Integrand size = 21, antiderivative size = 109 \[ \int \frac {1}{\sqrt [4]{a-b x} (2 a-b x)} \, dx=\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {a-b x}}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{a-b x}}\right )}{\sqrt [4]{a} b}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{a-b x}}{\sqrt {a}+\sqrt {a-b x}}\right )}{\sqrt [4]{a} b} \] Output:
2^(1/2)*arctan(1/2*(a^(1/2)-(-b*x+a)^(1/2))*2^(1/2)/a^(1/4)/(-b*x+a)^(1/4) )/a^(1/4)/b+2^(1/2)*arctanh(2^(1/2)*a^(1/4)*(-b*x+a)^(1/4)/(a^(1/2)+(-b*x+ a)^(1/2)))/a^(1/4)/b
Time = 0.10 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\sqrt [4]{a-b x} (2 a-b x)} \, dx=\frac {\sqrt {2} \left (\arctan \left (\frac {\sqrt {a}-\sqrt {a-b x}}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{a-b x}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{a-b x}}{\sqrt {a}+\sqrt {a-b x}}\right )\right )}{\sqrt [4]{a} b} \] Input:
Integrate[1/((a - b*x)^(1/4)*(2*a - b*x)),x]
Output:
(Sqrt[2]*(ArcTan[(Sqrt[a] - Sqrt[a - b*x])/(Sqrt[2]*a^(1/4)*(a - b*x)^(1/4 ))] + ArcTanh[(Sqrt[2]*a^(1/4)*(a - b*x)^(1/4))/(Sqrt[a] + Sqrt[a - b*x])] ))/(a^(1/4)*b)
Time = 0.31 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.77, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {73, 826, 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.
| \(\displaystyle \int \frac {1}{\sqrt [4]{a-b x} (2 a-b x)} \, dx\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {4 \int \frac {\sqrt {a-b x}}{2 a-b x}d\sqrt [4]{a-b x}}{b}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle -\frac {4 \left (\frac {1}{2} \int \frac {\sqrt {a}+\sqrt {a-b x}}{2 a-b x}d\sqrt [4]{a-b x}-\frac {1}{2} \int \frac {\sqrt {a}-\sqrt {a-b x}}{2 a-b x}d\sqrt [4]{a-b x}\right )}{b}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {4 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {a}-\sqrt {2} \sqrt [4]{a-b x} \sqrt [4]{a}+\sqrt {a-b x}}d\sqrt [4]{a-b x}+\frac {1}{2} \int \frac {1}{\sqrt {a}+\sqrt {2} \sqrt [4]{a-b x} \sqrt [4]{a}+\sqrt {a-b x}}d\sqrt [4]{a-b x}\right )-\frac {1}{2} \int \frac {\sqrt {a}-\sqrt {a-b x}}{2 a-b x}d\sqrt [4]{a-b x}\right )}{b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {4 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\sqrt {a-b x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{a-b x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a}}-\frac {\int \frac {1}{-\sqrt {a-b x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{a-b x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a}}\right )-\frac {1}{2} \int \frac {\sqrt {a}-\sqrt {a-b x}}{2 a-b x}d\sqrt [4]{a-b x}\right )}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {4 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a-b x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a-b x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a}}\right )-\frac {1}{2} \int \frac {\sqrt {a}-\sqrt {a-b x}}{2 a-b x}d\sqrt [4]{a-b x}\right )}{b}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {4 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{a-b x}}{\sqrt {a}-\sqrt {2} \sqrt [4]{a-b x} \sqrt [4]{a}+\sqrt {a-b x}}d\sqrt [4]{a-b x}}{2 \sqrt {2} \sqrt [4]{a}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a}+\sqrt {2} \sqrt [4]{a-b x}\right )}{\sqrt {a}+\sqrt {2} \sqrt [4]{a-b x} \sqrt [4]{a}+\sqrt {a-b x}}d\sqrt [4]{a-b x}}{2 \sqrt {2} \sqrt [4]{a}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a-b x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a-b x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a}}\right )\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {4 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{a-b x}}{\sqrt {a}-\sqrt {2} \sqrt [4]{a-b x} \sqrt [4]{a}+\sqrt {a-b x}}d\sqrt [4]{a-b x}}{2 \sqrt {2} \sqrt [4]{a}}-\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a}+\sqrt {2} \sqrt [4]{a-b x}\right )}{\sqrt {a}+\sqrt {2} \sqrt [4]{a-b x} \sqrt [4]{a}+\sqrt {a-b x}}d\sqrt [4]{a-b x}}{2 \sqrt {2} \sqrt [4]{a}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a-b x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a-b x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a}}\right )\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {4 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{a-b x}}{\sqrt {a}-\sqrt {2} \sqrt [4]{a-b x} \sqrt [4]{a}+\sqrt {a-b x}}d\sqrt [4]{a-b x}}{2 \sqrt {2} \sqrt [4]{a}}-\frac {\int \frac {\sqrt [4]{a}+\sqrt {2} \sqrt [4]{a-b x}}{\sqrt {a}+\sqrt {2} \sqrt [4]{a-b x} \sqrt [4]{a}+\sqrt {a-b x}}d\sqrt [4]{a-b x}}{2 \sqrt [4]{a}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a-b x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a-b x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a}}\right )\right )}{b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {4 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a-b x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a-b x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{a-b x}+\sqrt {a-b x}+\sqrt {a}\right )}{2 \sqrt {2} \sqrt [4]{a}}-\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{a-b x}+\sqrt {a-b x}+\sqrt {a}\right )}{2 \sqrt {2} \sqrt [4]{a}}\right )\right )}{b}\) |
Input:
Int[1/((a - b*x)^(1/4)*(2*a - b*x)),x]
Output:
(-4*((-(ArcTan[1 - (Sqrt[2]*(a - b*x)^(1/4))/a^(1/4)]/(Sqrt[2]*a^(1/4))) + ArcTan[1 + (Sqrt[2]*(a - b*x)^(1/4))/a^(1/4)]/(Sqrt[2]*a^(1/4)))/2 + (Log [Sqrt[a] - Sqrt[2]*a^(1/4)*(a - b*x)^(1/4) + Sqrt[a - b*x]]/(2*Sqrt[2]*a^( 1/4)) - Log[Sqrt[a] + Sqrt[2]*a^(1/4)*(a - b*x)^(1/4) + Sqrt[a - b*x]]/(2* Sqrt[2]*a^(1/4)))/2))/b
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, 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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) 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]
Time = 1.67 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.04
| method | result | size |
| derivativedivides | \(-\frac {\sqrt {2}\, \left (\ln \left (\frac {\sqrt {-b x +a}-a^{\frac {1}{4}} \left (-b x +a \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {a}}{\sqrt {-b x +a}+a^{\frac {1}{4}} \left (-b x +a \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {a}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (-b x +a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \left (-b x +a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}+1\right )\right )}{2 b \,a^{\frac {1}{4}}}\) | \(113\) |
| default | \(-\frac {\sqrt {2}\, \left (\ln \left (\frac {\sqrt {-b x +a}-a^{\frac {1}{4}} \left (-b x +a \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {a}}{\sqrt {-b x +a}+a^{\frac {1}{4}} \left (-b x +a \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {a}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (-b x +a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \left (-b x +a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}+1\right )\right )}{2 b \,a^{\frac {1}{4}}}\) | \(113\) |
| pseudoelliptic | \(-\frac {\sqrt {2}\, \left (\ln \left (\frac {\sqrt {-b x +a}-a^{\frac {1}{4}} \left (-b x +a \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {a}}{\sqrt {-b x +a}+a^{\frac {1}{4}} \left (-b x +a \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {a}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (-b x +a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \left (-b x +a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}+1\right )\right )}{2 b \,a^{\frac {1}{4}}}\) | \(113\) |
Input:
int(1/(-b*x+a)^(1/4)/(-b*x+2*a),x,method=_RETURNVERBOSE)
Output:
-1/2/b/a^(1/4)*2^(1/2)*(ln(((-b*x+a)^(1/2)-a^(1/4)*(-b*x+a)^(1/4)*2^(1/2)+ a^(1/2))/((-b*x+a)^(1/2)+a^(1/4)*(-b*x+a)^(1/4)*2^(1/2)+a^(1/2)))+2*arctan (2^(1/2)/a^(1/4)*(-b*x+a)^(1/4)+1)-2*arctan(-2^(1/2)/a^(1/4)*(-b*x+a)^(1/4 )+1))
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.39 \[ \int \frac {1}{\sqrt [4]{a-b x} (2 a-b x)} \, dx=-\left (-\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (a b^{3} \left (-\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + {\left (-b x + a\right )}^{\frac {1}{4}}\right ) + i \, \left (-\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (i \, a b^{3} \left (-\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + {\left (-b x + a\right )}^{\frac {1}{4}}\right ) - i \, \left (-\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (-i \, a b^{3} \left (-\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + {\left (-b x + a\right )}^{\frac {1}{4}}\right ) + \left (-\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (-a b^{3} \left (-\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + {\left (-b x + a\right )}^{\frac {1}{4}}\right ) \] Input:
integrate(1/(-b*x+a)^(1/4)/(-b*x+2*a),x, algorithm="fricas")
Output:
-(-1/(a*b^4))^(1/4)*log(a*b^3*(-1/(a*b^4))^(3/4) + (-b*x + a)^(1/4)) + I*( -1/(a*b^4))^(1/4)*log(I*a*b^3*(-1/(a*b^4))^(3/4) + (-b*x + a)^(1/4)) - I*( -1/(a*b^4))^(1/4)*log(-I*a*b^3*(-1/(a*b^4))^(3/4) + (-b*x + a)^(1/4)) + (- 1/(a*b^4))^(1/4)*log(-a*b^3*(-1/(a*b^4))^(3/4) + (-b*x + a)^(1/4))
\[ \int \frac {1}{\sqrt [4]{a-b x} (2 a-b x)} \, dx=- \int \frac {1}{- 2 a \sqrt [4]{a - b x} + b x \sqrt [4]{a - b x}}\, dx \] Input:
integrate(1/(-b*x+a)**(1/4)/(-b*x+2*a),x)
Output:
-Integral(1/(-2*a*(a - b*x)**(1/4) + b*x*(a - b*x)**(1/4)), x)
Time = 0.12 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.36 \[ \int \frac {1}{\sqrt [4]{a-b x} (2 a-b x)} \, dx=-\frac {\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} + 2 \, {\left (-b x + a\right )}^{\frac {1}{4}}\right )}}{2 \, a^{\frac {1}{4}}}\right )}{a^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} - 2 \, {\left (-b x + a\right )}^{\frac {1}{4}}\right )}}{2 \, a^{\frac {1}{4}}}\right )}{a^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {2} {\left (-b x + a\right )}^{\frac {1}{4}} a^{\frac {1}{4}} + \sqrt {-b x + a} + \sqrt {a}\right )}{a^{\frac {1}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (-b x + a\right )}^{\frac {1}{4}} a^{\frac {1}{4}} + \sqrt {-b x + a} + \sqrt {a}\right )}{a^{\frac {1}{4}}}}{2 \, b} \] Input:
integrate(1/(-b*x+a)^(1/4)/(-b*x+2*a),x, algorithm="maxima")
Output:
-1/2*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4) + 2*(-b*x + a)^(1/4))/ a^(1/4))/a^(1/4) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4) - 2*(-b* x + a)^(1/4))/a^(1/4))/a^(1/4) - sqrt(2)*log(sqrt(2)*(-b*x + a)^(1/4)*a^(1 /4) + sqrt(-b*x + a) + sqrt(a))/a^(1/4) + sqrt(2)*log(-sqrt(2)*(-b*x + a)^ (1/4)*a^(1/4) + sqrt(-b*x + a) + sqrt(a))/a^(1/4))/b
Time = 0.13 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.43 \[ \int \frac {1}{\sqrt [4]{a-b x} (2 a-b x)} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} + 2 \, {\left (-b x + a\right )}^{\frac {1}{4}}\right )}}{2 \, a^{\frac {1}{4}}}\right )}{a^{\frac {1}{4}} b} - \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} - 2 \, {\left (-b x + a\right )}^{\frac {1}{4}}\right )}}{2 \, a^{\frac {1}{4}}}\right )}{a^{\frac {1}{4}} b} + \frac {\sqrt {2} \log \left (\sqrt {2} {\left (-b x + a\right )}^{\frac {1}{4}} a^{\frac {1}{4}} + \sqrt {-b x + a} + \sqrt {a}\right )}{2 \, a^{\frac {1}{4}} b} - \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (-b x + a\right )}^{\frac {1}{4}} a^{\frac {1}{4}} + \sqrt {-b x + a} + \sqrt {a}\right )}{2 \, a^{\frac {1}{4}} b} \] Input:
integrate(1/(-b*x+a)^(1/4)/(-b*x+2*a),x, algorithm="giac")
Output:
-sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4) + 2*(-b*x + a)^(1/4))/a^(1/4) )/(a^(1/4)*b) - sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4) - 2*(-b*x + a )^(1/4))/a^(1/4))/(a^(1/4)*b) + 1/2*sqrt(2)*log(sqrt(2)*(-b*x + a)^(1/4)*a ^(1/4) + sqrt(-b*x + a) + sqrt(a))/(a^(1/4)*b) - 1/2*sqrt(2)*log(-sqrt(2)* (-b*x + a)^(1/4)*a^(1/4) + sqrt(-b*x + a) + sqrt(a))/(a^(1/4)*b)
Time = 0.16 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.47 \[ \int \frac {1}{\sqrt [4]{a-b x} (2 a-b x)} \, dx=\frac {2\,\mathrm {atanh}\left (\frac {{\left (a-b\,x\right )}^{1/4}}{{\left (-a\right )}^{1/4}}\right )}{{\left (-a\right )}^{1/4}\,b}-\frac {2\,\mathrm {atan}\left (\frac {{\left (a-b\,x\right )}^{1/4}}{{\left (-a\right )}^{1/4}}\right )}{{\left (-a\right )}^{1/4}\,b} \] Input:
int(1/((2*a - b*x)*(a - b*x)^(1/4)),x)
Output:
(2*atanh((a - b*x)^(1/4)/(-a)^(1/4)))/((-a)^(1/4)*b) - (2*atan((a - b*x)^( 1/4)/(-a)^(1/4)))/((-a)^(1/4)*b)
Time = 0.16 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\sqrt [4]{a-b x} (2 a-b x)} \, dx=\frac {\sqrt {2}\, \left (-2 \mathit {atan} \left (\frac {2 \left (-b x +a \right )^{\frac {1}{4}}-a^{\frac {1}{4}} \sqrt {2}}{a^{\frac {1}{4}} \sqrt {2}}\right )-2 \mathit {atan} \left (\frac {2 \left (-b x +a \right )^{\frac {1}{4}}+a^{\frac {1}{4}} \sqrt {2}}{a^{\frac {1}{4}} \sqrt {2}}\right )-\mathrm {log}\left (-a^{\frac {1}{4}} \left (-b x +a \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {-b x +a}+\sqrt {a}\right )+\mathrm {log}\left (a^{\frac {1}{4}} \left (-b x +a \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {-b x +a}+\sqrt {a}\right )\right )}{2 a^{\frac {1}{4}} b} \] Input:
int(1/(-b*x+a)^(1/4)/(-b*x+2*a),x)
Output:
(a**(3/4)*sqrt(2)*( - 2*atan((2*(a - b*x)**(1/4) - a**(1/4)*sqrt(2))/(a**( 1/4)*sqrt(2))) - 2*atan((2*(a - b*x)**(1/4) + a**(1/4)*sqrt(2))/(a**(1/4)* sqrt(2))) - log( - a**(1/4)*(a - b*x)**(1/4)*sqrt(2) + sqrt(a - b*x) + sqr t(a)) + log(a**(1/4)*(a - b*x)**(1/4)*sqrt(2) + sqrt(a - b*x) + sqrt(a)))) /(2*a*b)