3.31.99 \(\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx\)

Optimal. Leaf size=553 \[ -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )}{a^2}+\frac {\left (a^4+\sqrt {a^4+1}+1\right ) \tan ^{-1}\left (\frac {a \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {-\sqrt {a^4+1}-1}}\right )}{a^2 \sqrt {a^4+1} \sqrt {-\sqrt {a^4+1}-1}}+\frac {\left (-a^4+\sqrt {a^4+1}-1\right ) \tan ^{-1}\left (\frac {a \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {\sqrt {a^4+1}-1}}\right )}{a^2 \sqrt {a^4+1} \sqrt {\sqrt {a^4+1}-1}}+\frac {\left (-\sqrt {2} \sqrt {-\sqrt {a^4+1}-a^2} a^2+\sqrt {2} \sqrt {a^4+1} \sqrt {-\sqrt {a^4+1}-a^2}+\sqrt {2} \sqrt {-\sqrt {a^4+1}-a^2}\right ) \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {-\sqrt {a^4+1}-a^2} \left (\sqrt {x^4+1}+x^2-1\right )}\right )}{2 a^2}+\frac {\left (\sqrt {2} \sqrt {\sqrt {a^4+1}-a^2} a^2+\sqrt {2} \sqrt {a^4+1} \sqrt {\sqrt {a^4+1}-a^2}-\sqrt {2} \sqrt {\sqrt {a^4+1}-a^2}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {a^4+1}-a^2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )}{2 a^2}+\frac {\sqrt {\sqrt {x^4+1}+x^2}}{a} \]

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Rubi [F]  time = 0.10, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sqrt[x^2 + Sqrt[1 + x^4]]/(1 + a*x),x]

[Out]

Defer[Int][Sqrt[x^2 + Sqrt[1 + x^4]]/(1 + a*x), x]

Rubi steps

\begin {align*} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx &=\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.06, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[Sqrt[x^2 + Sqrt[1 + x^4]]/(1 + a*x),x]

[Out]

Integrate[Sqrt[x^2 + Sqrt[1 + x^4]]/(1 + a*x), x]

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IntegrateAlgebraic [A]  time = 5.75, size = 553, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x^2+\sqrt {1+x^4}}}{a}+\frac {\left (1+a^4+\sqrt {1+a^4}\right ) \tan ^{-1}\left (\frac {a \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-1-\sqrt {1+a^4}}}\right )}{a^2 \sqrt {1+a^4} \sqrt {-1-\sqrt {1+a^4}}}+\frac {\left (-1-a^4+\sqrt {1+a^4}\right ) \tan ^{-1}\left (\frac {a \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-1+\sqrt {1+a^4}}}\right )}{a^2 \sqrt {1+a^4} \sqrt {-1+\sqrt {1+a^4}}}+\frac {\left (\sqrt {2} \sqrt {-a^2-\sqrt {1+a^4}}-\sqrt {2} a^2 \sqrt {-a^2-\sqrt {1+a^4}}+\sqrt {2} \sqrt {1+a^4} \sqrt {-a^2-\sqrt {1+a^4}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-a^2-\sqrt {1+a^4}} \left (-1+x^2+\sqrt {1+x^4}\right )}\right )}{2 a^2}+\frac {\left (-\sqrt {2} \sqrt {-a^2+\sqrt {1+a^4}}+\sqrt {2} a^2 \sqrt {-a^2+\sqrt {1+a^4}}+\sqrt {2} \sqrt {1+a^4} \sqrt {-a^2+\sqrt {1+a^4}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {-a^2+\sqrt {1+a^4}} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{2 a^2}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[x^2 + Sqrt[1 + x^4]]/(1 + a*x),x]

[Out]

Sqrt[x^2 + Sqrt[1 + x^4]]/a + ((1 + a^4 + Sqrt[1 + a^4])*ArcTan[(a*Sqrt[x^2 + Sqrt[1 + x^4]])/Sqrt[-1 - Sqrt[1
+ a^4]]])/(a^2*Sqrt[1 + a^4]*Sqrt[-1 - Sqrt[1 + a^4]]) + ((-1 - a^4 + Sqrt[1 + a^4])*ArcTan[(a*Sqrt[x^2 + Sqr
t[1 + x^4]])/Sqrt[-1 + Sqrt[1 + a^4]]])/(a^2*Sqrt[1 + a^4]*Sqrt[-1 + Sqrt[1 + a^4]]) + ((Sqrt[2]*Sqrt[-a^2 - S
qrt[1 + a^4]] - Sqrt[2]*a^2*Sqrt[-a^2 - Sqrt[1 + a^4]] + Sqrt[2]*Sqrt[1 + a^4]*Sqrt[-a^2 - Sqrt[1 + a^4]])*Arc
Tan[(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(Sqrt[-a^2 - Sqrt[1 + a^4]]*(-1 + x^2 + Sqrt[1 + x^4]))])/(2*a^2) +
((-(Sqrt[2]*Sqrt[-a^2 + Sqrt[1 + a^4]]) + Sqrt[2]*a^2*Sqrt[-a^2 + Sqrt[1 + a^4]] + Sqrt[2]*Sqrt[1 + a^4]*Sqrt[
-a^2 + Sqrt[1 + a^4]])*ArcTan[(Sqrt[2]*Sqrt[-a^2 + Sqrt[1 + a^4]]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt
[1 + x^4])])/(2*a^2) - (Sqrt[2]*ArcTanh[(Sqrt[2]*x*Sqrt[x^2 + Sqrt[1 + x^4]])/(1 + x^2 + Sqrt[1 + x^4])])/a^2

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{a x + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x^2 + sqrt(x^4 + 1))/(a*x + 1), x)

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}+\sqrt {x^{4}+1}}}{a x +1}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{a x + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x^2 + sqrt(x^4 + 1))/(a*x + 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {\sqrt {x^4+1}+x^2}}{a\,x+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{a x + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x**2 + sqrt(x**4 + 1))/(a*x + 1), x)

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