\(\int \frac {(a+b \text {arcsinh}(c x))^2}{(\pi +c^2 \pi x^2)^{3/2}} \, dx\) [256]

Optimal result

Integrand size = 25, antiderivative size = 104 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {(a+b \text {arcsinh}(c x))^2}{c \pi ^{3/2}}+\frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {2 b (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c \pi ^{3/2}}-\frac {b^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c \pi ^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {5787, 5797, 3799, 2221, 2317, 2438} \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi c^2 x^2+\pi }}+\frac {(a+b \text {arcsinh}(c x))^2}{\pi ^{3/2} c}-\frac {2 b \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))}{\pi ^{3/2} c}-\frac {b^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{\pi ^{3/2} c} \]

[In]

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

[Out]

(a + b*ArcSinh[c*x])^2/(c*Pi^(3/2)) + (x*(a + b*ArcSinh[c*x])^2)/(Pi*Sqrt[Pi + c^2*Pi*x^2]) - (2*b*(a + b*ArcS
inh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(c*Pi^(3/2)) - (b^2*PolyLog[2, -E^(2*ArcSinh[c*x])])/(c*Pi^(3/2))

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m
+ 1)/(d*(m + 1))), x] + Dist[2*I, Int[(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x]
, x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 5787

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

Rule 5797

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

Rubi steps \begin{align*} \text {integral}& = \frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {(2 b c) \int \frac {x (a+b \text {arcsinh}(c x))}{1+c^2 x^2} \, dx}{\pi ^{3/2}} \\ & = \frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {(2 b) \text {Subst}(\int (a+b x) \tanh (x) \, dx,x,\text {arcsinh}(c x))}{c \pi ^{3/2}} \\ & = \frac {(a+b \text {arcsinh}(c x))^2}{c \pi ^{3/2}}+\frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {(4 b) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\text {arcsinh}(c x)\right )}{c \pi ^{3/2}} \\ & = \frac {(a+b \text {arcsinh}(c x))^2}{c \pi ^{3/2}}+\frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {2 b (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c \pi ^{3/2}}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c \pi ^{3/2}} \\ & = \frac {(a+b \text {arcsinh}(c x))^2}{c \pi ^{3/2}}+\frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {2 b (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c \pi ^{3/2}}+\frac {b^2 \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{c \pi ^{3/2}} \\ & = \frac {(a+b \text {arcsinh}(c x))^2}{c \pi ^{3/2}}+\frac {x (a+b \text {arcsinh}(c x))^2}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {2 b (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c \pi ^{3/2}}-\frac {b^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c \pi ^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.47 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {-b^2 \left (-c x+\sqrt {1+c^2 x^2}\right ) \text {arcsinh}(c x)^2+2 b \text {arcsinh}(c x) \left (a c x-b \sqrt {1+c^2 x^2} \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )\right )+a \left (a c x-b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )\right )+b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )}{c \pi ^{3/2} \sqrt {1+c^2 x^2}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(242\) vs. \(2(114)=228\).

Time = 0.26 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.34

method	result	size
default	\(\frac {a^{2} x}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}+b^{2} \left (-\frac {\left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \operatorname {arcsinh}\left (c x \right )^{2}}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}+1\right )}+\frac {2 \operatorname {arcsinh}\left (c x \right )^{2}}{c \,\pi ^{\frac {3}{2}}}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c \,\pi ^{\frac {3}{2}}}-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c \,\pi ^{\frac {3}{2}}}\right )+2 a b \left (\frac {2 \,\operatorname {arcsinh}\left (c x \right )}{c \,\pi ^{\frac {3}{2}}}-\frac {\left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \operatorname {arcsinh}\left (c x \right )}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}+1\right )}-\frac {\ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c \,\pi ^{\frac {3}{2}}}\right )\)	\(243\)
parts	\(\frac {a^{2} x}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}+b^{2} \left (-\frac {\left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \operatorname {arcsinh}\left (c x \right )^{2}}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}+1\right )}+\frac {2 \operatorname {arcsinh}\left (c x \right )^{2}}{c \,\pi ^{\frac {3}{2}}}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c \,\pi ^{\frac {3}{2}}}-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c \,\pi ^{\frac {3}{2}}}\right )+2 a b \left (\frac {2 \,\operatorname {arcsinh}\left (c x \right )}{c \,\pi ^{\frac {3}{2}}}-\frac {\left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \operatorname {arcsinh}\left (c x \right )}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}+1\right )}-\frac {\ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c \,\pi ^{\frac {3}{2}}}\right )\)	\(243\)

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integral(sqrt(pi + pi*c^2*x^2)*(b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)/(pi^2*c^4*x^4 + 2*pi^2*c^2*x^2
+ pi^2), x)

Sympy [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {\int \frac {a^{2}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {3}{2}}} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}} \,d x \]

[In]

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

[Out]

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