Integrand size = 16, antiderivative size = 69 \[ \int \frac {\sqrt [4]{a-b x^4}}{x} \, dx=\sqrt [4]{a-b x^4}-\frac {1}{2} \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )-\frac {1}{2} \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right ) \] Output:
(-b*x^4+a)^(1/4)-1/2*a^(1/4)*arctan((-b*x^4+a)^(1/4)/a^(1/4))-1/2*a^(1/4)* arctanh((-b*x^4+a)^(1/4)/a^(1/4))
Time = 0.06 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{a-b x^4}}{x} \, dx=\sqrt [4]{a-b x^4}-\frac {1}{2} \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )-\frac {1}{2} \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right ) \] Input:
Integrate[(a - b*x^4)^(1/4)/x,x]
Output:
(a - b*x^4)^(1/4) - (a^(1/4)*ArcTan[(a - b*x^4)^(1/4)/a^(1/4)])/2 - (a^(1/ 4)*ArcTanh[(a - b*x^4)^(1/4)/a^(1/4)])/2
Time = 0.19 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.22, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {798, 60, 73, 756, 216, 219}
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.
\(\displaystyle \int \frac {\sqrt [4]{a-b x^4}}{x} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{4} \int \frac {\sqrt [4]{a-b x^4}}{x^4}dx^4\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{4} \left (a \int \frac {1}{x^4 \left (a-b x^4\right )^{3/4}}dx^4+4 \sqrt [4]{a-b x^4}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{4} \left (4 \sqrt [4]{a-b x^4}-\frac {4 a \int \frac {1}{\frac {a}{b}-\frac {x^{16}}{b}}d\sqrt [4]{a-b x^4}}{b}\right )\) |
\(\Big \downarrow \) 756 |
\(\displaystyle \frac {1}{4} \left (4 \sqrt [4]{a-b x^4}-\frac {4 a \left (\frac {b \int \frac {1}{\sqrt {a}-x^8}d\sqrt [4]{a-b x^4}}{2 \sqrt {a}}+\frac {b \int \frac {1}{x^8+\sqrt {a}}d\sqrt [4]{a-b x^4}}{2 \sqrt {a}}\right )}{b}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{4} \left (4 \sqrt [4]{a-b x^4}-\frac {4 a \left (\frac {b \int \frac {1}{\sqrt {a}-x^8}d\sqrt [4]{a-b x^4}}{2 \sqrt {a}}+\frac {b \arctan \left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 a^{3/4}}\right )}{b}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{4} \left (4 \sqrt [4]{a-b x^4}-\frac {4 a \left (\frac {b \arctan \left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 a^{3/4}}+\frac {b \text {arctanh}\left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 a^{3/4}}\right )}{b}\right )\) |
Input:
Int[(a - b*x^4)^(1/4)/x,x]
Output:
(4*(a - b*x^4)^(1/4) - (4*a*((b*ArcTan[(a - b*x^4)^(1/4)/a^(1/4)])/(2*a^(3 /4)) + (b*ArcTanh[(a - b*x^4)^(1/4)/a^(1/4)])/(2*a^(3/4))))/b)/4
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, 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[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[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[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
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]]
Time = 1.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.03
method | result | size |
pseudoelliptic | \(\left (-b \,x^{4}+a \right )^{\frac {1}{4}}-\frac {\ln \left (\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}}+a^{\frac {1}{4}}}{\left (-b \,x^{4}+a \right )^{\frac {1}{4}}-a^{\frac {1}{4}}}\right ) a^{\frac {1}{4}}}{4}-\frac {a^{\frac {1}{4}} \arctan \left (\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{2}\) | \(71\) |
Input:
int((-b*x^4+a)^(1/4)/x,x,method=_RETURNVERBOSE)
Output:
(-b*x^4+a)^(1/4)-1/4*ln(((-b*x^4+a)^(1/4)+a^(1/4))/((-b*x^4+a)^(1/4)-a^(1/ 4)))*a^(1/4)-1/2*a^(1/4)*arctan((-b*x^4+a)^(1/4)/a^(1/4))
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt [4]{a-b x^4}}{x} \, dx=-\frac {1}{4} \, a^{\frac {1}{4}} \log \left ({\left (-b x^{4} + a\right )}^{\frac {1}{4}} + a^{\frac {1}{4}}\right ) - \frac {1}{4} i \, a^{\frac {1}{4}} \log \left ({\left (-b x^{4} + a\right )}^{\frac {1}{4}} + i \, a^{\frac {1}{4}}\right ) + \frac {1}{4} i \, a^{\frac {1}{4}} \log \left ({\left (-b x^{4} + a\right )}^{\frac {1}{4}} - i \, a^{\frac {1}{4}}\right ) + \frac {1}{4} \, a^{\frac {1}{4}} \log \left ({\left (-b x^{4} + a\right )}^{\frac {1}{4}} - a^{\frac {1}{4}}\right ) + {\left (-b x^{4} + a\right )}^{\frac {1}{4}} \] Input:
integrate((-b*x^4+a)^(1/4)/x,x, algorithm="fricas")
Output:
-1/4*a^(1/4)*log((-b*x^4 + a)^(1/4) + a^(1/4)) - 1/4*I*a^(1/4)*log((-b*x^4 + a)^(1/4) + I*a^(1/4)) + 1/4*I*a^(1/4)*log((-b*x^4 + a)^(1/4) - I*a^(1/4 )) + 1/4*a^(1/4)*log((-b*x^4 + a)^(1/4) - a^(1/4)) + (-b*x^4 + a)^(1/4)
Result contains complex when optimal does not.
Time = 0.68 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt [4]{a-b x^4}}{x} \, dx=- \frac {\sqrt [4]{b} x e^{\frac {i \pi }{4}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {a}{b x^{4}}} \right )}}{4 \Gamma \left (\frac {3}{4}\right )} \] Input:
integrate((-b*x**4+a)**(1/4)/x,x)
Output:
-b**(1/4)*x*exp(I*pi/4)*gamma(-1/4)*hyper((-1/4, -1/4), (3/4,), a/(b*x**4) )/(4*gamma(3/4))
Time = 0.10 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt [4]{a-b x^4}}{x} \, dx=-\frac {1}{2} \, a^{\frac {1}{4}} \arctan \left (\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right ) + \frac {1}{4} \, a^{\frac {1}{4}} \log \left (\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}} - a^{\frac {1}{4}}}{{\left (-b x^{4} + a\right )}^{\frac {1}{4}} + a^{\frac {1}{4}}}\right ) + {\left (-b x^{4} + a\right )}^{\frac {1}{4}} \] Input:
integrate((-b*x^4+a)^(1/4)/x,x, algorithm="maxima")
Output:
-1/2*a^(1/4)*arctan((-b*x^4 + a)^(1/4)/a^(1/4)) + 1/4*a^(1/4)*log(((-b*x^4 + a)^(1/4) - a^(1/4))/((-b*x^4 + a)^(1/4) + a^(1/4))) + (-b*x^4 + a)^(1/4 )
Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (51) = 102\).
Time = 0.13 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.75 \[ \int \frac {\sqrt [4]{a-b x^4}}{x} \, dx=-\frac {1}{4} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) - \frac {1}{4} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) - \frac {1}{8} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} \log \left (\sqrt {2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {-b x^{4} + a} + \sqrt {-a}\right ) + \frac {1}{8} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} \log \left (-\sqrt {2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {-b x^{4} + a} + \sqrt {-a}\right ) + {\left (-b x^{4} + a\right )}^{\frac {1}{4}} \] Input:
integrate((-b*x^4+a)^(1/4)/x,x, algorithm="giac")
Output:
-1/4*sqrt(2)*(-a)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(-b*x^4 + a)^(1/4))/(-a)^(1/4)) - 1/4*sqrt(2)*(-a)^(1/4)*arctan(-1/2*sqrt(2)*(sqr t(2)*(-a)^(1/4) - 2*(-b*x^4 + a)^(1/4))/(-a)^(1/4)) - 1/8*sqrt(2)*(-a)^(1/ 4)*log(sqrt(2)*(-b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(-b*x^4 + a) + sqrt(-a) ) + 1/8*sqrt(2)*(-a)^(1/4)*log(-sqrt(2)*(-b*x^4 + a)^(1/4)*(-a)^(1/4) + sq rt(-b*x^4 + a) + sqrt(-a)) + (-b*x^4 + a)^(1/4)
Time = 0.33 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt [4]{a-b x^4}}{x} \, dx={\left (a-b\,x^4\right )}^{1/4}-\frac {a^{1/4}\,\mathrm {atanh}\left (\frac {{\left (a-b\,x^4\right )}^{1/4}}{a^{1/4}}\right )}{2}-\frac {a^{1/4}\,\mathrm {atan}\left (\frac {{\left (a-b\,x^4\right )}^{1/4}}{a^{1/4}}\right )}{2} \] Input:
int((a - b*x^4)^(1/4)/x,x)
Output:
(a - b*x^4)^(1/4) - (a^(1/4)*atanh((a - b*x^4)^(1/4)/a^(1/4)))/2 - (a^(1/4 )*atan((a - b*x^4)^(1/4)/a^(1/4)))/2
\[ \int \frac {\sqrt [4]{a-b x^4}}{x} \, dx=\left (-b \,x^{4}+a \right )^{\frac {1}{4}}+\left (\int \frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{-b \,x^{5}+a x}d x \right ) a \] Input:
int((-b*x^4+a)^(1/4)/x,x)
Output:
(a - b*x**4)**(1/4) + int((a - b*x**4)**(1/4)/(a*x - b*x**5),x)*a