frac {\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )}{2}} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i e x \right ) \left (-2 n \mathrm {csgn}\left (i e x \right )^{2}+2 n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )+2 n \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )-2 n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right )+\mathrm {csgn}\left (i e x \right )^{2}-\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )-\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )\right )}{2}}}{d n e}-\frac {i e^{2 n} b \dilog \left (1-{\mathrm e}^{i \left (c +d \,x^{n}\right )}\right ) \left (-1\right )^{\frac {\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )}{2}} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i e x \right ) \left (-2 n \mathrm {csgn}\left (i e x \right )^{2}+2 n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )+2 n \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )-2 n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right )+\mathrm {csgn}\left (i e x \right )^{2}-\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )-\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )\right )}{2}}}{d^{2} n e}+\frac {i e^{2 n} b \dilog \left ({\mathrm e}^{i \left (c +d \,x^{n}\right )}+1\right ) \left (-1\right )^{\frac {\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )}{2}} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i e x \right ) \left (-2 n \mathrm {csgn}\left (i e x \right )^{2}+2 n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )+2 n \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )-2 n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right )+\mathrm {csgn}\left (i e x \right )^{2}-\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )-\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )\right )}{2}}}{d^{2} n e}\)	\(699\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(-1+2*n)*(a+b*csc(c+d*x^n)),x,method=_RETURNVERBOSE)

[Out]

1/2*a/n*x*exp(1/2*(-1+2*n)*(-I*csgn(I*e)*csgn(I*x)*csgn(I*e*x)*Pi+I*csgn(I*e)*csgn(I*e*x)^2*Pi+I*csgn(I*x)*csg
n(I*e*x)^2*Pi-I*csgn(I*e*x)^3*Pi+2*ln(e)+2*ln(x)))+1/d/n/e*(e^n)^2*b*ln(1-exp(I*(c+d*x^n)))*x^n*(-1)^(1/2*csgn
(I*e)*csgn(I*x)*csgn(I*e*x))*exp(1/2*I*Pi*csgn(I*e*x)*(-2*n*csgn(I*e*x)^2+2*n*csgn(I*e)*csgn(I*e*x)+2*n*csgn(I
*x)*csgn(I*e*x)-2*n*csgn(I*e)*csgn(I*x)+csgn(I*e*x)^2-csgn(I*e)*csgn(I*e*x)-csgn(I*x)*csgn(I*e*x)))-1/d/n/e*(e
^n)^2*b*ln(exp(I*(c+d*x^n))+1)*x^n*(-1)^(1/2*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*exp(1/2*I*Pi*csgn(I*e*x)*(-2*n*c
sgn(I*e*x)^2+2*n*csgn(I*e)*csgn(I*e*x)+2*n*csgn(I*x)*csgn(I*e*x)-2*n*csgn(I*e)*csgn(I*x)+csgn(I*e*x)^2-csgn(I*
e)*csgn(I*e*x)-csgn(I*x)*csgn(I*e*x)))-I/d^2/n/e*(e^n)^2*b*dilog(1-exp(I*(c+d*x^n)))*(-1)^(1/2*csgn(I*e)*csgn(
I*x)*csgn(I*e*x))*exp(1/2*I*Pi*csgn(I*e*x)*(-2*n*csgn(I*e*x)^2+2*n*csgn(I*e)*csgn(I*e*x)+2*n*csgn(I*x)*csgn(I*
e*x)-2*n*csgn(I*e)*csgn(I*x)+csgn(I*e*x)^2-csgn(I*e)*csgn(I*e*x)-csgn(I*x)*csgn(I*e*x)))+I/d^2/n/e*(e^n)^2*b*d
ilog(exp(I*(c+d*x^n))+1)*(-1)^(1/2*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*exp(1/2*I*Pi*csgn(I*e*x)*(-2*n*csgn(I*e*x)
^2+2*n*csgn(I*e)*csgn(I*e*x)+2*n*csgn(I*x)*csgn(I*e*x)-2*n*csgn(I*e)*csgn(I*x)+csgn(I*e*x)^2-csgn(I*e)*csgn(I*
e*x)-csgn(I*x)*csgn(I*e*x)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(-1+2*n)*(a+b*csc(c+d*x^n)),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(2*n-1>0)', see `assume?` for m
ore details)

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 371 vs. \(2 (131) = 262\).
time = 3.14, size = 371, normalized size = 2.63 \begin {gather*} \frac {a d^{2} x^{2 \, n} e^{\left (2 \, n - 1\right )} - b d x^{n} e^{\left (2 \, n - 1\right )} \log \left (\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right ) + 1\right ) - b d x^{n} e^{\left (2 \, n - 1\right )} \log \left (\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right ) + 1\right ) - b c e^{\left (2 \, n - 1\right )} \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 ) - b c e^{\left (2 \, n - 1\right )} \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 ) - i \, b {\rm Li}_2\left (\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right )\right ) e^{\left (2 \, n - 1\right )} + i \, b {\rm Li}_2\left (\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right )\right ) e^{\left (2 \, n - 1\right )} - i \, b {\rm Li}_2\left (-\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right )\right ) e^{\left (2 \, n - 1\right )} + i \, b {\rm Li}_2\left (-\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right )\right ) e^{\left (2 \, n - 1\right )} + {\left (b d x^{n} e^{\left (2 \, n - 1\right )} + b c e^{\left (2 \, n - 1\right )}\right )} \log \left (-\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right ) + 1\right ) + {\left (b d x^{n} e^{\left (2 \, n - 1\right )} + b c e^{\left (2 \, n - 1\right )}\right )} \log \left (-\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right ) + 1\right )}{2 \, d^{2} n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(-1+2*n)*(a+b*csc(c+d*x^n)),x, algorithm="fricas")

[Out]

1/2*(a*d^2*x^(2*n)*e^(2*n - 1) - b*d*x^n*e^(2*n - 1)*log(cos(d*x^n + c) + I*sin(d*x^n + c) + 1) - b*d*x^n*e^(2
*n - 1)*log(cos(d*x^n + c) - I*sin(d*x^n + c) + 1) - b*c*e^(2*n - 1)*log(-1/2*cos(d*x^n + c) + 1/2*I*sin(d*x^n
+ c) + 1/2) - b*c*e^(2*n - 1)*log(-1/2*cos(d*x^n + c) - 1/2*I*sin(d*x^n + c) + 1/2) - I*b*dilog(cos(d*x^n + c
) + I*sin(d*x^n + c))*e^(2*n - 1) + I*b*dilog(cos(d*x^n + c) - I*sin(d*x^n + c))*e^(2*n - 1) - I*b*dilog(-cos(
d*x^n + c) + I*sin(d*x^n + c))*e^(2*n - 1) + I*b*dilog(-cos(d*x^n + c) - I*sin(d*x^n + c))*e^(2*n - 1) + (b*d*
x^n*e^(2*n - 1) + b*c*e^(2*n - 1))*log(-cos(d*x^n + c) + I*sin(d*x^n + c) + 1) + (b*d*x^n*e^(2*n - 1) + b*c*e^
(2*n - 1))*log(-cos(d*x^n + c) - I*sin(d*x^n + c) + 1))/(d^2*n)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (e x\right )^{2 n - 1} \left (a + b \csc {\left (c + d x^{n} \right )}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**(-1+2*n)*(a+b*csc(c+d*x**n)),x)

[Out]

Integral((e*x)**(2*n - 1)*(a + b*csc(c + d*x**n)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(-1+2*n)*(a+b*csc(c+d*x^n)),x, algorithm="giac")

[Out]

integrate((b*csc(d*x^n + c) + a)*(e*x)^(2*n - 1), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \left (a+\frac {b}{\sin \left (c+d\,x^n\right )}\right )\,{\left (e\,x\right )}^{2\,n-1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/sin(c + d*x^n))*(e*x)^(2*n - 1),x)

[Out]

int((a + b/sin(c + d*x^n))*(e*x)^(2*n - 1), x)

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