3.320 \(\int (e x)^m \csc ^3(d (a+b \log (c x^n))) \, dx\)
Optimal. Leaf size=122 \[ -\frac{8 e^{3 i a d} (e x)^{m+1} \left (c x^n\right )^{3 i b d} \text{Hypergeometric2F1}\left (3,-\frac{-3 b d n+i (m+1)}{2 b d n},-\frac{-5 b d n+i (m+1)}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (-3 b d n+i (m+1))} \]
[Out]
(-8*E^((3*I)*a*d)*(e*x)^(1 + m)*(c*x^n)^((3*I)*b*d)*Hypergeometric2F1[3, -(I*(1 + m) - 3*b*d*n)/(2*b*d*n), -(I
*(1 + m) - 5*b*d*n)/(2*b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)])/(e*(I*(1 + m) - 3*b*d*n))
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Rubi [A] time = 0.109617, antiderivative size = 122, 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} \[ -\frac{8 e^{3 i a d} (e x)^{m+1} \left (c x^n\right )^{3 i b d} \, _2F_1\left (3,-\frac{i (m+1)-3 b d n}{2 b d n};-\frac{i (m+1)-5 b d n}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (-3 b d n+i (m+1))} \]
Antiderivative was successfully verified.
[In]
Int[(e*x)^m*Csc[d*(a + b*Log[c*x^n])]^3,x]
[Out]
(-8*E^((3*I)*a*d)*(e*x)^(1 + m)*(c*x^n)^((3*I)*b*d)*Hypergeometric2F1[3, -(I*(1 + m) - 3*b*d*n)/(2*b*d*n), -(I
*(1 + m) - 5*b*d*n)/(2*b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)])/(e*(I*(1 + m) - 3*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 ^3\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 ^3(d (a+b \log (x))) \, dx,x,c x^n\right )}{e n}\\ &=\frac{\left (8 i e^{3 i a d} (e x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+3 i b d+\frac{1+m}{n}}}{\left (1-e^{2 i a d} x^{2 i b d}\right )^3} \, dx,x,c x^n\right )}{e n}\\ &=-\frac{8 e^{3 i a d} (e x)^{1+m} \left (c x^n\right )^{3 i b d} \, _2F_1\left (3,-\frac{i (1+m)-3 b d n}{2 b d n};-\frac{i (1+m)-5 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)-3 b d e n}\\ \end{align*}
Mathematica [B] time = 2.28485, size = 367, normalized size = 3.01 \[ \frac{x (e x)^m \left (8 (-i b d n+m+1) x^{i b d n} \left (\sin \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )-i \cos \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )\right ) \text{Hypergeometric2F1}\left (1,\frac{b d n-i m-i}{2 b d n},-\frac{i (3 i b d n+m+1)}{2 b d n},x^{2 i b d n} \left (\cos \left (2 d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )+i \sin \left (2 d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )\right )\right )-4 (m+1) \csc \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )-2 (m+1) \sin \left (\frac{1}{2} b d n \log (x)\right ) \sec \left (\frac{1}{2} d \left (a+b \log \left (c x^n\right )\right )\right ) \sec \left (\frac{1}{2} d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )+2 (m+1) \sin \left (\frac{1}{2} b d n \log (x)\right ) \csc \left (\frac{1}{2} d \left (a+b \log \left (c x^n\right )\right )\right ) \csc \left (\frac{1}{2} d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )+b d n \sec ^2\left (\frac{1}{2} d \left (a+b \log \left (c x^n\right )\right )\right )-b d n \csc ^2\left (\frac{1}{2} d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{8 b^2 d^2 n^2} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(e*x)^m*Csc[d*(a + b*Log[c*x^n])]^3,x]
[Out]
(x*(e*x)^m*(-(b*d*n*Csc[(d*(a + b*Log[c*x^n]))/2]^2) - 4*(1 + m)*Csc[d*(a - b*n*Log[x] + b*Log[c*x^n])] + b*d*
n*Sec[(d*(a + b*Log[c*x^n]))/2]^2 + 2*(1 + m)*Csc[(d*(a + b*Log[c*x^n]))/2]*Csc[(d*(a - b*n*Log[x] + b*Log[c*x
^n]))/2]*Sin[(b*d*n*Log[x])/2] - 2*(1 + m)*Sec[(d*(a + b*Log[c*x^n]))/2]*Sec[(d*(a - b*n*Log[x] + b*Log[c*x^n]
))/2]*Sin[(b*d*n*Log[x])/2] + 8*(1 + m - I*b*d*n)*x^(I*b*d*n)*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] + b*Log[c*x^n])] + I*Sin[2
*d*(a - b*n*Log[x] + b*Log[c*x^n])])]*((-I)*Cos[d*(a - b*n*Log[x] + b*Log[c*x^n])] + Sin[d*(a - b*n*Log[x] + b
*Log[c*x^n])])))/(8*b^2*d^2*n^2)
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Maple [F] time = 5.298, 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 ) ^{3}\, 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)))^3,x)
[Out]
int((e*x)^m*csc(d*(a+b*ln(c*x^n)))^3,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \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)))^3,x, algorithm="maxima")
[Out]
-((b*d*e^m*n*cos(b*d*log(c)) - e^m*m*sin(b*d*log(c)) - e^m*sin(b*d*log(c)))*x*x^m*cos(b*d*log(x^n) + a*d) - (b
*d*e^m*n*sin(b*d*log(c)) + e^m*m*cos(b*d*log(c)) + e^m*cos(b*d*log(c)))*x*x^m*sin(b*d*log(x^n) + a*d) - (((cos
(3*b*d*log(c))*sin(4*b*d*log(c)) - cos(4*b*d*log(c))*sin(3*b*d*log(c)))*e^m*m - (b*d*cos(4*b*d*log(c))*cos(3*b
*d*log(c)) + b*d*sin(4*b*d*log(c))*sin(3*b*d*log(c)))*e^m*n + (cos(3*b*d*log(c))*sin(4*b*d*log(c)) - cos(4*b*d
*log(c))*sin(3*b*d*log(c)))*e^m)*x*x^m*cos(3*b*d*log(x^n) + 3*a*d) - ((cos(b*d*log(c))*sin(4*b*d*log(c)) - cos
(4*b*d*log(c))*sin(b*d*log(c)))*e^m*m + (b*d*cos(4*b*d*log(c))*cos(b*d*log(c)) + b*d*sin(4*b*d*log(c))*sin(b*d
*log(c)))*e^m*n + (cos(b*d*log(c))*sin(4*b*d*log(c)) - cos(4*b*d*log(c))*sin(b*d*log(c)))*e^m)*x*x^m*cos(b*d*l
og(x^n) + a*d) - ((cos(4*b*d*log(c))*cos(3*b*d*log(c)) + sin(4*b*d*log(c))*sin(3*b*d*log(c)))*e^m*m + (b*d*cos
(3*b*d*log(c))*sin(4*b*d*log(c)) - b*d*cos(4*b*d*log(c))*sin(3*b*d*log(c)))*e^m*n + (cos(4*b*d*log(c))*cos(3*b
*d*log(c)) + sin(4*b*d*log(c))*sin(3*b*d*log(c)))*e^m)*x*x^m*sin(3*b*d*log(x^n) + 3*a*d) + ((cos(4*b*d*log(c))
*cos(b*d*log(c)) + sin(4*b*d*log(c))*sin(b*d*log(c)))*e^m*m - (b*d*cos(b*d*log(c))*sin(4*b*d*log(c)) - b*d*cos
(4*b*d*log(c))*sin(b*d*log(c)))*e^m*n + (cos(4*b*d*log(c))*cos(b*d*log(c)) + sin(4*b*d*log(c))*sin(b*d*log(c))
)*e^m)*x*x^m*sin(b*d*log(x^n) + a*d))*cos(4*b*d*log(x^n) + 4*a*d) - (2*((cos(2*b*d*log(c))*sin(3*b*d*log(c)) -
cos(3*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m + (b*d*cos(3*b*d*log(c))*cos(2*b*d*log(c)) + b*d*sin(3*b*d*log(c))
*sin(2*b*d*log(c)))*e^m*n + (cos(2*b*d*log(c))*sin(3*b*d*log(c)) - cos(3*b*d*log(c))*sin(2*b*d*log(c)))*e^m)*x
*x^m*cos(2*b*d*log(x^n) + 2*a*d) - 2*((cos(3*b*d*log(c))*cos(2*b*d*log(c)) + sin(3*b*d*log(c))*sin(2*b*d*log(c
)))*e^m*m - (b*d*cos(2*b*d*log(c))*sin(3*b*d*log(c)) - b*d*cos(3*b*d*log(c))*sin(2*b*d*log(c)))*e^m*n + (cos(3
*b*d*log(c))*cos(2*b*d*log(c)) + sin(3*b*d*log(c))*sin(2*b*d*log(c)))*e^m)*x*x^m*sin(2*b*d*log(x^n) + 2*a*d) -
(b*d*e^m*n*cos(3*b*d*log(c)) + e^m*m*sin(3*b*d*log(c)) + e^m*sin(3*b*d*log(c)))*x*x^m)*cos(3*b*d*log(x^n) + 3
*a*d) - 2*(((cos(b*d*log(c))*sin(2*b*d*log(c)) - cos(2*b*d*log(c))*sin(b*d*log(c)))*e^m*m + (b*d*cos(2*b*d*log
(c))*cos(b*d*log(c)) + b*d*sin(2*b*d*log(c))*sin(b*d*log(c)))*e^m*n + (cos(b*d*log(c))*sin(2*b*d*log(c)) - cos
(2*b*d*log(c))*sin(b*d*log(c)))*e^m)*x*x^m*cos(b*d*log(x^n) + a*d) - ((cos(2*b*d*log(c))*cos(b*d*log(c)) + sin
(2*b*d*log(c))*sin(b*d*log(c)))*e^m*m - (b*d*cos(b*d*log(c))*sin(2*b*d*log(c)) - b*d*cos(2*b*d*log(c))*sin(b*d
*log(c)))*e^m*n + (cos(2*b*d*log(c))*cos(b*d*log(c)) + sin(2*b*d*log(c))*sin(b*d*log(c)))*e^m)*x*x^m*sin(b*d*l
og(x^n) + a*d))*cos(2*b*d*log(x^n) + 2*a*d) + 2*(b^6*d^6*e^m*n^6 + (b^4*d^4*e^m*m^2 + 2*b^4*d^4*e^m*m + b^4*d^
4*e^m)*n^4 + ((b^6*d^6*cos(4*b*d*log(c))^2 + b^6*d^6*sin(4*b*d*log(c))^2)*e^m*n^6 + ((b^4*d^4*cos(4*b*d*log(c)
)^2 + b^4*d^4*sin(4*b*d*log(c))^2)*e^m*m^2 + 2*(b^4*d^4*cos(4*b*d*log(c))^2 + b^4*d^4*sin(4*b*d*log(c))^2)*e^m
*m + (b^4*d^4*cos(4*b*d*log(c))^2 + b^4*d^4*sin(4*b*d*log(c))^2)*e^m)*n^4)*cos(4*b*d*log(x^n) + 4*a*d)^2 + 4*(
(b^6*d^6*cos(2*b*d*log(c))^2 + b^6*d^6*sin(2*b*d*log(c))^2)*e^m*n^6 + ((b^4*d^4*cos(2*b*d*log(c))^2 + b^4*d^4*
sin(2*b*d*log(c))^2)*e^m*m^2 + 2*(b^4*d^4*cos(2*b*d*log(c))^2 + b^4*d^4*sin(2*b*d*log(c))^2)*e^m*m + (b^4*d^4*
cos(2*b*d*log(c))^2 + b^4*d^4*sin(2*b*d*log(c))^2)*e^m)*n^4)*cos(2*b*d*log(x^n) + 2*a*d)^2 + ((b^6*d^6*cos(4*b
*d*log(c))^2 + b^6*d^6*sin(4*b*d*log(c))^2)*e^m*n^6 + ((b^4*d^4*cos(4*b*d*log(c))^2 + b^4*d^4*sin(4*b*d*log(c)
)^2)*e^m*m^2 + 2*(b^4*d^4*cos(4*b*d*log(c))^2 + b^4*d^4*sin(4*b*d*log(c))^2)*e^m*m + (b^4*d^4*cos(4*b*d*log(c)
)^2 + b^4*d^4*sin(4*b*d*log(c))^2)*e^m)*n^4)*sin(4*b*d*log(x^n) + 4*a*d)^2 + 4*((b^6*d^6*cos(2*b*d*log(c))^2 +
b^6*d^6*sin(2*b*d*log(c))^2)*e^m*n^6 + ((b^4*d^4*cos(2*b*d*log(c))^2 + b^4*d^4*sin(2*b*d*log(c))^2)*e^m*m^2 +
2*(b^4*d^4*cos(2*b*d*log(c))^2 + b^4*d^4*sin(2*b*d*log(c))^2)*e^m*m + (b^4*d^4*cos(2*b*d*log(c))^2 + b^4*d^4*
sin(2*b*d*log(c))^2)*e^m)*n^4)*sin(2*b*d*log(x^n) + 2*a*d)^2 + 2*(b^6*d^6*e^m*n^6*cos(4*b*d*log(c)) + (b^4*d^4
*e^m*m^2*cos(4*b*d*log(c)) + 2*b^4*d^4*e^m*m*cos(4*b*d*log(c)) + b^4*d^4*e^m*cos(4*b*d*log(c)))*n^4 - 2*((b^6*
d^6*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^6*d^6*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*n^6 + ((b^4*d^4*cos
(4*b*d*log(c))*cos(2*b*d*log(c)) + b^4*d^4*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m^2 + 2*(b^4*d^4*cos(4*b*d
*log(c))*cos(2*b*d*log(c)) + b^4*d^4*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m + (b^4*d^4*cos(4*b*d*log(c))*c
os(2*b*d*log(c)) + b^4*d^4*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m)*n^4)*cos(2*b*d*log(x^n) + 2*a*d) - 2*((b^
6*d^6*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^6*d^6*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*n^6 + ((b^4*d^4*c
os(2*b*d*log(c))*sin(4*b*d*log(c)) - b^4*d^4*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m^2 + 2*(b^4*d^4*cos(2*b
*d*log(c))*sin(4*b*d*log(c)) - b^4*d^4*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m + (b^4*d^4*cos(2*b*d*log(c))
*sin(4*b*d*log(c)) - b^4*d^4*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m)*n^4)*sin(2*b*d*log(x^n) + 2*a*d))*cos(4
*b*d*log(x^n) + 4*a*d) - 4*(b^6*d^6*e^m*n^6*cos(2*b*d*log(c)) + (b^4*d^4*e^m*m^2*cos(2*b*d*log(c)) + 2*b^4*d^4
*e^m*m*cos(2*b*d*log(c)) + b^4*d^4*e^m*cos(2*b*d*log(c)))*n^4)*cos(2*b*d*log(x^n) + 2*a*d) - 2*(b^6*d^6*e^m*n^
6*sin(4*b*d*log(c)) + (b^4*d^4*e^m*m^2*sin(4*b*d*log(c)) + 2*b^4*d^4*e^m*m*sin(4*b*d*log(c)) + b^4*d^4*e^m*sin
(4*b*d*log(c)))*n^4 - 2*((b^6*d^6*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^6*d^6*cos(4*b*d*log(c))*sin(2*b*d*lo
g(c)))*e^m*n^6 + ((b^4*d^4*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^4*d^4*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*
e^m*m^2 + 2*(b^4*d^4*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^4*d^4*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m
+ (b^4*d^4*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^4*d^4*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m)*n^4)*cos(2*
b*d*log(x^n) + 2*a*d) + 2*((b^6*d^6*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^6*d^6*sin(4*b*d*log(c))*sin(2*b*d*
log(c)))*e^m*n^6 + ((b^4*d^4*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^4*d^4*sin(4*b*d*log(c))*sin(2*b*d*log(c))
)*e^m*m^2 + 2*(b^4*d^4*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^4*d^4*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*
m + (b^4*d^4*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^4*d^4*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m)*n^4)*sin(
2*b*d*log(x^n) + 2*a*d))*sin(4*b*d*log(x^n) + 4*a*d) + 4*(b^6*d^6*e^m*n^6*sin(2*b*d*log(c)) + (b^4*d^4*e^m*m^2
*sin(2*b*d*log(c)) + 2*b^4*d^4*e^m*m*sin(2*b*d*log(c)) + b^4*d^4*e^m*sin(2*b*d*log(c)))*n^4)*sin(2*b*d*log(x^n
) + 2*a*d))*integrate(1/4*(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^4*d^4*n^4*cos(b*d*log(c))*cos(b*d*log(x^n) + a*d) - 2*b^4*d^4*n^4*sin(b*d*log(c))*sin(b*d*log(x^n
) + a*d) + b^4*d^4*n^4 + (b^4*d^4*cos(b*d*log(c))^2 + b^4*d^4*sin(b*d*log(c))^2)*n^4*cos(b*d*log(x^n) + a*d)^2
+ (b^4*d^4*cos(b*d*log(c))^2 + b^4*d^4*sin(b*d*log(c))^2)*n^4*sin(b*d*log(x^n) + a*d)^2), x) + 2*(b^6*d^6*e^m
*n^6 + (b^4*d^4*e^m*m^2 + 2*b^4*d^4*e^m*m + b^4*d^4*e^m)*n^4 + ((b^6*d^6*cos(4*b*d*log(c))^2 + b^6*d^6*sin(4*b
*d*log(c))^2)*e^m*n^6 + ((b^4*d^4*cos(4*b*d*log(c))^2 + b^4*d^4*sin(4*b*d*log(c))^2)*e^m*m^2 + 2*(b^4*d^4*cos(
4*b*d*log(c))^2 + b^4*d^4*sin(4*b*d*log(c))^2)*e^m*m + (b^4*d^4*cos(4*b*d*log(c))^2 + b^4*d^4*sin(4*b*d*log(c)
)^2)*e^m)*n^4)*cos(4*b*d*log(x^n) + 4*a*d)^2 + 4*((b^6*d^6*cos(2*b*d*log(c))^2 + b^6*d^6*sin(2*b*d*log(c))^2)*
e^m*n^6 + ((b^4*d^4*cos(2*b*d*log(c))^2 + b^4*d^4*sin(2*b*d*log(c))^2)*e^m*m^2 + 2*(b^4*d^4*cos(2*b*d*log(c))^
2 + b^4*d^4*sin(2*b*d*log(c))^2)*e^m*m + (b^4*d^4*cos(2*b*d*log(c))^2 + b^4*d^4*sin(2*b*d*log(c))^2)*e^m)*n^4)
*cos(2*b*d*log(x^n) + 2*a*d)^2 + ((b^6*d^6*cos(4*b*d*log(c))^2 + b^6*d^6*sin(4*b*d*log(c))^2)*e^m*n^6 + ((b^4*
d^4*cos(4*b*d*log(c))^2 + b^4*d^4*sin(4*b*d*log(c))^2)*e^m*m^2 + 2*(b^4*d^4*cos(4*b*d*log(c))^2 + b^4*d^4*sin(
4*b*d*log(c))^2)*e^m*m + (b^4*d^4*cos(4*b*d*log(c))^2 + b^4*d^4*sin(4*b*d*log(c))^2)*e^m)*n^4)*sin(4*b*d*log(x
^n) + 4*a*d)^2 + 4*((b^6*d^6*cos(2*b*d*log(c))^2 + b^6*d^6*sin(2*b*d*log(c))^2)*e^m*n^6 + ((b^4*d^4*cos(2*b*d*
log(c))^2 + b^4*d^4*sin(2*b*d*log(c))^2)*e^m*m^2 + 2*(b^4*d^4*cos(2*b*d*log(c))^2 + b^4*d^4*sin(2*b*d*log(c))^
2)*e^m*m + (b^4*d^4*cos(2*b*d*log(c))^2 + b^4*d^4*sin(2*b*d*log(c))^2)*e^m)*n^4)*sin(2*b*d*log(x^n) + 2*a*d)^2
+ 2*(b^6*d^6*e^m*n^6*cos(4*b*d*log(c)) + (b^4*d^4*e^m*m^2*cos(4*b*d*log(c)) + 2*b^4*d^4*e^m*m*cos(4*b*d*log(c
)) + b^4*d^4*e^m*cos(4*b*d*log(c)))*n^4 - 2*((b^6*d^6*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^6*d^6*sin(4*b*d*
log(c))*sin(2*b*d*log(c)))*e^m*n^6 + ((b^4*d^4*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^4*d^4*sin(4*b*d*log(c))
*sin(2*b*d*log(c)))*e^m*m^2 + 2*(b^4*d^4*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^4*d^4*sin(4*b*d*log(c))*sin(2
*b*d*log(c)))*e^m*m + (b^4*d^4*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^4*d^4*sin(4*b*d*log(c))*sin(2*b*d*log(c
)))*e^m)*n^4)*cos(2*b*d*log(x^n) + 2*a*d) - 2*((b^6*d^6*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^6*d^6*cos(4*b*
d*log(c))*sin(2*b*d*log(c)))*e^m*n^6 + ((b^4*d^4*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^4*d^4*cos(4*b*d*log(c
))*sin(2*b*d*log(c)))*e^m*m^2 + 2*(b^4*d^4*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^4*d^4*cos(4*b*d*log(c))*sin
(2*b*d*log(c)))*e^m*m + (b^4*d^4*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^4*d^4*cos(4*b*d*log(c))*sin(2*b*d*log
(c)))*e^m)*n^4)*sin(2*b*d*log(x^n) + 2*a*d))*cos(4*b*d*log(x^n) + 4*a*d) - 4*(b^6*d^6*e^m*n^6*cos(2*b*d*log(c)
) + (b^4*d^4*e^m*m^2*cos(2*b*d*log(c)) + 2*b^4*d^4*e^m*m*cos(2*b*d*log(c)) + b^4*d^4*e^m*cos(2*b*d*log(c)))*n^
4)*cos(2*b*d*log(x^n) + 2*a*d) - 2*(b^6*d^6*e^m*n^6*sin(4*b*d*log(c)) + (b^4*d^4*e^m*m^2*sin(4*b*d*log(c)) + 2
*b^4*d^4*e^m*m*sin(4*b*d*log(c)) + b^4*d^4*e^m*sin(4*b*d*log(c)))*n^4 - 2*((b^6*d^6*cos(2*b*d*log(c))*sin(4*b*
d*log(c)) - b^6*d^6*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*n^6 + ((b^4*d^4*cos(2*b*d*log(c))*sin(4*b*d*log(c
)) - b^4*d^4*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m^2 + 2*(b^4*d^4*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b
^4*d^4*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m + (b^4*d^4*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^4*d^4*cos
(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m)*n^4)*cos(2*b*d*log(x^n) + 2*a*d) + 2*((b^6*d^6*cos(4*b*d*log(c))*cos(2*
b*d*log(c)) + b^6*d^6*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*n^6 + ((b^4*d^4*cos(4*b*d*log(c))*cos(2*b*d*log
(c)) + b^4*d^4*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m^2 + 2*(b^4*d^4*cos(4*b*d*log(c))*cos(2*b*d*log(c)) +
b^4*d^4*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m + (b^4*d^4*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^4*d^4*s
in(4*b*d*log(c))*sin(2*b*d*log(c)))*e^m)*n^4)*sin(2*b*d*log(x^n) + 2*a*d))*sin(4*b*d*log(x^n) + 4*a*d) + 4*(b^
6*d^6*e^m*n^6*sin(2*b*d*log(c)) + (b^4*d^4*e^m*m^2*sin(2*b*d*log(c)) + 2*b^4*d^4*e^m*m*sin(2*b*d*log(c)) + b^4
*d^4*e^m*sin(2*b*d*log(c)))*n^4)*sin(2*b*d*log(x^n) + 2*a*d))*integrate(-1/4*(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^4*d^4*n^4*cos(b*d*log(c))*cos(b*d*log(x^n) + a
*d) - 2*b^4*d^4*n^4*sin(b*d*log(c))*sin(b*d*log(x^n) + a*d) - b^4*d^4*n^4 - (b^4*d^4*cos(b*d*log(c))^2 + b^4*d
^4*sin(b*d*log(c))^2)*n^4*cos(b*d*log(x^n) + a*d)^2 - (b^4*d^4*cos(b*d*log(c))^2 + b^4*d^4*sin(b*d*log(c))^2)*
n^4*sin(b*d*log(x^n) + a*d)^2), x) - (((cos(4*b*d*log(c))*cos(3*b*d*log(c)) + sin(4*b*d*log(c))*sin(3*b*d*log(
c)))*e^m*m + (b*d*cos(3*b*d*log(c))*sin(4*b*d*log(c)) - b*d*cos(4*b*d*log(c))*sin(3*b*d*log(c)))*e^m*n + (cos(
4*b*d*log(c))*cos(3*b*d*log(c)) + sin(4*b*d*log(c))*sin(3*b*d*log(c)))*e^m)*x*x^m*cos(3*b*d*log(x^n) + 3*a*d)
- ((cos(4*b*d*log(c))*cos(b*d*log(c)) + sin(4*b*d*log(c))*sin(b*d*log(c)))*e^m*m - (b*d*cos(b*d*log(c))*sin(4*
b*d*log(c)) - b*d*cos(4*b*d*log(c))*sin(b*d*log(c)))*e^m*n + (cos(4*b*d*log(c))*cos(b*d*log(c)) + sin(4*b*d*lo
g(c))*sin(b*d*log(c)))*e^m)*x*x^m*cos(b*d*log(x^n) + a*d) + ((cos(3*b*d*log(c))*sin(4*b*d*log(c)) - cos(4*b*d*
log(c))*sin(3*b*d*log(c)))*e^m*m - (b*d*cos(4*b*d*log(c))*cos(3*b*d*log(c)) + b*d*sin(4*b*d*log(c))*sin(3*b*d*
log(c)))*e^m*n + (cos(3*b*d*log(c))*sin(4*b*d*log(c)) - cos(4*b*d*log(c))*sin(3*b*d*log(c)))*e^m)*x*x^m*sin(3*
b*d*log(x^n) + 3*a*d) - ((cos(b*d*log(c))*sin(4*b*d*log(c)) - cos(4*b*d*log(c))*sin(b*d*log(c)))*e^m*m + (b*d*
cos(4*b*d*log(c))*cos(b*d*log(c)) + b*d*sin(4*b*d*log(c))*sin(b*d*log(c)))*e^m*n + (cos(b*d*log(c))*sin(4*b*d*
log(c)) - cos(4*b*d*log(c))*sin(b*d*log(c)))*e^m)*x*x^m*sin(b*d*log(x^n) + a*d))*sin(4*b*d*log(x^n) + 4*a*d) -
(2*((cos(3*b*d*log(c))*cos(2*b*d*log(c)) + sin(3*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m - (b*d*cos(2*b*d*log(c)
)*sin(3*b*d*log(c)) - b*d*cos(3*b*d*log(c))*sin(2*b*d*log(c)))*e^m*n + (cos(3*b*d*log(c))*cos(2*b*d*log(c)) +
sin(3*b*d*log(c))*sin(2*b*d*log(c)))*e^m)*x*x^m*cos(2*b*d*log(x^n) + 2*a*d) + 2*((cos(2*b*d*log(c))*sin(3*b*d*
log(c)) - cos(3*b*d*log(c))*sin(2*b*d*log(c)))*e^m*m + (b*d*cos(3*b*d*log(c))*cos(2*b*d*log(c)) + b*d*sin(3*b*
d*log(c))*sin(2*b*d*log(c)))*e^m*n + (cos(2*b*d*log(c))*sin(3*b*d*log(c)) - cos(3*b*d*log(c))*sin(2*b*d*log(c)
))*e^m)*x*x^m*sin(2*b*d*log(x^n) + 2*a*d) + (b*d*e^m*n*sin(3*b*d*log(c)) - e^m*m*cos(3*b*d*log(c)) - e^m*cos(3
*b*d*log(c)))*x*x^m)*sin(3*b*d*log(x^n) + 3*a*d) - 2*(((cos(2*b*d*log(c))*cos(b*d*log(c)) + sin(2*b*d*log(c))*
sin(b*d*log(c)))*e^m*m - (b*d*cos(b*d*log(c))*sin(2*b*d*log(c)) - b*d*cos(2*b*d*log(c))*sin(b*d*log(c)))*e^m*n
+ (cos(2*b*d*log(c))*cos(b*d*log(c)) + sin(2*b*d*log(c))*sin(b*d*log(c)))*e^m)*x*x^m*cos(b*d*log(x^n) + a*d)
+ ((cos(b*d*log(c))*sin(2*b*d*log(c)) - cos(2*b*d*log(c))*sin(b*d*log(c)))*e^m*m + (b*d*cos(2*b*d*log(c))*cos(
b*d*log(c)) + b*d*sin(2*b*d*log(c))*sin(b*d*log(c)))*e^m*n + (cos(b*d*log(c))*sin(2*b*d*log(c)) - cos(2*b*d*lo
g(c))*sin(b*d*log(c)))*e^m)*x*x^m*sin(b*d*log(x^n) + a*d))*sin(2*b*d*log(x^n) + 2*a*d))/(4*b^2*d^2*n^2*cos(2*b
*d*log(c))*cos(2*b*d*log(x^n) + 2*a*d) - 4*b^2*d^2*n^2*sin(2*b*d*log(c))*sin(2*b*d*log(x^n) + 2*a*d) - b^2*d^2
*n^2 - (b^2*d^2*cos(4*b*d*log(c))^2 + b^2*d^2*sin(4*b*d*log(c))^2)*n^2*cos(4*b*d*log(x^n) + 4*a*d)^2 - 4*(b^2*
d^2*cos(2*b*d*log(c))^2 + b^2*d^2*sin(2*b*d*log(c))^2)*n^2*cos(2*b*d*log(x^n) + 2*a*d)^2 - (b^2*d^2*cos(4*b*d*
log(c))^2 + b^2*d^2*sin(4*b*d*log(c))^2)*n^2*sin(4*b*d*log(x^n) + 4*a*d)^2 - 4*(b^2*d^2*cos(2*b*d*log(c))^2 +
b^2*d^2*sin(2*b*d*log(c))^2)*n^2*sin(2*b*d*log(x^n) + 2*a*d)^2 - 2*(b^2*d^2*n^2*cos(4*b*d*log(c)) - 2*(b^2*d^2
*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^2*d^2*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*n^2*cos(2*b*d*log(x^n) + 2
*a*d) - 2*(b^2*d^2*cos(2*b*d*log(c))*sin(4*b*d*log(c)) - b^2*d^2*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*n^2*sin(
2*b*d*log(x^n) + 2*a*d))*cos(4*b*d*log(x^n) + 4*a*d) + 2*(b^2*d^2*n^2*sin(4*b*d*log(c)) - 2*(b^2*d^2*cos(2*b*d
*log(c))*sin(4*b*d*log(c)) - b^2*d^2*cos(4*b*d*log(c))*sin(2*b*d*log(c)))*n^2*cos(2*b*d*log(x^n) + 2*a*d) + 2*
(b^2*d^2*cos(4*b*d*log(c))*cos(2*b*d*log(c)) + b^2*d^2*sin(4*b*d*log(c))*sin(2*b*d*log(c)))*n^2*sin(2*b*d*log(
x^n) + 2*a*d))*sin(4*b*d*log(x^n) + 4*a*d))
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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 )^{3}, 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)))^3,x, algorithm="fricas")
[Out]
integral((e*x)^m*csc(b*d*log(c*x^n) + a*d)^3, 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)))**3,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 )^{3}\,{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)))^3,x, algorithm="giac")
[Out]
integrate((e*x)^m*csc((b*log(c*x^n) + a)*d)^3, x)