3.9.49 \(\int \frac {e^{-\tanh ^{-1}(a+b x)}}{x^3} \, dx\) [849]

Optimal. Leaf size=162 \[ \frac {(1-2 a) b \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 (1-a) (1+a)^2 x}-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x}}{2 \left (1-a^2\right ) x^2}-\frac {(1-2 a) b^2 \tanh ^{-1}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {1-a-b x}}\right )}{(1-a) (1+a)^2 \sqrt {1-a^2}} \]

[Out]

-(1-2*a)*b^2*arctanh((1-a)^(1/2)*(b*x+a+1)^(1/2)/(1+a)^(1/2)/(-b*x-a+1)^(1/2))/(1-a)/(1+a)^2/(-a^2+1)^(1/2)-1/
2*(-b*x-a+1)^(3/2)*(b*x+a+1)^(1/2)/(-a^2+1)/x^2+1/2*(1-2*a)*b*(-b*x-a+1)^(1/2)*(b*x+a+1)^(1/2)/(1-a)/(1+a)^2/x

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Rubi [A]
time = 0.07, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6298, 98, 96, 95, 214} \begin {gather*} -\frac {(1-2 a) b^2 \tanh ^{-1}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{(1-a) (a+1)^2 \sqrt {1-a^2}}-\frac {(-a-b x+1)^{3/2} \sqrt {a+b x+1}}{2 \left (1-a^2\right ) x^2}+\frac {(1-2 a) b \sqrt {-a-b x+1} \sqrt {a+b x+1}}{2 (1-a) (a+1)^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(E^ArcTanh[a + b*x]*x^3),x]

[Out]

((1 - 2*a)*b*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(2*(1 - a)*(1 + a)^2*x) - ((1 - a - b*x)^(3/2)*Sqrt[1 + a +
b*x])/(2*(1 - a^2)*x^2) - ((1 - 2*a)*b^2*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x
])])/((1 - a)*(1 + a)^2*Sqrt[1 - a^2])

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[n*((d*e - c*f)/((m + 1)*(b*e - a*f
))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 214

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

Rule 6298

Int[E^(ArcTanh[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[(d + e*x)^m*((1
+ a*c + b*c*x)^(n/2)/(1 - a*c - b*c*x)^(n/2)), x] /; FreeQ[{a, b, c, d, e, m, n}, x]

Rubi steps

\begin {align*} \int \frac {e^{-\tanh ^{-1}(a+b x)}}{x^3} \, dx &=\int \frac {\sqrt {1-a-b x}}{x^3 \sqrt {1+a+b x}} \, dx\\ &=-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x}}{2 \left (1-a^2\right ) x^2}-\frac {((1-2 a) b) \int \frac {\sqrt {1-a-b x}}{x^2 \sqrt {1+a+b x}} \, dx}{2 \left (1-a^2\right )}\\ &=\frac {(1-2 a) b \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 (1-a) (1+a)^2 x}-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x}}{2 \left (1-a^2\right ) x^2}+\frac {\left ((1-2 a) b^2\right ) \int \frac {1}{x \sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{2 (1-a) (1+a)^2}\\ &=\frac {(1-2 a) b \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 (1-a) (1+a)^2 x}-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x}}{2 \left (1-a^2\right ) x^2}+\frac {\left ((1-2 a) b^2\right ) \text {Subst}\left (\int \frac {1}{-1-a-(-1+a) x^2} \, dx,x,\frac {\sqrt {1+a+b x}}{\sqrt {1-a-b x}}\right )}{(1-a) (1+a)^2}\\ &=\frac {(1-2 a) b \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 (1-a) (1+a)^2 x}-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x}}{2 \left (1-a^2\right ) x^2}-\frac {(1-2 a) b^2 \tanh ^{-1}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {1-a-b x}}\right )}{(1-a) (1+a)^2 \sqrt {1-a^2}}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 122, normalized size = 0.75 \begin {gather*} -\frac {\left (-1+a^2+2 b x-a b x\right ) \sqrt {1-a^2-2 a b x-b^2 x^2}}{2 (-1+a) (1+a)^2 x^2}+\frac {(-1+2 a) b^2 \tanh ^{-1}\left (\frac {\sqrt {-1-a} \sqrt {1-a-b x}}{\sqrt {-1+a} \sqrt {1+a+b x}}\right )}{(-1-a)^{5/2} (-1+a)^{3/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^ArcTanh[a + b*x]*x^3),x]

[Out]

-1/2*((-1 + a^2 + 2*b*x - a*b*x)*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2])/((-1 + a)*(1 + a)^2*x^2) + ((-1 + 2*a)*b^2
*ArcTanh[(Sqrt[-1 - a]*Sqrt[1 - a - b*x])/(Sqrt[-1 + a]*Sqrt[1 + a + b*x])])/((-1 - a)^(5/2)*(-1 + a)^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1045\) vs. \(2(140)=280\).
time = 0.07, size = 1046, normalized size = 6.46

method	result	size
risch	\(\frac {\left (b^{2} x^{2}+2 a b x +a^{2}-1\right ) \left (-a b x +a^{2}+2 b x -1\right )}{2 \left (1+a \right ) x^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \left (a^{2}-1\right )}-\frac {b^{2} \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right ) a}{\left (a^{2}-1\right ) \left (1+a \right ) \sqrt {-a^{2}+1}}+\frac {b^{2} \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{2 \left (a^{2}-1\right ) \left (1+a \right ) \sqrt {-a^{2}+1}}\)	\(225\)
default	\(-\frac {b \left (-\frac {\left (-b^{2} x^{2}-2 a b x -a^{2}+1\right )^{\frac {3}{2}}}{\left (-a^{2}+1\right ) x}-\frac {b a \left (\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}-\frac {b a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{\sqrt {b^{2}}}-\sqrt {-a^{2}+1}\, \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )\right )}{-a^{2}+1}-\frac {2 b^{2} \left (-\frac {\left (-2 b^{2} x -2 b a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{4 b^{2}}-\frac {\left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{8 b^{2} \sqrt {b^{2}}}\right )}{-a^{2}+1}\right )}{\left (1+a \right )^{2}}-\frac {b^{2} \left (\sqrt {-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )}+\frac {b \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1+a}{b}-\frac {1}{b}\right )}{\sqrt {-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )}}\right )}{\sqrt {b^{2}}}\right )}{\left (1+a \right )^{3}}+\frac {b^{2} \left (\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}-\frac {b a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{\sqrt {b^{2}}}-\sqrt {-a^{2}+1}\, \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )\right )}{\left (1+a \right )^{3}}+\frac {-\frac {\left (-b^{2} x^{2}-2 a b x -a^{2}+1\right )^{\frac {3}{2}}}{2 \left (-a^{2}+1\right ) x^{2}}+\frac {b a \left (-\frac {\left (-b^{2} x^{2}-2 a b x -a^{2}+1\right )^{\frac {3}{2}}}{\left (-a^{2}+1\right ) x}-\frac {b a \left (\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}-\frac {b a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{\sqrt {b^{2}}}-\sqrt {-a^{2}+1}\, \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )\right )}{-a^{2}+1}-\frac {2 b^{2} \left (-\frac {\left (-2 b^{2} x -2 b a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{4 b^{2}}-\frac {\left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{8 b^{2} \sqrt {b^{2}}}\right )}{-a^{2}+1}\right )}{-2 a^{2}+2}-\frac {b^{2} \left (\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}-\frac {b a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{\sqrt {b^{2}}}-\sqrt {-a^{2}+1}\, \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )\right )}{2 \left (-a^{2}+1\right )}}{1+a}\)	\(1046\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a+1)*(1-(b*x+a)^2)^(1/2)/x^3,x,method=_RETURNVERBOSE)

[Out]

-1/(1+a)^2*b*(-1/(-a^2+1)/x*(-b^2*x^2-2*a*b*x-a^2+1)^(3/2)-b*a/(-a^2+1)*((-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-b*a/(b
^2)^(1/2)*arctan((b^2)^(1/2)*(x+1/b*a)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))-(-a^2+1)^(1/2)*ln((-2*a^2+2-2*a*b*x+2*(
-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))/x))-2*b^2/(-a^2+1)*(-1/4*(-2*b^2*x-2*a*b)/b^2*(-b^2*x^2-2*a*b*x-
a^2+1)^(1/2)-1/8*(-4*b^2*(-a^2+1)-4*b^2*a^2)/b^2/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+1/b*a)/(-b^2*x^2-2*a*b*x-a^
2+1)^(1/2))))-1/(1+a)^3*b^2*((-b^2*(x+(1+a)/b)^2+2*b*(x+(1+a)/b))^(1/2)+b/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+(1
+a)/b-1/b)/(-b^2*(x+(1+a)/b)^2+2*b*(x+(1+a)/b))^(1/2)))+1/(1+a)^3*b^2*((-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-b*a/(b^2
)^(1/2)*arctan((b^2)^(1/2)*(x+1/b*a)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))-(-a^2+1)^(1/2)*ln((-2*a^2+2-2*a*b*x+2*(-a
^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))/x))+1/(1+a)*(-1/2/(-a^2+1)/x^2*(-b^2*x^2-2*a*b*x-a^2+1)^(3/2)+1/2*
b*a/(-a^2+1)*(-1/(-a^2+1)/x*(-b^2*x^2-2*a*b*x-a^2+1)^(3/2)-b*a/(-a^2+1)*((-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-b*a/(b
^2)^(1/2)*arctan((b^2)^(1/2)*(x+1/b*a)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))-(-a^2+1)^(1/2)*ln((-2*a^2+2-2*a*b*x+2*(
-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))/x))-2*b^2/(-a^2+1)*(-1/4*(-2*b^2*x-2*a*b)/b^2*(-b^2*x^2-2*a*b*x-
a^2+1)^(1/2)-1/8*(-4*b^2*(-a^2+1)-4*b^2*a^2)/b^2/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+1/b*a)/(-b^2*x^2-2*a*b*x-a^
2+1)^(1/2))))-1/2*b^2/(-a^2+1)*((-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-b*a/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+1/b*a)/(-
b^2*x^2-2*a*b*x-a^2+1)^(1/2))-(-a^2+1)^(1/2)*ln((-2*a^2+2-2*a*b*x+2*(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1)^(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/(b*x+a+1)*(1-(b*x+a)^2)^(1/2)/x^3,x, algorithm="maxima")

[Out]

integrate(sqrt(-(b*x + a)^2 + 1)/((b*x + a + 1)*x^3), x)

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Fricas [A]
time = 0.38, size = 356, normalized size = 2.20 \begin {gather*} \left [-\frac {\sqrt {-a^{2} + 1} {\left (2 \, a - 1\right )} b^{2} x^{2} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x + 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right ) + 2 \, {\left (a^{4} - {\left (a^{3} - 2 \, a^{2} - a + 2\right )} b x - 2 \, a^{2} + 1\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{4 \, {\left (a^{5} + a^{4} - 2 \, a^{3} - 2 \, a^{2} + a + 1\right )} x^{2}}, \frac {\sqrt {a^{2} - 1} {\left (2 \, a - 1\right )} b^{2} x^{2} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) - {\left (a^{4} - {\left (a^{3} - 2 \, a^{2} - a + 2\right )} b x - 2 \, a^{2} + 1\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \, {\left (a^{5} + a^{4} - 2 \, a^{3} - 2 \, a^{2} + a + 1\right )} x^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a+1)*(1-(b*x+a)^2)^(1/2)/x^3,x, algorithm="fricas")

[Out]

[-1/4*(sqrt(-a^2 + 1)*(2*a - 1)*b^2*x^2*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)*b*x + 2*sqrt(-b^2*x^2 -
2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(-a^2 + 1) - 4*a^2 + 2)/x^2) + 2*(a^4 - (a^3 - 2*a^2 - a + 2)*b*x -
2*a^2 + 1)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/((a^5 + a^4 - 2*a^3 - 2*a^2 + a + 1)*x^2), 1/2*(sqrt(a^2 - 1)*(
2*a - 1)*b^2*x^2*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(a^2 - 1)/((a^2 - 1)*b^2*x^2
+ a^4 + 2*(a^3 - a)*b*x - 2*a^2 + 1)) - (a^4 - (a^3 - 2*a^2 - a + 2)*b*x - 2*a^2 + 1)*sqrt(-b^2*x^2 - 2*a*b*x
- a^2 + 1))/((a^5 + a^4 - 2*a^3 - 2*a^2 + a + 1)*x^2)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}{x^{3} \left (a + b x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a+1)*(1-(b*x+a)**2)**(1/2)/x**3,x)

[Out]

Integral(sqrt(-(a + b*x - 1)*(a + b*x + 1))/(x**3*(a + b*x + 1)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 750 vs. \(2 (134) = 268\).
time = 0.42, size = 750, normalized size = 4.63 \begin {gather*} \frac {{\left (2 \, a b^{3} - b^{3}\right )} \arctan \left (\frac {\frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a}{b^{2} x + a b} - 1}{\sqrt {a^{2} - 1}}\right )}{{\left (a^{3} {\left | b \right |} + a^{2} {\left | b \right |} - a {\left | b \right |} - {\left | b \right |}\right )} \sqrt {a^{2} - 1}} + \frac {\frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a^{4} b^{3}}{{\left (b^{2} x + a b\right )}^{2}} + 2 \, a^{4} b^{3} - \frac {5 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a^{3} b^{3}}{b^{2} x + a b} - \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a^{3} b^{3}}{{\left (b^{2} x + a b\right )}^{2}} - \frac {3 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{3} a^{3} b^{3}}{{\left (b^{2} x + a b\right )}^{3}} - 2 \, a^{3} b^{3} + \frac {6 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a^{2} b^{3}}{b^{2} x + a b} + \frac {3 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a^{2} b^{3}}{{\left (b^{2} x + a b\right )}^{2}} + \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{3} a^{2} b^{3}}{{\left (b^{2} x + a b\right )}^{3}} - a^{2} b^{3} + \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a b^{3}}{b^{2} x + a b} - \frac {4 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a b^{3}}{{\left (b^{2} x + a b\right )}^{2}} + \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{3} a b^{3}}{{\left (b^{2} x + a b\right )}^{3}} - \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} b^{3}}{{\left (b^{2} x + a b\right )}^{2}}}{{\left (a^{5} {\left | b \right |} + a^{4} {\left | b \right |} - a^{3} {\left | b \right |} - a^{2} {\left | b \right |}\right )} {\left (\frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a}{{\left (b^{2} x + a b\right )}^{2}} + a - \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}}{b^{2} x + a b}\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a+1)*(1-(b*x+a)^2)^(1/2)/x^3,x, algorithm="giac")

[Out]

(2*a*b^3 - b^3)*arctan(((sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a/(b^2*x + a*b) - 1)/sqrt(a^2 - 1))/((
a^3*abs(b) + a^2*abs(b) - a*abs(b) - abs(b))*sqrt(a^2 - 1)) + (2*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) +
b)^2*a^4*b^3/(b^2*x + a*b)^2 + 2*a^4*b^3 - 5*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)*a^3*b^3/(b^2*x +
a*b) - 2*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a^3*b^3/(b^2*x + a*b)^2 - 3*(sqrt(-b^2*x^2 - 2*a*b*
x - a^2 + 1)*abs(b) + b)^3*a^3*b^3/(b^2*x + a*b)^3 - 2*a^3*b^3 + 6*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b)
+ b)*a^2*b^3/(b^2*x + a*b) + 3*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a^2*b^3/(b^2*x + a*b)^2 + 2*(
sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^3*a^2*b^3/(b^2*x + a*b)^3 - a^2*b^3 + 2*(sqrt(-b^2*x^2 - 2*a*b*
x - a^2 + 1)*abs(b) + b)*a*b^3/(b^2*x + a*b) - 4*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a*b^3/(b^2*
x + a*b)^2 + 2*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^3*a*b^3/(b^2*x + a*b)^3 - 2*(sqrt(-b^2*x^2 - 2*
a*b*x - a^2 + 1)*abs(b) + b)^2*b^3/(b^2*x + a*b)^2)/((a^5*abs(b) + a^4*abs(b) - a^3*abs(b) - a^2*abs(b))*((sqr
t(-b^2*x^2 - 2*a*b*x - a^2 + 1)*abs(b) + b)^2*a/(b^2*x + a*b)^2 + a - 2*(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*ab
s(b) + b)/(b^2*x + a*b))^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {1-{\left (a+b\,x\right )}^2}}{x^3\,\left (a+b\,x+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - (a + b*x)^2)^(1/2)/(x^3*(a + b*x + 1)),x)

[Out]

int((1 - (a + b*x)^2)^(1/2)/(x^3*(a + b*x + 1)), x)

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