Integrand size = 14, antiderivative size = 72 \[ \int \frac {a+b \text {arctanh}\left (c x^n\right )}{x^4} \, dx=-\frac {a+b \text {arctanh}\left (c x^n\right )}{3 x^3}-\frac {b c n x^{-3+n} \operatorname {Hypergeometric2F1}\left (1,-\frac {3-n}{2 n},-\frac {3 (1-n)}{2 n},c^2 x^{2 n}\right )}{3 (3-n)} \] Output:
-1/3*(a+b*arctanh(c*x^n))/x^3-b*c*n*x^(-3+n)*hypergeom([1, -1/2*(3-n)/n],[ 1/2*(-3+3*n)/n],c^2*x^(2*n))/(9-3*n)
Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.01 \[ \int \frac {a+b \text {arctanh}\left (c x^n\right )}{x^4} \, dx=-\frac {a}{3 x^3}-\frac {b \text {arctanh}\left (c x^n\right )}{3 x^3}+\frac {b c n x^{-3+n} \operatorname {Hypergeometric2F1}\left (1,\frac {-3+n}{2 n},1+\frac {-3+n}{2 n},c^2 x^{2 n}\right )}{3 (-3+n)} \] Input:
Integrate[(a + b*ArcTanh[c*x^n])/x^4,x]
Output:
-1/3*a/x^3 - (b*ArcTanh[c*x^n])/(3*x^3) + (b*c*n*x^(-3 + n)*Hypergeometric 2F1[1, (-3 + n)/(2*n), 1 + (-3 + n)/(2*n), c^2*x^(2*n)])/(3*(-3 + n))
Time = 0.23 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6452, 888}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {a+b \text {arctanh}\left (c x^n\right )}{x^4} \, dx\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle \frac {1}{3} b c n \int \frac {x^{n-4}}{1-c^2 x^{2 n}}dx-\frac {a+b \text {arctanh}\left (c x^n\right )}{3 x^3}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle -\frac {a+b \text {arctanh}\left (c x^n\right )}{3 x^3}-\frac {b c n x^{n-3} \operatorname {Hypergeometric2F1}\left (1,-\frac {3-n}{2 n},-\frac {3 (1-n)}{2 n},c^2 x^{2 n}\right )}{3 (3-n)}\) |
Input:
Int[(a + b*ArcTanh[c*x^n])/x^4,x]
Output:
-1/3*(a + b*ArcTanh[c*x^n])/x^3 - (b*c*n*x^(-3 + n)*Hypergeometric2F1[1, - 1/2*(3 - n)/n, (-3*(1 - n))/(2*n), c^2*x^(2*n)])/(3*(3 - n))
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x ], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 ] && IntegerQ[m])) && NeQ[m, -1]
\[\int \frac {a +b \,\operatorname {arctanh}\left (c \,x^{n}\right )}{x^{4}}d x\]
Input:
int((a+b*arctanh(c*x^n))/x^4,x)
Output:
int((a+b*arctanh(c*x^n))/x^4,x)
\[ \int \frac {a+b \text {arctanh}\left (c x^n\right )}{x^4} \, dx=\int { \frac {b \operatorname {artanh}\left (c x^{n}\right ) + a}{x^{4}} \,d x } \] Input:
integrate((a+b*arctanh(c*x^n))/x^4,x, algorithm="fricas")
Output:
integral((b*arctanh(c*x^n) + a)/x^4, x)
\[ \int \frac {a+b \text {arctanh}\left (c x^n\right )}{x^4} \, dx=\int \frac {a + b \operatorname {atanh}{\left (c x^{n} \right )}}{x^{4}}\, dx \] Input:
integrate((a+b*atanh(c*x**n))/x**4,x)
Output:
Integral((a + b*atanh(c*x**n))/x**4, x)
\[ \int \frac {a+b \text {arctanh}\left (c x^n\right )}{x^4} \, dx=\int { \frac {b \operatorname {artanh}\left (c x^{n}\right ) + a}{x^{4}} \,d x } \] Input:
integrate((a+b*arctanh(c*x^n))/x^4,x, algorithm="maxima")
Output:
-1/6*(3*n*integrate(1/3/(c*x^4*x^n + x^4), x) + 3*n*integrate(1/3/(c*x^4*x ^n - x^4), x) + (log(c*x^n + 1) - log(-c*x^n + 1))/x^3)*b - 1/3*a/x^3
\[ \int \frac {a+b \text {arctanh}\left (c x^n\right )}{x^4} \, dx=\int { \frac {b \operatorname {artanh}\left (c x^{n}\right ) + a}{x^{4}} \,d x } \] Input:
integrate((a+b*arctanh(c*x^n))/x^4,x, algorithm="giac")
Output:
integrate((b*arctanh(c*x^n) + a)/x^4, x)
Timed out. \[ \int \frac {a+b \text {arctanh}\left (c x^n\right )}{x^4} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c\,x^n\right )}{x^4} \,d x \] Input:
int((a + b*atanh(c*x^n))/x^4,x)
Output:
int((a + b*atanh(c*x^n))/x^4, x)
\[ \int \frac {a+b \text {arctanh}\left (c x^n\right )}{x^4} \, dx=\frac {3 \left (\int \frac {\mathit {atanh} \left (x^{n} c \right )}{x^{4}}d x \right ) b \,x^{3}-a}{3 x^{3}} \] Input:
int((a+b*atanh(c*x^n))/x^4,x)
Output:
(3*int(atanh(x**n*c)/x**4,x)*b*x**3 - a)/(3*x**3)