3.849 \(\int \frac {e^{-\tanh ^{-1}(a+b x)}}{x^3} \, dx\)

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

[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 = 0.10, 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 = {6163, 96, 94, 93, 208} \[ -\frac {(1-2 a) b^2 \tanh ^{-1}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{(1-a) (a+1)^2 \sqrt {1-a^2}}-\frac {(-a-b x+1)^{3/2} \sqrt {a+b x+1}}{2 \left (1-a^2\right ) x^2}+\frac {(1-2 a) b \sqrt {-a-b x+1} \sqrt {a+b x+1}}{2 (1-a) (a+1)^2 x} \]

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 93

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 94

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[p, 1] &&  !SumSimplerQ[m, 1])

Rule 96

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 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]

Rule 6163

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 ) \operatorname {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 = 0.13, size = 122, normalized size = 0.75 \[ \frac {(2 a-1) b^2 \tanh ^{-1}\left (\frac {\sqrt {-a-1} \sqrt {-a-b x+1}}{\sqrt {a-1} \sqrt {a+b x+1}}\right )}{(-a-1)^{5/2} (a-1)^{3/2}}-\frac {\left (a^2-a b x+2 b x-1\right ) \sqrt {-a^2-2 a b x-b^2 x^2+1}}{2 (a-1) (a+1)^2 x^2} \]

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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fricas [A]  time = 0.66, size = 356, normalized size = 2.20 \[ \left [-\frac {\sqrt {-a^{2} + 1} {\left (2 \, a - 1\right )} b^{2} x^{2} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x + 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right ) + 2 \, {\left (a^{4} - {\left (a^{3} - 2 \, a^{2} - a + 2\right )} b x - 2 \, a^{2} + 1\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{4 \, {\left (a^{5} + a^{4} - 2 \, a^{3} - 2 \, a^{2} + a + 1\right )} x^{2}}, \frac {\sqrt {a^{2} - 1} {\left (2 \, a - 1\right )} b^{2} x^{2} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) - {\left (a^{4} - {\left (a^{3} - 2 \, a^{2} - a + 2\right )} b x - 2 \, a^{2} + 1\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \, {\left (a^{5} + a^{4} - 2 \, a^{3} - 2 \, a^{2} + a + 1\right )} x^{2}}\right ] \]

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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giac [B]  time = 0.29, size = 750, normalized size = 4.63 \[ \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}} \]

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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maple [B]  time = 0.05, size = 1116, normalized size = 6.89 \[ \frac {b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (1+a \right )^{3}}-\frac {b^{3} 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 )}{\left (1+a \right )^{3} \sqrt {b^{2}}}-\frac {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 )}{\left (1+a \right )^{3}}+\frac {b \left (-b^{2} x^{2}-2 a b x -a^{2}+1\right )^{\frac {3}{2}}}{\left (1+a \right )^{2} \left (-a^{2}+1\right ) x}+\frac {2 b^{2} a \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (1+a \right )^{2} \left (-a^{2}+1\right )}-\frac {b^{3} a^{2} \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 )}{\left (1+a \right )^{2} \left (-a^{2}+1\right ) \sqrt {b^{2}}}-\frac {b^{2} a \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 )}{\left (1+a \right )^{2} \sqrt {-a^{2}+1}}+\frac {b^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, x}{\left (1+a \right )^{2} \left (-a^{2}+1\right )}+\frac {b^{3} \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 )}{\left (1+a \right )^{2} \left (-a^{2}+1\right ) \sqrt {b^{2}}}-\frac {b^{2} \sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}}{\left (1+a \right )^{3}}-\frac {b^{3} \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1+a}{b}-\frac {1}{b}\right )}{\sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}}\right )}{\left (1+a \right )^{3} \sqrt {b^{2}}}-\frac {\left (-b^{2} x^{2}-2 a b x -a^{2}+1\right )^{\frac {3}{2}}}{2 \left (1+a \right ) \left (-a^{2}+1\right ) x^{2}}-\frac {a b \left (-b^{2} x^{2}-2 a b x -a^{2}+1\right )^{\frac {3}{2}}}{2 \left (1+a \right ) \left (-a^{2}+1\right )^{2} x}-\frac {a^{2} b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{\left (1+a \right ) \left (-a^{2}+1\right )^{2}}+\frac {a^{3} b^{3} \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 )}{2 \left (1+a \right ) \left (-a^{2}+1\right )^{2} \sqrt {b^{2}}}+\frac {a^{2} 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 (1+a \right ) \left (-a^{2}+1\right )^{\frac {3}{2}}}-\frac {a \,b^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, x}{2 \left (1+a \right ) \left (-a^{2}+1\right )^{2}}-\frac {a \,b^{3} \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 )}{2 \left (1+a \right ) \left (-a^{2}+1\right )^{2} \sqrt {b^{2}}}-\frac {b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 \left (1+a \right ) \left (-a^{2}+1\right )}+\frac {b^{3} 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 )}{2 \left (1+a \right ) \left (-a^{2}+1\right ) \sqrt {b^{2}}}+\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 (1+a \right ) \sqrt {-a^{2}+1}} \]

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)

[Out]

1/(1+a)^3*b^2*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-1/(1+a)^3*b^3*a/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+a/b)/(-b^2*x^2-
2*a*b*x-a^2+1)^(1/2))-1/(1+a)^3*b^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)^2*b/(-a^2+1)/x*(-b^2*x^2-2*a*b*x-a^2+1)^(3/2)+2/(1+a)^2*b^2*a/(-a^2+1)*(-b^2*x^2-2*a*b*x
-a^2+1)^(1/2)-1/(1+a)^2*b^3*a^2/(-a^2+1)/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+a/b)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)
)-1/(1+a)^2*b^2*a/(-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)^2*b^3/(-a^2+1)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x+1/(1+a)^2*b^3/(-a^2+1)/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+a
/b)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))-1/(1+a)^3*b^2*(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(1/2)-1/(1+a)^3*b^3/(b^
2)^(1/2)*arctan((b^2)^(1/2)*(x+(1+a)/b-1/b)/(-(x+(1+a)/b)^2*b^2+2*b*(x+(1+a)/b))^(1/2))-1/2/(1+a)/(-a^2+1)/x^2
*(-b^2*x^2-2*a*b*x-a^2+1)^(3/2)-1/2/(1+a)*a*b/(-a^2+1)^2/x*(-b^2*x^2-2*a*b*x-a^2+1)^(3/2)-1/(1+a)*a^2*b^2/(-a^
2+1)^2*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+1/2/(1+a)*a^3*b^3/(-a^2+1)^2/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+a/b)/(-b^
2*x^2-2*a*b*x-a^2+1)^(1/2))+1/2/(1+a)*a^2*b^2/(-a^2+1)^(3/2)*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)-1/2/(1+a)*a*b^3/(-a^2+1)^2*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x-1/2/(1+a)*a*b^3/(-a^2+1)^2
/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+a/b)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))-1/2/(1+a)*b^2/(-a^2+1)*(-b^2*x^2-2*a*b
*x-a^2+1)^(1/2)+1/2/(1+a)*b^3/(-a^2+1)*a/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+a/b)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)
)+1/2/(1+a)*b^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 \[ \int \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1}}{{\left (b x + a + 1\right )} x^{3}}\,{d x} \]

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

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

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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