\(\int (e x)^m \csc ^2(d (a+b \log (c x^n))) \, dx\) [321]

Optimal result

Integrand size = 21, antiderivative size = 119 \[ \int (e x)^m \csc ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {4 e^{2 i a d} (e x)^{1+m} \left (c x^n\right )^{2 i b d} \operatorname {Hypergeometric2F1}\left (2,-\frac {i (1+m)-2 b d n}{2 b d n},-\frac {i (1+m)-4 b d n}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (1+m+2 i b d n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4606, 4602, 371} \[ \int (e x)^m \csc ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {4 e^{2 i a d} (e x)^{m+1} \left (c x^n\right )^{2 i b d} \operatorname {Hypergeometric2F1}\left (2,-\frac {i (m+1)-2 b d n}{2 b d n},-\frac {i (m+1)-4 b d n}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (2 i b d n+m+1)} \]

[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, -1/2*(I*(1 + m) - 2*b*d*n)/(b*d*n), -
1/2*(I*(1 + m) - 4*b*d*n)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)])/(e*(1 + m + (2*I)*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*} \text {integral}& = \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 ^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 ) \text {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} \operatorname {Hypergeometric2F1}\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 [A] (verified)

Time = 13.66 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.89 \[ \int (e x)^m \csc ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {x (e x)^m \left ((1+m+2 i b d n) \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )+i (1+m+2 i b d n) \operatorname {Hypergeometric2F1}\left (1,-\frac {i (1+m)}{2 b d n},1-\frac {i (1+m)}{2 b d n},e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+i e^{2 i a d} (1+m) \left (c x^n\right )^{2 i b d} \operatorname {Hypergeometric2F1}\left (1,-\frac {i (1+m+2 i b d n)}{2 b d n},-\frac {i (1+m+4 i b d n)}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )\right )}{b d n (1+m+2 i b d n)} \]

[In]

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

[Out]

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

Maple [F]

\[\int \left (e x \right )^{m} {\csc \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{2}d x\]

[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)

Fricas [F]

\[ \int (e x)^m \csc ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \csc \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2} \,d x } \]

[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)

Sympy [F]

\[ \int (e x)^m \csc ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \left (e x\right )^{m} \csc ^{2}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int (e x)^m \csc ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \csc \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int (e x)^m \csc ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \csc \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int (e x)^m \csc ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \frac {{\left (e\,x\right )}^m}{{\sin \left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2} \,d x \]

[In]

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

[Out]

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