3.5.37 \(\int \frac {x^3}{\sqrt {1+c^2 x^2} (a+b \sinh ^{-1}(c x))^2} \, dx\) [437]

Optimal. Leaf size=142 \[ -\frac {x^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{4 b^2 c^4}+\frac {3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^4}+\frac {3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{4 b^2 c^4}-\frac {3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^4} \]

[Out]

-x^3/b/c/(a+b*arcsinh(c*x))-3/4*Chi((a+b*arcsinh(c*x))/b)*cosh(a/b)/b^2/c^4+3/4*Chi(3*(a+b*arcsinh(c*x))/b)*co
sh(3*a/b)/b^2/c^4+3/4*Shi((a+b*arcsinh(c*x))/b)*sinh(a/b)/b^2/c^4-3/4*Shi(3*(a+b*arcsinh(c*x))/b)*sinh(3*a/b)/
b^2/c^4

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Rubi [A]
time = 0.23, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5818, 5780, 5556, 3384, 3379, 3382} \begin {gather*} -\frac {3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{4 b^2 c^4}+\frac {3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^4}+\frac {3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{4 b^2 c^4}-\frac {3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^4}-\frac {x^3}{b c \left (a+b \sinh ^{-1}(c x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2),x]

[Out]

-(x^3/(b*c*(a + b*ArcSinh[c*x]))) - (3*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(4*b^2*c^4) + (3*Cosh[(
3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(4*b^2*c^4) + (3*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])
/b])/(4*b^2*c^4) - (3*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(4*b^2*c^4)

Rule 3379

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 3382

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]

Rule 3384

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 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5780

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Sinh
[-a/b + x/b]^m*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 5818

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

Rubi steps

\begin {align*} \int \frac {x^3}{\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=-\frac {x^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {3 \int \frac {x^2}{a+b \sinh ^{-1}(c x)} \, dx}{b c}\\ &=-\frac {x^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {3 \text {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^4}\\ &=-\frac {x^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {3 \text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 (a+b x)}+\frac {\cosh (3 x)}{4 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^4}\\ &=-\frac {x^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {3 \text {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^4}+\frac {3 \text {Subst}\left (\int \frac {\cosh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^4}\\ &=-\frac {x^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {\left (3 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^4}+\frac {\left (3 \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^4}+\frac {\left (3 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^4}-\frac {\left (3 \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^4}\\ &=-\frac {x^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{4 b^2 c^4}+\frac {3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right )}{4 b^2 c^4}+\frac {3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{4 b^2 c^4}-\frac {3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right )}{4 b^2 c^4}\\ \end {align*}

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Mathematica [A]
time = 0.20, size = 113, normalized size = 0.80 \begin {gather*} -\frac {x^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {3 \left (-\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )-\sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )\right )}{4 b^2 c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2),x]

[Out]

-(x^3/(b*c*(a + b*ArcSinh[c*x]))) + (3*(-(Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]]) + Cosh[(3*a)/b]*CoshInte
gral[3*(a/b + ArcSinh[c*x])] + Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] - Sinh[(3*a)/b]*SinhIntegral[3*(a/b
+ ArcSinh[c*x])]))/(4*b^2*c^4)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(363\) vs. \(2(134)=268\).
time = 6.97, size = 364, normalized size = 2.56

method	result	size
default	\(-\frac {-4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c^{3} x^{3}-\sqrt {c^{2} x^{2}+1}+3 c x}{8 c^{4} b \left (a +b \arcsinh \left (c x \right )\right )}-\frac {3 \,{\mathrm e}^{\frac {3 a}{b}} \expIntegral \left (1, 3 \arcsinh \left (c x \right )+\frac {3 a}{b}\right )}{8 c^{4} b^{2}}+\frac {-\frac {3 \sqrt {c^{2} x^{2}+1}}{8}+\frac {3 c x}{8}}{c^{4} b \left (a +b \arcsinh \left (c x \right )\right )}+\frac {3 \,{\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \arcsinh \left (c x \right )+\frac {a}{b}\right )}{8 c^{4} b^{2}}+\frac {\frac {3 \arcsinh \left (c x \right ) {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\arcsinh \left (c x \right )-\frac {a}{b}\right ) b}{8}+\frac {3 \,{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\arcsinh \left (c x \right )-\frac {a}{b}\right ) a}{8}+\frac {3 b c x}{8}+\frac {3 \sqrt {c^{2} x^{2}+1}\, b}{8}}{c^{4} b^{2} \left (a +b \arcsinh \left (c x \right )\right )}-\frac {4 b \,c^{3} x^{3}+4 \sqrt {c^{2} x^{2}+1}\, b \,c^{2} x^{2}+3 \arcsinh \left (c x \right ) {\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \arcsinh \left (c x \right )-\frac {3 a}{b}\right ) b +3 \,{\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \arcsinh \left (c x \right )-\frac {3 a}{b}\right ) a +3 b c x +\sqrt {c^{2} x^{2}+1}\, b}{8 c^{4} b^{2} \left (a +b \arcsinh \left (c x \right )\right )}\)	\(364\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(-4*c^2*x^2*(c^2*x^2+1)^(1/2)+4*c^3*x^3-(c^2*x^2+1)^(1/2)+3*c*x)/c^4/b/(a+b*arcsinh(c*x))-3/8/c^4/b^2*exp
(3*a/b)*Ei(1,3*arcsinh(c*x)+3*a/b)+3/8*(-(c^2*x^2+1)^(1/2)+c*x)/c^4/b/(a+b*arcsinh(c*x))+3/8/c^4/b^2*exp(a/b)*
Ei(1,arcsinh(c*x)+a/b)+3/8/c^4/b^2*(arcsinh(c*x)*exp(-a/b)*Ei(1,-arcsinh(c*x)-a/b)*b+exp(-a/b)*Ei(1,-arcsinh(c
*x)-a/b)*a+b*c*x+(c^2*x^2+1)^(1/2)*b)/(a+b*arcsinh(c*x))-1/8/c^4/b^2*(4*b*c^3*x^3+4*(c^2*x^2+1)^(1/2)*b*c^2*x^
2+3*arcsinh(c*x)*exp(-3*a/b)*Ei(1,-3*arcsinh(c*x)-3*a/b)*b+3*exp(-3*a/b)*Ei(1,-3*arcsinh(c*x)-3*a/b)*a+3*b*c*x
+(c^2*x^2+1)^(1/2)*b)/(a+b*arcsinh(c*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(x^3/(a+b*arcsinh(c*x))^2/(c^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \sqrt {c^{2} x^{2} + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3/((a + b*asinh(c*x))**2*sqrt(c**2*x**2 + 1)), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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