∫ csc4(a + b log(cxn))dx

3.299 \(\int \csc ^4(a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=84 \[ \frac{16 e^{4 i a} x \left (c x^n\right )^{4 i b} \text{Hypergeometric2F1}\left (4,\frac{1}{2} \left (4-\frac{i}{b n}\right ),\frac{1}{2} \left (6-\frac{i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+4 i b n} \]

[Out]

(16*E^((4*I)*a)*x*(c*x^n)^((4*I)*b)*Hypergeometric2F1[4, (4 - I/(b*n))/2, (6 - I/(b*n))/2, E^((2*I)*a)*(c*x^n)

^((2*I)*b)])/(1 + (4*I)*b*n)

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Rubi [A] time = 0.0610124, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {4504, 4506, 364} \[ \frac{16 e^{4 i a} x \left (c x^n\right )^{4 i b} \, _2F_1\left (4,\frac{1}{2} \left (4-\frac{i}{b n}\right );\frac{1}{2} \left (6-\frac{i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+4 i b n} \]

Antiderivative was successfully veriﬁed.

[In]

Int[Csc[a + b*Log[c*x^n]]^4,x]

[Out]

(16*E^((4*I)*a)*x*(c*x^n)^((4*I)*b)*Hypergeometric2F1[4, (4 - I/(b*n))/2, (6 - I/(b*n))/2, E^((2*I)*a)*(c*x^n)

^((2*I)*b)])/(1 + (4*I)*b*n)

Rule 4504

Int[Csc[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x

^(1/n - 1)*Csc[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, 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 \csc ^4\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1}{n}} \csc ^4(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (16 e^{4 i a} x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+4 i b+\frac{1}{n}}}{\left (1-e^{2 i a} x^{2 i b}\right )^4} \, dx,x,c x^n\right )}{n}\\ &=\frac{16 e^{4 i a} x \left (c x^n\right )^{4 i b} \, _2F_1\left (4,\frac{1}{2} \left (4-\frac{i}{b n}\right );\frac{1}{2} \left (6-\frac{i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+4 i b n}\\ \end{align*}

Mathematica [B] time = 13.4751, size = 782, normalized size = 9.31 \[ -\frac{e^{-\frac{a+b \left (\log \left (c x^n\right )-n \log (x)\right )}{b n}} \csc \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right ) \left (x (2 b n-i) e^{\frac{a}{b n}+\frac{\log \left (c x^n\right )-n \log (x)}{n}} \left (\cos \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )+i \sin \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right ) \text{Hypergeometric2F1}\left (1,-\frac{i}{2 b n},1-\frac{i}{2 b n},\exp \left (2 i \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )+b n \log (x)\right )\right )\right )\right )+e^{\left (\frac{1}{b n}+2 i\right ) \left (a+b \log \left (c x^n\right )\right )} \sin \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right ) \text{Hypergeometric2F1}\left (1,1-\frac{i}{2 b n},2-\frac{i}{2 b n},e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{6 b^3 n^3 (2 b n-i)}-\frac{2 e^{-\frac{a+b \left (\log \left (c x^n\right )-n \log (x)\right )}{b n}} \csc \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right ) \left (x (2 b n-i) e^{\frac{a}{b n}+\frac{\log \left (c x^n\right )-n \log (x)}{n}} \left (\cos \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )+i \sin \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right ) \text{Hypergeometric2F1}\left (1,-\frac{i}{2 b n},1-\frac{i}{2 b n},\exp \left (2 i \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )+b n \log (x)\right )\right )\right )\right )+e^{\left (\frac{1}{b n}+2 i\right ) \left (a+b \log \left (c x^n\right )\right )} \sin \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right ) \text{Hypergeometric2F1}\left (1,1-\frac{i}{2 b n},2-\frac{i}{2 b n},e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{3 b n (2 b n-i)}+\frac{x \left (4 b^2 n^2+1\right ) \sin (b n \log (x)) \csc \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right ) \csc \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )+b n \log (x)\right )}{6 b^3 n^3}-\frac{x \csc \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right ) \csc ^2\left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )+b n \log (x)\right ) \left (\sin \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )+2 b n \cos \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )}{6 b^2 n^2}+\frac{x \sin (b n \log (x)) \csc \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right ) \csc ^3\left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )+b n \log (x)\right )}{3 b n} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Csc[a + b*Log[c*x^n]]^4,x]

[Out]

((1 + 4*b^2*n^2)*x*Csc[a + b*(-(n*Log[x]) + Log[c*x^n])]*Csc[a + b*n*Log[x] + b*(-(n*Log[x]) + Log[c*x^n])]*Si

n[b*n*Log[x]])/(6*b^3*n^3) + (x*Csc[a + b*(-(n*Log[x]) + Log[c*x^n])]*Csc[a + b*n*Log[x] + b*(-(n*Log[x]) + Lo

g[c*x^n])]^3*Sin[b*n*Log[x]])/(3*b*n) - (x*Csc[a + b*(-(n*Log[x]) + Log[c*x^n])]*Csc[a + b*n*Log[x] + b*(-(n*L

og[x]) + Log[c*x^n])]^2*(2*b*n*Cos[a + b*(-(n*Log[x]) + Log[c*x^n])] + Sin[a + b*(-(n*Log[x]) + Log[c*x^n])]))

/(6*b^2*n^2) - (Csc[a + b*(-(n*Log[x]) + Log[c*x^n])]*(E^((2*I + 1/(b*n))*(a + b*Log[c*x^n]))*Hypergeometric2F

1[1, 1 - (I/2)/(b*n), 2 - (I/2)/(b*n), E^((2*I)*(a + b*Log[c*x^n]))]*Sin[a + b*(-(n*Log[x]) + Log[c*x^n])] + E

^(a/(b*n) + (-(n*Log[x]) + Log[c*x^n])/n)*(-I + 2*b*n)*x*(Cos[a + b*(-(n*Log[x]) + Log[c*x^n])] + I*Hypergeome

tric2F1[1, (-I/2)/(b*n), 1 - (I/2)/(b*n), E^((2*I)*(a + b*n*Log[x] + b*(-(n*Log[x]) + Log[c*x^n])))]*Sin[a + b

*(-(n*Log[x]) + Log[c*x^n])])))/(6*b^3*E^((a + b*(-(n*Log[x]) + Log[c*x^n]))/(b*n))*n^3*(-I + 2*b*n)) - (2*Csc

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

), 2 - (I/2)/(b*n), E^((2*I)*(a + b*Log[c*x^n]))]*Sin[a + b*(-(n*Log[x]) + Log[c*x^n])] + E^(a/(b*n) + (-(n*Lo

g[x]) + Log[c*x^n])/n)*(-I + 2*b*n)*x*(Cos[a + b*(-(n*Log[x]) + Log[c*x^n])] + I*Hypergeometric2F1[1, (-I/2)/(

b*n), 1 - (I/2)/(b*n), E^((2*I)*(a + b*n*Log[x] + b*(-(n*Log[x]) + Log[c*x^n])))]*Sin[a + b*(-(n*Log[x]) + Log

[c*x^n])])))/(3*b*E^((a + b*(-(n*Log[x]) + Log[c*x^n]))/(b*n))*n*(-I + 2*b*n))

________________________________________________________________________________________

Maple [F] time = 1.441, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{4}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(csc(a+b*ln(c*x^n))^4,x)

[Out]

int(csc(a+b*ln(c*x^n))^4,x)

________________________________________________________________________________________

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(csc(a+b*log(c*x^n))^4,x, algorithm="maxima")

[Out]

1/3*(6*(b*cos(4*b*log(c))^2 + b*sin(4*b*log(c))^2)*n*x*cos(4*b*log(x^n) + 4*a)^2 + 6*(b*cos(2*b*log(c))^2 + b*

sin(2*b*log(c))^2)*n*x*cos(2*b*log(x^n) + 2*a)^2 + 6*(b*cos(4*b*log(c))^2 + b*sin(4*b*log(c))^2)*n*x*sin(4*b*l

og(x^n) + 4*a)^2 + 6*(b*cos(2*b*log(c))^2 + b*sin(2*b*log(c))^2)*n*x*sin(2*b*log(x^n) + 2*a)^2 - (2*b*n*cos(2*

b*log(c)) - sin(2*b*log(c)))*x*cos(2*b*log(x^n) + 2*a) + (2*b*n*sin(2*b*log(c)) + cos(2*b*log(c)))*x*sin(2*b*l

og(x^n) + 2*a) - ((2*(b*cos(6*b*log(c))*cos(4*b*log(c)) + b*sin(6*b*log(c))*sin(4*b*log(c)))*n - cos(4*b*log(c

))*sin(6*b*log(c)) + cos(6*b*log(c))*sin(4*b*log(c)))*x*cos(4*b*log(x^n) + 4*a) + 2*(6*(b^2*cos(2*b*log(c))*si

n(6*b*log(c)) - b^2*cos(6*b*log(c))*sin(2*b*log(c)))*n^2 - (b*cos(6*b*log(c))*cos(2*b*log(c)) + b*sin(6*b*log(

c))*sin(2*b*log(c)))*n + cos(2*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(2*b*log(c)))*x*cos(2*b*log(x^n)

+ 2*a) + (2*(b*cos(4*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(4*b*log(c)))*n + cos(6*b*log(c))*cos(4

*b*log(c)) + sin(6*b*log(c))*sin(4*b*log(c)))*x*sin(4*b*log(x^n) + 4*a) - 2*(6*(b^2*cos(6*b*log(c))*cos(2*b*lo

g(c)) + b^2*sin(6*b*log(c))*sin(2*b*log(c)))*n^2 + (b*cos(2*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(

2*b*log(c)))*n + cos(6*b*log(c))*cos(2*b*log(c)) + sin(6*b*log(c))*sin(2*b*log(c)))*x*sin(2*b*log(x^n) + 2*a)

- (4*b^2*n^2*sin(6*b*log(c)) + sin(6*b*log(c)))*x)*cos(6*b*log(x^n) + 6*a) + (3*(12*(b^2*cos(2*b*log(c))*sin(4

*b*log(c)) - b^2*cos(4*b*log(c))*sin(2*b*log(c)))*n^2 - 4*(b*cos(4*b*log(c))*cos(2*b*log(c)) + b*sin(4*b*log(c

))*sin(2*b*log(c)))*n + cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*x*cos(2*b*log(x^n)

+ 2*a) - 3*(12*(b^2*cos(4*b*log(c))*cos(2*b*log(c)) + b^2*sin(4*b*log(c))*sin(2*b*log(c)))*n^2 + 4*(b*cos(2*b*

log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(2*b*log(c)))*n + cos(4*b*log(c))*cos(2*b*log(c)) + sin(4*b*log

(c))*sin(2*b*log(c)))*x*sin(2*b*log(x^n) + 2*a) - 2*(6*b^2*n^2*sin(4*b*log(c)) - b*n*cos(4*b*log(c)) + sin(4*b

*log(c)))*x)*cos(4*b*log(x^n) + 4*a) + 18*(4*b^8*n^8 + b^6*n^6 + (4*(b^8*cos(6*b*log(c))^2 + b^8*sin(6*b*log(c

))^2)*n^8 + (b^6*cos(6*b*log(c))^2 + b^6*sin(6*b*log(c))^2)*n^6)*cos(6*b*log(x^n) + 6*a)^2 + 9*(4*(b^8*cos(4*b

*log(c))^2 + b^8*sin(4*b*log(c))^2)*n^8 + (b^6*cos(4*b*log(c))^2 + b^6*sin(4*b*log(c))^2)*n^6)*cos(4*b*log(x^n

) + 4*a)^2 + 9*(4*(b^8*cos(2*b*log(c))^2 + b^8*sin(2*b*log(c))^2)*n^8 + (b^6*cos(2*b*log(c))^2 + b^6*sin(2*b*l

og(c))^2)*n^6)*cos(2*b*log(x^n) + 2*a)^2 + (4*(b^8*cos(6*b*log(c))^2 + b^8*sin(6*b*log(c))^2)*n^8 + (b^6*cos(6

*b*log(c))^2 + b^6*sin(6*b*log(c))^2)*n^6)*sin(6*b*log(x^n) + 6*a)^2 + 9*(4*(b^8*cos(4*b*log(c))^2 + b^8*sin(4

*b*log(c))^2)*n^8 + (b^6*cos(4*b*log(c))^2 + b^6*sin(4*b*log(c))^2)*n^6)*sin(4*b*log(x^n) + 4*a)^2 + 9*(4*(b^8

*cos(2*b*log(c))^2 + b^8*sin(2*b*log(c))^2)*n^8 + (b^6*cos(2*b*log(c))^2 + b^6*sin(2*b*log(c))^2)*n^6)*sin(2*b

*log(x^n) + 2*a)^2 - 2*(4*b^8*n^8*cos(6*b*log(c)) + b^6*n^6*cos(6*b*log(c)) + 3*(4*(b^8*cos(6*b*log(c))*cos(4*

b*log(c)) + b^8*sin(6*b*log(c))*sin(4*b*log(c)))*n^8 + (b^6*cos(6*b*log(c))*cos(4*b*log(c)) + b^6*sin(6*b*log(

c))*sin(4*b*log(c)))*n^6)*cos(4*b*log(x^n) + 4*a) - 3*(4*(b^8*cos(6*b*log(c))*cos(2*b*log(c)) + b^8*sin(6*b*lo

g(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(6*b*log(c))*cos(2*b*log(c)) + b^6*sin(6*b*log(c))*sin(2*b*log(c)))*n^6)*

cos(2*b*log(x^n) + 2*a) + 3*(4*(b^8*cos(4*b*log(c))*sin(6*b*log(c)) - b^8*cos(6*b*log(c))*sin(4*b*log(c)))*n^8

+ (b^6*cos(4*b*log(c))*sin(6*b*log(c)) - b^6*cos(6*b*log(c))*sin(4*b*log(c)))*n^6)*sin(4*b*log(x^n) + 4*a) -

3*(4*(b^8*cos(2*b*log(c))*sin(6*b*log(c)) - b^8*cos(6*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(2*b*log(c))*si

n(6*b*log(c)) - b^6*cos(6*b*log(c))*sin(2*b*log(c)))*n^6)*sin(2*b*log(x^n) + 2*a))*cos(6*b*log(x^n) + 6*a) + 6

*(4*b^8*n^8*cos(4*b*log(c)) + b^6*n^6*cos(4*b*log(c)) - 3*(4*(b^8*cos(4*b*log(c))*cos(2*b*log(c)) + b^8*sin(4*

b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(4*b*log(c))*cos(2*b*log(c)) + b^6*sin(4*b*log(c))*sin(2*b*log(c)))*n

^6)*cos(2*b*log(x^n) + 2*a) - 3*(4*(b^8*cos(2*b*log(c))*sin(4*b*log(c)) - b^8*cos(4*b*log(c))*sin(2*b*log(c)))

*n^8 + (b^6*cos(2*b*log(c))*sin(4*b*log(c)) - b^6*cos(4*b*log(c))*sin(2*b*log(c)))*n^6)*sin(2*b*log(x^n) + 2*a

))*cos(4*b*log(x^n) + 4*a) - 6*(4*b^8*n^8*cos(2*b*log(c)) + b^6*n^6*cos(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) +

2*(4*b^8*n^8*sin(6*b*log(c)) + b^6*n^6*sin(6*b*log(c)) + 3*(4*(b^8*cos(4*b*log(c))*sin(6*b*log(c)) - b^8*cos(

6*b*log(c))*sin(4*b*log(c)))*n^8 + (b^6*cos(4*b*log(c))*sin(6*b*log(c)) - b^6*cos(6*b*log(c))*sin(4*b*log(c)))

*n^6)*cos(4*b*log(x^n) + 4*a) - 3*(4*(b^8*cos(2*b*log(c))*sin(6*b*log(c)) - b^8*cos(6*b*log(c))*sin(2*b*log(c)

))*n^8 + (b^6*cos(2*b*log(c))*sin(6*b*log(c)) - b^6*cos(6*b*log(c))*sin(2*b*log(c)))*n^6)*cos(2*b*log(x^n) + 2

*a) - 3*(4*(b^8*cos(6*b*log(c))*cos(4*b*log(c)) + b^8*sin(6*b*log(c))*sin(4*b*log(c)))*n^8 + (b^6*cos(6*b*log(

c))*cos(4*b*log(c)) + b^6*sin(6*b*log(c))*sin(4*b*log(c)))*n^6)*sin(4*b*log(x^n) + 4*a) + 3*(4*(b^8*cos(6*b*lo

g(c))*cos(2*b*log(c)) + b^8*sin(6*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(6*b*log(c))*cos(2*b*log(c)) + b^6*

sin(6*b*log(c))*sin(2*b*log(c)))*n^6)*sin(2*b*log(x^n) + 2*a))*sin(6*b*log(x^n) + 6*a) - 6*(4*b^8*n^8*sin(4*b*

log(c)) + b^6*n^6*sin(4*b*log(c)) - 3*(4*(b^8*cos(2*b*log(c))*sin(4*b*log(c)) - b^8*cos(4*b*log(c))*sin(2*b*lo

g(c)))*n^8 + (b^6*cos(2*b*log(c))*sin(4*b*log(c)) - b^6*cos(4*b*log(c))*sin(2*b*log(c)))*n^6)*cos(2*b*log(x^n)

+ 2*a) + 3*(4*(b^8*cos(4*b*log(c))*cos(2*b*log(c)) + b^8*sin(4*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(4*b*

log(c))*cos(2*b*log(c)) + b^6*sin(4*b*log(c))*sin(2*b*log(c)))*n^6)*sin(2*b*log(x^n) + 2*a))*sin(4*b*log(x^n)

+ 4*a) + 6*(4*b^8*n^8*sin(2*b*log(c)) + b^6*n^6*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a))*integrate(1/36*(cos(

b*log(x^n) + a)*sin(b*log(c)) + cos(b*log(c))*sin(b*log(x^n) + a))/(2*b^6*n^6*cos(b*log(c))*cos(b*log(x^n) + a

) - 2*b^6*n^6*sin(b*log(c))*sin(b*log(x^n) + a) + b^6*n^6 + (b^6*cos(b*log(c))^2 + b^6*sin(b*log(c))^2)*n^6*co

s(b*log(x^n) + a)^2 + (b^6*cos(b*log(c))^2 + b^6*sin(b*log(c))^2)*n^6*sin(b*log(x^n) + a)^2), x) - 18*(4*b^8*n

^8 + b^6*n^6 + (4*(b^8*cos(6*b*log(c))^2 + b^8*sin(6*b*log(c))^2)*n^8 + (b^6*cos(6*b*log(c))^2 + b^6*sin(6*b*l

og(c))^2)*n^6)*cos(6*b*log(x^n) + 6*a)^2 + 9*(4*(b^8*cos(4*b*log(c))^2 + b^8*sin(4*b*log(c))^2)*n^8 + (b^6*cos

(4*b*log(c))^2 + b^6*sin(4*b*log(c))^2)*n^6)*cos(4*b*log(x^n) + 4*a)^2 + 9*(4*(b^8*cos(2*b*log(c))^2 + b^8*sin

(2*b*log(c))^2)*n^8 + (b^6*cos(2*b*log(c))^2 + b^6*sin(2*b*log(c))^2)*n^6)*cos(2*b*log(x^n) + 2*a)^2 + (4*(b^8

*cos(6*b*log(c))^2 + b^8*sin(6*b*log(c))^2)*n^8 + (b^6*cos(6*b*log(c))^2 + b^6*sin(6*b*log(c))^2)*n^6)*sin(6*b

*log(x^n) + 6*a)^2 + 9*(4*(b^8*cos(4*b*log(c))^2 + b^8*sin(4*b*log(c))^2)*n^8 + (b^6*cos(4*b*log(c))^2 + b^6*s

in(4*b*log(c))^2)*n^6)*sin(4*b*log(x^n) + 4*a)^2 + 9*(4*(b^8*cos(2*b*log(c))^2 + b^8*sin(2*b*log(c))^2)*n^8 +

(b^6*cos(2*b*log(c))^2 + b^6*sin(2*b*log(c))^2)*n^6)*sin(2*b*log(x^n) + 2*a)^2 - 2*(4*b^8*n^8*cos(6*b*log(c))

+ b^6*n^6*cos(6*b*log(c)) + 3*(4*(b^8*cos(6*b*log(c))*cos(4*b*log(c)) + b^8*sin(6*b*log(c))*sin(4*b*log(c)))*n

^8 + (b^6*cos(6*b*log(c))*cos(4*b*log(c)) + b^6*sin(6*b*log(c))*sin(4*b*log(c)))*n^6)*cos(4*b*log(x^n) + 4*a)

- 3*(4*(b^8*cos(6*b*log(c))*cos(2*b*log(c)) + b^8*sin(6*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(6*b*log(c))*

cos(2*b*log(c)) + b^6*sin(6*b*log(c))*sin(2*b*log(c)))*n^6)*cos(2*b*log(x^n) + 2*a) + 3*(4*(b^8*cos(4*b*log(c)

)*sin(6*b*log(c)) - b^8*cos(6*b*log(c))*sin(4*b*log(c)))*n^8 + (b^6*cos(4*b*log(c))*sin(6*b*log(c)) - b^6*cos(

6*b*log(c))*sin(4*b*log(c)))*n^6)*sin(4*b*log(x^n) + 4*a) - 3*(4*(b^8*cos(2*b*log(c))*sin(6*b*log(c)) - b^8*co

s(6*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(2*b*log(c))*sin(6*b*log(c)) - b^6*cos(6*b*log(c))*sin(2*b*log(c)

))*n^6)*sin(2*b*log(x^n) + 2*a))*cos(6*b*log(x^n) + 6*a) + 6*(4*b^8*n^8*cos(4*b*log(c)) + b^6*n^6*cos(4*b*log(

c)) - 3*(4*(b^8*cos(4*b*log(c))*cos(2*b*log(c)) + b^8*sin(4*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(4*b*log(

c))*cos(2*b*log(c)) + b^6*sin(4*b*log(c))*sin(2*b*log(c)))*n^6)*cos(2*b*log(x^n) + 2*a) - 3*(4*(b^8*cos(2*b*lo

g(c))*sin(4*b*log(c)) - b^8*cos(4*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(2*b*log(c))*sin(4*b*log(c)) - b^6*

cos(4*b*log(c))*sin(2*b*log(c)))*n^6)*sin(2*b*log(x^n) + 2*a))*cos(4*b*log(x^n) + 4*a) - 6*(4*b^8*n^8*cos(2*b*

log(c)) + b^6*n^6*cos(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + 2*(4*b^8*n^8*sin(6*b*log(c)) + b^6*n^6*sin(6*b*lo

g(c)) + 3*(4*(b^8*cos(4*b*log(c))*sin(6*b*log(c)) - b^8*cos(6*b*log(c))*sin(4*b*log(c)))*n^8 + (b^6*cos(4*b*lo

g(c))*sin(6*b*log(c)) - b^6*cos(6*b*log(c))*sin(4*b*log(c)))*n^6)*cos(4*b*log(x^n) + 4*a) - 3*(4*(b^8*cos(2*b*

log(c))*sin(6*b*log(c)) - b^8*cos(6*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(2*b*log(c))*sin(6*b*log(c)) - b^

6*cos(6*b*log(c))*sin(2*b*log(c)))*n^6)*cos(2*b*log(x^n) + 2*a) - 3*(4*(b^8*cos(6*b*log(c))*cos(4*b*log(c)) +

b^8*sin(6*b*log(c))*sin(4*b*log(c)))*n^8 + (b^6*cos(6*b*log(c))*cos(4*b*log(c)) + b^6*sin(6*b*log(c))*sin(4*b*

log(c)))*n^6)*sin(4*b*log(x^n) + 4*a) + 3*(4*(b^8*cos(6*b*log(c))*cos(2*b*log(c)) + b^8*sin(6*b*log(c))*sin(2*

b*log(c)))*n^8 + (b^6*cos(6*b*log(c))*cos(2*b*log(c)) + b^6*sin(6*b*log(c))*sin(2*b*log(c)))*n^6)*sin(2*b*log(

x^n) + 2*a))*sin(6*b*log(x^n) + 6*a) - 6*(4*b^8*n^8*sin(4*b*log(c)) + b^6*n^6*sin(4*b*log(c)) - 3*(4*(b^8*cos(

2*b*log(c))*sin(4*b*log(c)) - b^8*cos(4*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(2*b*log(c))*sin(4*b*log(c))

- b^6*cos(4*b*log(c))*sin(2*b*log(c)))*n^6)*cos(2*b*log(x^n) + 2*a) + 3*(4*(b^8*cos(4*b*log(c))*cos(2*b*log(c)

) + b^8*sin(4*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(4*b*log(c))*cos(2*b*log(c)) + b^6*sin(4*b*log(c))*sin(

2*b*log(c)))*n^6)*sin(2*b*log(x^n) + 2*a))*sin(4*b*log(x^n) + 4*a) + 6*(4*b^8*n^8*sin(2*b*log(c)) + b^6*n^6*si

n(2*b*log(c)))*sin(2*b*log(x^n) + 2*a))*integrate(-1/36*(cos(b*log(x^n) + a)*sin(b*log(c)) + cos(b*log(c))*sin

(b*log(x^n) + a))/(2*b^6*n^6*cos(b*log(c))*cos(b*log(x^n) + a) - 2*b^6*n^6*sin(b*log(c))*sin(b*log(x^n) + a) -

b^6*n^6 - (b^6*cos(b*log(c))^2 + b^6*sin(b*log(c))^2)*n^6*cos(b*log(x^n) + a)^2 - (b^6*cos(b*log(c))^2 + b^6*

sin(b*log(c))^2)*n^6*sin(b*log(x^n) + a)^2), x) + ((2*(b*cos(4*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*s

in(4*b*log(c)))*n + cos(6*b*log(c))*cos(4*b*log(c)) + sin(6*b*log(c))*sin(4*b*log(c)))*x*cos(4*b*log(x^n) + 4*

a) - 2*(6*(b^2*cos(6*b*log(c))*cos(2*b*log(c)) + b^2*sin(6*b*log(c))*sin(2*b*log(c)))*n^2 + (b*cos(2*b*log(c))

*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(2*b*log(c)))*n + cos(6*b*log(c))*cos(2*b*log(c)) + sin(6*b*log(c))*si

n(2*b*log(c)))*x*cos(2*b*log(x^n) + 2*a) - (2*(b*cos(6*b*log(c))*cos(4*b*log(c)) + b*sin(6*b*log(c))*sin(4*b*l

og(c)))*n - cos(4*b*log(c))*sin(6*b*log(c)) + cos(6*b*log(c))*sin(4*b*log(c)))*x*sin(4*b*log(x^n) + 4*a) - 2*(

6*(b^2*cos(2*b*log(c))*sin(6*b*log(c)) - b^2*cos(6*b*log(c))*sin(2*b*log(c)))*n^2 - (b*cos(6*b*log(c))*cos(2*b

*log(c)) + b*sin(6*b*log(c))*sin(2*b*log(c)))*n + cos(2*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(2*b*lo

g(c)))*x*sin(2*b*log(x^n) + 2*a) + (4*b^2*n^2*cos(6*b*log(c)) + cos(6*b*log(c)))*x)*sin(6*b*log(x^n) + 6*a) +

(3*(12*(b^2*cos(4*b*log(c))*cos(2*b*log(c)) + b^2*sin(4*b*log(c))*sin(2*b*log(c)))*n^2 + 4*(b*cos(2*b*log(c))*

sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(2*b*log(c)))*n + cos(4*b*log(c))*cos(2*b*log(c)) + sin(4*b*log(c))*sin

(2*b*log(c)))*x*cos(2*b*log(x^n) + 2*a) + 3*(12*(b^2*cos(2*b*log(c))*sin(4*b*log(c)) - b^2*cos(4*b*log(c))*sin

(2*b*log(c)))*n^2 - 4*(b*cos(4*b*log(c))*cos(2*b*log(c)) + b*sin(4*b*log(c))*sin(2*b*log(c)))*n + cos(2*b*log(

c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*x*sin(2*b*log(x^n) + 2*a) - 2*(6*b^2*n^2*cos(4*b*log(c)

) + b*n*sin(4*b*log(c)) + cos(4*b*log(c)))*x)*sin(4*b*log(x^n) + 4*a))/(6*b^3*n^3*cos(2*b*log(c))*cos(2*b*log(

x^n) + 2*a) - 6*b^3*n^3*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) - b^3*n^3 - (b^3*cos(6*b*log(c))^2 + b^3*sin(6

*b*log(c))^2)*n^3*cos(6*b*log(x^n) + 6*a)^2 - 9*(b^3*cos(4*b*log(c))^2 + b^3*sin(4*b*log(c))^2)*n^3*cos(4*b*lo

g(x^n) + 4*a)^2 - 9*(b^3*cos(2*b*log(c))^2 + b^3*sin(2*b*log(c))^2)*n^3*cos(2*b*log(x^n) + 2*a)^2 - (b^3*cos(6

*b*log(c))^2 + b^3*sin(6*b*log(c))^2)*n^3*sin(6*b*log(x^n) + 6*a)^2 - 9*(b^3*cos(4*b*log(c))^2 + b^3*sin(4*b*l

og(c))^2)*n^3*sin(4*b*log(x^n) + 4*a)^2 - 9*(b^3*cos(2*b*log(c))^2 + b^3*sin(2*b*log(c))^2)*n^3*sin(2*b*log(x^

n) + 2*a)^2 + 2*(b^3*n^3*cos(6*b*log(c)) + 3*(b^3*cos(6*b*log(c))*cos(4*b*log(c)) + b^3*sin(6*b*log(c))*sin(4*

b*log(c)))*n^3*cos(4*b*log(x^n) + 4*a) - 3*(b^3*cos(6*b*log(c))*cos(2*b*log(c)) + b^3*sin(6*b*log(c))*sin(2*b*

log(c)))*n^3*cos(2*b*log(x^n) + 2*a) + 3*(b^3*cos(4*b*log(c))*sin(6*b*log(c)) - b^3*cos(6*b*log(c))*sin(4*b*lo

g(c)))*n^3*sin(4*b*log(x^n) + 4*a) - 3*(b^3*cos(2*b*log(c))*sin(6*b*log(c)) - b^3*cos(6*b*log(c))*sin(2*b*log(

c)))*n^3*sin(2*b*log(x^n) + 2*a))*cos(6*b*log(x^n) + 6*a) - 6*(b^3*n^3*cos(4*b*log(c)) - 3*(b^3*cos(4*b*log(c)

)*cos(2*b*log(c)) + b^3*sin(4*b*log(c))*sin(2*b*log(c)))*n^3*cos(2*b*log(x^n) + 2*a) - 3*(b^3*cos(2*b*log(c))*

sin(4*b*log(c)) - b^3*cos(4*b*log(c))*sin(2*b*log(c)))*n^3*sin(2*b*log(x^n) + 2*a))*cos(4*b*log(x^n) + 4*a) -

2*(b^3*n^3*sin(6*b*log(c)) + 3*(b^3*cos(4*b*log(c))*sin(6*b*log(c)) - b^3*cos(6*b*log(c))*sin(4*b*log(c)))*n^3

*cos(4*b*log(x^n) + 4*a) - 3*(b^3*cos(2*b*log(c))*sin(6*b*log(c)) - b^3*cos(6*b*log(c))*sin(2*b*log(c)))*n^3*c

os(2*b*log(x^n) + 2*a) - 3*(b^3*cos(6*b*log(c))*cos(4*b*log(c)) + b^3*sin(6*b*log(c))*sin(4*b*log(c)))*n^3*sin

(4*b*log(x^n) + 4*a) + 3*(b^3*cos(6*b*log(c))*cos(2*b*log(c)) + b^3*sin(6*b*log(c))*sin(2*b*log(c)))*n^3*sin(2

*b*log(x^n) + 2*a))*sin(6*b*log(x^n) + 6*a) + 6*(b^3*n^3*sin(4*b*log(c)) - 3*(b^3*cos(2*b*log(c))*sin(4*b*log(

c)) - b^3*cos(4*b*log(c))*sin(2*b*log(c)))*n^3*cos(2*b*log(x^n) + 2*a) + 3*(b^3*cos(4*b*log(c))*cos(2*b*log(c)

) + b^3*sin(4*b*log(c))*sin(2*b*log(c)))*n^3*sin(2*b*log(x^n) + 2*a))*sin(4*b*log(x^n) + 4*a))

________________________________________________________________________________________

Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\csc \left (b \log \left (c x^{n}\right ) + a\right )^{4}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(a+b*log(c*x^n))^4,x, algorithm="fricas")

[Out]

integral(csc(b*log(c*x^n) + a)^4, x)

________________________________________________________________________________________

Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc ^{4}{\left (a + b \log{\left (c x^{n} \right )} \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(a+b*ln(c*x**n))**4,x)

[Out]

Integral(csc(a + b*log(c*x**n))**4, x)

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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc \left (b \log \left (c x^{n}\right ) + a\right )^{4}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(a+b*log(c*x^n))^4,x, algorithm="giac")

[Out]

integrate(csc(b*log(c*x^n) + a)^4, x)

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