3.367 \(\int \frac{(1+c^2 x^2)^{3/2}}{a+b \sinh ^{-1}(c x)} \, dx\)

Optimal. Leaf size=144 \[ \frac{\cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{2 b c}+\frac{\cosh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 b c}-\frac{\sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{2 b c}-\frac{\sinh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 b c}+\frac{3 \log \left (a+b \sinh ^{-1}(c x)\right )}{8 b c} \]

[Out]

(Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(2*b*c) + (Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcSi
nh[c*x]))/b])/(8*b*c) + (3*Log[a + b*ArcSinh[c*x]])/(8*b*c) - (Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*
x]))/b])/(2*b*c) - (Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(8*b*c)

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Rubi [A]  time = 0.286632, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {5699, 3312, 3303, 3298, 3301} \[ \frac{\cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right )}{2 b c}+\frac{\cosh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b c}-\frac{\sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right )}{2 b c}-\frac{\sinh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b c}+\frac{3 \log \left (a+b \sinh ^{-1}(c x)\right )}{8 b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cosh[(2*a)/b]*CoshIntegral[(2*a)/b + 2*ArcSinh[c*x]])/(2*b*c) + (Cosh[(4*a)/b]*CoshIntegral[(4*a)/b + 4*ArcSi
nh[c*x]])/(8*b*c) + (3*Log[a + b*ArcSinh[c*x]])/(8*b*c) - (Sinh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcSinh[c*x]
])/(2*b*c) - (Sinh[(4*a)/b]*SinhIntegral[(4*a)/b + 4*ArcSinh[c*x]])/(8*b*c)

Rule 5699

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p/c, Subst[Int[
(a + b*x)^n*Cosh[x]^(2*p + 1), x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IG
tQ[2*p, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

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{\left (1+c^2 x^2\right )^{3/2}}{a+b \sinh ^{-1}(c x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cosh ^4(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{c}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{3}{8 (a+b x)}+\frac{\cosh (2 x)}{2 (a+b x)}+\frac{\cosh (4 x)}{8 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c}\\ &=\frac{3 \log \left (a+b \sinh ^{-1}(c x)\right )}{8 b c}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (4 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 c}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (2 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 c}\\ &=\frac{3 \log \left (a+b \sinh ^{-1}(c x)\right )}{8 b c}+\frac{\cosh \left (\frac{2 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 c}+\frac{\cosh \left (\frac{4 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 c}-\frac{\sinh \left (\frac{2 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 c}-\frac{\sinh \left (\frac{4 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 c}\\ &=\frac{\cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right )}{2 b c}+\frac{\cosh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b c}+\frac{3 \log \left (a+b \sinh ^{-1}(c x)\right )}{8 b c}-\frac{\sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right )}{2 b c}-\frac{\sinh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b c}\\ \end{align*}

Mathematica [A]  time = 0.25348, size = 109, normalized size = 0.76 \[ \frac{4 \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (2 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+\cosh \left (\frac{4 a}{b}\right ) \text{Chi}\left (4 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-4 \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (2 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-\sinh \left (\frac{4 a}{b}\right ) \text{Shi}\left (4 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+3 \log \left (a+b \sinh ^{-1}(c x)\right )}{8 b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSinh[c*x])] + Cosh[(4*a)/b]*CoshIntegral[4*(a/b + ArcSinh[c*x])] + 3
*Log[a + b*ArcSinh[c*x]] - 4*Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] - Sinh[(4*a)/b]*SinhIntegral[4
*(a/b + ArcSinh[c*x])])/(8*b*c)

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Maple [A]  time = 0.144, size = 139, normalized size = 1. \begin{align*}{\frac{3\,\ln \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) }{8\,cb}}-{\frac{1}{16\,cb}{{\rm e}^{4\,{\frac{a}{b}}}}{\it Ei} \left ( 1,4\,{\it Arcsinh} \left ( cx \right ) +4\,{\frac{a}{b}} \right ) }-{\frac{1}{4\,cb}{{\rm e}^{2\,{\frac{a}{b}}}}{\it Ei} \left ( 1,2\,{\it Arcsinh} \left ( cx \right ) +2\,{\frac{a}{b}} \right ) }-{\frac{1}{4\,cb}{{\rm e}^{-2\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-2\,{\it Arcsinh} \left ( cx \right ) -2\,{\frac{a}{b}} \right ) }-{\frac{1}{16\,cb}{{\rm e}^{-4\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-4\,{\it Arcsinh} \left ( cx \right ) -4\,{\frac{a}{b}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*ln(a+b*arcsinh(c*x))/b/c-1/16/c/b*exp(4*a/b)*Ei(1,4*arcsinh(c*x)+4*a/b)-1/4/c/b*exp(2*a/b)*Ei(1,2*arcsinh(
c*x)+2*a/b)-1/4/c/b*exp(-2*a/b)*Ei(1,-2*arcsinh(c*x)-2*a/b)-1/16/c/b*exp(-4*a/b)*Ei(1,-4*arcsinh(c*x)-4*a/b)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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