3.322 \(\int (e x)^m \csc (d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=123 \[ \frac{2 e^{i a d} (e x)^{m+1} \left (c x^n\right )^{i b d} \text{Hypergeometric2F1}\left (1,-\frac{-b d n+i m+i}{2 b d n},-\frac{-3 b d n+i (m+1)}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (-b d n+i (m+1))} \]

[Out]

(2*E^(I*a*d)*(e*x)^(1 + m)*(c*x^n)^(I*b*d)*Hypergeometric2F1[1, -(I + I*m - b*d*n)/(2*b*d*n), -(I*(1 + m) - 3*
b*d*n)/(2*b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)])/(e*(I*(1 + m) - b*d*n))

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Rubi [A]  time = 0.0770617, antiderivative size = 118, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {4510, 4506, 364} \[ \frac{2 e^{i a d} (e x)^{m+1} \left (c x^n\right )^{i b d} \, _2F_1\left (1,\frac{1}{2} \left (1-\frac{i (m+1)}{b d n}\right );-\frac{i (m+1)-3 b d n}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (-b d n+i (m+1))} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*Csc[d*(a + b*Log[c*x^n])],x]

[Out]

(2*E^(I*a*d)*(e*x)^(1 + m)*(c*x^n)^(I*b*d)*Hypergeometric2F1[1, (1 - (I*(1 + m))/(b*d*n))/2, -(I*(1 + m) - 3*b
*d*n)/(2*b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)])/(e*(I*(1 + m) - b*d*n))

Rule 4510

Int[Csc[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(e*x)^(m + 1)
/(e*n*(c*x^n)^((m + 1)/n)), Subst[Int[x^((m + 1)/n - 1)*Csc[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a
, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])

Rule 4506

Int[Csc[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(-2*I)^p*E^(I*a*d*p), Int[(
(e*x)^m*x^(I*b*d*p))/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p, x], x] /; FreeQ[{a, b, d, e, m}, x] && IntegerQ[p]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int (e x)^m \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac{\left ((e x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1+m}{n}} \csc (d (a+b \log (x))) \, dx,x,c x^n\right )}{e n}\\ &=-\frac{\left (2 i e^{i a d} (e x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+i b d+\frac{1+m}{n}}}{1-e^{2 i a d} x^{2 i b d}} \, dx,x,c x^n\right )}{e n}\\ &=\frac{2 e^{i a d} (e x)^{1+m} \left (c x^n\right )^{i b d} \, _2F_1\left (1,\frac{1}{2} \left (1-\frac{i (1+m)}{b d n}\right );-\frac{i (1+m)-3 b d n}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{i (e+e m)-b d e n}\\ \end{align*}

Mathematica [A]  time = 0.425706, size = 181, normalized size = 1.47 \[ \frac{2 (e x)^m x^{1+i b d n} \left (\sin \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )-i \cos \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )\right ) \text{Hypergeometric2F1}\left (1,\frac{b d n-i m-i}{2 b d n},-\frac{i (3 i b d n+m+1)}{2 b d n},x^{2 i b d n} \left (\cos \left (2 d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )+i \sin \left (2 d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )\right )\right )}{i b d n+m+1} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(e*x)^m*Csc[d*(a + b*Log[c*x^n])],x]

[Out]

(2*x^(1 + I*b*d*n)*(e*x)^m*Hypergeometric2F1[1, (-I - I*m + b*d*n)/(2*b*d*n), ((-I/2)*(1 + m + (3*I)*b*d*n))/(
b*d*n), x^((2*I)*b*d*n)*(Cos[2*d*(a + b*(-(n*Log[x]) + Log[c*x^n]))] + I*Sin[2*d*(a + b*(-(n*Log[x]) + Log[c*x
^n]))])]*((-I)*Cos[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))] + Sin[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]))/(1 + m
+ I*b*d*n)

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Maple [F]  time = 0.94, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m}\csc \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*csc(d*(a+b*ln(c*x^n))),x)

[Out]

int((e*x)^m*csc(d*(a+b*ln(c*x^n))),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \csc \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*csc(d*(a+b*log(c*x^n))),x, algorithm="maxima")

[Out]

integrate((e*x)^m*csc((b*log(c*x^n) + a)*d), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (e x\right )^{m} \csc \left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*csc(d*(a+b*log(c*x^n))),x, algorithm="fricas")

[Out]

integral((e*x)^m*csc(b*d*log(c*x^n) + a*d), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \csc{\left (a d + b d \log{\left (c x^{n} \right )} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*csc(d*(a+b*ln(c*x**n))),x)

[Out]

Integral((e*x)**m*csc(a*d + b*d*log(c*x**n)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \csc \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*csc(d*(a+b*log(c*x^n))),x, algorithm="giac")

[Out]

integrate((e*x)^m*csc((b*log(c*x^n) + a)*d), x)