3.28.92 \(\int \frac {1}{\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx\) [2792]

Optimal. Leaf size=269 \[ \frac {(6 b-16 a x) \sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+(-9 b+8 a x) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\sqrt {-b+a^2 x^2} \left (-16 \sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+8 \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{12 a^2 x+12 a \sqrt {-b+a^2 x^2}}+\frac {3 b \tanh ^{-1}\left (\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{4 a} \]

[Out]

((-16*a*x+6*b)*(1+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2)+(8*a*x-9*b)*(a*x+(a^2*x^2-b)^(1/2))^(1/2)*(1+(a*x+(a^2*
x^2-b)^(1/2))^(1/2))^(1/2)+(a^2*x^2-b)^(1/2)*(-16*(1+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2)+8*(a*x+(a^2*x^2-b)^(
1/2))^(1/2)*(1+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2)))/(12*a^2*x+12*a*(a^2*x^2-b)^(1/2))+3/4*b*arctanh((1+(a*x+
(a^2*x^2-b)^(1/2))^(1/2))^(1/2))/a

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Rubi [F]
time = 0.05, 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 {1}{\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]],x]

[Out]

Defer[Int][1/Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]], x]

Rubi steps

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

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Mathematica [A]
time = 0.40, size = 168, normalized size = 0.62 \begin {gather*} \frac {\frac {\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}} \left (b \left (6-9 \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )+8 \left (a x+\sqrt {-b+a^2 x^2}\right ) \left (-2+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )\right )}{a x+\sqrt {-b+a^2 x^2}}+9 b \tanh ^{-1}\left (\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{12 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]],x]

[Out]

((Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]*(b*(6 - 9*Sqrt[a*x + Sqrt[-b + a^2*x^2]]) + 8*(a*x + Sqrt[-b + a^2*
x^2])*(-2 + Sqrt[a*x + Sqrt[-b + a^2*x^2]])))/(a*x + Sqrt[-b + a^2*x^2]) + 9*b*ArcTanh[Sqrt[1 + Sqrt[a*x + Sqr
t[-b + a^2*x^2]]]])/(12*a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(1+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2),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(1/(1+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.37, size = 152, normalized size = 0.57 \begin {gather*} \frac {9 \, b \log \left (\sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1} + 1\right ) - 9 \, b \log \left (\sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1} - 1\right ) + 2 \, {\left (6 \, a x - {\left (9 \, a x - 9 \, \sqrt {a^{2} x^{2} - b} - 8\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} - 6 \, \sqrt {a^{2} x^{2} - b} - 16\right )} \sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1}}{24 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(9*b*log(sqrt(sqrt(a*x + sqrt(a^2*x^2 - b)) + 1) + 1) - 9*b*log(sqrt(sqrt(a*x + sqrt(a^2*x^2 - b)) + 1) -
1) + 2*(6*a*x - (9*a*x - 9*sqrt(a^2*x^2 - b) - 8)*sqrt(a*x + sqrt(a^2*x^2 - b)) - 6*sqrt(a^2*x^2 - b) - 16)*s
qrt(sqrt(a*x + sqrt(a^2*x^2 - b)) + 1))/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+(a*x+(a**2*x**2-b)**(1/2))**(1/2))**(1/2),x)

[Out]

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

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Giac [F(-1)] Timed out
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(1/(1+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2),x, algorithm="giac")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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