\(\int \frac {(a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^{3/2}} \, dx\) [312]

Optimal result

Integrand size = 25, antiderivative size = 179 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{c d \sqrt {d+c^2 d x^2}}-\frac {2 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c d \sqrt {d+c^2 d x^2}}-\frac {b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c d \sqrt {d+c^2 d x^2}} \] Output:

x*(a+b*arcsinh(c*x))^2/d/(c^2*d*x^2+d)^(1/2)+(c^2*x^2+1)^(1/2)*(a+b*arcsin 
h(c*x))^2/c/d/(c^2*d*x^2+d)^(1/2)-2*b*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x)) 
*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)/c/d/(c^2*d*x^2+d)^(1/2)-b^2*(c^2*x^2+1)^( 
1/2)*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)/c/d/(c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {-b^2 \left (-c x+\sqrt {1+c^2 x^2}\right ) \text {arcsinh}(c x)^2+2 b \text {arcsinh}(c x) \left (a c x-b \sqrt {1+c^2 x^2} \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )\right )+a \left (a c x-b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )\right )+b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )}{c d \sqrt {d+c^2 d x^2}} \] Input:

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

Output:

(-(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*Sqr 
t[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*d*Sqrt[d + c^2*d*x^2])
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.64 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.74, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {6202, 6212, 3042, 26, 4201, 2620, 2715, 2838}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (c^2 d x^2+d\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 6202

\(\displaystyle \frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}-\frac {2 b c \sqrt {c^2 x^2+1} \int \frac {x (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{d \sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 6212

\(\displaystyle \frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}-\frac {2 b \sqrt {c^2 x^2+1} \int \frac {c x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{c d \sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}-\frac {2 b \sqrt {c^2 x^2+1} \int -i (a+b \text {arcsinh}(c x)) \tan (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{c d \sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \int (a+b \text {arcsinh}(c x)) \tan (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{c d \sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 4201

\(\displaystyle \frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \int \frac {e^{2 \text {arcsinh}(c x)} (a+b \text {arcsinh}(c x))}{1+e^{2 \text {arcsinh}(c x)}}d\text {arcsinh}(c x)-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b \int \log \left (1+e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{4} b \int e^{-2 \text {arcsinh}(c x)} \log \left (1+e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {x (a+b \text {arcsinh}(c x))^2}{d \sqrt {c^2 d x^2+d}}+\frac {2 i b \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c d \sqrt {c^2 d x^2+d}}\)

Input:

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

Output:

(x*(a + b*ArcSinh[c*x])^2)/(d*Sqrt[d + c^2*d*x^2]) + ((2*I)*b*Sqrt[1 + c^2 
*x^2]*(((-1/2*I)*(a + b*ArcSinh[c*x])^2)/b + (2*I)*(((a + b*ArcSinh[c*x])* 
Log[1 + E^(2*ArcSinh[c*x])])/2 + (b*PolyLog[2, -E^(2*ArcSinh[c*x])])/4)))/ 
(c*d*Sqrt[d + c^2*d*x^2])
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2620

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] - Si 
mp[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 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[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 2838

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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4201

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] + Simp[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 6202

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] - Simp 
[b*c*(n/d)*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]   Int[x*((a + b*ArcSinh[ 
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 6212

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), 
 x_Symbol] :> Simp[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]
 

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.92

method	result	size
default	\(\frac {a^{2} x}{d \sqrt {c^{2} d \,x^{2}+d}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2} x}{d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2}}{d^{2} c \sqrt {c^{2} x^{2}+1}}-\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{2} c}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{2} c}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )}{\sqrt {c^{2} x^{2}+1}\, d^{2} c}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) x}{d^{2} \left (c^{2} x^{2}+1\right )}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{2} c}\)	\(343\)
parts	\(\frac {a^{2} x}{d \sqrt {c^{2} d \,x^{2}+d}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2} x}{d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2}}{d^{2} c \sqrt {c^{2} x^{2}+1}}-\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{2} c}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{2} c}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )}{\sqrt {c^{2} x^{2}+1}\, d^{2} c}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) x}{d^{2} \left (c^{2} x^{2}+1\right )}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{2} c}\)	\(343\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(c**2*x**2 + 1)*a**2*c*x + 2*int(asinh(c*x)/(sqrt(c**2*x**2 + 1)*c**2 
*x**2 + sqrt(c**2*x**2 + 1)),x)*a*b*c**3*x**2 + 2*int(asinh(c*x)/(sqrt(c** 
2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*a*b*c + int(asinh(c*x)**2/ 
(sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b**2*c**3*x**2 + 
int(asinh(c*x)**2/(sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x) 
*b**2*c + a**2*c**2*x**2 + a**2)/(sqrt(d)*c*d*(c**2*x**2 + 1))