3.282 \(\int \frac {e^{\cosh ^{-1}(a+b x)}}{x^4} \, dx\)

Optimal. Leaf size=189 \[ -\frac {a b^3 \tan ^{-1}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {a+b x-1}}\right )}{\left (1-a^2\right )^{5/2}}+\frac {a b^2 \sqrt {a+b x-1} \sqrt {a+b x+1}}{2 \left (1-a^2\right )^2 x}+\frac {(a+b x-1)^{3/2} (a+b x+1)^{3/2}}{3 \left (1-a^2\right ) x^3}-\frac {a b \sqrt {a+b x-1} (a+b x+1)^{3/2}}{2 (1-a) (a+1)^2 x^2}-\frac {a}{3 x^3}-\frac {b}{2 x^2} \]

[Out]

-1/3*a/x^3-1/2*b/x^2+1/3*(b*x+a-1)^(3/2)*(b*x+a+1)^(3/2)/(-a^2+1)/x^3-a*b^3*arctan((1-a)^(1/2)*(b*x+a+1)^(1/2)
/(1+a)^(1/2)/(b*x+a-1)^(1/2))/(-a^2+1)^(5/2)-1/2*a*b*(b*x+a+1)^(3/2)*(b*x+a-1)^(1/2)/(1-a)/(1+a)^2/x^2+1/2*a*b
^2*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)/(-a^2+1)^2/x

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Rubi [A]  time = 0.16, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5909, 14, 96, 94, 93, 205} \[ \frac {a b^2 \sqrt {a+b x-1} \sqrt {a+b x+1}}{2 \left (1-a^2\right )^2 x}-\frac {a b^3 \tan ^{-1}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {a+b x-1}}\right )}{\left (1-a^2\right )^{5/2}}+\frac {(a+b x-1)^{3/2} (a+b x+1)^{3/2}}{3 \left (1-a^2\right ) x^3}-\frac {a b \sqrt {a+b x-1} (a+b x+1)^{3/2}}{2 (1-a) (a+1)^2 x^2}-\frac {a}{3 x^3}-\frac {b}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCosh[a + b*x]/x^4,x]

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 205

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

Rule 5909

Int[E^(ArcCosh[u_]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*(u + Sqrt[-1 + u]*Sqrt[1 + u])^n, x] /; RationalQ[m
] && IntegerQ[n] && PolynomialQ[u, x]

Rubi steps

\begin {align*} \int \frac {e^{\cosh ^{-1}(a+b x)}}{x^4} \, dx &=\int \frac {a+b x+\sqrt {-1+a+b x} \sqrt {1+a+b x}}{x^4} \, dx\\ &=\int \left (\frac {a}{x^4}+\frac {b}{x^3}+\frac {\sqrt {-1+a+b x} \sqrt {1+a+b x}}{x^4}\right ) \, dx\\ &=-\frac {a}{3 x^3}-\frac {b}{2 x^2}+\int \frac {\sqrt {-1+a+b x} \sqrt {1+a+b x}}{x^4} \, dx\\ &=-\frac {a}{3 x^3}-\frac {b}{2 x^2}+\frac {(-1+a+b x)^{3/2} (1+a+b x)^{3/2}}{3 \left (1-a^2\right ) x^3}+\frac {(a b) \int \frac {\sqrt {-1+a+b x} \sqrt {1+a+b x}}{x^3} \, dx}{1-a^2}\\ &=-\frac {a}{3 x^3}-\frac {b}{2 x^2}-\frac {a b \sqrt {-1+a+b x} (1+a+b x)^{3/2}}{2 (1-a) (1+a)^2 x^2}+\frac {(-1+a+b x)^{3/2} (1+a+b x)^{3/2}}{3 \left (1-a^2\right ) x^3}+\frac {\left (a b^2\right ) \int \frac {\sqrt {1+a+b x}}{x^2 \sqrt {-1+a+b x}} \, dx}{2 (1-a) (1+a)^2}\\ &=-\frac {a}{3 x^3}-\frac {b}{2 x^2}+\frac {a b^2 \sqrt {-1+a+b x} \sqrt {1+a+b x}}{2 \left (1-a^2\right )^2 x}-\frac {a b \sqrt {-1+a+b x} (1+a+b x)^{3/2}}{2 (1-a) (1+a)^2 x^2}+\frac {(-1+a+b x)^{3/2} (1+a+b x)^{3/2}}{3 \left (1-a^2\right ) x^3}+\frac {\left (a b^3\right ) \int \frac {1}{x \sqrt {-1+a+b x} \sqrt {1+a+b x}} \, dx}{2 \left (1-a^2\right )^2}\\ &=-\frac {a}{3 x^3}-\frac {b}{2 x^2}+\frac {a b^2 \sqrt {-1+a+b x} \sqrt {1+a+b x}}{2 \left (1-a^2\right )^2 x}-\frac {a b \sqrt {-1+a+b x} (1+a+b x)^{3/2}}{2 (1-a) (1+a)^2 x^2}+\frac {(-1+a+b x)^{3/2} (1+a+b x)^{3/2}}{3 \left (1-a^2\right ) x^3}+\frac {\left (a b^3\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 )}{\left (1-a^2\right )^2}\\ &=-\frac {a}{3 x^3}-\frac {b}{2 x^2}+\frac {a b^2 \sqrt {-1+a+b x} \sqrt {1+a+b x}}{2 \left (1-a^2\right )^2 x}-\frac {a b \sqrt {-1+a+b x} (1+a+b x)^{3/2}}{2 (1-a) (1+a)^2 x^2}+\frac {(-1+a+b x)^{3/2} (1+a+b x)^{3/2}}{3 \left (1-a^2\right ) x^3}-\frac {a b^3 \tan ^{-1}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {-1+a+b x}}\right )}{\left (1-a^2\right )^{5/2}}\\ \end {align*}

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Mathematica [C]  time = 0.17, size = 179, normalized size = 0.95 \[ \frac {1}{6} \left (-\frac {3 i a b^3 \log \left (\frac {4 \left (1-a^2\right )^{3/2} \left (\sqrt {1-a^2} \sqrt {a+b x-1} \sqrt {a+b x+1}+i a^2+i a b x-i\right )}{a b^3 x}\right )}{\left (1-a^2\right )^{5/2}}+\frac {\sqrt {a+b x-1} \sqrt {a+b x+1} \left (-2 a^4-a^3 b x+a^2 \left (b^2 x^2+4\right )+a b x+2 b^2 x^2-2\right )}{\left (a^2-1\right )^2 x^3}-\frac {2 a}{x^3}-\frac {3 b}{x^2}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcCosh[a + b*x]/x^4,x]

[Out]

((-2*a)/x^3 - (3*b)/x^2 + (Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]*(-2 - 2*a^4 + a*b*x - a^3*b*x + 2*b^2*x^2 + a^
2*(4 + b^2*x^2)))/((-1 + a^2)^2*x^3) - ((3*I)*a*b^3*Log[(4*(1 - a^2)^(3/2)*(-I + I*a^2 + I*a*b*x + Sqrt[1 - a^
2]*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]))/(a*b^3*x)])/(1 - a^2)^(5/2))/6

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fricas [A]  time = 2.07, size = 431, normalized size = 2.28 \[ \left [\frac {3 \, \sqrt {a^{2} - 1} a b^{3} x^{3} \log \left (\frac {a^{2} b x + a^{3} + {\left (a^{2} - \sqrt {a^{2} - 1} a - 1\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} - {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1} - a}{x}\right ) - 2 \, a^{7} + {\left (a^{4} + a^{2} - 2\right )} b^{3} x^{3} + 6 \, a^{5} - 6 \, a^{3} - 3 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} b x - {\left (2 \, a^{6} - {\left (a^{4} + a^{2} - 2\right )} b^{2} x^{2} - 6 \, a^{4} + {\left (a^{5} - 2 \, a^{3} + a\right )} b x + 6 \, a^{2} - 2\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} + 2 \, a}{6 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3}}, \frac {6 \, \sqrt {-a^{2} + 1} a b^{3} x^{3} \arctan \left (-\frac {\sqrt {-a^{2} + 1} b x - \sqrt {-a^{2} + 1} \sqrt {b x + a + 1} \sqrt {b x + a - 1}}{a^{2} - 1}\right ) - 2 \, a^{7} + {\left (a^{4} + a^{2} - 2\right )} b^{3} x^{3} + 6 \, a^{5} - 6 \, a^{3} - 3 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} b x - {\left (2 \, a^{6} - {\left (a^{4} + a^{2} - 2\right )} b^{2} x^{2} - 6 \, a^{4} + {\left (a^{5} - 2 \, a^{3} + a\right )} b x + 6 \, a^{2} - 2\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} + 2 \, a}{6 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(3*sqrt(a^2 - 1)*a*b^3*x^3*log((a^2*b*x + a^3 + (a^2 - sqrt(a^2 - 1)*a - 1)*sqrt(b*x + a + 1)*sqrt(b*x +
a - 1) - (a*b*x + a^2 - 1)*sqrt(a^2 - 1) - a)/x) - 2*a^7 + (a^4 + a^2 - 2)*b^3*x^3 + 6*a^5 - 6*a^3 - 3*(a^6 -
3*a^4 + 3*a^2 - 1)*b*x - (2*a^6 - (a^4 + a^2 - 2)*b^2*x^2 - 6*a^4 + (a^5 - 2*a^3 + a)*b*x + 6*a^2 - 2)*sqrt(b*
x + a + 1)*sqrt(b*x + a - 1) + 2*a)/((a^6 - 3*a^4 + 3*a^2 - 1)*x^3), 1/6*(6*sqrt(-a^2 + 1)*a*b^3*x^3*arctan(-(
sqrt(-a^2 + 1)*b*x - sqrt(-a^2 + 1)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1))/(a^2 - 1)) - 2*a^7 + (a^4 + a^2 - 2)*
b^3*x^3 + 6*a^5 - 6*a^3 - 3*(a^6 - 3*a^4 + 3*a^2 - 1)*b*x - (2*a^6 - (a^4 + a^2 - 2)*b^2*x^2 - 6*a^4 + (a^5 -
2*a^3 + a)*b*x + 6*a^2 - 2)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1) + 2*a)/((a^6 - 3*a^4 + 3*a^2 - 1)*x^3)]

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giac [B]  time = 0.70, size = 487, normalized size = 2.58 \[ -\frac {\frac {6 \, a b^{4} \arctan \left (\frac {{\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} - 2 \, a}{2 \, \sqrt {-a^{2} + 1}}\right )}{{\left (a^{4} - 2 \, a^{2} + 1\right )} \sqrt {-a^{2} + 1}} - \frac {4 \, {\left (12 \, a^{4} b^{4} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{8} - 16 \, a^{5} b^{4} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{6} - 3 \, a b^{4} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{10} + 6 \, a^{2} b^{4} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{8} - 56 \, a^{3} b^{4} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{6} + 48 \, a^{4} b^{4} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{4} + 12 \, b^{4} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{8} - 48 \, a b^{4} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{6} + 192 \, a^{2} b^{4} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{4} - 96 \, a^{3} b^{4} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} - 144 \, a b^{4} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} + 32 \, a^{2} b^{4} + 64 \, b^{4}\right )}}{{\left (a^{4} - 2 \, a^{2} + 1\right )} {\left ({\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{4} - 4 \, a {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} + 4\right )}^{3}} + \frac {3 \, {\left (b x + a + 1\right )} b^{4} - a b^{4} - 3 \, b^{4}}{b^{3} x^{3}}}{6 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(6*a*b^4*arctan(1/2*((sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^2 - 2*a)/sqrt(-a^2 + 1))/((a^4 - 2*a^2 + 1)*
sqrt(-a^2 + 1)) - 4*(12*a^4*b^4*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^8 - 16*a^5*b^4*(sqrt(b*x + a + 1) - sq
rt(b*x + a - 1))^6 - 3*a*b^4*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^10 + 6*a^2*b^4*(sqrt(b*x + a + 1) - sqrt(
b*x + a - 1))^8 - 56*a^3*b^4*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^6 + 48*a^4*b^4*(sqrt(b*x + a + 1) - sqrt(
b*x + a - 1))^4 + 12*b^4*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^8 - 48*a*b^4*(sqrt(b*x + a + 1) - sqrt(b*x +
a - 1))^6 + 192*a^2*b^4*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^4 - 96*a^3*b^4*(sqrt(b*x + a + 1) - sqrt(b*x +
a - 1))^2 - 144*a*b^4*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^2 + 32*a^2*b^4 + 64*b^4)/((a^4 - 2*a^2 + 1)*((s
qrt(b*x + a + 1) - sqrt(b*x + a - 1))^4 - 4*a*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^2 + 4)^3) + (3*(b*x + a
+ 1)*b^4 - a*b^4 - 3*b^4)/(b^3*x^3))/b

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maple [B]  time = 0.01, size = 374, normalized size = 1.98 \[ -\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \left (3 \sqrt {a^{2}-1}\, \ln \left (\frac {2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+2 a^{2}-2}{x}\right ) x^{3} a \,b^{3}-x^{2} a^{4} b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+x \,a^{5} b \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, x^{2} a^{2} b^{2}+2 a^{6} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-2 x \,a^{3} b \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, x^{2} b^{2}-6 a^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, x a b +6 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{2}-2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\right )}{6 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (a^{2}-1\right )^{3} x^{3}}-\frac {a}{3 x^{3}}-\frac {b}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)*(3*(a^2-1)^(1/2)*ln(2*(a*b*x+(a^2-1)^(1/2)*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+
a^2-1)/x)*x^3*a*b^3-x^2*a^4*b^2*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+x*a^5*b*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)-(b^2*x^2+2
*a*b*x+a^2-1)^(1/2)*x^2*a^2*b^2+2*a^6*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)-2*x*a^3*b*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+2*
(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*x^2*b^2-6*a^4*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*x*a*b+
6*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*a^2-2*(b^2*x^2+2*a*b*x+a^2-1)^(1/2))/(b^2*x^2+2*a*b*x+a^2-1)^(1/2)/(a^2-1)^3/x
^3-1/3*a/x^3-1/2*b/x^2

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a-1>0)', see `assume?` for mor
e details)Is a-1 zero or nonzero?

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mupad [B]  time = 18.89, size = 1537, normalized size = 8.13 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^2*((3*b^3)/32 - (a^2*b^3)/32))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^
2*(a^4 - 2*a^2 + 1)) - b^3/(192*(a^2 - 1)) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^4*((9*a^2*b^3)/8 - b^3/2 +
(5*a^4*b^3)/8))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^4*(3*a^2 - 3*a^4 + a^6 - 1)) + (((a - 1)^(1/2) - (a +
b*x - 1)^(1/2))^8*((a^2*b^3)/32 - (21*b^3)/64 + (3*a^4*b^3)/64))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^8*(3*a
^2 - 3*a^4 + a^6 - 1)) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^6*((103*b^3)/96 - (121*a^2*b^3)/32 + (11*a^4*b
^3)/32 + (67*a^6*b^3)/96))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^6*(6*a^4 - 4*a^2 - 4*a^6 + a^8 + 1)) - (((a
- 1)^(1/2) - (a + b*x - 1)^(1/2))^3*(a - 1)^(1/2)*(a + 1)^(1/2)*((17*a*b^3)/32 + (17*a^3*b^3)/96))/(((a + 1)^(
1/2) - (a + b*x + 1)^(1/2))^3*(3*a^2 - 3*a^4 + a^6 - 1)) - (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^7*(a - 1)^(1
/2)*(a + 1)^(1/2)*((3*a^3*b^3)/16 - (63*a*b^3)/32 + (9*a^5*b^3)/32))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^7*
(6*a^4 - 4*a^2 - 4*a^6 + a^8 + 1)) - (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^5*(a - 1)^(1/2)*(a + 1)^(1/2)*((17
*a^3*b^3)/16 - (79*a*b^3)/32 + (29*a^5*b^3)/32))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^5*(6*a^4 - 4*a^2 - 4*a
^6 + a^8 + 1)) + (a*b^3*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))*(a - 1)^(1/2)*(a + 1)^(1/2))/(32*((a + 1)^(1/2)
- (a + b*x + 1)^(1/2))*(a^4 - 2*a^2 + 1)))/(((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^3/((a + 1)^(1/2) - (a + b*x
+ 1)^(1/2))^3 + ((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^9/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^9 + (((a - 1)^(1
/2) - (a + b*x - 1)^(1/2))^5*(15*a^2 - 3))/((a^2 - 1)*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^5) + (((a - 1)^(1/
2) - (a + b*x - 1)^(1/2))^7*(15*a^2 - 3))/((a^2 - 1)*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^7) + (((a - 1)^(1/2
) - (a + b*x - 1)^(1/2))^6*(12*a - 20*a^3)*(a - 1)^(1/2)*(a + 1)^(1/2))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))
^6*(a^4 - 2*a^2 + 1)) - (6*a*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^4*(a - 1)^(1/2)*(a + 1)^(1/2))/((a^2 - 1)*(
(a + 1)^(1/2) - (a + b*x + 1)^(1/2))^4) - (6*a*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^8*(a - 1)^(1/2)*(a + 1)^(
1/2))/((a^2 - 1)*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^8)) - (a/3 + (b*x)/2)/x^3 + (((a - 1)^(1/2) - (a + b*x
- 1)^(1/2))*((b^3*(1792*a^4 - 2048*a^2 + 256))/(4096*(a^2 - 1)^3) - (b^3*(25*a^2 - 9))/(64*(a^2 - 1)^2) + (8*a
*(a - 1)^(3/2)*(a + 1)^(3/2)*((a*b^3*(a - 1)^(1/2)*(a + 1)^(1/2))/(8*(a^2 - 1)^2) - (a*b^3*(a - 1)^(3/2)*(a +
1)^(3/2))/(8*(a^2 - 1)^3)))/(a^2 - 1)^2))/((a + 1)^(1/2) - (a + b*x + 1)^(1/2)) + (((a - 1)^(1/2) - (a + b*x -
1)^(1/2))^2*((a*b^3*(a - 1)^(1/2)*(a + 1)^(1/2))/(16*(a^2 - 1)^2) - (a*b^3*(a - 1)^(3/2)*(a + 1)^(3/2))/(16*(
a^2 - 1)^3)))/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^2 - (b^3*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^3)/(192*(a^
2 - 1)*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^3) + (a*b^3*log(((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^2/((a + 1)^
(1/2) - (a + b*x + 1)^(1/2))^2 - a^2 - (a^2*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^2)/((a + 1)^(1/2) - (a + b*x
+ 1)^(1/2))^2 + (2*a*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))*(a - 1)^(1/2)*(a + 1)^(1/2))/((a + 1)^(1/2) - (a +
b*x + 1)^(1/2)) + 1)*(a - 1)^(1/2)*(a + 1)^(1/2))/(6*a^2 - 6*a^4 + 2*a^6 - 2) - (a*b^3*log(((a - 1)^(1/2) - (
a + b*x - 1)^(1/2))/((a + 1)^(1/2) - (a + b*x + 1)^(1/2)))*(a - 1)^(1/2)*(a + 1)^(1/2))/(6*a^2 - 6*a^4 + 2*a^6
- 2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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