3.1.58 \(\int \sqrt {\pi +c^2 \pi x^2} (a+b \sinh ^{-1}(c x)) \, dx\) [58]

Optimal. Leaf size=67 \[ -\frac {1}{4} b c \sqrt {\pi } x^2+\frac {1}{2} x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {\sqrt {\pi } \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c} \]

[Out]

-1/4*b*c*x^2*Pi^(1/2)+1/4*(a+b*arcsinh(c*x))^2*Pi^(1/2)/b/c+1/2*x*(a+b*arcsinh(c*x))*(Pi*c^2*x^2+Pi)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {5785, 5783, 30} \begin {gather*} \frac {1}{2} x \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )+\frac {\sqrt {\pi } \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c}-\frac {1}{4} \sqrt {\pi } b c x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(b*c*Sqrt[Pi]*x^2) + (x*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/2 + (Sqrt[Pi]*(a + b*ArcSinh[c*x])^2)
/(4*b*c)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 5783

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ
[e, c^2*d] && NeQ[n, -1]

Rule 5785

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

Rubi steps

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

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Mathematica [A]
time = 0.09, size = 69, normalized size = 1.03 \begin {gather*} \frac {\sqrt {\pi } \left (4 a c x \sqrt {1+c^2 x^2}+2 b \sinh ^{-1}(c x)^2-b \cosh \left (2 \sinh ^{-1}(c x)\right )+2 \sinh ^{-1}(c x) \left (2 a+b \sinh \left (2 \sinh ^{-1}(c x)\right )\right )\right )}{8 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[Pi]*(4*a*c*x*Sqrt[1 + c^2*x^2] + 2*b*ArcSinh[c*x]^2 - b*Cosh[2*ArcSinh[c*x]] + 2*ArcSinh[c*x]*(2*a + b*S
inh[2*ArcSinh[c*x]])))/(8*c)

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Maple [A]
time = 1.00, size = 100, normalized size = 1.49

method	result	size
default	\(\frac {a x \sqrt {\pi \,c^{2} x^{2}+\pi }}{2}+\frac {a \pi \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{2 \sqrt {\pi \,c^{2}}}+\frac {b \sqrt {\pi }\, \left (2 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x c -c^{2} x^{2}+\arcsinh \left (c x \right )^{2}-1\right )}{4 c}\)	\(100\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*x*(Pi*c^2*x^2+Pi)^(1/2)+1/2*a*Pi*ln(Pi*c^2*x/(Pi*c^2)^(1/2)+(Pi*c^2*x^2+Pi)^(1/2))/(Pi*c^2)^(1/2)+1/4*b*
Pi^(1/2)*(2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x*c-c^2*x^2+arcsinh(c*x)^2-1)/c

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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

[Out]

integral(sqrt(pi + pi*c^2*x^2)*(b*arcsinh(c*x) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(pi)*(Integral(a*sqrt(c**2*x**2 + 1), x) + Integral(b*sqrt(c**2*x**2 + 1)*asinh(c*x), 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((a+b*arcsinh(c*x))*(pi*c^2*x^2+pi)^(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 \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {\Pi \,c^2\,x^2+\Pi } \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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