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

Optimal. Leaf size=119 \[ -\frac {4 e^{2 i a d} (e x)^{1+m} \left (c x^n\right )^{2 i b d} \, _2F_1\left (2,-\frac {i (1+m)-2 b d n}{2 b d n};-\frac {i (1+m)-4 b d n}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (1+m+2 i b d n)} \]

[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)

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Rubi [A]
time = 0.07, antiderivative size = 119, normalized size of antiderivative = 1.00, 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 = {4606, 4602, 371} \begin {gather*} -\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)} \end {gather*}

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, -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*} \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 ) \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} \, _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*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(534\) vs. \(2(119)=238\).
time = 6.57, size = 534, normalized size = 4.49 \begin {gather*} \frac {x (e x)^m \csc \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right ) \csc \left (b d n \log (x)+d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right ) \sin (b d n \log (x))}{b d n}-\frac {(1+m) x^{-m} (e x)^m \csc \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right ) \left (\frac {x^{1+m} \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \sin (b d n \log (x))}{1+m}-\frac {i e^{-\frac {(1+2 m) \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{b n}} \left (i e^{\frac {a+2 a m+b (1+m) n \log (x)+b (1+2 m) \left (-n \log (x)+\log \left (c x^n\right )\right )}{b n}} (1+m+2 i b d n) \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{\frac {a+2 a m+b (1+m) n \log (x)+b (1+2 m) \left (-n \log (x)+\log \left (c x^n\right )\right )}{b n}} (1+m+2 i b d n) \, _2F_1\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 )-e^{\frac {a (1+2 m+2 i b d n)}{b n}+(1+m+2 i b d n) \log (x)+\frac {(1+2 m+2 i b d n) \left (-n \log (x)+\log \left (c x^n\right )\right )}{n}} (1+m) \, _2F_1\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 d \left (a+b \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 )}{(1+m) (1+m+2 i b d n)}\right )}{b d n} \end {gather*}

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, ((-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]))] - 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)*Hypergeometri
c2F1[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)*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 = 0.08, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{m} \left (\csc ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

(2*x*cos(2*b*d*log(x^n) + 2*a*d)*e^(m*log(x) + m)*sin(2*b*d*log(c)) + 2*x*cos(2*b*d*log(c))*e^(m*log(x) + m)*s
in(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 + (b^2*d^2*cos(2*b*d*
log(c))^2 + b^2*d^2*sin(2*b*d*log(c))^2)*m)*n^2*cos(2*b*d*log(x^n) + 2*a*d)^2*e^m + (b^2*d^2*cos(2*b*d*log(c))
^2 + b^2*d^2*sin(2*b*d*log(c))^2 + (b^2*d^2*cos(2*b*d*log(c))^2 + b^2*d^2*sin(2*b*d*log(c))^2)*m)*n^2*e^m*sin(
2*b*d*log(x^n) + 2*a*d)^2 - 2*(b^2*d^2*m*cos(2*b*d*log(c)) + b^2*d^2*cos(2*b*d*log(c)))*n^2*cos(2*b*d*log(x^n)
+ 2*a*d)*e^m + 2*(b^2*d^2*m*sin(2*b*d*log(c)) + b^2*d^2*sin(2*b*d*log(c)))*n^2*e^m*sin(2*b*d*log(x^n) + 2*a*d
) + (b^2*d^2*m + b^2*d^2)*n^2*e^m)*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 + (b^2*d^2*cos(2*b*d*log(c))^2 + b^2*d^2*sin(
2*b*d*log(c))^2)*m)*n^2*cos(2*b*d*log(x^n) + 2*a*d)^2*e^m + (b^2*d^2*cos(2*b*d*log(c))^2 + b^2*d^2*sin(2*b*d*l
og(c))^2 + (b^2*d^2*cos(2*b*d*log(c))^2 + b^2*d^2*sin(2*b*d*log(c))^2)*m)*n^2*e^m*sin(2*b*d*log(x^n) + 2*a*d)^
2 - 2*(b^2*d^2*m*cos(2*b*d*log(c)) + b^2*d^2*cos(2*b*d*log(c)))*n^2*cos(2*b*d*log(x^n) + 2*a*d)*e^m + 2*(b^2*d
^2*m*sin(2*b*d*log(c)) + b^2*d^2*sin(2*b*d*log(c)))*n^2*e^m*sin(2*b*d*log(x^n) + 2*a*d) + (b^2*d^2*m + b^2*d^2
)*n^2*e^m)*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*l
og(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)

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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)))^2,x, algorithm="fricas")

[Out]

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

[Out]

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

[Out]

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

[Out]

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

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