3.31.99 \(\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx\) [3099]

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

[Out]

(x^2+(x^4+1)^(1/2))^(1/2)/a+(1+a^4+(a^4+1)^(1/2))*arctan(a*(x^2+(x^4+1)^(1/2))^(1/2)/(-1-(a^4+1)^(1/2))^(1/2))
/a^2/(a^4+1)^(1/2)/(-1-(a^4+1)^(1/2))^(1/2)+(-1-a^4+(a^4+1)^(1/2))*arctan(a*(x^2+(x^4+1)^(1/2))^(1/2)/(-1+(a^4
+1)^(1/2))^(1/2))/a^2/(a^4+1)^(1/2)/(-1+(a^4+1)^(1/2))^(1/2)+1/2*(2^(1/2)*(-a^2-(a^4+1)^(1/2))^(1/2)-2^(1/2)*a
^2*(-a^2-(a^4+1)^(1/2))^(1/2)+2^(1/2)*(a^4+1)^(1/2)*(-a^2-(a^4+1)^(1/2))^(1/2))*arctan(2^(1/2)*x*(x^2+(x^4+1)^
(1/2))^(1/2)/(-a^2-(a^4+1)^(1/2))^(1/2)/(-1+x^2+(x^4+1)^(1/2)))/a^2+1/2*(-2^(1/2)*(-a^2+(a^4+1)^(1/2))^(1/2)+2
^(1/2)*a^2*(-a^2+(a^4+1)^(1/2))^(1/2)+2^(1/2)*(a^4+1)^(1/2)*(-a^2+(a^4+1)^(1/2))^(1/2))*arctan(2^(1/2)*(-a^2+(
a^4+1)^(1/2))^(1/2)*x*(x^2+(x^4+1)^(1/2))^(1/2)/(1+x^2+(x^4+1)^(1/2)))/a^2-2^(1/2)*arctanh(2^(1/2)*x*(x^2+(x^4
+1)^(1/2))^(1/2)/(1+x^2+(x^4+1)^(1/2)))/a^2

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Rubi [F]
time = 0.08, 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 [A]
time = 4.09, size = 436, normalized size = 0.79 \begin {gather*} \frac {2 a \sqrt {x^2+\sqrt {1+x^4}}+\frac {2 \left (1+a^4+\sqrt {1+a^4}\right ) \text {ArcTan}\left (\frac {a \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-1-\sqrt {1+a^4}}}\right )}{\sqrt {1+a^4} \sqrt {-1-\sqrt {1+a^4}}}+\frac {2 \left (-1-a^4+\sqrt {1+a^4}\right ) \text {ArcTan}\left (\frac {a \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-1+\sqrt {1+a^4}}}\right )}{\sqrt {1+a^4} \sqrt {-1+\sqrt {1+a^4}}}+\sqrt {2} \sqrt {-a^2-\sqrt {1+a^4}} \left (1-a^2+\sqrt {1+a^4}\right ) \text {ArcTan}\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 )+\sqrt {2} \sqrt {-a^2+\sqrt {1+a^4}} \left (-1+a^2+\sqrt {1+a^4}\right ) \text {ArcTan}\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 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{2 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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*} \text {Failed to integrate} \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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1078 vs. \(2 (441) = 882\).
time = 185.36, size = 1078, normalized size = 1.95 \begin {gather*} -\frac {a^{2} \sqrt {\frac {a^{4} \sqrt {\frac {a^{4} + 1}{a^{8}}} + 1}{a^{4}}} \log \left (\frac {4 \, {\left ({\left (a^{5} - {\left (a^{4} + 1\right )} x^{3} + \sqrt {x^{4} + 1} {\left ({\left (a^{4} + 1\right )} x + {\left (a^{7} - a^{4} x\right )} \sqrt {\frac {a^{4} + 1}{a^{8}}}\right )} - {\left (a^{7} x^{2} + a^{6} x - a^{4} x^{3} + a^{5}\right )} \sqrt {\frac {a^{4} + 1}{a^{8}}} + a\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + {\left (a^{6} + {\left (a^{4} + 1\right )} x^{2} + a^{2} - {\left (a^{4} - {\left (a^{8} + a^{4}\right )} \sqrt {\frac {a^{4} + 1}{a^{8}}} + 1\right )} \sqrt {x^{4} + 1} - {\left (a^{6} + a^{4} x^{2}\right )} \sqrt {\frac {a^{4} + 1}{a^{8}}}\right )} \sqrt {\frac {a^{4} \sqrt {\frac {a^{4} + 1}{a^{8}}} + 1}{a^{4}}}\right )}}{a^{2} x^{2} + 2 \, a x + 1}\right ) - a^{2} \sqrt {\frac {a^{4} \sqrt {\frac {a^{4} + 1}{a^{8}}} + 1}{a^{4}}} \log \left (\frac {4 \, {\left ({\left (a^{5} - {\left (a^{4} + 1\right )} x^{3} + \sqrt {x^{4} + 1} {\left ({\left (a^{4} + 1\right )} x + {\left (a^{7} - a^{4} x\right )} \sqrt {\frac {a^{4} + 1}{a^{8}}}\right )} - {\left (a^{7} x^{2} + a^{6} x - a^{4} x^{3} + a^{5}\right )} \sqrt {\frac {a^{4} + 1}{a^{8}}} + a\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (a^{6} + {\left (a^{4} + 1\right )} x^{2} + a^{2} - {\left (a^{4} - {\left (a^{8} + a^{4}\right )} \sqrt {\frac {a^{4} + 1}{a^{8}}} + 1\right )} \sqrt {x^{4} + 1} - {\left (a^{6} + a^{4} x^{2}\right )} \sqrt {\frac {a^{4} + 1}{a^{8}}}\right )} \sqrt {\frac {a^{4} \sqrt {\frac {a^{4} + 1}{a^{8}}} + 1}{a^{4}}}\right )}}{a^{2} x^{2} + 2 \, a x + 1}\right ) + a^{2} \sqrt {-\frac {a^{4} \sqrt {\frac {a^{4} + 1}{a^{8}}} - 1}{a^{4}}} \log \left (\frac {4 \, {\left ({\left (a^{5} - {\left (a^{4} + 1\right )} x^{3} + \sqrt {x^{4} + 1} {\left ({\left (a^{4} + 1\right )} x - {\left (a^{7} - a^{4} x\right )} \sqrt {\frac {a^{4} + 1}{a^{8}}}\right )} + {\left (a^{7} x^{2} + a^{6} x - a^{4} x^{3} + a^{5}\right )} \sqrt {\frac {a^{4} + 1}{a^{8}}} + a\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + {\left (a^{6} + {\left (a^{4} + 1\right )} x^{2} + a^{2} - {\left (a^{4} + {\left (a^{8} + a^{4}\right )} \sqrt {\frac {a^{4} + 1}{a^{8}}} + 1\right )} \sqrt {x^{4} + 1} + {\left (a^{6} + a^{4} x^{2}\right )} \sqrt {\frac {a^{4} + 1}{a^{8}}}\right )} \sqrt {-\frac {a^{4} \sqrt {\frac {a^{4} + 1}{a^{8}}} - 1}{a^{4}}}\right )}}{a^{2} x^{2} + 2 \, a x + 1}\right ) - a^{2} \sqrt {-\frac {a^{4} \sqrt {\frac {a^{4} + 1}{a^{8}}} - 1}{a^{4}}} \log \left (\frac {4 \, {\left ({\left (a^{5} - {\left (a^{4} + 1\right )} x^{3} + \sqrt {x^{4} + 1} {\left ({\left (a^{4} + 1\right )} x - {\left (a^{7} - a^{4} x\right )} \sqrt {\frac {a^{4} + 1}{a^{8}}}\right )} + {\left (a^{7} x^{2} + a^{6} x - a^{4} x^{3} + a^{5}\right )} \sqrt {\frac {a^{4} + 1}{a^{8}}} + a\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (a^{6} + {\left (a^{4} + 1\right )} x^{2} + a^{2} - {\left (a^{4} + {\left (a^{8} + a^{4}\right )} \sqrt {\frac {a^{4} + 1}{a^{8}}} + 1\right )} \sqrt {x^{4} + 1} + {\left (a^{6} + a^{4} x^{2}\right )} \sqrt {\frac {a^{4} + 1}{a^{8}}}\right )} \sqrt {-\frac {a^{4} \sqrt {\frac {a^{4} + 1}{a^{8}}} - 1}{a^{4}}}\right )}}{a^{2} x^{2} + 2 \, a x + 1}\right ) - 4 \, \sqrt {x^{2} + \sqrt {x^{4} + 1}} a - \sqrt {2} \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} - 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\right )}{4 \, a^{2}} \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]

-1/4*(a^2*sqrt((a^4*sqrt((a^4 + 1)/a^8) + 1)/a^4)*log(4*((a^5 - (a^4 + 1)*x^3 + sqrt(x^4 + 1)*((a^4 + 1)*x + (
a^7 - a^4*x)*sqrt((a^4 + 1)/a^8)) - (a^7*x^2 + a^6*x - a^4*x^3 + a^5)*sqrt((a^4 + 1)/a^8) + a)*sqrt(x^2 + sqrt
(x^4 + 1)) + (a^6 + (a^4 + 1)*x^2 + a^2 - (a^4 - (a^8 + a^4)*sqrt((a^4 + 1)/a^8) + 1)*sqrt(x^4 + 1) - (a^6 + a
^4*x^2)*sqrt((a^4 + 1)/a^8))*sqrt((a^4*sqrt((a^4 + 1)/a^8) + 1)/a^4))/(a^2*x^2 + 2*a*x + 1)) - a^2*sqrt((a^4*s
qrt((a^4 + 1)/a^8) + 1)/a^4)*log(4*((a^5 - (a^4 + 1)*x^3 + sqrt(x^4 + 1)*((a^4 + 1)*x + (a^7 - a^4*x)*sqrt((a^
4 + 1)/a^8)) - (a^7*x^2 + a^6*x - a^4*x^3 + a^5)*sqrt((a^4 + 1)/a^8) + a)*sqrt(x^2 + sqrt(x^4 + 1)) - (a^6 + (
a^4 + 1)*x^2 + a^2 - (a^4 - (a^8 + a^4)*sqrt((a^4 + 1)/a^8) + 1)*sqrt(x^4 + 1) - (a^6 + a^4*x^2)*sqrt((a^4 + 1
)/a^8))*sqrt((a^4*sqrt((a^4 + 1)/a^8) + 1)/a^4))/(a^2*x^2 + 2*a*x + 1)) + a^2*sqrt(-(a^4*sqrt((a^4 + 1)/a^8) -
1)/a^4)*log(4*((a^5 - (a^4 + 1)*x^3 + sqrt(x^4 + 1)*((a^4 + 1)*x - (a^7 - a^4*x)*sqrt((a^4 + 1)/a^8)) + (a^7*
x^2 + a^6*x - a^4*x^3 + a^5)*sqrt((a^4 + 1)/a^8) + a)*sqrt(x^2 + sqrt(x^4 + 1)) + (a^6 + (a^4 + 1)*x^2 + a^2 -
(a^4 + (a^8 + a^4)*sqrt((a^4 + 1)/a^8) + 1)*sqrt(x^4 + 1) + (a^6 + a^4*x^2)*sqrt((a^4 + 1)/a^8))*sqrt(-(a^4*s
qrt((a^4 + 1)/a^8) - 1)/a^4))/(a^2*x^2 + 2*a*x + 1)) - a^2*sqrt(-(a^4*sqrt((a^4 + 1)/a^8) - 1)/a^4)*log(4*((a^
5 - (a^4 + 1)*x^3 + sqrt(x^4 + 1)*((a^4 + 1)*x - (a^7 - a^4*x)*sqrt((a^4 + 1)/a^8)) + (a^7*x^2 + a^6*x - a^4*x
^3 + a^5)*sqrt((a^4 + 1)/a^8) + a)*sqrt(x^2 + sqrt(x^4 + 1)) - (a^6 + (a^4 + 1)*x^2 + a^2 - (a^4 + (a^8 + a^4)
*sqrt((a^4 + 1)/a^8) + 1)*sqrt(x^4 + 1) + (a^6 + a^4*x^2)*sqrt((a^4 + 1)/a^8))*sqrt(-(a^4*sqrt((a^4 + 1)/a^8)
- 1)/a^4))/(a^2*x^2 + 2*a*x + 1)) - 4*sqrt(x^2 + sqrt(x^4 + 1))*a - sqrt(2)*log(4*x^4 + 4*sqrt(x^4 + 1)*x^2 -
2*(sqrt(2)*x^3 + sqrt(2)*sqrt(x^4 + 1)*x)*sqrt(x^2 + sqrt(x^4 + 1)) + 1))/a^2

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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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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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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