3.808 \(\int \frac{1}{x (a+b x^4) \sqrt{c+d x^4}} \, dx\)

Optimal. Leaf size=85 \[ \frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 a \sqrt{b c-a d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right )}{2 a \sqrt{c}} \]

[Out]

-ArcTanh[Sqrt[c + d*x^4]/Sqrt[c]]/(2*a*Sqrt[c]) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])
/(2*a*Sqrt[b*c - a*d])

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Rubi [A]  time = 0.0731661, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {446, 86, 63, 208} \[ \frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 a \sqrt{b c-a d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right )}{2 a \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x^4)*Sqrt[c + d*x^4]),x]

[Out]

-ArcTanh[Sqrt[c + d*x^4]/Sqrt[c]]/(2*a*Sqrt[c]) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])
/(2*a*Sqrt[b*c - a*d])

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 86

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), In
t[(e + f*x)^p/(a + b*x), x], x] - Dist[d/(b*c - a*d), Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d,
e, f, p}, x] &&  !IntegerQ[p]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{x \left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x (a+b x) \sqrt{c+d x}} \, dx,x,x^4\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,x^4\right )}{4 a}-\frac{b \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,x^4\right )}{4 a}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x^4}\right )}{2 a d}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x^4}\right )}{2 a d}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right )}{2 a \sqrt{c}}+\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 a \sqrt{b c-a d}}\\ \end{align*}

Mathematica [A]  time = 0.0769461, size = 81, normalized size = 0.95 \[ \frac{\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{\sqrt{b c-a d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^4}}{\sqrt{c}}\right )}{\sqrt{c}}}{2 a} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a + b*x^4)*Sqrt[c + d*x^4]),x]

[Out]

(-(ArcTanh[Sqrt[c + d*x^4]/Sqrt[c]]/Sqrt[c]) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/Sq
rt[b*c - a*d])/(2*a)

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Maple [B]  time = 0.013, size = 347, normalized size = 4.1 \begin{align*}{\frac{1}{4\,a}\ln \left ({ \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{d\sqrt{-ab}}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}+{\frac{1}{4\,a}\ln \left ({ \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{d\sqrt{-ab}}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}-{\frac{1}{2\,a}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{4}+c} \right ) } \right ){\frac{1}{\sqrt{c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(b*x^4+a)/(d*x^4+c)^(1/2),x)

[Out]

1/4/a/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x^2-(-a*b)^(1/2)/b)+2*(-(a*d-b*c)/b)^(1/2)*(
(x^2-(-a*b)^(1/2)/b)^2*d+2*d*(-a*b)^(1/2)/b*(x^2-(-a*b)^(1/2)/b)-(a*d-b*c)/b)^(1/2))/(x^2-(-a*b)^(1/2)/b))+1/4
/a/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x^2+(-a*b)^(1/2)/b)+2*(-(a*d-b*c)/b)^(1/2)*((x^
2+(-a*b)^(1/2)/b)^2*d-2*d*(-a*b)^(1/2)/b*(x^2+(-a*b)^(1/2)/b)-(a*d-b*c)/b)^(1/2))/(x^2+(-a*b)^(1/2)/b))-1/2/a/
c^(1/2)*ln((2*c+2*c^(1/2)*(d*x^4+c)^(1/2))/x^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )} \sqrt{d x^{4} + c} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)*sqrt(d*x^4 + c)*x), x)

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Fricas [A]  time = 1.65892, size = 948, normalized size = 11.15 \begin{align*} \left [\frac{c \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b d x^{4} + 2 \, b c - a d + 2 \, \sqrt{d x^{4} + c}{\left (b c - a d\right )} \sqrt{\frac{b}{b c - a d}}}{b x^{4} + a}\right ) + \sqrt{c} \log \left (\frac{d x^{4} - 2 \, \sqrt{d x^{4} + c} \sqrt{c} + 2 \, c}{x^{4}}\right )}{4 \, a c}, \frac{2 \, c \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{\sqrt{d x^{4} + c}{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}{b d x^{4} + b c}\right ) + \sqrt{c} \log \left (\frac{d x^{4} - 2 \, \sqrt{d x^{4} + c} \sqrt{c} + 2 \, c}{x^{4}}\right )}{4 \, a c}, \frac{c \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b d x^{4} + 2 \, b c - a d + 2 \, \sqrt{d x^{4} + c}{\left (b c - a d\right )} \sqrt{\frac{b}{b c - a d}}}{b x^{4} + a}\right ) + 2 \, \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{4} + c} \sqrt{-c}}{c}\right )}{4 \, a c}, \frac{c \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{\sqrt{d x^{4} + c}{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}{b d x^{4} + b c}\right ) + \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{4} + c} \sqrt{-c}}{c}\right )}{2 \, a c}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="fricas")

[Out]

[1/4*(c*sqrt(b/(b*c - a*d))*log((b*d*x^4 + 2*b*c - a*d + 2*sqrt(d*x^4 + c)*(b*c - a*d)*sqrt(b/(b*c - a*d)))/(b
*x^4 + a)) + sqrt(c)*log((d*x^4 - 2*sqrt(d*x^4 + c)*sqrt(c) + 2*c)/x^4))/(a*c), 1/4*(2*c*sqrt(-b/(b*c - a*d))*
arctan(-sqrt(d*x^4 + c)*(b*c - a*d)*sqrt(-b/(b*c - a*d))/(b*d*x^4 + b*c)) + sqrt(c)*log((d*x^4 - 2*sqrt(d*x^4
+ c)*sqrt(c) + 2*c)/x^4))/(a*c), 1/4*(c*sqrt(b/(b*c - a*d))*log((b*d*x^4 + 2*b*c - a*d + 2*sqrt(d*x^4 + c)*(b*
c - a*d)*sqrt(b/(b*c - a*d)))/(b*x^4 + a)) + 2*sqrt(-c)*arctan(sqrt(d*x^4 + c)*sqrt(-c)/c))/(a*c), 1/2*(c*sqrt
(-b/(b*c - a*d))*arctan(-sqrt(d*x^4 + c)*(b*c - a*d)*sqrt(-b/(b*c - a*d))/(b*d*x^4 + b*c)) + sqrt(-c)*arctan(s
qrt(d*x^4 + c)*sqrt(-c)/c))/(a*c)]

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Sympy [A]  time = 16.3892, size = 66, normalized size = 0.78 \begin{align*} - \frac{\operatorname{atan}{\left (\frac{\sqrt{c + d x^{4}}}{\sqrt{\frac{a d - b c}{b}}} \right )}}{2 a \sqrt{\frac{a d - b c}{b}}} + \frac{\operatorname{atan}{\left (\frac{\sqrt{c + d x^{4}}}{\sqrt{- c}} \right )}}{2 a \sqrt{- c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x**4+a)/(d*x**4+c)**(1/2),x)

[Out]

-atan(sqrt(c + d*x**4)/sqrt((a*d - b*c)/b))/(2*a*sqrt((a*d - b*c)/b)) + atan(sqrt(c + d*x**4)/sqrt(-c))/(2*a*s
qrt(-c))

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Giac [A]  time = 1.06789, size = 107, normalized size = 1.26 \begin{align*} -\frac{1}{2} \, d{\left (\frac{b \arctan \left (\frac{\sqrt{d x^{4} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a d} - \frac{\arctan \left (\frac{\sqrt{d x^{4} + c}}{\sqrt{-c}}\right )}{a \sqrt{-c} d}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="giac")

[Out]

-1/2*d*(b*arctan(sqrt(d*x^4 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*a*d) - arctan(sqrt(d*x^4 + c)/s
qrt(-c))/(a*sqrt(-c)*d))