3.297 \(\int \csc ^3(a+b \log (c x^n)) \, dx\)
Optimal. Leaf size=84 \[ -\frac{8 e^{3 i a} x \left (c x^n\right )^{3 i b} \text{Hypergeometric2F1}\left (3,\frac{1}{2} \left (3-\frac{i}{b n}\right ),\frac{1}{2} \left (5-\frac{i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{-3 b n+i} \]
[Out]
(-8*E^((3*I)*a)*x*(c*x^n)^((3*I)*b)*Hypergeometric2F1[3, (3 - I/(b*n))/2, (5 - I/(b*n))/2, E^((2*I)*a)*(c*x^n)
^((2*I)*b)])/(I - 3*b*n)
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Rubi [A] time = 0.0622205, 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} \[ -\frac{8 e^{3 i a} x \left (c x^n\right )^{3 i b} \, _2F_1\left (3,\frac{1}{2} \left (3-\frac{i}{b n}\right );\frac{1}{2} \left (5-\frac{i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{-3 b n+i} \]
Antiderivative was successfully verified.
[In]
Int[Csc[a + b*Log[c*x^n]]^3,x]
[Out]
(-8*E^((3*I)*a)*x*(c*x^n)^((3*I)*b)*Hypergeometric2F1[3, (3 - I/(b*n))/2, (5 - I/(b*n))/2, E^((2*I)*a)*(c*x^n)
^((2*I)*b)])/(I - 3*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 ^3\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 ^3(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (8 i e^{3 i a} x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+3 i b+\frac{1}{n}}}{\left (1-e^{2 i a} x^{2 i b}\right )^3} \, dx,x,c x^n\right )}{n}\\ &=-\frac{8 e^{3 i a} x \left (c x^n\right )^{3 i b} \, _2F_1\left (3,\frac{1}{2} \left (3-\frac{i}{b n}\right );\frac{1}{2} \left (5-\frac{i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{i-3 b n}\\ \end{align*}
Mathematica [A] time = 5.6157, size = 117, normalized size = 1.39 \[ -\frac{x \left (\left (b n \cot \left (a+b \log \left (c x^n\right )\right )+1\right ) \csc \left (a+b \log \left (c x^n\right )\right )+2 e^{i a} (b n+i) \left (c x^n\right )^{i b} \text{Hypergeometric2F1}\left (1,\frac{1}{2}-\frac{i}{2 b n},\frac{3}{2}-\frac{i}{2 b n},e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{2 b^2 n^2} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Csc[a + b*Log[c*x^n]]^3,x]
[Out]
-(x*((1 + b*n*Cot[a + b*Log[c*x^n]])*Csc[a + b*Log[c*x^n]] + 2*E^(I*a)*(I + b*n)*(c*x^n)^(I*b)*Hypergeometric2
F1[1, 1/2 - (I/2)/(b*n), 3/2 - (I/2)/(b*n), E^((2*I)*(a + b*Log[c*x^n]))]))/(2*b^2*n^2)
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Maple [F] time = 2.277, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(a+b*ln(c*x^n))^3,x)
[Out]
int(csc(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(csc(a+b*log(c*x^n))^3,x, algorithm="maxima")
[Out]
-((b*n*cos(b*log(c)) - sin(b*log(c)))*x*cos(b*log(x^n) + a) - (b*n*sin(b*log(c)) + cos(b*log(c)))*x*sin(b*log(
x^n) + a) + (((b*cos(4*b*log(c))*cos(3*b*log(c)) + b*sin(4*b*log(c))*sin(3*b*log(c)))*n - cos(3*b*log(c))*sin(
4*b*log(c)) + cos(4*b*log(c))*sin(3*b*log(c)))*x*cos(3*b*log(x^n) + 3*a) + ((b*cos(4*b*log(c))*cos(b*log(c)) +
b*sin(4*b*log(c))*sin(b*log(c)))*n + cos(b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(b*log(c)))*x*cos(b*l
og(x^n) + a) + ((b*cos(3*b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(3*b*log(c)))*n + cos(4*b*log(c))*co
s(3*b*log(c)) + sin(4*b*log(c))*sin(3*b*log(c)))*x*sin(3*b*log(x^n) + 3*a) + ((b*cos(b*log(c))*sin(4*b*log(c))
- b*cos(4*b*log(c))*sin(b*log(c)))*n - cos(4*b*log(c))*cos(b*log(c)) - sin(4*b*log(c))*sin(b*log(c)))*x*sin(b
*log(x^n) + a))*cos(4*b*log(x^n) + 4*a) - (2*((b*cos(3*b*log(c))*cos(2*b*log(c)) + b*sin(3*b*log(c))*sin(2*b*l
og(c)))*n + cos(2*b*log(c))*sin(3*b*log(c)) - cos(3*b*log(c))*sin(2*b*log(c)))*x*cos(2*b*log(x^n) + 2*a) + 2*(
(b*cos(2*b*log(c))*sin(3*b*log(c)) - b*cos(3*b*log(c))*sin(2*b*log(c)))*n - cos(3*b*log(c))*cos(2*b*log(c)) -
sin(3*b*log(c))*sin(2*b*log(c)))*x*sin(2*b*log(x^n) + 2*a) - (b*n*cos(3*b*log(c)) + sin(3*b*log(c)))*x)*cos(3*
b*log(x^n) + 3*a) - 2*(((b*cos(2*b*log(c))*cos(b*log(c)) + b*sin(2*b*log(c))*sin(b*log(c)))*n + cos(b*log(c))*
sin(2*b*log(c)) - cos(2*b*log(c))*sin(b*log(c)))*x*cos(b*log(x^n) + a) + ((b*cos(b*log(c))*sin(2*b*log(c)) - b
*cos(2*b*log(c))*sin(b*log(c)))*n - cos(2*b*log(c))*cos(b*log(c)) - sin(2*b*log(c))*sin(b*log(c)))*x*sin(b*log
(x^n) + a))*cos(2*b*log(x^n) + 2*a) + 2*(b^6*n^6 + b^4*n^4 + ((b^6*cos(4*b*log(c))^2 + b^6*sin(4*b*log(c))^2)*
n^6 + (b^4*cos(4*b*log(c))^2 + b^4*sin(4*b*log(c))^2)*n^4)*cos(4*b*log(x^n) + 4*a)^2 + 4*((b^6*cos(2*b*log(c))
^2 + b^6*sin(2*b*log(c))^2)*n^6 + (b^4*cos(2*b*log(c))^2 + b^4*sin(2*b*log(c))^2)*n^4)*cos(2*b*log(x^n) + 2*a)
^2 + ((b^6*cos(4*b*log(c))^2 + b^6*sin(4*b*log(c))^2)*n^6 + (b^4*cos(4*b*log(c))^2 + b^4*sin(4*b*log(c))^2)*n^
4)*sin(4*b*log(x^n) + 4*a)^2 + 4*((b^6*cos(2*b*log(c))^2 + b^6*sin(2*b*log(c))^2)*n^6 + (b^4*cos(2*b*log(c))^2
+ b^4*sin(2*b*log(c))^2)*n^4)*sin(2*b*log(x^n) + 2*a)^2 + 2*(b^6*n^6*cos(4*b*log(c)) + b^4*n^4*cos(4*b*log(c)
) - 2*((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 + (b^4*cos(4*b*log(c))*
cos(2*b*log(c)) + b^4*sin(4*b*log(c))*sin(2*b*log(c)))*n^4)*cos(2*b*log(x^n) + 2*a) - 2*((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 + (b^4*cos(2*b*log(c))*sin(4*b*log(c)) - b^4*cos(4*
b*log(c))*sin(2*b*log(c)))*n^4)*sin(2*b*log(x^n) + 2*a))*cos(4*b*log(x^n) + 4*a) - 4*(b^6*n^6*cos(2*b*log(c))
+ b^4*n^4*cos(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) - 2*(b^6*n^6*sin(4*b*log(c)) + b^4*n^4*sin(4*b*log(c)) - 2*
((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 + (b^4*cos(2*b*log(c))*sin(4*
b*log(c)) - b^4*cos(4*b*log(c))*sin(2*b*log(c)))*n^4)*cos(2*b*log(x^n) + 2*a) + 2*((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 + (b^4*cos(4*b*log(c))*cos(2*b*log(c)) + b^4*sin(4*b*log(
c))*sin(2*b*log(c)))*n^4)*sin(2*b*log(x^n) + 2*a))*sin(4*b*log(x^n) + 4*a) + 4*(b^6*n^6*sin(2*b*log(c)) + b^4*
n^4*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a))*integrate(1/4*(cos(b*log(x^n) + a)*sin(b*log(c)) + cos(b*log(c))
*sin(b*log(x^n) + a))/(2*b^4*n^4*cos(b*log(c))*cos(b*log(x^n) + a) - 2*b^4*n^4*sin(b*log(c))*sin(b*log(x^n) +
a) + b^4*n^4 + (b^4*cos(b*log(c))^2 + b^4*sin(b*log(c))^2)*n^4*cos(b*log(x^n) + a)^2 + (b^4*cos(b*log(c))^2 +
b^4*sin(b*log(c))^2)*n^4*sin(b*log(x^n) + a)^2), x) + 2*(b^6*n^6 + b^4*n^4 + ((b^6*cos(4*b*log(c))^2 + b^6*sin
(4*b*log(c))^2)*n^6 + (b^4*cos(4*b*log(c))^2 + b^4*sin(4*b*log(c))^2)*n^4)*cos(4*b*log(x^n) + 4*a)^2 + 4*((b^6
*cos(2*b*log(c))^2 + b^6*sin(2*b*log(c))^2)*n^6 + (b^4*cos(2*b*log(c))^2 + b^4*sin(2*b*log(c))^2)*n^4)*cos(2*b
*log(x^n) + 2*a)^2 + ((b^6*cos(4*b*log(c))^2 + b^6*sin(4*b*log(c))^2)*n^6 + (b^4*cos(4*b*log(c))^2 + b^4*sin(4
*b*log(c))^2)*n^4)*sin(4*b*log(x^n) + 4*a)^2 + 4*((b^6*cos(2*b*log(c))^2 + b^6*sin(2*b*log(c))^2)*n^6 + (b^4*c
os(2*b*log(c))^2 + b^4*sin(2*b*log(c))^2)*n^4)*sin(2*b*log(x^n) + 2*a)^2 + 2*(b^6*n^6*cos(4*b*log(c)) + b^4*n^
4*cos(4*b*log(c)) - 2*((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 + (b^4*
cos(4*b*log(c))*cos(2*b*log(c)) + b^4*sin(4*b*log(c))*sin(2*b*log(c)))*n^4)*cos(2*b*log(x^n) + 2*a) - 2*((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 + (b^4*cos(2*b*log(c))*sin(4*b*log(
c)) - b^4*cos(4*b*log(c))*sin(2*b*log(c)))*n^4)*sin(2*b*log(x^n) + 2*a))*cos(4*b*log(x^n) + 4*a) - 4*(b^6*n^6*
cos(2*b*log(c)) + b^4*n^4*cos(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) - 2*(b^6*n^6*sin(4*b*log(c)) + b^4*n^4*sin(
4*b*log(c)) - 2*((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 + (b^4*cos(2*
b*log(c))*sin(4*b*log(c)) - b^4*cos(4*b*log(c))*sin(2*b*log(c)))*n^4)*cos(2*b*log(x^n) + 2*a) + 2*((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 + (b^4*cos(4*b*log(c))*cos(2*b*log(c)) +
b^4*sin(4*b*log(c))*sin(2*b*log(c)))*n^4)*sin(2*b*log(x^n) + 2*a))*sin(4*b*log(x^n) + 4*a) + 4*(b^6*n^6*sin(2*
b*log(c)) + b^4*n^4*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a))*integrate(-1/4*(cos(b*log(x^n) + a)*sin(b*log(c)
) + cos(b*log(c))*sin(b*log(x^n) + a))/(2*b^4*n^4*cos(b*log(c))*cos(b*log(x^n) + a) - 2*b^4*n^4*sin(b*log(c))*
sin(b*log(x^n) + a) - b^4*n^4 - (b^4*cos(b*log(c))^2 + b^4*sin(b*log(c))^2)*n^4*cos(b*log(x^n) + a)^2 - (b^4*c
os(b*log(c))^2 + b^4*sin(b*log(c))^2)*n^4*sin(b*log(x^n) + a)^2), x) - (((b*cos(3*b*log(c))*sin(4*b*log(c)) -
b*cos(4*b*log(c))*sin(3*b*log(c)))*n + cos(4*b*log(c))*cos(3*b*log(c)) + sin(4*b*log(c))*sin(3*b*log(c)))*x*co
s(3*b*log(x^n) + 3*a) + ((b*cos(b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(b*log(c)))*n - cos(4*b*log(c
))*cos(b*log(c)) - sin(4*b*log(c))*sin(b*log(c)))*x*cos(b*log(x^n) + a) - ((b*cos(4*b*log(c))*cos(3*b*log(c))
+ b*sin(4*b*log(c))*sin(3*b*log(c)))*n - cos(3*b*log(c))*sin(4*b*log(c)) + cos(4*b*log(c))*sin(3*b*log(c)))*x*
sin(3*b*log(x^n) + 3*a) - ((b*cos(4*b*log(c))*cos(b*log(c)) + b*sin(4*b*log(c))*sin(b*log(c)))*n + cos(b*log(c
))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(b*log(c)))*x*sin(b*log(x^n) + a))*sin(4*b*log(x^n) + 4*a) + (2*((b*co
s(2*b*log(c))*sin(3*b*log(c)) - b*cos(3*b*log(c))*sin(2*b*log(c)))*n - cos(3*b*log(c))*cos(2*b*log(c)) - sin(3
*b*log(c))*sin(2*b*log(c)))*x*cos(2*b*log(x^n) + 2*a) - 2*((b*cos(3*b*log(c))*cos(2*b*log(c)) + b*sin(3*b*log(
c))*sin(2*b*log(c)))*n + cos(2*b*log(c))*sin(3*b*log(c)) - cos(3*b*log(c))*sin(2*b*log(c)))*x*sin(2*b*log(x^n)
+ 2*a) - (b*n*sin(3*b*log(c)) - cos(3*b*log(c)))*x)*sin(3*b*log(x^n) + 3*a) + 2*(((b*cos(b*log(c))*sin(2*b*lo
g(c)) - b*cos(2*b*log(c))*sin(b*log(c)))*n - cos(2*b*log(c))*cos(b*log(c)) - sin(2*b*log(c))*sin(b*log(c)))*x*
cos(b*log(x^n) + a) - ((b*cos(2*b*log(c))*cos(b*log(c)) + b*sin(2*b*log(c))*sin(b*log(c)))*n + cos(b*log(c))*s
in(2*b*log(c)) - cos(2*b*log(c))*sin(b*log(c)))*x*sin(b*log(x^n) + a))*sin(2*b*log(x^n) + 2*a))/(4*b^2*n^2*cos
(2*b*log(c))*cos(2*b*log(x^n) + 2*a) - 4*b^2*n^2*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) - (b^2*cos(4*b*log(c)
)^2 + b^2*sin(4*b*log(c))^2)*n^2*cos(4*b*log(x^n) + 4*a)^2 - 4*(b^2*cos(2*b*log(c))^2 + b^2*sin(2*b*log(c))^2)
*n^2*cos(2*b*log(x^n) + 2*a)^2 - (b^2*cos(4*b*log(c))^2 + b^2*sin(4*b*log(c))^2)*n^2*sin(4*b*log(x^n) + 4*a)^2
- 4*(b^2*cos(2*b*log(c))^2 + b^2*sin(2*b*log(c))^2)*n^2*sin(2*b*log(x^n) + 2*a)^2 - b^2*n^2 - 2*(b^2*n^2*cos(
4*b*log(c)) - 2*(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*cos(2*b*log(x^
n) + 2*a) - 2*(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*sin(2*b*log(x^n)
+ 2*a))*cos(4*b*log(x^n) + 4*a) + 2*(b^2*n^2*sin(4*b*log(c)) - 2*(b^2*cos(2*b*log(c))*sin(4*b*log(c)) - b^2*c
os(4*b*log(c))*sin(2*b*log(c)))*n^2*cos(2*b*log(x^n) + 2*a) + 2*(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*sin(2*b*log(x^n) + 2*a))*sin(4*b*log(x^n) + 4*a))
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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 )^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(a+b*log(c*x^n))^3,x, algorithm="fricas")
[Out]
integral(csc(b*log(c*x^n) + a)^3, x)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc ^{3}{\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))**3,x)
[Out]
Integral(csc(a + b*log(c*x**n))**3, 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 )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(a+b*log(c*x^n))^3,x, algorithm="giac")
[Out]
integrate(csc(b*log(c*x^n) + a)^3, x)