3.265 \(\int \frac{\sqrt{1+a^2+2 a b x+b^2 x^2}}{\sinh ^{-1}(a+b x)^3} \, dx\)
Optimal. Leaf size=69 \[ \frac{\text{Chi}\left (2 \sinh ^{-1}(a+b x)\right )}{b}-\frac{\sqrt{(a+b x)^2+1} (a+b x)}{b \sinh ^{-1}(a+b x)}-\frac{(a+b x)^2+1}{2 b \sinh ^{-1}(a+b x)^2} \]
[Out]
-(1 + (a + b*x)^2)/(2*b*ArcSinh[a + b*x]^2) - ((a + b*x)*Sqrt[1 + (a + b*x)^2])/(b*ArcSinh[a + b*x]) + CoshInt
egral[2*ArcSinh[a + b*x]]/b
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Rubi [A] time = 0.106354, antiderivative size = 69, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used =
{5867, 5696, 5665, 3301} \[ \frac{\text{Chi}\left (2 \sinh ^{-1}(a+b x)\right )}{b}-\frac{\sqrt{(a+b x)^2+1} (a+b x)}{b \sinh ^{-1}(a+b x)}-\frac{(a+b x)^2+1}{2 b \sinh ^{-1}(a+b x)^2} \]
Antiderivative was successfully verified.
[In]
Int[Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]/ArcSinh[a + b*x]^3,x]
[Out]
-(1 + (a + b*x)^2)/(2*b*ArcSinh[a + b*x]^2) - ((a + b*x)*Sqrt[1 + (a + b*x)^2])/(b*ArcSinh[a + b*x]) + CoshInt
egral[2*ArcSinh[a + b*x]]/b
Rule 5867
Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)^(p_.), x_Symbol] :> D
ist[1/d, Subst[Int[(C/d^2 + (C*x^2)/d^2)^p*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, A,
B, C, n, p}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
Rule 5696
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(Sqrt[1 + c^2*x^2]
*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[(c*(2*p + 1)*d^IntPart[p]*(d + e*x^2)^Fr
acPart[p])/(b*(n + 1)*(1 + c^2*x^2)^FracPart[p]), Int[x*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n + 1),
x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && LtQ[n, -1]
Rule 5665
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[1 + c^2*x^2]*(a + b*ArcSi
nh[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a + b*x)^(n +
1), Sinh[x]^(m - 1)*(m + (m + 1)*Sinh[x]^2), x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0]
&& GeQ[n, -2] && LtQ[n, -1]
Rule 3301
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
Rubi steps
\begin{align*} \int \frac{\sqrt{1+a^2+2 a b x+b^2 x^2}}{\sinh ^{-1}(a+b x)^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{1+x^2}}{\sinh ^{-1}(x)^3} \, dx,x,a+b x\right )}{b}\\ &=-\frac{1+(a+b x)^2}{2 b \sinh ^{-1}(a+b x)^2}+\frac{\operatorname{Subst}\left (\int \frac{x}{\sinh ^{-1}(x)^2} \, dx,x,a+b x\right )}{b}\\ &=-\frac{1+(a+b x)^2}{2 b \sinh ^{-1}(a+b x)^2}-\frac{(a+b x) \sqrt{1+(a+b x)^2}}{b \sinh ^{-1}(a+b x)}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=-\frac{1+(a+b x)^2}{2 b \sinh ^{-1}(a+b x)^2}-\frac{(a+b x) \sqrt{1+(a+b x)^2}}{b \sinh ^{-1}(a+b x)}+\frac{\text{Chi}\left (2 \sinh ^{-1}(a+b x)\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.267333, size = 85, normalized size = 1.23 \[ -\frac{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2+1} \sinh ^{-1}(a+b x)+a^2-2 \sinh ^{-1}(a+b x)^2 \text{Chi}\left (2 \sinh ^{-1}(a+b x)\right )+2 a b x+b^2 x^2+1}{2 b \sinh ^{-1}(a+b x)^2} \]
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]/ArcSinh[a + b*x]^3,x]
[Out]
-(1 + a^2 + 2*a*b*x + b^2*x^2 + 2*(a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x] - 2*ArcSinh[a +
b*x]^2*CoshIntegral[2*ArcSinh[a + b*x]])/(2*b*ArcSinh[a + b*x]^2)
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Maple [A] time = 0.054, size = 63, normalized size = 0.9 \begin{align*}{\frac{4\,{\it Chi} \left ( 2\,{\it Arcsinh} \left ( bx+a \right ) \right ) \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{2}-2\,\sinh \left ( 2\,{\it Arcsinh} \left ( bx+a \right ) \right ){\it Arcsinh} \left ( bx+a \right ) -\cosh \left ( 2\,{\it Arcsinh} \left ( bx+a \right ) \right ) -1}{4\,b \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b^2*x^2+2*a*b*x+a^2+1)^(1/2)/arcsinh(b*x+a)^3,x)
[Out]
1/4/b*(4*Chi(2*arcsinh(b*x+a))*arcsinh(b*x+a)^2-2*sinh(2*arcsinh(b*x+a))*arcsinh(b*x+a)-cosh(2*arcsinh(b*x+a))
-1)/arcsinh(b*x+a)^2
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b^2*x^2+2*a*b*x+a^2+1)^(1/2)/arcsinh(b*x+a)^3,x, algorithm="maxima")
[Out]
-1/2*((b^4*x^4 + 4*a*b^3*x^3 + a^4 + (6*a^2*b^2 + b^2)*x^2 + a^2 + 2*(2*a^3*b + a*b)*x)*(b^2*x^2 + 2*a*b*x + a
^2 + 1)^2 + (3*b^5*x^5 + 15*a*b^4*x^4 + 3*a^5 + 5*(6*a^2*b^3 + b^3)*x^3 + 5*a^3 + 15*(2*a^3*b^2 + a*b^2)*x^2 +
(15*a^4*b + 15*a^2*b + 2*b)*x + 2*a)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2) + (3*b^6*x^6 + 18*a*b^5*x^5 + 3*a^6
+ (45*a^2*b^4 + 7*b^4)*x^4 + 7*a^4 + 4*(15*a^3*b^3 + 7*a*b^3)*x^3 + (45*a^4*b^2 + 42*a^2*b^2 + 5*b^2)*x^2 + 5*
a^2 + 2*(9*a^5*b + 14*a^3*b + 5*a*b)*x + 1)*(b^2*x^2 + 2*a*b*x + a^2 + 1) + ((2*b^4*x^4 + 8*a*b^3*x^3 + 2*a^4
+ (12*a^2*b^2 + b^2)*x^2 + a^2 + 2*(4*a^3*b + a*b)*x - 1)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^2 + (6*b^5*x^5 + 30*a*
b^4*x^4 + 6*a^5 + (60*a^2*b^3 + 7*b^3)*x^3 + 7*a^3 + 3*(20*a^3*b^2 + 7*a*b^2)*x^2 + (30*a^4*b + 21*a^2*b + b)*
x + a)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2) + (6*b^6*x^6 + 36*a*b^5*x^5 + 6*a^6 + (90*a^2*b^4 + 11*b^4)*x^4 + 1
1*a^4 + 4*(30*a^3*b^3 + 11*a*b^3)*x^3 + 6*(15*a^4*b^2 + 11*a^2*b^2 + b^2)*x^2 + 6*a^2 + 4*(9*a^5*b + 11*a^3*b
+ 3*a*b)*x + 1)*(b^2*x^2 + 2*a*b*x + a^2 + 1) + (2*b^7*x^7 + 14*a*b^6*x^6 + 2*a^7 + (42*a^2*b^5 + 5*b^5)*x^5 +
5*a^5 + 5*(14*a^3*b^4 + 5*a*b^4)*x^4 + 2*(35*a^4*b^3 + 25*a^2*b^3 + 2*b^3)*x^3 + 4*a^3 + 2*(21*a^5*b^2 + 25*a
^3*b^2 + 6*a*b^2)*x^2 + (14*a^6*b + 25*a^4*b + 12*a^2*b + b)*x + a)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*log(b*x
+ a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) + (b^7*x^7 + 7*a*b^6*x^6 + a^7 + 3*(7*a^2*b^5 + b^5)*x^5 + 3*a^5 + 5
*(7*a^3*b^4 + 3*a*b^4)*x^4 + (35*a^4*b^3 + 30*a^2*b^3 + 3*b^3)*x^3 + 3*a^3 + 3*(7*a^5*b^2 + 10*a^3*b^2 + 3*a*b
^2)*x^2 + (7*a^6*b + 15*a^4*b + 9*a^2*b + b)*x + a)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))/((b^7*x^6 + 6*a*b^6*x^5
+ a^6*b + 3*a^4*b + 3*(5*a^2*b^5 + b^5)*x^4 + 4*(5*a^3*b^4 + 3*a*b^4)*x^3 + 3*a^2*b + 3*(5*a^4*b^3 + 6*a^2*b^
3 + b^3)*x^2 + (b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b^2*x + a^3*b)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2) + 3*(b^5*x^4
+ 4*a*b^4*x^3 + a^4*b + a^2*b + (6*a^2*b^3 + b^3)*x^2 + 2*(2*a^3*b^2 + a*b^2)*x)*(b^2*x^2 + 2*a*b*x + a^2 + 1)
+ 6*(a^5*b^2 + 2*a^3*b^2 + a*b^2)*x + 3*(b^6*x^5 + 5*a*b^5*x^4 + a^5*b + 2*a^3*b + 2*(5*a^2*b^4 + b^4)*x^3 +
2*(5*a^3*b^3 + 3*a*b^3)*x^2 + a*b + (5*a^4*b^2 + 6*a^2*b^2 + b^2)*x)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) + b)*lo
g(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2) + integrate(1/2*((4*b^4*x^4 + 16*a*b^3*x^3 + 4*a^4 + 2*(12*a
^2*b^2 - b^2)*x^2 - 2*a^2 + 4*(4*a^3*b - a*b)*x + 3)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(5/2) + 2*(8*b^5*x^5 + 40*a
*b^4*x^4 + 8*a^5 + 4*(20*a^2*b^3 + b^3)*x^3 + 4*a^3 + 4*(20*a^3*b^2 + 3*a*b^2)*x^2 + (40*a^4*b + 12*a^2*b + b)
*x + a)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^2 + 2*(12*b^6*x^6 + 72*a*b^5*x^5 + 12*a^6 + 18*(10*a^2*b^4 + b^4)*x^4 +
18*a^4 + 24*(10*a^3*b^3 + 3*a*b^3)*x^3 + 6*(30*a^4*b^2 + 18*a^2*b^2 + b^2)*x^2 + 6*a^2 + 12*(6*a^5*b + 6*a^3*b
+ a*b)*x - 1)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2) + 2*(8*b^7*x^7 + 56*a*b^6*x^6 + 8*a^7 + 4*(42*a^2*b^5 + 5*b
^5)*x^5 + 20*a^5 + 20*(14*a^3*b^4 + 5*a*b^4)*x^4 + 5*(56*a^4*b^3 + 40*a^2*b^3 + 3*b^3)*x^3 + 15*a^3 + (168*a^5
*b^2 + 200*a^3*b^2 + 45*a*b^2)*x^2 + (56*a^6*b + 100*a^4*b + 45*a^2*b + 3*b)*x + 3*a)*(b^2*x^2 + 2*a*b*x + a^2
+ 1) + (4*b^8*x^8 + 32*a*b^7*x^7 + 4*a^8 + 14*(8*a^2*b^6 + b^6)*x^6 + 14*a^6 + 28*(8*a^3*b^5 + 3*a*b^5)*x^5 +
(280*a^4*b^4 + 210*a^2*b^4 + 17*b^4)*x^4 + 17*a^4 + 4*(56*a^5*b^3 + 70*a^3*b^3 + 17*a*b^3)*x^3 + 2*(56*a^6*b^
2 + 105*a^4*b^2 + 51*a^2*b^2 + 4*b^2)*x^2 + 8*a^2 + 4*(8*a^7*b + 21*a^5*b + 17*a^3*b + 4*a*b)*x + 1)*sqrt(b^2*
x^2 + 2*a*b*x + a^2 + 1))/((b^8*x^8 + 8*a*b^7*x^7 + a^8 + 4*(7*a^2*b^6 + b^6)*x^6 + 4*a^6 + 8*(7*a^3*b^5 + 3*a
*b^5)*x^5 + 2*(35*a^4*b^4 + 30*a^2*b^4 + 3*b^4)*x^4 + 6*a^4 + 8*(7*a^5*b^3 + 10*a^3*b^3 + 3*a*b^3)*x^3 + (b^4*
x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x + a^4)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^2 + 4*(7*a^6*b^2 + 15*a^4*b
^2 + 9*a^2*b^2 + b^2)*x^2 + 4*(b^5*x^5 + 5*a*b^4*x^4 + a^5 + (10*a^2*b^3 + b^3)*x^3 + a^3 + (10*a^3*b^2 + 3*a*
b^2)*x^2 + (5*a^4*b + 3*a^2*b)*x)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2) + 6*(b^6*x^6 + 6*a*b^5*x^5 + a^6 + (15*a
^2*b^4 + 2*b^4)*x^4 + 2*a^4 + 4*(5*a^3*b^3 + 2*a*b^3)*x^3 + (15*a^4*b^2 + 12*a^2*b^2 + b^2)*x^2 + a^2 + 2*(3*a
^5*b + 4*a^3*b + a*b)*x)*(b^2*x^2 + 2*a*b*x + a^2 + 1) + 4*a^2 + 8*(a^7*b + 3*a^5*b + 3*a^3*b + a*b)*x + 4*(b^
7*x^7 + 7*a*b^6*x^6 + a^7 + 3*(7*a^2*b^5 + b^5)*x^5 + 3*a^5 + 5*(7*a^3*b^4 + 3*a*b^4)*x^4 + (35*a^4*b^3 + 30*a
^2*b^3 + 3*b^3)*x^3 + 3*a^3 + 3*(7*a^5*b^2 + 10*a^3*b^2 + 3*a*b^2)*x^2 + (7*a^6*b + 15*a^4*b + 9*a^2*b + b)*x
+ a)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) + 1)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{\operatorname{arsinh}\left (b x + a\right )^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b^2*x^2+2*a*b*x+a^2+1)^(1/2)/arcsinh(b*x+a)^3,x, algorithm="fricas")
[Out]
integral(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/arcsinh(b*x + a)^3, x)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2} + 1}}{\operatorname{asinh}^{3}{\left (a + b x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b**2*x**2+2*a*b*x+a**2+1)**(1/2)/asinh(b*x+a)**3,x)
[Out]
Integral(sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)/asinh(a + b*x)**3, x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{\operatorname{arsinh}\left (b x + a\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b^2*x^2+2*a*b*x+a^2+1)^(1/2)/arcsinh(b*x+a)^3,x, algorithm="giac")
[Out]
integrate(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/arcsinh(b*x + a)^3, x)