Integrand size = 21, antiderivative size = 37 \[ \int \frac {1}{x \sqrt {a+b \left (c (d x)^m\right )^n}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \left (c (d x)^m\right )^n}}{\sqrt {a}}\right )}{\sqrt {a} m n} \] Output:
-2*arctanh((a+b*(c*(d*x)^m)^n)^(1/2)/a^(1/2))/a^(1/2)/m/n
Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {a+b \left (c (d x)^m\right )^n}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \left (c (d x)^m\right )^n}}{\sqrt {a}}\right )}{\sqrt {a} m n} \] Input:
Integrate[1/(x*Sqrt[a + b*(c*(d*x)^m)^n]),x]
Output:
(-2*ArcTanh[Sqrt[a + b*(c*(d*x)^m)^n]/Sqrt[a]])/(Sqrt[a]*m*n)
Time = 0.54 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {7282, 891, 27, 798, 73, 221}
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.
\(\displaystyle \int \frac {1}{x \sqrt {a+b \left (c (d x)^m\right )^n}} \, dx\) |
\(\Big \downarrow \) 7282 |
\(\displaystyle \frac {\int \frac {(d x)^{-m}}{\sqrt {b \left (c (d x)^m\right )^n+a}}d(d x)^m}{m}\) |
\(\Big \downarrow \) 891 |
\(\displaystyle \frac {\int \frac {c (d x)^{-m}}{\sqrt {b \left (c (d x)^m\right )^n+a}}d\left (c (d x)^m\right )}{c m}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {(d x)^{-m}}{\sqrt {b \left (c (d x)^m\right )^n+a}}d\left (c (d x)^m\right )}{m}\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\int \frac {(d x)^{-m}}{\sqrt {b \left (c (d x)^m\right )^n+a}}d\left (c (d x)^m\right )^n}{m n}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {2 \int \frac {1}{\frac {(d x)^{2 m}}{b}-\frac {a}{b}}d\sqrt {b \left (c (d x)^m\right )^n+a}}{b m n}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \left (c (d x)^m\right )^n}}{\sqrt {a}}\right )}{\sqrt {a} m n}\) |
Input:
Int[1/(x*Sqrt[a + b*(c*(d*x)^m)^n]),x]
Output:
(-2*ArcTanh[Sqrt[a + b*(c*(d*x)^m)^n]/Sqrt[a]])/(Sqrt[a]*m*n)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_)*(x_))^(n_))^(p_.), x_Symbol] :> Simp[1/c Subst[Int[(d*(x/c))^m*(a + b*x^n)^p, x], x, c*x], x] /; FreeQ[{a , b, c, d, m, n, p}, x]
Int[(u_)/(x_), x_Symbol] :> With[{lst = PowerVariableExpn[u, 0, x]}, Simp[1 /lst[[2]] Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, ( lst[[3]]*x)^lst[[2]]], x] /; !FalseQ[lst] && NeQ[lst[[2]], 0]] /; NonsumQ[ u] && !RationalFunctionQ[u, x]
Time = 0.15 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \left (c \left (d x \right )^{m}\right )^{n}}}{\sqrt {a}}\right )}{\sqrt {a}\, m n}\) | \(32\) |
default | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \left (c \left (d x \right )^{m}\right )^{n}}}{\sqrt {a}}\right )}{\sqrt {a}\, m n}\) | \(32\) |
Input:
int(1/x/(a+b*(c*(d*x)^m)^n)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2*arctanh((a+b*(c*(d*x)^m)^n)^(1/2)/a^(1/2))/a^(1/2)/m/n
Time = 0.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 3.05 \[ \int \frac {1}{x \sqrt {a+b \left (c (d x)^m\right )^n}} \, dx=\left [\frac {\log \left ({\left (b e^{\left (m n \log \left (d x\right ) + n \log \left (c\right )\right )} - 2 \, \sqrt {b e^{\left (m n \log \left (d x\right ) + n \log \left (c\right )\right )} + a} \sqrt {a} + 2 \, a\right )} e^{\left (-m n \log \left (d x\right ) - n \log \left (c\right )\right )}\right )}{\sqrt {a} m n}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b e^{\left (m n \log \left (d x\right ) + n \log \left (c\right )\right )} + a}}\right )}{a m n}\right ] \] Input:
integrate(1/x/(a+b*(c*(d*x)^m)^n)^(1/2),x, algorithm="fricas")
Output:
[log((b*e^(m*n*log(d*x) + n*log(c)) - 2*sqrt(b*e^(m*n*log(d*x) + n*log(c)) + a)*sqrt(a) + 2*a)*e^(-m*n*log(d*x) - n*log(c)))/(sqrt(a)*m*n), 2*sqrt(- a)*arctan(sqrt(-a)/sqrt(b*e^(m*n*log(d*x) + n*log(c)) + a))/(a*m*n)]
\[ \int \frac {1}{x \sqrt {a+b \left (c (d x)^m\right )^n}} \, dx=\int \frac {1}{x \sqrt {a + b \left (c \left (d x\right )^{m}\right )^{n}}}\, dx \] Input:
integrate(1/x/(a+b*(c*(d*x)**m)**n)**(1/2),x)
Output:
Integral(1/(x*sqrt(a + b*(c*(d*x)**m)**n)), x)
\[ \int \frac {1}{x \sqrt {a+b \left (c (d x)^m\right )^n}} \, dx=\int { \frac {1}{\sqrt {\left (\left (d x\right )^{m} c\right )^{n} b + a} x} \,d x } \] Input:
integrate(1/x/(a+b*(c*(d*x)^m)^n)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(((d*x)^m*c)^n*b + a)*x), x)
\[ \int \frac {1}{x \sqrt {a+b \left (c (d x)^m\right )^n}} \, dx=\int { \frac {1}{\sqrt {\left (\left (d x\right )^{m} c\right )^{n} b + a} x} \,d x } \] Input:
integrate(1/x/(a+b*(c*(d*x)^m)^n)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(((d*x)^m*c)^n*b + a)*x), x)
Timed out. \[ \int \frac {1}{x \sqrt {a+b \left (c (d x)^m\right )^n}} \, dx=\int \frac {1}{x\,\sqrt {a+b\,{\left (c\,{\left (d\,x\right )}^m\right )}^n}} \,d x \] Input:
int(1/(x*(a + b*(c*(d*x)^m)^n)^(1/2)),x)
Output:
int(1/(x*(a + b*(c*(d*x)^m)^n)^(1/2)), x)
\[ \int \frac {1}{x \sqrt {a+b \left (c (d x)^m\right )^n}} \, dx=\int \frac {\sqrt {x^{m n} d^{m n} c^{n} b +a}}{x^{m n} d^{m n} c^{n} b x +a x}d x \] Input:
int(1/x/(a+b*(c*(d*x)^m)^n)^(1/2),x)
Output:
int(sqrt(x**(m*n)*d**(m*n)*c**n*b + a)/(x**(m*n)*d**(m*n)*c**n*b*x + a*x), x)