\(\int (e x)^{-1+3 n} (a+b \csc (c+d x^n)) \, dx\) [75]

Optimal result

Integrand size = 22, antiderivative size = 221 \[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right ) \, dx=\frac {a (e x)^{3 n}}{3 e n}-\frac {2 b x^{-n} (e x)^{3 n} \text {arctanh}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {2 i b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {2 i b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {2 b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-e^{i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac {2 b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,e^{i \left (c+d x^n\right )}\right )}{d^3 e n} \]

[Out]

1/3*a*(e*x)^(3*n)/e/n-2*b*(e*x)^(3*n)*arctanh(exp(I*(c+d*x^n)))/d/e/n/(x^n)+2*I*b*(e*x)^(3*n)*polylog(2,-exp(I
*(c+d*x^n)))/d^2/e/n/(x^(2*n))-2*I*b*(e*x)^(3*n)*polylog(2,exp(I*(c+d*x^n)))/d^2/e/n/(x^(2*n))-2*b*(e*x)^(3*n)
*polylog(3,-exp(I*(c+d*x^n)))/d^3/e/n/(x^(3*n))+2*b*(e*x)^(3*n)*polylog(3,exp(I*(c+d*x^n)))/d^3/e/n/(x^(3*n))

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {14, 4294, 4290, 4268, 2611, 2320, 6724} \[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right ) \, dx=\frac {a (e x)^{3 n}}{3 e n}-\frac {2 b x^{-n} (e x)^{3 n} \text {arctanh}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}-\frac {2 b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-e^{i \left (d x^n+c\right )}\right )}{d^3 e n}+\frac {2 b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,e^{i \left (d x^n+c\right )}\right )}{d^3 e n}+\frac {2 i b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-e^{i \left (d x^n+c\right )}\right )}{d^2 e n}-\frac {2 i b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,e^{i \left (d x^n+c\right )}\right )}{d^2 e n} \]

[In]

Int[(e*x)^(-1 + 3*n)*(a + b*Csc[c + d*x^n]),x]

[Out]

(a*(e*x)^(3*n))/(3*e*n) - (2*b*(e*x)^(3*n)*ArcTanh[E^(I*(c + d*x^n))])/(d*e*n*x^n) + ((2*I)*b*(e*x)^(3*n)*Poly
Log[2, -E^(I*(c + d*x^n))])/(d^2*e*n*x^(2*n)) - ((2*I)*b*(e*x)^(3*n)*PolyLog[2, E^(I*(c + d*x^n))])/(d^2*e*n*x
^(2*n)) - (2*b*(e*x)^(3*n)*PolyLog[3, -E^(I*(c + d*x^n))])/(d^3*e*n*x^(3*n)) + (2*b*(e*x)^(3*n)*PolyLog[3, E^(
I*(c + d*x^n))])/(d^3*e*n*x^(3*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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

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]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps \begin{align*} \text {integral}& = \int \left (a (e x)^{-1+3 n}+b (e x)^{-1+3 n} \csc \left (c+d x^n\right )\right ) \, dx \\ & = \frac {a (e x)^{3 n}}{3 e n}+b \int (e x)^{-1+3 n} \csc \left (c+d x^n\right ) \, dx \\ & = \frac {a (e x)^{3 n}}{3 e n}+\frac {\left (b x^{-3 n} (e x)^{3 n}\right ) \int x^{-1+3 n} \csc \left (c+d x^n\right ) \, dx}{e} \\ & = \frac {a (e x)^{3 n}}{3 e n}+\frac {\left (b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int x^2 \csc (c+d x) \, dx,x,x^n\right )}{e n} \\ & = \frac {a (e x)^{3 n}}{3 e n}-\frac {2 b x^{-n} (e x)^{3 n} \text {arctanh}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}-\frac {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int x \log \left (1-e^{i (c+d x)}\right ) \, dx,x,x^n\right )}{d e n}+\frac {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int x \log \left (1+e^{i (c+d x)}\right ) \, dx,x,x^n\right )}{d e n} \\ & = \frac {a (e x)^{3 n}}{3 e n}-\frac {2 b x^{-n} (e x)^{3 n} \text {arctanh}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {2 i b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {2 i b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {\left (2 i b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right ) \, dx,x,x^n\right )}{d^2 e n}+\frac {\left (2 i b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right ) \, dx,x,x^n\right )}{d^2 e n} \\ & = \frac {a (e x)^{3 n}}{3 e n}-\frac {2 b x^{-n} (e x)^{3 n} \text {arctanh}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {2 i b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {2 i b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{d^3 e n} \\ & = \frac {a (e x)^{3 n}}{3 e n}-\frac {2 b x^{-n} (e x)^{3 n} \text {arctanh}\left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {2 i b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {2 i b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {2 b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-e^{i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac {2 b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,e^{i \left (c+d x^n\right )}\right )}{d^3 e n} \\ \end{align*}

Mathematica [F]

\[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right ) \, dx=\int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right ) \, dx \]

[In]

Integrate[(e*x)^(-1 + 3*n)*(a + b*Csc[c + d*x^n]),x]

[Out]

Integrate[(e*x)^(-1 + 3*n)*(a + b*Csc[c + d*x^n]), x]

Maple [F]

\[\int \left (e x \right )^{-1+3 n} \left (a +b \csc \left (c +d \,x^{n}\right )\right )d x\]

[In]

int((e*x)^(-1+3*n)*(a+b*csc(c+d*x^n)),x)

[Out]

int((e*x)^(-1+3*n)*(a+b*csc(c+d*x^n)),x)

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 557 vs. \(2 (211) = 422\).

Time = 0.28 (sec) , antiderivative size = 557, normalized size of antiderivative = 2.52 \[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right ) \, dx=\frac {2 \, a d^{3} e^{3 \, n - 1} x^{3 \, n} - 3 \, b d^{2} e^{3 \, n - 1} x^{2 \, n} \log \left (\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right ) + 1\right ) - 3 \, b d^{2} e^{3 \, n - 1} x^{2 \, n} \log \left (\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right ) + 1\right ) - 6 i \, b d e^{3 \, n - 1} x^{n} {\rm Li}_2\left (\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right )\right ) + 6 i \, b d e^{3 \, n - 1} x^{n} {\rm Li}_2\left (\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right )\right ) - 6 i \, b d e^{3 \, n - 1} x^{n} {\rm Li}_2\left (-\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right )\right ) + 6 i \, b d e^{3 \, n - 1} x^{n} {\rm Li}_2\left (-\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right )\right ) + 3 \, b c^{2} e^{3 \, 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 ) + 3 \, b c^{2} e^{3 \, 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 ) + 6 \, b e^{3 \, n - 1} {\rm polylog}\left (3, \cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right )\right ) + 6 \, b e^{3 \, n - 1} {\rm polylog}\left (3, \cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right )\right ) - 6 \, b e^{3 \, n - 1} {\rm polylog}\left (3, -\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right )\right ) - 6 \, b e^{3 \, n - 1} {\rm polylog}\left (3, -\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right )\right ) + 3 \, {\left (b d^{2} e^{3 \, n - 1} x^{2 \, n} - b c^{2} e^{3 \, n - 1}\right )} \log \left (-\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right ) + 1\right ) + 3 \, {\left (b d^{2} e^{3 \, n - 1} x^{2 \, n} - b c^{2} e^{3 \, n - 1}\right )} \log \left (-\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right ) + 1\right )}{6 \, d^{3} n} \]

[In]

integrate((e*x)^(-1+3*n)*(a+b*csc(c+d*x^n)),x, algorithm="fricas")

[Out]

1/6*(2*a*d^3*e^(3*n - 1)*x^(3*n) - 3*b*d^2*e^(3*n - 1)*x^(2*n)*log(cos(d*x^n + c) + I*sin(d*x^n + c) + 1) - 3*
b*d^2*e^(3*n - 1)*x^(2*n)*log(cos(d*x^n + c) - I*sin(d*x^n + c) + 1) - 6*I*b*d*e^(3*n - 1)*x^n*dilog(cos(d*x^n
+ c) + I*sin(d*x^n + c)) + 6*I*b*d*e^(3*n - 1)*x^n*dilog(cos(d*x^n + c) - I*sin(d*x^n + c)) - 6*I*b*d*e^(3*n
- 1)*x^n*dilog(-cos(d*x^n + c) + I*sin(d*x^n + c)) + 6*I*b*d*e^(3*n - 1)*x^n*dilog(-cos(d*x^n + c) - I*sin(d*x
^n + c)) + 3*b*c^2*e^(3*n - 1)*log(-1/2*cos(d*x^n + c) + 1/2*I*sin(d*x^n + c) + 1/2) + 3*b*c^2*e^(3*n - 1)*log
(-1/2*cos(d*x^n + c) - 1/2*I*sin(d*x^n + c) + 1/2) + 6*b*e^(3*n - 1)*polylog(3, cos(d*x^n + c) + I*sin(d*x^n +
c)) + 6*b*e^(3*n - 1)*polylog(3, cos(d*x^n + c) - I*sin(d*x^n + c)) - 6*b*e^(3*n - 1)*polylog(3, -cos(d*x^n +
c) + I*sin(d*x^n + c)) - 6*b*e^(3*n - 1)*polylog(3, -cos(d*x^n + c) - I*sin(d*x^n + c)) + 3*(b*d^2*e^(3*n - 1
)*x^(2*n) - b*c^2*e^(3*n - 1))*log(-cos(d*x^n + c) + I*sin(d*x^n + c) + 1) + 3*(b*d^2*e^(3*n - 1)*x^(2*n) - b*
c^2*e^(3*n - 1))*log(-cos(d*x^n + c) - I*sin(d*x^n + c) + 1))/(d^3*n)

Sympy [F]

\[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right ) \, dx=\int \left (e x\right )^{3 n - 1} \left (a + b \csc {\left (c + d x^{n} \right )}\right )\, dx \]

[In]

integrate((e*x)**(-1+3*n)*(a+b*csc(c+d*x**n)),x)

[Out]

Integral((e*x)**(3*n - 1)*(a + b*csc(c + d*x**n)), x)

Maxima [F]

\[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right ) \, dx=\int { {\left (b \csc \left (d x^{n} + c\right ) + a\right )} \left (e x\right )^{3 \, n - 1} \,d x } \]

[In]

integrate((e*x)^(-1+3*n)*(a+b*csc(c+d*x^n)),x, algorithm="maxima")

[Out]

(e^(3*n + 1)*integrate(x^(3*n)*sin(d*x^n + c)/(e^2*x*cos(d*x^n + c)^2 + e^2*x*sin(d*x^n + c)^2 + 2*e^2*x*cos(d
*x^n + c) + e^2*x), x) + e^(3*n + 1)*integrate(x^(3*n)*sin(d*x^n + c)/(e^2*x*cos(d*x^n + c)^2 + e^2*x*sin(d*x^
n + c)^2 - 2*e^2*x*cos(d*x^n + c) + e^2*x), x))*b + 1/3*(e*x)^(3*n)*a/(e*n)

Giac [F]

\[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right ) \, dx=\int { {\left (b \csc \left (d x^{n} + c\right ) + a\right )} \left (e x\right )^{3 \, n - 1} \,d x } \]

[In]

integrate((e*x)^(-1+3*n)*(a+b*csc(c+d*x^n)),x, algorithm="giac")

[Out]

integrate((b*csc(d*x^n + c) + a)*(e*x)^(3*n - 1), x)

Mupad [F(-1)]

Timed out. \[ \int (e x)^{-1+3 n} \left (a+b \csc \left (c+d x^n\right )\right ) \, dx=\int \left (a+\frac {b}{\sin \left (c+d\,x^n\right )}\right )\,{\left (e\,x\right )}^{3\,n-1} \,d x \]

[In]

int((a + b/sin(c + d*x^n))*(e*x)^(3*n - 1),x)

[Out]

int((a + b/sin(c + d*x^n))*(e*x)^(3*n - 1), x)