Integrand size = 17, antiderivative size = 163 \[ \int \frac {1}{a x+b x \log ^4\left (c x^n\right )} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b} n}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b} n}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt {a}+\sqrt {b} \log ^2\left (c x^n\right )}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b} n} \] Output:
-1/4*arctan(1-2^(1/2)*b^(1/4)*ln(c*x^n)/a^(1/4))*2^(1/2)/a^(3/4)/b^(1/4)/n +1/4*arctan(1+2^(1/2)*b^(1/4)*ln(c*x^n)/a^(1/4))*2^(1/2)/a^(3/4)/b^(1/4)/n +1/4*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*ln(c*x^n)/(a^(1/2)+b^(1/2)*ln(c*x^n)^ 2))*2^(1/2)/a^(3/4)/b^(1/4)/n
Time = 0.13 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.02 \[ \int \frac {1}{a x+b x \log ^4\left (c x^n\right )} \, dx=\frac {-2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )+2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )-\log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt {b} \log ^2\left (c x^n\right )\right )+\log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt {b} \log ^2\left (c x^n\right )\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b} n} \] Input:
Integrate[(a*x + b*x*Log[c*x^n]^4)^(-1),x]
Output:
(-2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Log[c*x^n])/a^(1/4)] + 2*ArcTan[1 + (Sqrt[ 2]*b^(1/4)*Log[c*x^n])/a^(1/4)] - Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Lo g[c*x^n] + Sqrt[b]*Log[c*x^n]^2] + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*L og[c*x^n] + Sqrt[b]*Log[c*x^n]^2])/(4*Sqrt[2]*a^(3/4)*b^(1/4)*n)
Time = 0.49 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.44, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {3039, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {1}{a x+b x \log ^4\left (c x^n\right )} \, dx\) |
| \(\Big \downarrow \) 3039 |
| \(\displaystyle \frac {\int \frac {1}{b \log ^4\left (c x^n\right )+a}d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {a}-\sqrt {b} \log ^2\left (c x^n\right )}{b \log ^4\left (c x^n\right )+a}d\log \left (c x^n\right )}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} \log ^2\left (c x^n\right )+\sqrt {a}}{b \log ^4\left (c x^n\right )+a}d\log \left (c x^n\right )}{2 \sqrt {a}}}{n}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {1}{\log ^2\left (c x^n\right )-\frac {\sqrt {2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\log \left (c x^n\right )}{2 \sqrt {b}}+\frac {\int \frac {1}{\log ^2\left (c x^n\right )+\frac {\sqrt {2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\log \left (c x^n\right )}{2 \sqrt {b}}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {a}-\sqrt {b} \log ^2\left (c x^n\right )}{b \log ^4\left (c x^n\right )+a}d\log \left (c x^n\right )}{2 \sqrt {a}}}{n}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {a}-\sqrt {b} \log ^2\left (c x^n\right )}{b \log ^4\left (c x^n\right )+a}d\log \left (c x^n\right )}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}}{n}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {a}-\sqrt {b} \log ^2\left (c x^n\right )}{b \log ^4\left (c x^n\right )+a}d\log \left (c x^n\right )}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}}{n}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{b} \left (\log ^2\left (c x^n\right )-\frac {\sqrt {2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\log \left (c x^n\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (\log ^2\left (c x^n\right )+\frac {\sqrt {2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\log \left (c x^n\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}}{n}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{b} \left (\log ^2\left (c x^n\right )-\frac {\sqrt {2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\log \left (c x^n\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (\log ^2\left (c x^n\right )+\frac {\sqrt {2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\log \left (c x^n\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}}{n}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \log \left (c x^n\right )}{\log ^2\left (c x^n\right )-\frac {\sqrt {2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\log \left (c x^n\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt [4]{a}}{\log ^2\left (c x^n\right )+\frac {\sqrt {2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\log \left (c x^n\right )}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}}{n}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt {a}+\sqrt {b} \log ^2\left (c x^n\right )\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt {a}+\sqrt {b} \log ^2\left (c x^n\right )\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}}{n}\) |
Input:
Int[(a*x + b*x*Log[c*x^n]^4)^(-1),x]
Output:
((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*Log[c*x^n])/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1 /4))) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Log[c*x^n])/a^(1/4)]/(Sqrt[2]*a^(1/4)* b^(1/4)))/(2*Sqrt[a]) + (-1/2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Log[c* x^n] + Sqrt[b]*Log[c*x^n]^2]/(Sqrt[2]*a^(1/4)*b^(1/4)) + Log[Sqrt[a] + Sqr t[2]*a^(1/4)*b^(1/4)*Log[c*x^n] + Sqrt[b]*Log[c*x^n]^2]/(2*Sqrt[2]*a^(1/4) *b^(1/4)))/(2*Sqrt[a]))/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_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst [[3]] Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /; !FalseQ[lst]] /; NonsumQ[u]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.03 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.69
| method | result | size |
| risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (256 a^{3} b \,n^{4} \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (\ln \left (x^{n}\right )+4 a n \textit {\_R} +\frac {i \left (\pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-\pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right )-\pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+\pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )-2 i \ln \left (c \right )\right )}{2}\right )\) | \(113\) |
| default | \(\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {\ln \left (c \,x^{n}\right )^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \ln \left (c \,x^{n}\right ) \sqrt {2}+\sqrt {\frac {a}{b}}}{\ln \left (c \,x^{n}\right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \ln \left (c \,x^{n}\right ) \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \ln \left (c \,x^{n}\right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \ln \left (c \,x^{n}\right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )\right )}{8 n a}\) | \(136\) |
Input:
int(1/(a*x+b*x*ln(c*x^n)^4),x,method=_RETURNVERBOSE)
Output:
sum(_R*ln(ln(x^n)+4*a*n*_R+1/2*I*(Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-Pi*csgn(I *c*x^n)*csgn(I*c)*csgn(I*x^n)-Pi*csgn(I*c*x^n)^3+Pi*csgn(I*c*x^n)^2*csgn(I *c)-2*I*ln(c))),_R=RootOf(256*_Z^4*a^3*b*n^4+1))
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.98 \[ \int \frac {1}{a x+b x \log ^4\left (c x^n\right )} \, dx=\frac {1}{4} \, \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} \log \left (a n \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} + n \log \left (x\right ) + \log \left (c\right )\right ) + \frac {1}{4} i \, \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} \log \left (i \, a n \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} + n \log \left (x\right ) + \log \left (c\right )\right ) - \frac {1}{4} i \, \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} \log \left (-i \, a n \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} + n \log \left (x\right ) + \log \left (c\right )\right ) - \frac {1}{4} \, \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} \log \left (-a n \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} + n \log \left (x\right ) + \log \left (c\right )\right ) \] Input:
integrate(1/(a*x+b*x*log(c*x^n)^4),x, algorithm="fricas")
Output:
1/4*(-1/(a^3*b*n^4))^(1/4)*log(a*n*(-1/(a^3*b*n^4))^(1/4) + n*log(x) + log (c)) + 1/4*I*(-1/(a^3*b*n^4))^(1/4)*log(I*a*n*(-1/(a^3*b*n^4))^(1/4) + n*l og(x) + log(c)) - 1/4*I*(-1/(a^3*b*n^4))^(1/4)*log(-I*a*n*(-1/(a^3*b*n^4)) ^(1/4) + n*log(x) + log(c)) - 1/4*(-1/(a^3*b*n^4))^(1/4)*log(-a*n*(-1/(a^3 *b*n^4))^(1/4) + n*log(x) + log(c))
Time = 15.32 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.82 \[ \int \frac {1}{a x+b x \log ^4\left (c x^n\right )} \, dx=\begin {cases} \frac {\tilde {\infty } \log {\left (x \right )}}{\log {\left (c \right )}^{4}} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\- \frac {1}{3 b n \log {\left (c x^{n} \right )}^{3}} & \text {for}\: a = 0 \\\frac {\log {\left (x \right )}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (x \right )}}{a + b \log {\left (c \right )}^{4}} & \text {for}\: n = 0 \\- \frac {\sqrt [4]{- \frac {a}{b}} \log {\left (- \sqrt [4]{- \frac {a}{b}} + \log {\left (c x^{n} \right )} \right )}}{4 a n} + \frac {\sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt [4]{- \frac {a}{b}} + \log {\left (c x^{n} \right )} \right )}}{4 a n} + \frac {\sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\log {\left (c x^{n} \right )}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{2 a n} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a*x+b*x*ln(c*x**n)**4),x)
Output:
Piecewise((zoo*log(x)/log(c)**4, Eq(a, 0) & Eq(b, 0) & Eq(n, 0)), (-1/(3*b *n*log(c*x**n)**3), Eq(a, 0)), (log(x)/a, Eq(b, 0)), (log(x)/(a + b*log(c) **4), Eq(n, 0)), (-(-a/b)**(1/4)*log(-(-a/b)**(1/4) + log(c*x**n))/(4*a*n) + (-a/b)**(1/4)*log((-a/b)**(1/4) + log(c*x**n))/(4*a*n) + (-a/b)**(1/4)* atan(log(c*x**n)/(-a/b)**(1/4))/(2*a*n), True))
\[ \int \frac {1}{a x+b x \log ^4\left (c x^n\right )} \, dx=\int { \frac {1}{b x \log \left (c x^{n}\right )^{4} + a x} \,d x } \] Input:
integrate(1/(a*x+b*x*log(c*x^n)^4),x, algorithm="maxima")
Output:
integrate(1/(b*x*log(c*x^n)^4 + a*x), x)
Time = 0.12 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.04 \[ \int \frac {1}{a x+b x \log ^4\left (c x^n\right )} \, dx=-\frac {1}{2} \, \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} \arctan \left (\frac {\pi b {\left (\mathrm {sgn}\left (c\right ) - 1\right )} + 2 \, \left (-a b^{3}\right )^{\frac {1}{4}}}{2 \, {\left (b n \log \left (x\right ) + b \log \left ({\left | c \right |}\right )\right )}}\right ) + \frac {1}{8} \, \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {1}{4} \, {\left (\pi b n {\left (\mathrm {sgn}\left (x\right ) - 1\right )} + \pi b {\left (\mathrm {sgn}\left (c\right ) - 1\right )}\right )}^{2} + {\left (b n \log \left ({\left | x \right |}\right ) + b \log \left ({\left | c \right |}\right ) + \left (-a b^{3}\right )^{\frac {1}{4}}\right )}^{2}\right ) - \frac {1}{8} \, \left (-\frac {1}{a^{3} b n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {1}{4} \, {\left (\pi b n {\left (\mathrm {sgn}\left (x\right ) - 1\right )} + \pi b {\left (\mathrm {sgn}\left (c\right ) - 1\right )}\right )}^{2} + {\left (b n \log \left ({\left | x \right |}\right ) + b \log \left ({\left | c \right |}\right ) - \left (-a b^{3}\right )^{\frac {1}{4}}\right )}^{2}\right ) \] Input:
integrate(1/(a*x+b*x*log(c*x^n)^4),x, algorithm="giac")
Output:
-1/2*(-1/(a^3*b*n^4))^(1/4)*arctan(1/2*(pi*b*(sgn(c) - 1) + 2*(-a*b^3)^(1/ 4))/(b*n*log(x) + b*log(abs(c)))) + 1/8*(-1/(a^3*b*n^4))^(1/4)*log(1/4*(pi *b*n*(sgn(x) - 1) + pi*b*(sgn(c) - 1))^2 + (b*n*log(abs(x)) + b*log(abs(c) ) + (-a*b^3)^(1/4))^2) - 1/8*(-1/(a^3*b*n^4))^(1/4)*log(1/4*(pi*b*n*(sgn(x ) - 1) + pi*b*(sgn(c) - 1))^2 + (b*n*log(abs(x)) + b*log(abs(c)) - (-a*b^3 )^(1/4))^2)
Time = 28.22 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.58 \[ \int \frac {1}{a x+b x \log ^4\left (c x^n\right )} \, dx=-\frac {\ln \left ({\left (-a\right )}^{1/4}+b^{1/4}\,\ln \left (c\,x^n\right )\right )-\ln \left ({\left (-a\right )}^{1/4}-b^{1/4}\,\ln \left (c\,x^n\right )\right )+\ln \left ({\left (-a\right )}^{1/4}-b^{1/4}\,\ln \left (c\,x^n\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}-\ln \left ({\left (-a\right )}^{1/4}+b^{1/4}\,\ln \left (c\,x^n\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{3/4}\,b^{1/4}\,n} \] Input:
int(1/(a*x + b*x*log(c*x^n)^4),x)
Output:
-(log((-a)^(1/4) + b^(1/4)*log(c*x^n)) - log((-a)^(1/4) - b^(1/4)*log(c*x^ n)) + log((-a)^(1/4) - b^(1/4)*log(c*x^n)*1i)*1i - log((-a)^(1/4) + b^(1/4 )*log(c*x^n)*1i)*1i)/(4*(-a)^(3/4)*b^(1/4)*n)
Time = 0.16 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.89 \[ \int \frac {1}{a x+b x \log ^4\left (c x^n\right )} \, dx=\frac {\sqrt {2}\, \left (-2 \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {b}\, \mathrm {log}\left (x^{n} c \right )}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )+2 \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {b}\, \mathrm {log}\left (x^{n} c \right )}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )-\mathrm {log}\left (-b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (x^{n} c \right )+\sqrt {a}+\sqrt {b}\, \mathrm {log}\left (x^{n} c \right )^{2}\right )+\mathrm {log}\left (b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (x^{n} c \right )+\sqrt {a}+\sqrt {b}\, \mathrm {log}\left (x^{n} c \right )^{2}\right )\right )}{8 b^{\frac {1}{4}} a^{\frac {3}{4}} n} \] Input:
int(1/(a*x+b*x*log(c*x^n)^4),x)
Output:
(b**(3/4)*a**(1/4)*sqrt(2)*( - 2*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt( b)*log(x**n*c))/(b**(1/4)*a**(1/4)*sqrt(2))) + 2*atan((b**(1/4)*a**(1/4)*s qrt(2) + 2*sqrt(b)*log(x**n*c))/(b**(1/4)*a**(1/4)*sqrt(2))) - log( - b**( 1/4)*a**(1/4)*sqrt(2)*log(x**n*c) + sqrt(a) + sqrt(b)*log(x**n*c)**2) + lo g(b**(1/4)*a**(1/4)*sqrt(2)*log(x**n*c) + sqrt(a) + sqrt(b)*log(x**n*c)**2 )))/(8*a*b*n)