3.4.22 \(\int (e x)^m \csc (d (a+b \log (c x^n))) \, dx\) [322]

Optimal. Leaf size=123 \[ \frac {2 e^{i a d} (e x)^{1+m} \left (c x^n\right )^{i b d} \, _2F_1\left (1,-\frac {i+i m-b d n}{2 b d n};-\frac {i (1+m)-3 b d n}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (i (1+m)-b d n)} \]

[Out]

2*exp(I*a*d)*(e*x)^(1+m)*(c*x^n)^(I*b*d)*hypergeom([1, 1/2*(-I-I*m+b*d*n)/b/d/n],[1/2*(-I*(1+m)+3*b*d*n)/b/d/n
],exp(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 = 0.06, 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 = {4606, 4602, 371} \begin {gather*} \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))} \end {gather*}

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

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 4602

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 4606

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])

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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.35, size = 181, normalized size = 1.47 \begin {gather*} \frac {2 x^{1+i b d n} (e x)^m \, _2F_1\left (1,\frac {-i-i m+b d n}{2 b d n};-\frac {i (1+m+3 i b d n)}{2 b d n};x^{2 i b d n} \left (\cos \left (2 d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )+i \sin \left (2 d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )\right )\right ) \left (-i \cos \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )+\sin \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )\right )}{1+m+i b d n} \end {gather*}

Antiderivative was successfully verified.

[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), ((-1/2*I)*(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 = 0.08, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{m} \csc \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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

[Out]

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

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

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 = 0.00, 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)^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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m/sin(d*(a + b*log(c*x^n))),x)

[Out]

int((e*x)^m/sin(d*(a + b*log(c*x^n))), x)

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