\(\int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx\) [62]

Optimal result

Integrand size = 26, antiderivative size = 62 \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=-\frac {b c \sqrt {\pi }}{6 x^2}-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 \pi x^3}+\frac {1}{3} b c^3 \sqrt {\pi } \log (x) \]

[Out]

-1/3*(Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(c*x))/Pi/x^3-1/6*b*c*Pi^(1/2)/x^2+1/3*b*c^3*ln(x)*Pi^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {5800, 14} \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 \pi x^3}+\frac {1}{3} \sqrt {\pi } b c^3 \log (x)-\frac {\sqrt {\pi } b c}{6 x^2} \]

[In]

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

[Out]

-1/6*(b*c*Sqrt[Pi])/x^2 - ((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*Pi*x^3) + (b*c^3*Sqrt[Pi]*Log[x])/
3

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 5800

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] - Dist[b*c*(n/(f*(m + 1)))*Simp[(
d + e*x^2)^p/(1 + c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]
/; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 \pi x^3}+\frac {1}{3} \left (b c \sqrt {\pi }\right ) \int \frac {1+c^2 x^2}{x^3} \, dx \\ & = -\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 \pi x^3}+\frac {1}{3} \left (b c \sqrt {\pi }\right ) \int \left (\frac {1}{x^3}+\frac {c^2}{x}\right ) \, dx \\ & = -\frac {b c \sqrt {\pi }}{6 x^2}-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 \pi x^3}+\frac {1}{3} b c^3 \sqrt {\pi } \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.55 \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\frac {\sqrt {\pi } \left (-b c x-3 b c^3 x^3-2 a \sqrt {1+c^2 x^2}-2 a c^2 x^2 \sqrt {1+c^2 x^2}-2 b \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)+2 b c^3 x^3 \log (x)\right )}{6 x^3} \]

[In]

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

[Out]

(Sqrt[Pi]*(-(b*c*x) - 3*b*c^3*x^3 - 2*a*Sqrt[1 + c^2*x^2] - 2*a*c^2*x^2*Sqrt[1 + c^2*x^2] - 2*b*(1 + c^2*x^2)^
(3/2)*ArcSinh[c*x] + 2*b*c^3*x^3*Log[x]))/(6*x^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(500\) vs. \(2(50)=100\).

Time = 0.16 (sec) , antiderivative size = 501, normalized size of antiderivative = 8.08

method	result	size
default	\(-\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3 \pi \,x^{3}}-\frac {2 b \,c^{3} \sqrt {\pi }\, \operatorname {arcsinh}\left (c x \right )}{3}+\frac {b \sqrt {\pi }\, x^{4} \operatorname {arcsinh}\left (c x \right ) c^{7}}{3 c^{4} x^{4}+3 c^{2} x^{2}+1}-\frac {b \sqrt {\pi }\, x^{3} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) c^{6}}{3 c^{4} x^{4}+3 c^{2} x^{2}+1}+\frac {b \sqrt {\pi }\, x^{4} c^{7}}{18 c^{4} x^{4}+18 c^{2} x^{2}+6}-\frac {b \sqrt {\pi }\, x^{2} \left (c^{2} x^{2}+1\right ) c^{5}}{6 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right )}+\frac {b \sqrt {\pi }\, x^{2} \operatorname {arcsinh}\left (c x \right ) c^{5}}{3 c^{4} x^{4}+3 c^{2} x^{2}+1}-\frac {2 b \sqrt {\pi }\, x \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) c^{4}}{3 c^{4} x^{4}+3 c^{2} x^{2}+1}-\frac {b \sqrt {\pi }\, \left (c^{2} x^{2}+1\right ) c^{3}}{3 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right )}+\frac {b \,c^{3} \sqrt {\pi }\, \operatorname {arcsinh}\left (c x \right )}{9 c^{4} x^{4}+9 c^{2} x^{2}+3}-\frac {4 b \sqrt {\pi }\, \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) c^{2}}{3 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) x}-\frac {b \sqrt {\pi }\, \left (c^{2} x^{2}+1\right ) c}{6 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) x^{2}}-\frac {b \sqrt {\pi }\, \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right )}{3 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) x^{3}}+\frac {b \,c^{3} \sqrt {\pi }\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right )}{3}\)	\(501\)
parts	\(-\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3 \pi \,x^{3}}-\frac {2 b \,c^{3} \sqrt {\pi }\, \operatorname {arcsinh}\left (c x \right )}{3}+\frac {b \sqrt {\pi }\, x^{4} \operatorname {arcsinh}\left (c x \right ) c^{7}}{3 c^{4} x^{4}+3 c^{2} x^{2}+1}-\frac {b \sqrt {\pi }\, x^{3} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) c^{6}}{3 c^{4} x^{4}+3 c^{2} x^{2}+1}+\frac {b \sqrt {\pi }\, x^{4} c^{7}}{18 c^{4} x^{4}+18 c^{2} x^{2}+6}-\frac {b \sqrt {\pi }\, x^{2} \left (c^{2} x^{2}+1\right ) c^{5}}{6 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right )}+\frac {b \sqrt {\pi }\, x^{2} \operatorname {arcsinh}\left (c x \right ) c^{5}}{3 c^{4} x^{4}+3 c^{2} x^{2}+1}-\frac {2 b \sqrt {\pi }\, x \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) c^{4}}{3 c^{4} x^{4}+3 c^{2} x^{2}+1}-\frac {b \sqrt {\pi }\, \left (c^{2} x^{2}+1\right ) c^{3}}{3 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right )}+\frac {b \,c^{3} \sqrt {\pi }\, \operatorname {arcsinh}\left (c x \right )}{9 c^{4} x^{4}+9 c^{2} x^{2}+3}-\frac {4 b \sqrt {\pi }\, \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) c^{2}}{3 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) x}-\frac {b \sqrt {\pi }\, \left (c^{2} x^{2}+1\right ) c}{6 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) x^{2}}-\frac {b \sqrt {\pi }\, \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right )}{3 \left (3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) x^{3}}+\frac {b \,c^{3} \sqrt {\pi }\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right )}{3}\)	\(501\)

[In]

int((a+b*arcsinh(c*x))*(Pi*c^2*x^2+Pi)^(1/2)/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*a/Pi/x^3*(Pi*c^2*x^2+Pi)^(3/2)-2/3*b*c^3*Pi^(1/2)*arcsinh(c*x)+b*Pi^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^4*arc
sinh(c*x)*c^7-b*Pi^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^3*(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^6+1/6*b*Pi^(1/2)/(3*c^4*
x^4+3*c^2*x^2+1)*x^4*c^7-1/6*b*Pi^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^2*(c^2*x^2+1)*c^5+b*Pi^(1/2)/(3*c^4*x^4+3*c^
2*x^2+1)*x^2*arcsinh(c*x)*c^5-2*b*Pi^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x*(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^4-1/3*b*
Pi^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*(c^2*x^2+1)*c^3+1/3*b*Pi^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*arcsinh(c*x)*c^3-4/3*b
*Pi^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/x*(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^2-1/6*b*Pi^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/
x^2*(c^2*x^2+1)*c-1/3*b*Pi^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/x^3*(c^2*x^2+1)^(1/2)*arcsinh(c*x)+1/3*b*c^3*Pi^(1/2)
*ln((c*x+(c^2*x^2+1)^(1/2))^2-1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (50) = 100\).

Time = 0.30 (sec) , antiderivative size = 217, normalized size of antiderivative = 3.50 \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=-\frac {2 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (b c^{4} x^{4} + 2 \, b c^{2} x^{2} + b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - \sqrt {\pi } {\left (b c^{5} x^{5} + b c^{3} x^{3}\right )} \log \left (\frac {\pi + \pi c^{2} x^{6} + \pi c^{2} x^{2} + \pi x^{4} + \sqrt {\pi } \sqrt {\pi + \pi c^{2} x^{2}} \sqrt {c^{2} x^{2} + 1} {\left (x^{4} - 1\right )}}{c^{2} x^{4} + x^{2}}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (2 \, a c^{4} x^{4} + 4 \, a c^{2} x^{2} - {\left (b c x^{3} - b c x\right )} \sqrt {c^{2} x^{2} + 1} + 2 \, a\right )}}{6 \, {\left (c^{2} x^{5} + x^{3}\right )}} \]

[In]

integrate((a+b*arcsinh(c*x))*(pi*c^2*x^2+pi)^(1/2)/x^4,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\sqrt {\pi } \left (\int \frac {a \sqrt {c^{2} x^{2} + 1}}{x^{4}}\, dx + \int \frac {b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x^{4}}\, dx\right ) \]

[In]

integrate((a+b*asinh(c*x))*(pi*c**2*x**2+pi)**(1/2)/x**4,x)

[Out]

sqrt(pi)*(Integral(a*sqrt(c**2*x**2 + 1)/x**4, x) + Integral(b*sqrt(c**2*x**2 + 1)*asinh(c*x)/x**4, x))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (50) = 100\).

Time = 0.22 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.15 \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=-\frac {{\left (\pi ^{\frac {3}{2}} \left (-1\right )^{2 \, \pi + 2 \, \pi c^{2} x^{2}} c^{2} \log \left (2 \, \pi c^{2} + \frac {2 \, \pi }{x^{2}}\right ) - \pi ^{\frac {3}{2}} c^{2} \log \left (x^{2} + \frac {1}{c^{2}}\right ) + \frac {\pi \sqrt {\pi + \pi c^{4} x^{4} + 2 \, \pi c^{2} x^{2}}}{x^{2}}\right )} b c}{6 \, \pi } - \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} b \operatorname {arsinh}\left (c x\right )}{3 \, \pi x^{3}} - \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} a}{3 \, \pi x^{3}} \]

[In]

integrate((a+b*arcsinh(c*x))*(pi*c^2*x^2+pi)^(1/2)/x^4,x, algorithm="maxima")

[Out]

-1/6*(pi^(3/2)*(-1)^(2*pi + 2*pi*c^2*x^2)*c^2*log(2*pi*c^2 + 2*pi/x^2) - pi^(3/2)*c^2*log(x^2 + 1/c^2) + pi*sq
rt(pi + pi*c^4*x^4 + 2*pi*c^2*x^2)/x^2)*b*c/pi - 1/3*(pi + pi*c^2*x^2)^(3/2)*b*arcsinh(c*x)/(pi*x^3) - 1/3*(pi
+ pi*c^2*x^2)^(3/2)*a/(pi*x^3)

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*arcsinh(c*x))*(pi*c^2*x^2+pi)^(1/2)/x^4,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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {\Pi \,c^2\,x^2+\Pi }}{x^4} \,d x \]

[In]

int(((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(1/2))/x^4,x)

[Out]

int(((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(1/2))/x^4, x)