3.321 \(\int (e x)^m \csc ^2(d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=119 \[ -\frac{4 e^{2 i a d} (e x)^{m+1} \left (c x^n\right )^{2 i b d} \text{Hypergeometric2F1}\left (2,-\frac{-2 b d n+i (m+1)}{2 b d n},-\frac{-4 b d n+i (m+1)}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (2 i b d n+m+1)} \]

[Out]

(-4*E^((2*I)*a*d)*(e*x)^(1 + m)*(c*x^n)^((2*I)*b*d)*Hypergeometric2F1[2, -(I*(1 + m) - 2*b*d*n)/(2*b*d*n), -(I
*(1 + m) - 4*b*d*n)/(2*b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)])/(e*(1 + m + (2*I)*b*d*n))

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Rubi [A]  time = 0.0966439, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4510, 4506, 364} \[ -\frac{4 e^{2 i a d} (e x)^{m+1} \left (c x^n\right )^{2 i b d} \, _2F_1\left (2,-\frac{i (m+1)-2 b d n}{2 b d n};-\frac{i (m+1)-4 b d n}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (2 i b d n+m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*E^((2*I)*a*d)*(e*x)^(1 + m)*(c*x^n)^((2*I)*b*d)*Hypergeometric2F1[2, -(I*(1 + m) - 2*b*d*n)/(2*b*d*n), -(I
*(1 + m) - 4*b*d*n)/(2*b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)])/(e*(1 + m + (2*I)*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 ^2\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 ^2(d (a+b \log (x))) \, dx,x,c x^n\right )}{e n}\\ &=-\frac{\left (4 e^{2 i a d} (e x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+2 i b d+\frac{1+m}{n}}}{\left (1-e^{2 i a d} x^{2 i b d}\right )^2} \, dx,x,c x^n\right )}{e n}\\ &=-\frac{4 e^{2 i a d} (e x)^{1+m} \left (c x^n\right )^{2 i b d} \, _2F_1\left (2,-\frac{i (1+m)-2 b d n}{2 b d n};-\frac{i (1+m)-4 b d n}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (1+m+2 i b d n)}\\ \end{align*}

Mathematica [B]  time = 6.54271, size = 534, normalized size = 4.49 \[ \frac{x (e x)^m \sin (b d n \log (x)) \csc \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \csc \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )+b d n \log (x)\right )}{b d n}-\frac{(m+1) x^{-m} (e x)^m \csc \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \left (\frac{x^{m+1} \sin (b d n \log (x)) \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{m+1}-\frac{i \sin \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \exp \left (-\frac{(2 m+1) \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )}{b n}\right ) \left (-(2 i b d n+m+1) \exp \left (\frac{2 a m+a+b (2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )+b (m+1) n \log (x)}{b n}\right ) \text{Hypergeometric2F1}\left (1,-\frac{i (m+1)}{2 b d n},1-\frac{i (m+1)}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )-(m+1) \exp \left (\frac{a (2 i b d n+2 m+1)}{b n}+\frac{(2 i b d n+2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )}{n}+\log (x) (2 i b d n+m+1)\right ) \text{Hypergeometric2F1}\left (1,-\frac{i (2 i b d n+m+1)}{2 b d n},-\frac{i (4 i b d n+m+1)}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+i (2 i b d n+m+1) \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \exp \left (\frac{2 a m+a+b (2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )+b (m+1) n \log (x)}{b n}\right )\right )}{(m+1) (2 i b d n+m+1)}\right )}{b d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(e*x)^m*Csc[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*Csc[b*d*n*Log[x] + d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*
Sin[b*d*n*Log[x]])/(b*d*n) - ((1 + m)*(e*x)^m*Csc[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*((x^(1 + m)*Csc[d*(a +
b*Log[c*x^n])]*Sin[b*d*n*Log[x]])/(1 + m) - (I*(I*E^((a + 2*a*m + b*(1 + m)*n*Log[x] + b*(1 + 2*m)*(-(n*Log[x
]) + Log[c*x^n]))/(b*n))*(1 + m + (2*I)*b*d*n)*Cot[d*(a + b*Log[c*x^n])] - E^((a + 2*a*m + b*(1 + m)*n*Log[x]
+ b*(1 + 2*m)*(-(n*Log[x]) + Log[c*x^n]))/(b*n))*(1 + m + (2*I)*b*d*n)*Hypergeometric2F1[1, ((-I/2)*(1 + m))/(
b*d*n), 1 - ((I/2)*(1 + m))/(b*d*n), E^((2*I)*d*(a + b*Log[c*x^n]))] - E^((a*(1 + 2*m + (2*I)*b*d*n))/(b*n) +
(1 + m + (2*I)*b*d*n)*Log[x] + ((1 + 2*m + (2*I)*b*d*n)*(-(n*Log[x]) + Log[c*x^n]))/n)*(1 + m)*Hypergeometric2
F1[1, ((-I/2)*(1 + m + (2*I)*b*d*n))/(b*d*n), ((-I/2)*(1 + m + (4*I)*b*d*n))/(b*d*n), E^((2*I)*d*(a + b*Log[c*
x^n]))])*Sin[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))])/(E^(((1 + 2*m)*(a + b*(-(n*Log[x]) + Log[c*x^n])))/(b*n))*
(1 + m)*(1 + m + (2*I)*b*d*n))))/(b*d*n*x^m)

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Maple [F]  time = 2.25, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( \csc \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \right ) ^{2}\, 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)))^2,x)

[Out]

int((e*x)^m*csc(d*(a+b*ln(c*x^n)))^2,x)

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)))^2,x, algorithm="maxima")

[Out]

Timed out

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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 )^{2}, 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)))^2,x, algorithm="fricas")

[Out]

integral((e*x)^m*csc(b*d*log(c*x^n) + a*d)^2, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)))**2,x)

[Out]

Timed out

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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 )^{2}\,{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)))^2,x, algorithm="giac")

[Out]

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