3.172 \(\int \frac{1}{(a+b \sinh ^{-1}(c+d x))^3} \, dx\)

Optimal. Leaf size=125 \[ \frac{\cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 b^3 d}-\frac{\sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 b^3 d}-\frac{c+d x}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac{\sqrt{(c+d x)^2+1}}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2} \]

[Out]

-Sqrt[1 + (c + d*x)^2]/(2*b*d*(a + b*ArcSinh[c + d*x])^2) - (c + d*x)/(2*b^2*d*(a + b*ArcSinh[c + d*x])) + (Co
sh[a/b]*CoshIntegral[(a + b*ArcSinh[c + d*x])/b])/(2*b^3*d) - (Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c + d*x])
/b])/(2*b^3*d)

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Rubi [A]  time = 0.177495, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {5863, 5655, 5774, 5657, 3303, 3298, 3301} \[ \frac{\cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 b^3 d}-\frac{\sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 b^3 d}-\frac{c+d x}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac{\sqrt{(c+d x)^2+1}}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSinh[c + d*x])^(-3),x]

[Out]

-Sqrt[1 + (c + d*x)^2]/(2*b*d*(a + b*ArcSinh[c + d*x])^2) - (c + d*x)/(2*b^2*d*(a + b*ArcSinh[c + d*x])) + (Co
sh[a/b]*CoshIntegral[(a + b*ArcSinh[c + d*x])/b])/(2*b^3*d) - (Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c + d*x])
/b])/(2*b^3*d)

Rule 5863

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcSinh[x])^n, x
], x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]

Rule 5655

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^(n + 1
))/(b*c*(n + 1)), x] - Dist[c/(b*(n + 1)), Int[(x*(a + b*ArcSinh[c*x])^(n + 1))/Sqrt[1 + c^2*x^2], x], x] /; F
reeQ[{a, b, c}, x] && LtQ[n, -1]

Rule 5774

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

Rule 5657

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Cosh[a/b - x/b], x], x,
a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3298

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(I*SinhIntegral[(c*f*fz)
/d + f*fz*x])/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

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{1}{\left (a+b \sinh ^{-1}(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b \sinh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=-\frac{\sqrt{1+(c+d x)^2}}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{x}{\sqrt{1+x^2} \left (a+b \sinh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{2 b d}\\ &=-\frac{\sqrt{1+(c+d x)^2}}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac{c+d x}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{a+b \sinh ^{-1}(x)} \, dx,x,c+d x\right )}{2 b^2 d}\\ &=-\frac{\sqrt{1+(c+d x)^2}}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac{c+d x}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}-\frac{x}{b}\right )}{x} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{2 b^3 d}\\ &=-\frac{\sqrt{1+(c+d x)^2}}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac{c+d x}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac{\cosh \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{x}{b}\right )}{x} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{2 b^3 d}-\frac{\sinh \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{x}{b}\right )}{x} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{2 b^3 d}\\ &=-\frac{\sqrt{1+(c+d x)^2}}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac{c+d x}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac{\cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 b^3 d}-\frac{\sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 b^3 d}\\ \end{align*}

Mathematica [A]  time = 0.275074, size = 100, normalized size = 0.8 \[ -\frac{\frac{b^2 \sqrt{(c+d x)^2+1}}{\left (a+b \sinh ^{-1}(c+d x)\right )^2}-\cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )+\sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )+\frac{b (c+d x)}{a+b \sinh ^{-1}(c+d x)}}{2 b^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSinh[c + d*x])^(-3),x]

[Out]

-((b^2*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x])^2 + (b*(c + d*x))/(a + b*ArcSinh[c + d*x]) - Cosh[a/b]*
CoshIntegral[a/b + ArcSinh[c + d*x]] + Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]])/(2*b^3*d)

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Maple [A]  time = 0.053, size = 190, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ( -{\frac{b{\it Arcsinh} \left ( dx+c \right ) +a-b}{4\,{b}^{2} \left ({b}^{2} \left ({\it Arcsinh} \left ( dx+c \right ) \right ) ^{2}+2\,ab{\it Arcsinh} \left ( dx+c \right ) +{a}^{2} \right ) } \left ( -\sqrt{1+ \left ( dx+c \right ) ^{2}}+dx+c \right ) }-{\frac{1}{4\,{b}^{3}}{{\rm e}^{{\frac{a}{b}}}}{\it Ei} \left ( 1,{\it Arcsinh} \left ( dx+c \right ) +{\frac{a}{b}} \right ) }-{\frac{1}{4\,b \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{2}} \left ( dx+c+\sqrt{1+ \left ( dx+c \right ) ^{2}} \right ) }-{\frac{1}{4\,{b}^{2} \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) } \left ( dx+c+\sqrt{1+ \left ( dx+c \right ) ^{2}} \right ) }-{\frac{1}{4\,{b}^{3}}{{\rm e}^{-{\frac{a}{b}}}}{\it Ei} \left ( 1,-{\it Arcsinh} \left ( dx+c \right ) -{\frac{a}{b}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*arcsinh(d*x+c))^3,x)

[Out]

1/d*(-1/4*(-(1+(d*x+c)^2)^(1/2)+d*x+c)*(b*arcsinh(d*x+c)+a-b)/b^2/(b^2*arcsinh(d*x+c)^2+2*a*b*arcsinh(d*x+c)+a
^2)-1/4/b^3*exp(a/b)*Ei(1,arcsinh(d*x+c)+a/b)-1/4/b*(d*x+c+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))^2-1/4/b^2
*(d*x+c+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))-1/4/b^3*exp(-a/b)*Ei(1,-arcsinh(d*x+c)-a/b))

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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(1/(a+b*arcsinh(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/2*((a*d^7 + b*d^7)*x^7 + 7*(a*c*d^6 + b*c*d^6)*x^6 + 3*((7*c^2*d^5 + d^5)*a + (7*c^2*d^5 + d^5)*b)*x^5 + 5*
((7*c^3*d^4 + 3*c*d^4)*a + (7*c^3*d^4 + 3*c*d^4)*b)*x^4 + ((35*c^4*d^3 + 30*c^2*d^3 + 3*d^3)*a + (35*c^4*d^3 +
30*c^2*d^3 + 3*d^3)*b)*x^3 + 3*((7*c^5*d^2 + 10*c^3*d^2 + 3*c*d^2)*a + (7*c^5*d^2 + 10*c^3*d^2 + 3*c*d^2)*b)*
x^2 + ((a*d^4 + b*d^4)*x^4 + 4*(a*c*d^3 + b*c*d^3)*x^3 + (6*a*c^2*d^2 + (6*c^2*d^2 + d^2)*b)*x^2 + (c^4 - 1)*a
+ (c^4 + c^2)*b + 2*(2*a*c^3*d + (2*c^3*d + c*d)*b)*x)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + (3*(a*d^5 + b*d^
5)*x^5 + 15*(a*c*d^4 + b*c*d^4)*x^4 + (3*(10*c^2*d^3 + d^3)*a + 5*(6*c^2*d^3 + d^3)*b)*x^3 + 3*((10*c^3*d^2 +
3*c*d^2)*a + 5*(2*c^3*d^2 + c*d^2)*b)*x^2 + 3*(c^5 + c^3)*a + (3*c^5 + 5*c^3 + 2*c)*b + (3*(5*c^4*d + 3*c^2*d)
*a + (15*c^4*d + 15*c^2*d + 2*d)*b)*x)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + (c^7 + 3*c^5 + 3*c^3 + c)*a + (c^7 + 3*
c^5 + 3*c^3 + c)*b + ((7*c^6*d + 15*c^4*d + 9*c^2*d + d)*a + (7*c^6*d + 15*c^4*d + 9*c^2*d + d)*b)*x + (b*d^7*
x^7 + 7*b*c*d^6*x^6 + 3*(7*c^2*d^5 + d^5)*b*x^5 + 5*(7*c^3*d^4 + 3*c*d^4)*b*x^4 + (35*c^4*d^3 + 30*c^2*d^3 + 3
*d^3)*b*x^3 + 3*(7*c^5*d^2 + 10*c^3*d^2 + 3*c*d^2)*b*x^2 + (7*c^6*d + 15*c^4*d + 9*c^2*d + d)*b*x + (b*d^4*x^4
+ 4*b*c*d^3*x^3 + 6*b*c^2*d^2*x^2 + 4*b*c^3*d*x + (c^4 - 1)*b)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + 3*(b*d^5
*x^5 + 5*b*c*d^4*x^4 + (10*c^2*d^3 + d^3)*b*x^3 + (10*c^3*d^2 + 3*c*d^2)*b*x^2 + (5*c^4*d + 3*c^2*d)*b*x + (c^
5 + c^3)*b)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + (c^7 + 3*c^5 + 3*c^3 + c)*b + (3*b*d^6*x^6 + 18*b*c*d^5*x^5 + 3*(1
5*c^2*d^4 + 2*d^4)*b*x^4 + 12*(5*c^3*d^3 + 2*c*d^3)*b*x^3 + (45*c^4*d^2 + 36*c^2*d^2 + 4*d^2)*b*x^2 + 2*(9*c^5
*d + 12*c^3*d + 4*c*d)*b*x + (3*c^6 + 6*c^4 + 4*c^2 + 1)*b)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + s
qrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) + (3*(a*d^6 + b*d^6)*x^6 + 18*(a*c*d^5 + b*c*d^5)*x^5 + (3*(15*c^2*d^4 + 2*d
^4)*a + (45*c^2*d^4 + 7*d^4)*b)*x^4 + 4*(3*(5*c^3*d^3 + 2*c*d^3)*a + (15*c^3*d^3 + 7*c*d^3)*b)*x^3 + ((45*c^4*
d^2 + 36*c^2*d^2 + 4*d^2)*a + (45*c^4*d^2 + 42*c^2*d^2 + 5*d^2)*b)*x^2 + (3*c^6 + 6*c^4 + 4*c^2 + 1)*a + (3*c^
6 + 7*c^4 + 5*c^2 + 1)*b + 2*((9*c^5*d + 12*c^3*d + 4*c*d)*a + (9*c^5*d + 14*c^3*d + 5*c*d)*b)*x)*sqrt(d^2*x^2
+ 2*c*d*x + c^2 + 1))/(a^2*b^2*d^7*x^6 + 6*a^2*b^2*c*d^6*x^5 + 3*(5*c^2*d^5 + d^5)*a^2*b^2*x^4 + 4*(5*c^3*d^4
+ 3*c*d^4)*a^2*b^2*x^3 + 3*(5*c^4*d^3 + 6*c^2*d^3 + d^3)*a^2*b^2*x^2 + 6*(c^5*d^2 + 2*c^3*d^2 + c*d^2)*a^2*b^
2*x + (c^6*d + 3*c^4*d + 3*c^2*d + d)*a^2*b^2 + (b^4*d^7*x^6 + 6*b^4*c*d^6*x^5 + 3*(5*c^2*d^5 + d^5)*b^4*x^4 +
4*(5*c^3*d^4 + 3*c*d^4)*b^4*x^3 + 3*(5*c^4*d^3 + 6*c^2*d^3 + d^3)*b^4*x^2 + 6*(c^5*d^2 + 2*c^3*d^2 + c*d^2)*b
^4*x + (c^6*d + 3*c^4*d + 3*c^2*d + d)*b^4 + (b^4*d^4*x^3 + 3*b^4*c*d^3*x^2 + 3*b^4*c^2*d^2*x + b^4*c^3*d)*(d^
2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + 3*(b^4*d^5*x^4 + 4*b^4*c*d^4*x^3 + (6*c^2*d^3 + d^3)*b^4*x^2 + 2*(2*c^3*d^2
+ c*d^2)*b^4*x + (c^4*d + c^2*d)*b^4)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + 3*(b^4*d^6*x^5 + 5*b^4*c*d^5*x^4 + 2*(5
*c^2*d^4 + d^4)*b^4*x^3 + 2*(5*c^3*d^3 + 3*c*d^3)*b^4*x^2 + (5*c^4*d^2 + 6*c^2*d^2 + d^2)*b^4*x + (c^5*d + 2*c
^3*d + c*d)*b^4)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 + (a^2*
b^2*d^4*x^3 + 3*a^2*b^2*c*d^3*x^2 + 3*a^2*b^2*c^2*d^2*x + a^2*b^2*c^3*d)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) +
3*(a^2*b^2*d^5*x^4 + 4*a^2*b^2*c*d^4*x^3 + (6*c^2*d^3 + d^3)*a^2*b^2*x^2 + 2*(2*c^3*d^2 + c*d^2)*a^2*b^2*x +
(c^4*d + c^2*d)*a^2*b^2)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + 2*(a*b^3*d^7*x^6 + 6*a*b^3*c*d^6*x^5 + 3*(5*c^2*d^5 +
d^5)*a*b^3*x^4 + 4*(5*c^3*d^4 + 3*c*d^4)*a*b^3*x^3 + 3*(5*c^4*d^3 + 6*c^2*d^3 + d^3)*a*b^3*x^2 + 6*(c^5*d^2 +
2*c^3*d^2 + c*d^2)*a*b^3*x + (c^6*d + 3*c^4*d + 3*c^2*d + d)*a*b^3 + (a*b^3*d^4*x^3 + 3*a*b^3*c*d^3*x^2 + 3*a
*b^3*c^2*d^2*x + a*b^3*c^3*d)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + 3*(a*b^3*d^5*x^4 + 4*a*b^3*c*d^4*x^3 + (6*
c^2*d^3 + d^3)*a*b^3*x^2 + 2*(2*c^3*d^2 + c*d^2)*a*b^3*x + (c^4*d + c^2*d)*a*b^3)*(d^2*x^2 + 2*c*d*x + c^2 + 1
) + 3*(a*b^3*d^6*x^5 + 5*a*b^3*c*d^5*x^4 + 2*(5*c^2*d^4 + d^4)*a*b^3*x^3 + 2*(5*c^3*d^3 + 3*c*d^3)*a*b^3*x^2 +
(5*c^4*d^2 + 6*c^2*d^2 + d^2)*a*b^3*x + (c^5*d + 2*c^3*d + c*d)*a*b^3)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log
(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) + 3*(a^2*b^2*d^6*x^5 + 5*a^2*b^2*c*d^5*x^4 + 2*(5*c^2*d^4 + d^4)
*a^2*b^2*x^3 + 2*(5*c^3*d^3 + 3*c*d^3)*a^2*b^2*x^2 + (5*c^4*d^2 + 6*c^2*d^2 + d^2)*a^2*b^2*x + (c^5*d + 2*c^3*
d + c*d)*a^2*b^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) + integrate(1/2*(d^8*x^8 + 8*c*d^7*x^7 + c^8 + 4*(7*c^2*d
^6 + d^6)*x^6 + 4*c^6 + 8*(7*c^3*d^5 + 3*c*d^5)*x^5 + 2*(35*c^4*d^4 + 30*c^2*d^4 + 3*d^4)*x^4 + 6*c^4 + 8*(7*c
^5*d^3 + 10*c^3*d^3 + 3*c*d^3)*x^3 + (d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x + c^4 + 3)*(d^2*x^2 +
2*c*d*x + c^2 + 1)^2 + 4*(7*c^6*d^2 + 15*c^4*d^2 + 9*c^2*d^2 + d^2)*x^2 + (4*d^5*x^5 + 20*c*d^4*x^4 + 4*c^5 +
4*(10*c^2*d^3 + d^3)*x^3 + 4*c^3 + 4*(10*c^3*d^2 + 3*c*d^2)*x^2 + (20*c^4*d + 12*c^2*d + 3*d)*x + 3*c)*(d^2*x^
2 + 2*c*d*x + c^2 + 1)^(3/2) + 3*(2*d^6*x^6 + 12*c*d^5*x^5 + 2*c^6 + 2*(15*c^2*d^4 + 2*d^4)*x^4 + 4*c^4 + 8*(5
*c^3*d^3 + 2*c*d^3)*x^3 + (30*c^4*d^2 + 24*c^2*d^2 + d^2)*x^2 + c^2 + 2*(6*c^5*d + 8*c^3*d + c*d)*x - 1)*(d^2*
x^2 + 2*c*d*x + c^2 + 1) + 4*c^2 + 8*(c^7*d + 3*c^5*d + 3*c^3*d + c*d)*x + (4*d^7*x^7 + 28*c*d^6*x^6 + 4*c^7 +
12*(7*c^2*d^5 + d^5)*x^5 + 12*c^5 + 20*(7*c^3*d^4 + 3*c*d^4)*x^4 + (140*c^4*d^3 + 120*c^2*d^3 + 9*d^3)*x^3 +
9*c^3 + 3*(28*c^5*d^2 + 40*c^3*d^2 + 9*c*d^2)*x^2 + (28*c^6*d + 60*c^4*d + 27*c^2*d + d)*x + c)*sqrt(d^2*x^2 +
2*c*d*x + c^2 + 1) + 1)/(a*b^2*d^8*x^8 + 8*a*b^2*c*d^7*x^7 + 4*(7*c^2*d^6 + d^6)*a*b^2*x^6 + 8*(7*c^3*d^5 + 3
*c*d^5)*a*b^2*x^5 + 2*(35*c^4*d^4 + 30*c^2*d^4 + 3*d^4)*a*b^2*x^4 + 8*(7*c^5*d^3 + 10*c^3*d^3 + 3*c*d^3)*a*b^2
*x^3 + 4*(7*c^6*d^2 + 15*c^4*d^2 + 9*c^2*d^2 + d^2)*a*b^2*x^2 + 8*(c^7*d + 3*c^5*d + 3*c^3*d + c*d)*a*b^2*x +
(c^8 + 4*c^6 + 6*c^4 + 4*c^2 + 1)*a*b^2 + (a*b^2*d^4*x^4 + 4*a*b^2*c*d^3*x^3 + 6*a*b^2*c^2*d^2*x^2 + 4*a*b^2*c
^3*d*x + a*b^2*c^4)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + 4*(a*b^2*d^5*x^5 + 5*a*b^2*c*d^4*x^4 + (10*c^2*d^3 + d^3
)*a*b^2*x^3 + (10*c^3*d^2 + 3*c*d^2)*a*b^2*x^2 + (5*c^4*d + 3*c^2*d)*a*b^2*x + (c^5 + c^3)*a*b^2)*(d^2*x^2 + 2
*c*d*x + c^2 + 1)^(3/2) + 6*(a*b^2*d^6*x^6 + 6*a*b^2*c*d^5*x^5 + (15*c^2*d^4 + 2*d^4)*a*b^2*x^4 + 4*(5*c^3*d^3
+ 2*c*d^3)*a*b^2*x^3 + (15*c^4*d^2 + 12*c^2*d^2 + d^2)*a*b^2*x^2 + 2*(3*c^5*d + 4*c^3*d + c*d)*a*b^2*x + (c^6
+ 2*c^4 + c^2)*a*b^2)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + (b^3*d^8*x^8 + 8*b^3*c*d^7*x^7 + 4*(7*c^2*d^6 + d^6)*b^
3*x^6 + 8*(7*c^3*d^5 + 3*c*d^5)*b^3*x^5 + 2*(35*c^4*d^4 + 30*c^2*d^4 + 3*d^4)*b^3*x^4 + 8*(7*c^5*d^3 + 10*c^3*
d^3 + 3*c*d^3)*b^3*x^3 + 4*(7*c^6*d^2 + 15*c^4*d^2 + 9*c^2*d^2 + d^2)*b^3*x^2 + 8*(c^7*d + 3*c^5*d + 3*c^3*d +
c*d)*b^3*x + (c^8 + 4*c^6 + 6*c^4 + 4*c^2 + 1)*b^3 + (b^3*d^4*x^4 + 4*b^3*c*d^3*x^3 + 6*b^3*c^2*d^2*x^2 + 4*b
^3*c^3*d*x + b^3*c^4)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + 4*(b^3*d^5*x^5 + 5*b^3*c*d^4*x^4 + (10*c^2*d^3 + d^3)*
b^3*x^3 + (10*c^3*d^2 + 3*c*d^2)*b^3*x^2 + (5*c^4*d + 3*c^2*d)*b^3*x + (c^5 + c^3)*b^3)*(d^2*x^2 + 2*c*d*x + c
^2 + 1)^(3/2) + 6*(b^3*d^6*x^6 + 6*b^3*c*d^5*x^5 + (15*c^2*d^4 + 2*d^4)*b^3*x^4 + 4*(5*c^3*d^3 + 2*c*d^3)*b^3*
x^3 + (15*c^4*d^2 + 12*c^2*d^2 + d^2)*b^3*x^2 + 2*(3*c^5*d + 4*c^3*d + c*d)*b^3*x + (c^6 + 2*c^4 + c^2)*b^3)*(
d^2*x^2 + 2*c*d*x + c^2 + 1) + 4*(b^3*d^7*x^7 + 7*b^3*c*d^6*x^6 + 3*(7*c^2*d^5 + d^5)*b^3*x^5 + 5*(7*c^3*d^4 +
3*c*d^4)*b^3*x^4 + (35*c^4*d^3 + 30*c^2*d^3 + 3*d^3)*b^3*x^3 + 3*(7*c^5*d^2 + 10*c^3*d^2 + 3*c*d^2)*b^3*x^2 +
(7*c^6*d + 15*c^4*d + 9*c^2*d + d)*b^3*x + (c^7 + 3*c^5 + 3*c^3 + c)*b^3)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*
log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) + 4*(a*b^2*d^7*x^7 + 7*a*b^2*c*d^6*x^6 + 3*(7*c^2*d^5 + d^5)*
a*b^2*x^5 + 5*(7*c^3*d^4 + 3*c*d^4)*a*b^2*x^4 + (35*c^4*d^3 + 30*c^2*d^3 + 3*d^3)*a*b^2*x^3 + 3*(7*c^5*d^2 + 1
0*c^3*d^2 + 3*c*d^2)*a*b^2*x^2 + (7*c^6*d + 15*c^4*d + 9*c^2*d + d)*a*b^2*x + (c^7 + 3*c^5 + 3*c^3 + c)*a*b^2)
*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b^{3} \operatorname{arsinh}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname{arsinh}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname{arsinh}\left (d x + c\right ) + a^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*arcsinh(d*x+c))^3,x, algorithm="fricas")

[Out]

integral(1/(b^3*arcsinh(d*x + c)^3 + 3*a*b^2*arcsinh(d*x + c)^2 + 3*a^2*b*arcsinh(d*x + c) + a^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \operatorname{asinh}{\left (c + d x \right )}\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*asinh(d*x+c))**3,x)

[Out]

Integral((a + b*asinh(c + d*x))**(-3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*arcsinh(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((b*arcsinh(d*x + c) + a)^(-3), x)