3.298 \(\int \frac{\csc ^3(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=55 \[ -\frac{\tanh ^{-1}\left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}-\frac{\cot \left (a+b \log \left (c x^n\right )\right ) \csc \left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

[Out]

-ArcTanh[Cos[a + b*Log[c*x^n]]]/(2*b*n) - (Cot[a + b*Log[c*x^n]]*Csc[a + b*Log[c*x^n]])/(2*b*n)

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Rubi [A]  time = 0.0395494, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3768, 3770} \[ -\frac{\tanh ^{-1}\left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}-\frac{\cot \left (a+b \log \left (c x^n\right )\right ) \csc \left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*Log[c*x^n]]^3/x,x]

[Out]

-ArcTanh[Cos[a + b*Log[c*x^n]]]/(2*b*n) - (Cot[a + b*Log[c*x^n]]*Csc[a + b*Log[c*x^n]])/(2*b*n)

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\csc ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \csc ^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{\cot \left (a+b \log \left (c x^n\right )\right ) \csc \left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac{\operatorname{Subst}\left (\int \csc (a+b x) \, dx,x,\log \left (c x^n\right )\right )}{2 n}\\ &=-\frac{\tanh ^{-1}\left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}-\frac{\cot \left (a+b \log \left (c x^n\right )\right ) \csc \left (a+b \log \left (c x^n\right )\right )}{2 b n}\\ \end{align*}

Mathematica [A]  time = 0.0891395, size = 107, normalized size = 1.95 \[ \frac{\log \left (\sin \left (\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right )\right )}{2 b n}+\frac{\sec ^2\left (\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}-\frac{\log \left (\cos \left (\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right )\right )}{2 b n}-\frac{\csc ^2\left (\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*Log[c*x^n]]^3/x,x]

[Out]

-Csc[(a + b*Log[c*x^n])/2]^2/(8*b*n) - Log[Cos[(a + b*Log[c*x^n])/2]]/(2*b*n) + Log[Sin[(a + b*Log[c*x^n])/2]]
/(2*b*n) + Sec[(a + b*Log[c*x^n])/2]^2/(8*b*n)

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Maple [A]  time = 0.046, size = 66, normalized size = 1.2 \begin{align*} -{\frac{\csc \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \cot \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{2\,bn}}+{\frac{\ln \left ( \csc \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) -\cot \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) }{2\,bn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(a+b*ln(c*x^n))^3/x,x)

[Out]

-1/2*cot(a+b*ln(c*x^n))*csc(a+b*ln(c*x^n))/b/n+1/2/b/n*ln(csc(a+b*ln(c*x^n))-cot(a+b*ln(c*x^n)))

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Maxima [B]  time = 1.29191, size = 2927, normalized size = 53.22 \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,x, algorithm="maxima")

[Out]

1/4*(4*((cos(4*b*log(c))*cos(3*b*log(c)) + sin(4*b*log(c))*sin(3*b*log(c)))*cos(3*b*log(x^n) + 3*a) + (cos(4*b
*log(c))*cos(b*log(c)) + sin(4*b*log(c))*sin(b*log(c)))*cos(b*log(x^n) + a) + (cos(3*b*log(c))*sin(4*b*log(c))
- cos(4*b*log(c))*sin(3*b*log(c)))*sin(3*b*log(x^n) + 3*a) + (cos(b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))
*sin(b*log(c)))*sin(b*log(x^n) + a))*cos(4*b*log(x^n) + 4*a) - 4*(2*(cos(3*b*log(c))*cos(2*b*log(c)) + sin(3*b
*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + 2*(cos(2*b*log(c))*sin(3*b*log(c)) - cos(3*b*log(c))*sin(2
*b*log(c)))*sin(2*b*log(x^n) + 2*a) - cos(3*b*log(c)))*cos(3*b*log(x^n) + 3*a) - 8*((cos(2*b*log(c))*cos(b*log
(c)) + sin(2*b*log(c))*sin(b*log(c)))*cos(b*log(x^n) + a) + (cos(b*log(c))*sin(2*b*log(c)) - cos(2*b*log(c))*s
in(b*log(c)))*sin(b*log(x^n) + a))*cos(2*b*log(x^n) + 2*a) + 4*cos(b*log(c))*cos(b*log(x^n) + a) - ((cos(4*b*l
og(c))^2 + sin(4*b*log(c))^2)*cos(4*b*log(x^n) + 4*a)^2 + 4*(cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*cos(2*b*lo
g(x^n) + 2*a)^2 + (cos(4*b*log(c))^2 + sin(4*b*log(c))^2)*sin(4*b*log(x^n) + 4*a)^2 + 4*(cos(2*b*log(c))^2 + s
in(2*b*log(c))^2)*sin(2*b*log(x^n) + 2*a)^2 - 2*(2*(cos(4*b*log(c))*cos(2*b*log(c)) + sin(4*b*log(c))*sin(2*b*
log(c)))*cos(2*b*log(x^n) + 2*a) + 2*(cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*sin(2
*b*log(x^n) + 2*a) - cos(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) - 4*cos(2*b*log(c))*cos(2*b*log(x^n) + 2*a) + 2*
(2*(cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) - 2*(cos(4*b*lo
g(c))*cos(2*b*log(c)) + sin(4*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) - sin(4*b*log(c)))*sin(4*b*lo
g(x^n) + 4*a) + 4*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) + 1)*log((cos(a)^2 + sin(a)^2)*cos(b*log(c))^2 + (co
s(a)^2 + sin(a)^2)*sin(b*log(c))^2 + 2*(cos(b*log(c))*cos(a) - sin(b*log(c))*sin(a))*cos(b*log(x^n)) + cos(b*l
og(x^n))^2 - 2*(cos(a)*sin(b*log(c)) + cos(b*log(c))*sin(a))*sin(b*log(x^n)) + sin(b*log(x^n))^2) + ((cos(4*b*
log(c))^2 + sin(4*b*log(c))^2)*cos(4*b*log(x^n) + 4*a)^2 + 4*(cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*cos(2*b*l
og(x^n) + 2*a)^2 + (cos(4*b*log(c))^2 + sin(4*b*log(c))^2)*sin(4*b*log(x^n) + 4*a)^2 + 4*(cos(2*b*log(c))^2 +
sin(2*b*log(c))^2)*sin(2*b*log(x^n) + 2*a)^2 - 2*(2*(cos(4*b*log(c))*cos(2*b*log(c)) + sin(4*b*log(c))*sin(2*b
*log(c)))*cos(2*b*log(x^n) + 2*a) + 2*(cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*sin(
2*b*log(x^n) + 2*a) - cos(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) - 4*cos(2*b*log(c))*cos(2*b*log(x^n) + 2*a) + 2
*(2*(cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) - 2*(cos(4*b*l
og(c))*cos(2*b*log(c)) + sin(4*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) - sin(4*b*log(c)))*sin(4*b*l
og(x^n) + 4*a) + 4*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) + 1)*log((cos(a)^2 + sin(a)^2)*cos(b*log(c))^2 + (c
os(a)^2 + sin(a)^2)*sin(b*log(c))^2 - 2*(cos(b*log(c))*cos(a) - sin(b*log(c))*sin(a))*cos(b*log(x^n)) + cos(b*
log(x^n))^2 + 2*(cos(a)*sin(b*log(c)) + cos(b*log(c))*sin(a))*sin(b*log(x^n)) + sin(b*log(x^n))^2) - 4*((cos(3
*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(3*b*log(c)))*cos(3*b*log(x^n) + 3*a) + (cos(b*log(c))*sin(4*b
*log(c)) - cos(4*b*log(c))*sin(b*log(c)))*cos(b*log(x^n) + a) - (cos(4*b*log(c))*cos(3*b*log(c)) + sin(4*b*log
(c))*sin(3*b*log(c)))*sin(3*b*log(x^n) + 3*a) - (cos(4*b*log(c))*cos(b*log(c)) + sin(4*b*log(c))*sin(b*log(c))
)*sin(b*log(x^n) + a))*sin(4*b*log(x^n) + 4*a) + 4*(2*(cos(2*b*log(c))*sin(3*b*log(c)) - cos(3*b*log(c))*sin(2
*b*log(c)))*cos(2*b*log(x^n) + 2*a) - 2*(cos(3*b*log(c))*cos(2*b*log(c)) + sin(3*b*log(c))*sin(2*b*log(c)))*si
n(2*b*log(x^n) + 2*a) - sin(3*b*log(c)))*sin(3*b*log(x^n) + 3*a) + 8*((cos(b*log(c))*sin(2*b*log(c)) - cos(2*b
*log(c))*sin(b*log(c)))*cos(b*log(x^n) + a) - (cos(2*b*log(c))*cos(b*log(c)) + sin(2*b*log(c))*sin(b*log(c)))*
sin(b*log(x^n) + a))*sin(2*b*log(x^n) + 2*a) - 4*sin(b*log(c))*sin(b*log(x^n) + a))/((b*cos(4*b*log(c))^2 + b*
sin(4*b*log(c))^2)*n*cos(4*b*log(x^n) + 4*a)^2 - 4*b*n*cos(2*b*log(c))*cos(2*b*log(x^n) + 2*a) + 4*(b*cos(2*b*
log(c))^2 + b*sin(2*b*log(c))^2)*n*cos(2*b*log(x^n) + 2*a)^2 + (b*cos(4*b*log(c))^2 + b*sin(4*b*log(c))^2)*n*s
in(4*b*log(x^n) + 4*a)^2 + 4*b*n*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) + 4*(b*cos(2*b*log(c))^2 + b*sin(2*b*
log(c))^2)*n*sin(2*b*log(x^n) + 2*a)^2 + b*n + 2*(b*n*cos(4*b*log(c)) - 2*(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(x^n) + 2*a) - 2*(b*cos(2*b*log(c))*sin(4*b*log(c)) - b*cos(4
*b*log(c))*sin(2*b*log(c)))*n*sin(2*b*log(x^n) + 2*a))*cos(4*b*log(x^n) + 4*a) + 2*(2*(b*cos(2*b*log(c))*sin(4
*b*log(c)) - b*cos(4*b*log(c))*sin(2*b*log(c)))*n*cos(2*b*log(x^n) + 2*a) - b*n*sin(4*b*log(c)) - 2*(b*cos(4*b
*log(c))*cos(2*b*log(c)) + b*sin(4*b*log(c))*sin(2*b*log(c)))*n*sin(2*b*log(x^n) + 2*a))*sin(4*b*log(x^n) + 4*
a))

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Fricas [B]  time = 0.503356, size = 352, normalized size = 6.4 \begin{align*} -\frac{{\left (\cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \frac{1}{2}\right ) -{\left (\cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \frac{1}{2}\right ) - 2 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{4 \,{\left (b n \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - b n\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(a+b*log(c*x^n))^3/x,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc ^{3}{\left (a + b \log{\left (c x^{n} \right )} \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(a+b*ln(c*x**n))**3/x,x)

[Out]

Integral(csc(a + b*log(c*x**n))**3/x, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (b \log \left (c x^{n}\right ) + a\right )^{3}}{x}\,{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,x, algorithm="giac")

[Out]

integrate(csc(b*log(c*x^n) + a)^3/x, x)