3.1.60 \(\int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \sinh ^{-1}(c x))}{x^2} \, dx\) [60]

Optimal. Leaf size=61 \[ -\frac {\sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac {c \sqrt {\pi } \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+b c \sqrt {\pi } \log (x) \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {5805, 29, 5783} \begin {gather*} -\frac {\sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac {\sqrt {\pi } c \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+\sqrt {\pi } b c \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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 5805

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

Rubi steps

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

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Mathematica [A]
time = 0.11, size = 75, normalized size = 1.23 \begin {gather*} \frac {\sqrt {\pi } \left (-2 a \sqrt {1+c^2 x^2}+2 \left (a c x-b \sqrt {1+c^2 x^2}\right ) \sinh ^{-1}(c x)+b c x \sinh ^{-1}(c x)^2+2 b c x \log (c x)\right )}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(154\) vs. \(2(53)=106\).
time = 3.63, size = 155, normalized size = 2.54

method	result	size
default	\(-\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{\pi x}+a \,c^{2} x \sqrt {\pi \,c^{2} x^{2}+\pi }+\frac {a \,c^{2} \pi \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{\sqrt {\pi \,c^{2}}}+\frac {b c \sqrt {\pi }\, \arcsinh \left (c x \right )^{2}}{2}-b c \sqrt {\pi }\, \arcsinh \left (c x \right )-\frac {b \sqrt {\pi }\, \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{x}+b c \sqrt {\pi }\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right )\)	\(155\)

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^2,x,method=_RETURNVERBOSE)

[Out]

-a/Pi/x*(Pi*c^2*x^2+Pi)^(3/2)+a*c^2*x*(Pi*c^2*x^2+Pi)^(1/2)+a*c^2*Pi*ln(Pi*c^2*x/(Pi*c^2)^(1/2)+(Pi*c^2*x^2+Pi
)^(1/2))/(Pi*c^2)^(1/2)+1/2*b*c*Pi^(1/2)*arcsinh(c*x)^2-b*c*Pi^(1/2)*arcsinh(c*x)-b*Pi^(1/2)*arcsinh(c*x)/x*(c
^2*x^2+1)^(1/2)+b*c*Pi^(1/2)*ln((c*x+(c^2*x^2+1)^(1/2))^2-1)

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

[Out]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (54) = 108\).
time = 1.84, size = 110, normalized size = 1.80 \begin {gather*} - \frac {\sqrt {\pi } a c^{2} x}{\sqrt {c^{2} x^{2} + 1}} + \sqrt {\pi } a c \operatorname {asinh}{\left (c x \right )} - \frac {\sqrt {\pi } a}{x \sqrt {c^{2} x^{2} + 1}} + \sqrt {\pi } b c \log {\left (x \right )} + \frac {\sqrt {\pi } b c \operatorname {asinh}^{2}{\left (c x \right )}}{2} - \frac {\sqrt {\pi } b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x} \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**2,x)

[Out]

-sqrt(pi)*a*c**2*x/sqrt(c**2*x**2 + 1) + sqrt(pi)*a*c*asinh(c*x) - sqrt(pi)*a/(x*sqrt(c**2*x**2 + 1)) + sqrt(p
i)*b*c*log(x) + sqrt(pi)*b*c*asinh(c*x)**2/2 - sqrt(pi)*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^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 = -0.02 \begin {gather*} \int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {\Pi \,c^2\,x^2+\Pi }}{x^2} \,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^2,x)

[Out]

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

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