Integrand size = 33, antiderivative size = 40 \[ \int \frac {1}{\left (a+b x^4\right ) \sqrt {c x^2+d \sqrt {a+b x^4}}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {c x^2+d \sqrt {a+b x^4}}}\right )}{a \sqrt {c}} \] Output:
arctanh(c^(1/2)*x/(c*x^2+d*(b*x^4+a)^(1/2))^(1/2))/a/c^(1/2)
Time = 1.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (a+b x^4\right ) \sqrt {c x^2+d \sqrt {a+b x^4}}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {c x^2+d \sqrt {a+b x^4}}}{\sqrt {c} x}\right )}{a \sqrt {c}} \] Input:
Integrate[1/((a + b*x^4)*Sqrt[c*x^2 + d*Sqrt[a + b*x^4]]),x]
Output:
ArcTanh[Sqrt[c*x^2 + d*Sqrt[a + b*x^4]]/(Sqrt[c]*x)]/(a*Sqrt[c])
Time = 0.43 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2553, 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.
\(\displaystyle \int \frac {1}{\left (a+b x^4\right ) \sqrt {d \sqrt {a+b x^4}+c x^2}} \, dx\) |
\(\Big \downarrow \) 2553 |
\(\displaystyle \frac {\int \frac {1}{1-\frac {c x^2}{c x^2+d \sqrt {b x^4+a}}}d\frac {x}{\sqrt {c x^2+d \sqrt {b x^4+a}}}}{a}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {d \sqrt {a+b x^4}+c x^2}}\right )}{a \sqrt {c}}\) |
Input:
Int[1/((a + b*x^4)*Sqrt[c*x^2 + d*Sqrt[a + b*x^4]]),x]
Output:
ArcTanh[(Sqrt[c]*x)/Sqrt[c*x^2 + d*Sqrt[a + b*x^4]]]/(a*Sqrt[c])
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[1/(((a_) + (b_.)*(x_)^(n_.))*Sqrt[(c_.)*(x_)^2 + (d_.)*((a_) + (b_.)*(x _)^(n_.))^(p_.)]), x_Symbol] :> Simp[1/a Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[c*x^2 + d*(a + b*x^n)^(2/n)]], x] /; FreeQ[{a, b, c, d, n}, x] && Eq Q[p, 2/n]
\[\int \frac {1}{\left (b \,x^{4}+a \right ) \sqrt {c \,x^{2}+d \sqrt {b \,x^{4}+a}}}d x\]
Input:
int(1/(b*x^4+a)/(c*x^2+d*(b*x^4+a)^(1/2))^(1/2),x)
Output:
int(1/(b*x^4+a)/(c*x^2+d*(b*x^4+a)^(1/2))^(1/2),x)
Timed out. \[ \int \frac {1}{\left (a+b x^4\right ) \sqrt {c x^2+d \sqrt {a+b x^4}}} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x^4+a)/(c*x^2+d*(b*x^4+a)^(1/2))^(1/2),x, algorithm="fricas ")
Output:
Timed out
\[ \int \frac {1}{\left (a+b x^4\right ) \sqrt {c x^2+d \sqrt {a+b x^4}}} \, dx=\int \frac {1}{\left (a + b x^{4}\right ) \sqrt {c x^{2} + d \sqrt {a + b x^{4}}}}\, dx \] Input:
integrate(1/(b*x**4+a)/(c*x**2+d*(b*x**4+a)**(1/2))**(1/2),x)
Output:
Integral(1/((a + b*x**4)*sqrt(c*x**2 + d*sqrt(a + b*x**4))), x)
\[ \int \frac {1}{\left (a+b x^4\right ) \sqrt {c x^2+d \sqrt {a+b x^4}}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )} \sqrt {c x^{2} + \sqrt {b x^{4} + a} d}} \,d x } \] Input:
integrate(1/(b*x^4+a)/(c*x^2+d*(b*x^4+a)^(1/2))^(1/2),x, algorithm="maxima ")
Output:
integrate(1/((b*x^4 + a)*sqrt(c*x^2 + sqrt(b*x^4 + a)*d)), x)
\[ \int \frac {1}{\left (a+b x^4\right ) \sqrt {c x^2+d \sqrt {a+b x^4}}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )} \sqrt {c x^{2} + \sqrt {b x^{4} + a} d}} \,d x } \] Input:
integrate(1/(b*x^4+a)/(c*x^2+d*(b*x^4+a)^(1/2))^(1/2),x, algorithm="giac")
Output:
integrate(1/((b*x^4 + a)*sqrt(c*x^2 + sqrt(b*x^4 + a)*d)), x)
Timed out. \[ \int \frac {1}{\left (a+b x^4\right ) \sqrt {c x^2+d \sqrt {a+b x^4}}} \, dx=\int \frac {1}{\left (b\,x^4+a\right )\,\sqrt {d\,\sqrt {b\,x^4+a}+c\,x^2}} \,d x \] Input:
int(1/((a + b*x^4)*(d*(a + b*x^4)^(1/2) + c*x^2)^(1/2)),x)
Output:
int(1/((a + b*x^4)*(d*(a + b*x^4)^(1/2) + c*x^2)^(1/2)), x)
\[ \int \frac {1}{\left (a+b x^4\right ) \sqrt {c x^2+d \sqrt {a+b x^4}}} \, dx=-\left (\int \frac {\sqrt {\sqrt {b \,x^{4}+a}\, d +c \,x^{2}}\, x^{2}}{b^{2} d^{2} x^{8}-b \,c^{2} x^{8}+2 a b \,d^{2} x^{4}-a \,c^{2} x^{4}+a^{2} d^{2}}d x \right ) c +\left (\int \frac {\sqrt {b \,x^{4}+a}\, \sqrt {\sqrt {b \,x^{4}+a}\, d +c \,x^{2}}}{b^{2} d^{2} x^{8}-b \,c^{2} x^{8}+2 a b \,d^{2} x^{4}-a \,c^{2} x^{4}+a^{2} d^{2}}d x \right ) d \] Input:
int(1/(b*x^4+a)/(c*x^2+d*(b*x^4+a)^(1/2))^(1/2),x)
Output:
- int((sqrt(sqrt(a + b*x**4)*d + c*x**2)*x**2)/(a**2*d**2 + 2*a*b*d**2*x* *4 - a*c**2*x**4 + b**2*d**2*x**8 - b*c**2*x**8),x)*c + int((sqrt(a + b*x* *4)*sqrt(sqrt(a + b*x**4)*d + c*x**2))/(a**2*d**2 + 2*a*b*d**2*x**4 - a*c* *2*x**4 + b**2*d**2*x**8 - b*c**2*x**8),x)*d