Integrand size = 26, antiderivative size = 57 \[ \int \frac {1}{\left (a-(a-b) x^4\right ) \sqrt [4]{a+b x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}} \] Output:
1/2*arctan(a^(1/4)*x/(b*x^4+a)^(1/4))/a^(5/4)+1/2*arctanh(a^(1/4)*x/(b*x^4 +a)^(1/4))/a^(5/4)
Time = 0.60 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (a-(a-b) x^4\right ) \sqrt [4]{a+b x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}} \] Input:
Integrate[1/((a - (a - b)*x^4)*(a + b*x^4)^(1/4)),x]
Output:
(ArcTan[(a^(1/4)*x)/(a + b*x^4)^(1/4)] + ArcTanh[(a^(1/4)*x)/(a + b*x^4)^( 1/4)])/(2*a^(5/4))
Time = 0.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {902, 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.
\(\displaystyle \int \frac {1}{\left (a-x^4 (a-b)\right ) \sqrt [4]{a+b x^4}} \, dx\) |
\(\Big \downarrow \) 902 |
\(\displaystyle \int \frac {1}{a-\frac {x^4 (a b-a (b-a))}{a+b x^4}}d\frac {x}{\sqrt [4]{a+b x^4}}\) |
\(\Big \downarrow \) 756 |
\(\displaystyle \frac {\int \frac {1}{1-\frac {\sqrt {a} x^2}{\sqrt {b x^4+a}}}d\frac {x}{\sqrt [4]{b x^4+a}}}{2 a}+\frac {\int \frac {1}{\frac {\sqrt {a} x^2}{\sqrt {b x^4+a}}+1}d\frac {x}{\sqrt [4]{b x^4+a}}}{2 a}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\int \frac {1}{1-\frac {\sqrt {a} x^2}{\sqrt {b x^4+a}}}d\frac {x}{\sqrt [4]{b x^4+a}}}{2 a}+\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}}\) |
Input:
Int[1/((a - (a - b)*x^4)*(a + b*x^4)^(1/4)),x]
Output:
ArcTan[(a^(1/4)*x)/(a + b*x^4)^(1/4)]/(2*a^(5/4)) + ArcTanh[(a^(1/4)*x)/(a + b*x^4)^(1/4)]/(2*a^(5/4))
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[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Su bst[Int[1/(c - (b*c - a*d)*x^n), x], x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b , c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Time = 2.81 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.14
method | result | size |
pseudoelliptic | \(\frac {-2 \arctan \left (\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )+\ln \left (\frac {-a^{\frac {1}{4}} x -\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (b \,x^{4}+a \right )^{\frac {1}{4}}}\right )}{4 a^{\frac {5}{4}}}\) | \(65\) |
Input:
int(1/(a-(a-b)*x^4)/(b*x^4+a)^(1/4),x,method=_RETURNVERBOSE)
Output:
1/4*(-2*arctan(1/a^(1/4)/x*(b*x^4+a)^(1/4))+ln((-a^(1/4)*x-(b*x^4+a)^(1/4) )/(a^(1/4)*x-(b*x^4+a)^(1/4))))/a^(5/4)
Timed out. \[ \int \frac {1}{\left (a-(a-b) x^4\right ) \sqrt [4]{a+b x^4}} \, dx=\text {Timed out} \] Input:
integrate(1/(a-(a-b)*x^4)/(b*x^4+a)^(1/4),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{\left (a-(a-b) x^4\right ) \sqrt [4]{a+b x^4}} \, dx=- \int \frac {1}{a x^{4} \sqrt [4]{a + b x^{4}} - a \sqrt [4]{a + b x^{4}} - b x^{4} \sqrt [4]{a + b x^{4}}}\, dx \] Input:
integrate(1/(a-(a-b)*x**4)/(b*x**4+a)**(1/4),x)
Output:
-Integral(1/(a*x**4*(a + b*x**4)**(1/4) - a*(a + b*x**4)**(1/4) - b*x**4*( a + b*x**4)**(1/4)), x)
\[ \int \frac {1}{\left (a-(a-b) x^4\right ) \sqrt [4]{a+b x^4}} \, dx=\int { -\frac {1}{{\left ({\left (a - b\right )} x^{4} - a\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate(1/(a-(a-b)*x^4)/(b*x^4+a)^(1/4),x, algorithm="maxima")
Output:
-integrate(1/(((a - b)*x^4 - a)*(b*x^4 + a)^(1/4)), x)
\[ \int \frac {1}{\left (a-(a-b) x^4\right ) \sqrt [4]{a+b x^4}} \, dx=\int { -\frac {1}{{\left ({\left (a - b\right )} x^{4} - a\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate(1/(a-(a-b)*x^4)/(b*x^4+a)^(1/4),x, algorithm="giac")
Output:
integrate(-1/(((a - b)*x^4 - a)*(b*x^4 + a)^(1/4)), x)
Timed out. \[ \int \frac {1}{\left (a-(a-b) x^4\right ) \sqrt [4]{a+b x^4}} \, dx=\int \frac {1}{{\left (b\,x^4+a\right )}^{1/4}\,\left (a-x^4\,\left (a-b\right )\right )} \,d x \] Input:
int(1/((a + b*x^4)^(1/4)*(a - x^4*(a - b))),x)
Output:
int(1/((a + b*x^4)^(1/4)*(a - x^4*(a - b))), x)
\[ \int \frac {1}{\left (a-(a-b) x^4\right ) \sqrt [4]{a+b x^4}} \, dx=-\left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {1}{4}} a \,x^{4}-\left (b \,x^{4}+a \right )^{\frac {1}{4}} a -\left (b \,x^{4}+a \right )^{\frac {1}{4}} b \,x^{4}}d x \right ) \] Input:
int(1/(a-(a-b)*x^4)/(b*x^4+a)^(1/4),x)
Output:
- int(1/((a + b*x**4)**(1/4)*a*x**4 - (a + b*x**4)**(1/4)*a - (a + b*x**4 )**(1/4)*b*x**4),x)