Integrand size = 29, antiderivative size = 51 \[ \int \frac {1}{x \sqrt {a+b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q}}{\sqrt {a}}\right )}{\sqrt {a} m n p q} \]
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {a+b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q}}{\sqrt {a}}\right )}{\sqrt {a} m n p q} \]
Time = 0.81 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {7282, 7282, 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.
\(\displaystyle \int \frac {1}{x \sqrt {a+b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q}} \, dx\) |
\(\Big \downarrow \) 7282 |
\(\displaystyle \frac {\int \frac {(f x)^{-m}}{\sqrt {b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q+a}}d(f x)^m}{m}\) |
\(\Big \downarrow \) 7282 |
\(\displaystyle \frac {\int \frac {(f x)^{-m}}{\sqrt {b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q+a}}d\left (e (f x)^m\right )^n}{m n}\) |
\(\Big \downarrow \) 7282 |
\(\displaystyle \frac {\int \frac {(f x)^{-m}}{\sqrt {b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q+a}}d\left (d \left (e (f x)^m\right )^n\right )^p}{m n p}\) |
\(\Big \downarrow \) 891 |
\(\displaystyle \frac {\int \frac {c (f x)^{-m}}{\sqrt {b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q+a}}d\left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )}{c m n p}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {(f x)^{-m}}{\sqrt {b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q+a}}d\left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )}{m n p}\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\int \frac {(f x)^{-m}}{\sqrt {b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q+a}}d\left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q}{m n p q}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {2 \int \frac {1}{\frac {(f x)^{2 m}}{b}-\frac {a}{b}}d\sqrt {b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q+a}}{b m n p q}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q}}{\sqrt {a}}\right )}{\sqrt {a} m n p q}\) |
3.7.74.3.1 Defintions of rubi rules used
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 = 1.20 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b {\left (c {\left (d \left (e \left (f x \right )^{m}\right )^{n}\right )}^{p}\right )}^{q}}}{\sqrt {a}}\right )}{m n p q \sqrt {a}}\) | \(46\) |
default | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b {\left (c {\left (d \left (e \left (f x \right )^{m}\right )^{n}\right )}^{p}\right )}^{q}}}{\sqrt {a}}\right )}{m n p q \sqrt {a}}\) | \(46\) |
Time = 0.30 (sec) , antiderivative size = 182, normalized size of antiderivative = 3.57 \[ \int \frac {1}{x \sqrt {a+b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q}} \, dx=\left [\frac {\log \left ({\left (b e^{\left (m n p q \log \left (f x\right ) + n p q \log \left (e\right ) + p q \log \left (d\right ) + q \log \left (c\right )\right )} - 2 \, \sqrt {b e^{\left (m n p q \log \left (f x\right ) + n p q \log \left (e\right ) + p q \log \left (d\right ) + q \log \left (c\right )\right )} + a} \sqrt {a} + 2 \, a\right )} e^{\left (-m n p q \log \left (f x\right ) - n p q \log \left (e\right ) - p q \log \left (d\right ) - q \log \left (c\right )\right )}\right )}{\sqrt {a} m n p q}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {b e^{\left (m n p q \log \left (f x\right ) + n p q \log \left (e\right ) + p q \log \left (d\right ) + q \log \left (c\right )\right )} + a} \sqrt {-a}}{a}\right )}{a m n p q}\right ] \]
[log((b*e^(m*n*p*q*log(f*x) + n*p*q*log(e) + p*q*log(d) + q*log(c)) - 2*sq rt(b*e^(m*n*p*q*log(f*x) + n*p*q*log(e) + p*q*log(d) + q*log(c)) + a)*sqrt (a) + 2*a)*e^(-m*n*p*q*log(f*x) - n*p*q*log(e) - p*q*log(d) - q*log(c)))/( sqrt(a)*m*n*p*q), 2*sqrt(-a)*arctan(sqrt(b*e^(m*n*p*q*log(f*x) + n*p*q*log (e) + p*q*log(d) + q*log(c)) + a)*sqrt(-a)/a)/(a*m*n*p*q)]
\[ \int \frac {1}{x \sqrt {a+b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q}} \, dx=\int \frac {1}{x \sqrt {a + b \left (c \left (d \left (e \left (f x\right )^{m}\right )^{n}\right )^{p}\right )^{q}}}\, dx \]
\[ \int \frac {1}{x \sqrt {a+b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q}} \, dx=\int { \frac {1}{\sqrt {\left (\left (\left (\left (f x\right )^{m} e\right )^{n} d\right )^{p} c\right )^{q} b + a} x} \,d x } \]
\[ \int \frac {1}{x \sqrt {a+b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q}} \, dx=\int { \frac {1}{\sqrt {\left (\left (\left (\left (f x\right )^{m} e\right )^{n} d\right )^{p} c\right )^{q} b + a} x} \,d x } \]
Timed out. \[ \int \frac {1}{x \sqrt {a+b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q}} \, dx=\int \frac {1}{x\,\sqrt {a+b\,{\left (c\,{\left (d\,{\left (e\,{\left (f\,x\right )}^m\right )}^n\right )}^p\right )}^q}} \,d x \]