3.173 \(\int \frac{\tan ^4(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=45 \[ \frac{\tan ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac{\tan \left (a+b \log \left (c x^n\right )\right )}{b n}+\log (x) \]

[Out]

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

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Rubi [A]  time = 0.0375052, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3473, 8} \[ \frac{\tan ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac{\tan \left (a+b \log \left (c x^n\right )\right )}{b n}+\log (x) \]

Antiderivative was successfully verified.

[In]

Int[Tan[a + b*Log[c*x^n]]^4/x,x]

[Out]

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

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\tan ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \tan ^4(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\tan ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac{\operatorname{Subst}\left (\int \tan ^2(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{\tan \left (a+b \log \left (c x^n\right )\right )}{b n}+\frac{\tan ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac{\operatorname{Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\log (x)-\frac{\tan \left (a+b \log \left (c x^n\right )\right )}{b n}+\frac{\tan ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}\\ \end{align*}

Mathematica [A]  time = 0.100872, size = 62, normalized size = 1.38 \[ \frac{\tan ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac{\tan ^{-1}\left (\tan \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac{\tan \left (a+b \log \left (c x^n\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[Tan[a + b*Log[c*x^n]]]/(b*n) - Tan[a + b*Log[c*x^n]]/(b*n) + Tan[a + b*Log[c*x^n]]^3/(3*b*n)

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Maple [A]  time = 0.018, size = 61, normalized size = 1.4 \begin{align*}{\frac{ \left ( \tan \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{3}}{3\,bn}}-{\frac{\tan \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{bn}}+{\frac{\arctan \left ( \tan \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) }{bn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*tan(a+b*ln(c*x^n))^3/b/n-tan(a+b*ln(c*x^n))/b/n+1/n/b*arctan(tan(a+b*ln(c*x^n)))

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Maxima [B]  time = 1.35695, size = 2931, normalized size = 65.13 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*(b*cos(6*b*log(c))^2 + b*sin(6*b*log(c))^2)*n*cos(6*b*log(x^n) + 6*a)^2*log(x) + 27*(b*cos(4*b*log(c))^
2 + b*sin(4*b*log(c))^2)*n*cos(4*b*log(x^n) + 4*a)^2*log(x) + 27*(b*cos(2*b*log(c))^2 + b*sin(2*b*log(c))^2)*n
*cos(2*b*log(x^n) + 2*a)^2*log(x) + 3*(b*cos(6*b*log(c))^2 + b*sin(6*b*log(c))^2)*n*log(x)*sin(6*b*log(x^n) +
6*a)^2 + 27*(b*cos(4*b*log(c))^2 + b*sin(4*b*log(c))^2)*n*log(x)*sin(4*b*log(x^n) + 4*a)^2 + 27*(b*cos(2*b*log
(c))^2 + b*sin(2*b*log(c))^2)*n*log(x)*sin(2*b*log(x^n) + 2*a)^2 + 3*b*n*log(x) + 2*(3*b*n*cos(6*b*log(c))*log
(x) + 3*(3*(b*cos(6*b*log(c))*cos(4*b*log(c)) + b*sin(6*b*log(c))*sin(4*b*log(c)))*n*log(x) - 2*cos(4*b*log(c)
)*sin(6*b*log(c)) + 2*cos(6*b*log(c))*sin(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) + 3*(3*(b*cos(6*b*log(c))*cos(2
*b*log(c)) + b*sin(6*b*log(c))*sin(2*b*log(c)))*n*log(x) - 2*cos(2*b*log(c))*sin(6*b*log(c)) + 2*cos(6*b*log(c
))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + 3*(3*(b*cos(4*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(
4*b*log(c)))*n*log(x) + 2*cos(6*b*log(c))*cos(4*b*log(c)) + 2*sin(6*b*log(c))*sin(4*b*log(c)))*sin(4*b*log(x^n
) + 4*a) + 3*(3*(b*cos(2*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(2*b*log(c)))*n*log(x) + 2*cos(6*b*l
og(c))*cos(2*b*log(c)) + 2*sin(6*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) - 4*sin(6*b*log(c)))*cos(6
*b*log(x^n) + 6*a) + 6*(3*b*n*cos(4*b*log(c))*log(x) + 9*(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)*log(x) + 9*(b*cos(2*b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))
*sin(2*b*log(c)))*n*log(x)*sin(2*b*log(x^n) + 2*a) - 2*sin(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) + 6*(3*b*n*cos
(2*b*log(c))*log(x) - 2*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) - 2*(3*b*n*log(x)*sin(6*b*log(c)) + 3*(3*(b*c
os(4*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(4*b*log(c)))*n*log(x) + 2*cos(6*b*log(c))*cos(4*b*log(c
)) + 2*sin(6*b*log(c))*sin(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) + 3*(3*(b*cos(2*b*log(c))*sin(6*b*log(c)) - b*
cos(6*b*log(c))*sin(2*b*log(c)))*n*log(x) + 2*cos(6*b*log(c))*cos(2*b*log(c)) + 2*sin(6*b*log(c))*sin(2*b*log(
c)))*cos(2*b*log(x^n) + 2*a) - 3*(3*(b*cos(6*b*log(c))*cos(4*b*log(c)) + b*sin(6*b*log(c))*sin(4*b*log(c)))*n*
log(x) - 2*cos(4*b*log(c))*sin(6*b*log(c)) + 2*cos(6*b*log(c))*sin(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) - 3*(3
*(b*cos(6*b*log(c))*cos(2*b*log(c)) + b*sin(6*b*log(c))*sin(2*b*log(c)))*n*log(x) - 2*cos(2*b*log(c))*sin(6*b*
log(c)) + 2*cos(6*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) + 4*cos(6*b*log(c)))*sin(6*b*log(x^n) + 6
*a) - 6*(9*(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)*l
og(x) + 3*b*n*log(x)*sin(4*b*log(c)) - 9*(b*cos(4*b*log(c))*cos(2*b*log(c)) + b*sin(4*b*log(c))*sin(2*b*log(c)
))*n*log(x)*sin(2*b*log(x^n) + 2*a) + 2*cos(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) - 6*(3*b*n*log(x)*sin(2*b*log
(c)) + 2*cos(2*b*log(c)))*sin(2*b*log(x^n) + 2*a))/((b*cos(6*b*log(c))^2 + b*sin(6*b*log(c))^2)*n*cos(6*b*log(
x^n) + 6*a)^2 + 9*(b*cos(4*b*log(c))^2 + b*sin(4*b*log(c))^2)*n*cos(4*b*log(x^n) + 4*a)^2 + 6*b*n*cos(2*b*log(
c))*cos(2*b*log(x^n) + 2*a) + 9*(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*c
os(6*b*log(c))^2 + b*sin(6*b*log(c))^2)*n*sin(6*b*log(x^n) + 6*a)^2 + 9*(b*cos(4*b*log(c))^2 + b*sin(4*b*log(c
))^2)*n*sin(4*b*log(x^n) + 4*a)^2 - 6*b*n*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) + 9*(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(6*b*log(c)) + 3*(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(x^n) + 4*a) + 3*(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(x^n) + 2*a) + 3*(b*cos(4*b*log(c))*sin(6*b*log(c)) - b*cos(
6*b*log(c))*sin(4*b*log(c)))*n*sin(4*b*log(x^n) + 4*a) + 3*(b*cos(2*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(
c))*sin(2*b*log(c)))*n*sin(2*b*log(x^n) + 2*a))*cos(6*b*log(x^n) + 6*a) + 6*(b*n*cos(4*b*log(c)) + 3*(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) + 3*(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*
(3*(b*cos(4*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(4*b*log(c)))*n*cos(4*b*log(x^n) + 4*a) + 3*(b*co
s(2*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(2*b*log(c)))*n*cos(2*b*log(x^n) + 2*a) + b*n*sin(6*b*log
(c)) - 3*(b*cos(6*b*log(c))*cos(4*b*log(c)) + b*sin(6*b*log(c))*sin(4*b*log(c)))*n*sin(4*b*log(x^n) + 4*a) - 3
*(b*cos(6*b*log(c))*cos(2*b*log(c)) + b*sin(6*b*log(c))*sin(2*b*log(c)))*n*sin(2*b*log(x^n) + 2*a))*sin(6*b*lo
g(x^n) + 6*a) - 6*(3*(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)) - 3*(b*cos(4*b*log(c))*cos(2*b*log(c)) + b*sin(4*b*log(c))*sin(2*b*log(c)))*n*s
in(2*b*log(x^n) + 2*a))*sin(4*b*log(x^n) + 4*a))

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Fricas [B]  time = 0.486267, size = 416, normalized size = 9.24 \begin{align*} \frac{3 \, b n \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )^{2} \log \left (x\right ) + 6 \, b n \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) \log \left (x\right ) + 3 \, b n \log \left (x\right ) - 2 \,{\left (2 \, \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1\right )} \sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}{3 \,{\left (b n \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )^{2} + 2 \, b n \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + b n\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 32.2557, size = 66, normalized size = 1.47 \begin{align*} \begin{cases} \log{\left (x \right )} \tan ^{4}{\left (a \right )} & \text{for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log{\left (x \right )} \tan ^{4}{\left (a + b \log{\left (c \right )} \right )} & \text{for}\: n = 0 \\\log{\left (x \right )} + \frac{\tan ^{3}{\left (a + b n \log{\left (x \right )} + b \log{\left (c \right )} \right )}}{3 b n} - \frac{\tan{\left (a + b n \log{\left (x \right )} + b \log{\left (c \right )} \right )}}{b n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((log(x)*tan(a)**4, Eq(b, 0) & (Eq(b, 0) | Eq(n, 0))), (log(x)*tan(a + b*log(c))**4, Eq(n, 0)), (log(
x) + tan(a + b*n*log(x) + b*log(c))**3/(3*b*n) - tan(a + b*n*log(x) + b*log(c))/(b*n), True))

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Giac [B]  time = 5.26057, size = 1430, normalized size = 31.78 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*tan(b*n*log(x))^2*tan(b*log(c))^6*tan(a)^4 + 6*tan(b*n*log(x))^2*tan(b*log(c))^5*tan(a)^5 + 3*tan(b*n*
log(x))*tan(b*log(c))^6*tan(a)^5 + 3*tan(b*n*log(x))^2*tan(b*log(c))^4*tan(a)^6 + 3*tan(b*n*log(x))*tan(b*log(
c))^5*tan(a)^6 + tan(b*log(c))^6*tan(a)^6 - 12*tan(b*n*log(x))^2*tan(b*log(c))^5*tan(a)^3 - 6*tan(b*n*log(x))*
tan(b*log(c))^6*tan(a)^3 - 24*tan(b*n*log(x))^2*tan(b*log(c))^4*tan(a)^4 - 33*tan(b*n*log(x))*tan(b*log(c))^5*
tan(a)^4 - 3*tan(b*log(c))^6*tan(a)^4 - 12*tan(b*n*log(x))^2*tan(b*log(c))^3*tan(a)^5 - 33*tan(b*n*log(x))*tan
(b*log(c))^4*tan(a)^5 - 12*tan(b*log(c))^5*tan(a)^5 - 6*tan(b*n*log(x))*tan(b*log(c))^3*tan(a)^6 - 3*tan(b*log
(c))^4*tan(a)^6 - 3*tan(b*n*log(x))^2*tan(b*log(c))^6 - 18*tan(b*n*log(x))^2*tan(b*log(c))^5*tan(a) - 9*tan(b*
n*log(x))*tan(b*log(c))^6*tan(a) - 27*tan(b*n*log(x))^2*tan(b*log(c))^4*tan(a)^2 - 27*tan(b*n*log(x))*tan(b*lo
g(c))^5*tan(a)^2 - 3*tan(b*log(c))^6*tan(a)^2 - 24*tan(b*n*log(x))^2*tan(b*log(c))^3*tan(a)^3 - 6*tan(b*n*log(
x))*tan(b*log(c))^4*tan(a)^3 - 27*tan(b*n*log(x))^2*tan(b*log(c))^2*tan(a)^4 - 6*tan(b*n*log(x))*tan(b*log(c))
^3*tan(a)^4 + 21*tan(b*log(c))^4*tan(a)^4 - 18*tan(b*n*log(x))^2*tan(b*log(c))*tan(a)^5 - 27*tan(b*n*log(x))*t
an(b*log(c))^2*tan(a)^5 - 3*tan(b*n*log(x))^2*tan(a)^6 - 9*tan(b*n*log(x))*tan(b*log(c))*tan(a)^6 - 3*tan(b*lo
g(c))^2*tan(a)^6 + 9*tan(b*n*log(x))*tan(b*log(c))^5 + tan(b*log(c))^6 - 12*tan(b*n*log(x))^2*tan(b*log(c))^3*
tan(a) + 27*tan(b*n*log(x))*tan(b*log(c))^4*tan(a) + 12*tan(b*log(c))^5*tan(a) - 24*tan(b*n*log(x))^2*tan(b*lo
g(c))^2*tan(a)^2 + 6*tan(b*n*log(x))*tan(b*log(c))^3*tan(a)^2 + 21*tan(b*log(c))^4*tan(a)^2 - 12*tan(b*n*log(x
))^2*tan(b*log(c))*tan(a)^3 + 6*tan(b*n*log(x))*tan(b*log(c))^2*tan(a)^3 + 27*tan(b*n*log(x))*tan(b*log(c))*ta
n(a)^4 + 21*tan(b*log(c))^2*tan(a)^4 + 9*tan(b*n*log(x))*tan(a)^5 + 12*tan(b*log(c))*tan(a)^5 + tan(a)^6 + 3*t
an(b*n*log(x))^2*tan(b*log(c))^2 + 6*tan(b*n*log(x))*tan(b*log(c))^3 - 3*tan(b*log(c))^4 + 6*tan(b*n*log(x))^2
*tan(b*log(c))*tan(a) + 33*tan(b*n*log(x))*tan(b*log(c))^2*tan(a) + 3*tan(b*n*log(x))^2*tan(a)^2 + 33*tan(b*n*
log(x))*tan(b*log(c))*tan(a)^2 + 21*tan(b*log(c))^2*tan(a)^2 + 6*tan(b*n*log(x))*tan(a)^3 - 3*tan(a)^4 - 3*tan
(b*n*log(x))*tan(b*log(c)) - 3*tan(b*log(c))^2 - 3*tan(b*n*log(x))*tan(a) - 12*tan(b*log(c))*tan(a) - 3*tan(a)
^2 + 1)/((b*n*tan(b*log(c))^3 + 3*b*n*tan(b*log(c))^2*tan(a) + 3*b*n*tan(b*log(c))*tan(a)^2 + b*n*tan(a)^3)*(t
an(b*n*log(x))*tan(b*log(c)) + tan(b*n*log(x))*tan(a) + tan(b*log(c))*tan(a) - 1)^3) + log(x)