\(\int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \, dx\) [272]

Optimal result

Integrand size = 37, antiderivative size = 408 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \, dx=-\frac {b^2 x}{3 d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b (a+b \text {arcsinh}(c x))}{3 c d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}}+\frac {2 x (a+b \text {arcsinh}(c x))^2}{3 d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {x (a+b \text {arcsinh}(c x))^2}{3 d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )}+\frac {2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 c d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {4 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 c d^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}} \] Output:

-1/3*b^2*x/d^2/f^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+1/3*b*(a+b*arcsinh( 
c*x))/c/d^2/f^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)/(c^2*x^2+1)^(1/2)+2/3* 
x*(a+b*arcsinh(c*x))^2/d^2/f^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+1/3*x*( 
a+b*arcsinh(c*x))^2/d^2/f^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)/(c^2*x^2+1 
)+2/3*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^2/c/d^2/f^2/(d+I*c*d*x)^(1/2)/( 
f-I*c*f*x)^(1/2)-4/3*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^2/f^2/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)-2/3*b^2*(c 
^2*x^2+1)^(1/2)*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)/c/d^2/f^2/(d+I*c*d*x 
)^(1/2)/(f-I*c*f*x)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 7.79 (sec) , antiderivative size = 642, normalized size of antiderivative = 1.57 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*ArcSinh[c*x])^2/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) 
,x]
 

Output:

(4*a^2*c*x*(3 + 2*c^2*x^2) - b^2*(c*x - 6*c*x*ArcSinh[c*x]^2 + (4*I)*Pi*Ar 
cSinh[c*x]*Cosh[3*ArcSinh[c*x]] + 2*ArcSinh[c*x]^2*Cosh[3*ArcSinh[c*x]] - 
(2*I)*Pi*Cosh[3*ArcSinh[c*x]]*Log[1 - I/E^ArcSinh[c*x]] + 4*ArcSinh[c*x]*C 
osh[3*ArcSinh[c*x]]*Log[1 - I/E^ArcSinh[c*x]] + (2*I)*Pi*Cosh[3*ArcSinh[c* 
x]]*Log[1 + I/E^ArcSinh[c*x]] + 4*ArcSinh[c*x]*Cosh[3*ArcSinh[c*x]]*Log[1 
+ I/E^ArcSinh[c*x]] - (8*I)*Pi*Cosh[3*ArcSinh[c*x]]*Log[1 + E^ArcSinh[c*x] 
] - (2*I)*Pi*Cosh[3*ArcSinh[c*x]]*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] + 
 (8*I)*Pi*Cosh[3*ArcSinh[c*x]]*Log[Cosh[ArcSinh[c*x]/2]] + (2*I)*Pi*Cosh[3 
*ArcSinh[c*x]]*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]] + 2*Sqrt[1 + c^2*x^2] 
*(((-3*I)*Pi + 6*ArcSinh[c*x])*Log[1 - I/E^ArcSinh[c*x]] + I*((2*I)*ArcSin 
h[c*x] + 6*Pi*ArcSinh[c*x] - (3*I)*ArcSinh[c*x]^2 + 3*(Pi - (2*I)*ArcSinh[ 
c*x])*Log[1 + I/E^ArcSinh[c*x]] - 12*Pi*Log[1 + E^ArcSinh[c*x]] - 3*Pi*Log 
[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] + 12*Pi*Log[Cosh[ArcSinh[c*x]/2]] + 3* 
Pi*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]])) - 16*(1 + c^2*x^2)^(3/2)*PolyLo 
g[2, (-I)/E^ArcSinh[c*x]] - 16*(1 + c^2*x^2)^(3/2)*PolyLog[2, I/E^ArcSinh[ 
c*x]] + Sinh[3*ArcSinh[c*x]] - 2*ArcSinh[c*x]^2*Sinh[3*ArcSinh[c*x]]) + 2* 
a*b*(Sqrt[1 + c^2*x^2]*(2 - 3*Log[1 + c^2*x^2]) - Cosh[3*ArcSinh[c*x]]*Log 
[1 + c^2*x^2] + 2*ArcSinh[c*x]*(3*c*x + Sinh[3*ArcSinh[c*x]])))/(12*d^2*f^ 
2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(c + c^3*x^2))
 

Rubi [A] (verified)

Time = 1.96 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.56, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {6211, 6203, 6202, 6212, 3042, 26, 4201, 2620, 2715, 2838, 6213, 208}

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.

\(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \, dx\)

\(\Big \downarrow \) 6211

\(\displaystyle \frac {\left (c^2 x^2+1\right )^{5/2} \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^{5/2}}dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\)

\(\Big \downarrow \) 6203

\(\displaystyle \frac {\left (c^2 x^2+1\right )^{5/2} \left (-\frac {2}{3} b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx+\frac {2}{3} \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\)

\(\Big \downarrow \) 6202

\(\displaystyle \frac {\left (c^2 x^2+1\right )^{5/2} \left (-\frac {2}{3} b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}-2 b c \int \frac {x (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\)

\(\Big \downarrow \) 6212

\(\displaystyle \frac {\left (c^2 x^2+1\right )^{5/2} \left (-\frac {2}{3} b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}-\frac {2 b \int \frac {c x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{c}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\left (c^2 x^2+1\right )^{5/2} \left (-\frac {2}{3} b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}-\frac {2 b \int -i (a+b \text {arcsinh}(c x)) \tan (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{c}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {\left (c^2 x^2+1\right )^{5/2} \left (-\frac {2}{3} b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}+\frac {2 i b \int (a+b \text {arcsinh}(c x)) \tan (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{c}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\)

\(\Big \downarrow \) 4201

\(\displaystyle \frac {\left (c^2 x^2+1\right )^{5/2} \left (\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}+\frac {2 i b \left (2 i \int \frac {e^{2 \text {arcsinh}(c x)} (a+b \text {arcsinh}(c x))}{1+e^{2 \text {arcsinh}(c x)}}d\text {arcsinh}(c x)-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c}\right )-\frac {2}{3} b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {\left (c^2 x^2+1\right )^{5/2} \left (-\frac {2}{3} b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}+\frac {2 i b \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b \int \log \left (1+e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {\left (c^2 x^2+1\right )^{5/2} \left (-\frac {2}{3} b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}+\frac {2 i b \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{4} b \int e^{-2 \text {arcsinh}(c x)} \log \left (1+e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {\left (c^2 x^2+1\right )^{5/2} \left (-\frac {2}{3} b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}+\frac {2 i b \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\)

\(\Big \downarrow \) 6213

\(\displaystyle \frac {\left (c^2 x^2+1\right )^{5/2} \left (-\frac {2}{3} b c \left (\frac {b \int \frac {1}{\left (c^2 x^2+1\right )^{3/2}}dx}{2 c}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}+\frac {2 i b \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\)

\(\Big \downarrow \) 208

\(\displaystyle \frac {\left (c^2 x^2+1\right )^{5/2} \left (\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}+\frac {2 i b \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {2}{3} b c \left (\frac {b x}{2 c \sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 \left (c^2 x^2+1\right )}\right )\right )}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\)

Input:

Int[(a + b*ArcSinh[c*x])^2/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)),x]
 

Output:

((1 + c^2*x^2)^(5/2)*((x*(a + b*ArcSinh[c*x])^2)/(3*(1 + c^2*x^2)^(3/2)) - 
 (2*b*c*((b*x)/(2*c*Sqrt[1 + c^2*x^2]) - (a + b*ArcSinh[c*x])/(2*c^2*(1 + 
c^2*x^2))))/3 + (2*((x*(a + b*ArcSinh[c*x])^2)/Sqrt[1 + c^2*x^2] + ((2*I)* 
b*(((-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))/ 
3))/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))
 

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 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

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 6203

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

rule 6211

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_ 
) + (g_.)*(x_))^(q_), x_Symbol] :> Simp[(d + e*x)^q*((f + g*x)^q/(1 + c^2*x 
^2)^q)   Int[(d + e*x)^(p - q)*(1 + c^2*x^2)^q*(a + b*ArcSinh[c*x])^n, x], 
x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^ 
2 + e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 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]
 

rule 6213

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1764 vs. \(2 (362 ) = 724\).

Time = 6.17 (sec) , antiderivative size = 1765, normalized size of antiderivative = 4.33

method	result	size
default	\(\text {Expression too large to display}\)	\(1765\)
parts	\(\text {Expression too large to display}\)	\(1765\)

Input:

int((a+b*arcsinh(x*c))^2/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(5/2),x,method=_RET 
URNVERBOSE)
 

Output:

a^2*(1/3*I/c/d/f/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)+1/d*(I/c/d/f/(d+I*c*d 
*x)^(1/2)/(f-I*c*f*x)^(3/2)+2/d*(-1/3*I/d/c/f/(f-I*c*f*x)^(3/2)*(d+I*c*d*x 
)^(1/2)-1/3*I/c/d/f^2/(f-I*c*f*x)^(1/2)*(d+I*c*d*x)^(1/2))))+1/3*b^2*(2-8* 
polylog(2,I*(x*c+(c^2*x^2+1)^(1/2)))-8*polylog(2,-I*(x*c+(c^2*x^2+1)^(1/2) 
))+8*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*ln(1-I*(x*c+(c^2*x^2+1)^(1/2)))*x^5*c^ 
5+12*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*ln(1+I*(x*c+(c^2*x^2+1)^(1/2)))*x*c-2* 
arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x^3*c^3-3*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x* 
c+4*arcsinh(x*c)^2+2*arcsinh(x*c)-8*arcsinh(x*c)*ln(1-I*(x*c+(c^2*x^2+1)^( 
1/2)))-24*ln(1+I*(x*c+(c^2*x^2+1)^(1/2)))*arcsinh(x*c)*x^4*c^4-24*ln(1+I*( 
x*c+(c^2*x^2+1)^(1/2)))*arcsinh(x*c)*x^2*c^2+20*(c^2*x^2+1)^(1/2)*arcsinh( 
x*c)*ln(1+I*(x*c+(c^2*x^2+1)^(1/2)))*x^3*c^3+20*(c^2*x^2+1)^(1/2)*arcsinh( 
x*c)*ln(1-I*(x*c+(c^2*x^2+1)^(1/2)))*x^3*c^3+8*(c^2*x^2+1)^(1/2)*arcsinh(x 
*c)*ln(1+I*(x*c+(c^2*x^2+1)^(1/2)))*x^5*c^5-4*(c^2*x^2+1)^(1/2)*x^5*c^5+11 
*c^4*x^4-24*arcsinh(x*c)*ln(1-I*(x*c+(c^2*x^2+1)^(1/2)))*x^4*c^4-24*arcsin 
h(x*c)*ln(1-I*(x*c+(c^2*x^2+1)^(1/2)))*x^2*c^2+12*(c^2*x^2+1)^(1/2)*polylo 
g(2,-I*(x*c+(c^2*x^2+1)^(1/2)))*x*c+12*(c^2*x^2+1)^(1/2)*polylog(2,I*(x*c+ 
(c^2*x^2+1)^(1/2)))*x*c+8*(c^2*x^2+1)^(1/2)*polylog(2,I*(x*c+(c^2*x^2+1)^( 
1/2)))*x^5*c^5+8*(c^2*x^2+1)^(1/2)*polylog(2,-I*(x*c+(c^2*x^2+1)^(1/2)))*x 
^5*c^5+20*(c^2*x^2+1)^(1/2)*polylog(2,I*(x*c+(c^2*x^2+1)^(1/2)))*x^3*c^3+2 
0*(c^2*x^2+1)^(1/2)*polylog(2,-I*(x*c+(c^2*x^2+1)^(1/2)))*x^3*c^3-8*arc...
 

Fricas [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (i \, c d x + d\right )}^{\frac {5}{2}} {\left (-i \, c f x + f\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))^2/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(5/2),x, algo 
rithm="fricas")
 

Output:

1/3*((2*b^2*c^2*x^3 + 3*b^2*x)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*log(c* 
x + sqrt(c^2*x^2 + 1))^2 + 3*(c^4*d^3*f^3*x^4 + 2*c^2*d^3*f^3*x^2 + d^3*f^ 
3)*integral(1/3*(3*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*a^2 + 2*(3*sqrt(I* 
c*d*x + d)*sqrt(-I*c*f*x + f)*a*b - (2*b^2*c^3*x^3 + 3*b^2*c*x)*sqrt(c^2*x 
^2 + 1)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f))*log(c*x + sqrt(c^2*x^2 + 1)) 
)/(c^6*d^3*f^3*x^6 + 3*c^4*d^3*f^3*x^4 + 3*c^2*d^3*f^3*x^2 + d^3*f^3), x)) 
/(c^4*d^3*f^3*x^4 + 2*c^2*d^3*f^3*x^2 + d^3*f^3)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((a+b*asinh(c*x))**2/(d+I*c*d*x)**(5/2)/(f-I*c*f*x)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (i \, c d x + d\right )}^{\frac {5}{2}} {\left (-i \, c f x + f\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))^2/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(5/2),x, algo 
rithm="maxima")
 

Output:

1/3*a*b*c*(1/(c^4*d^(5/2)*f^(5/2)*x^2 + c^2*d^(5/2)*f^(5/2)) - 2*log(c^2*x 
^2 + 1)/(c^2*d^(5/2)*f^(5/2))) + 2/3*a*b*(x/((c^2*d*f*x^2 + d*f)^(3/2)*d*f 
) + 2*x/(sqrt(c^2*d*f*x^2 + d*f)*d^2*f^2))*arcsinh(c*x) + 1/3*a^2*(x/((c^2 
*d*f*x^2 + d*f)^(3/2)*d*f) + 2*x/(sqrt(c^2*d*f*x^2 + d*f)*d^2*f^2)) + b^2* 
integrate(log(c*x + sqrt(c^2*x^2 + 1))^2/((I*c*d*x + d)^(5/2)*(-I*c*f*x + 
f)^(5/2)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+b*arcsinh(c*x))^2/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(5/2),x, algo 
rithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{5/2}\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{5/2}} \,d x \] Input:

int((a + b*asinh(c*x))^2/((d + c*d*x*1i)^(5/2)*(f - c*f*x*1i)^(5/2)),x)
 

Output:

int((a + b*asinh(c*x))^2/((d + c*d*x*1i)^(5/2)*(f - c*f*x*1i)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \, dx=\frac {6 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c^{4} x^{4}+2 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, c^{2} x^{2}+\sqrt {c i x +1}\, \sqrt {-c i x +1}}d x \right ) a b \,c^{2} x^{2}+6 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c^{4} x^{4}+2 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, c^{2} x^{2}+\sqrt {c i x +1}\, \sqrt {-c i x +1}}d x \right ) a b +3 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c^{4} x^{4}+2 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, c^{2} x^{2}+\sqrt {c i x +1}\, \sqrt {-c i x +1}}d x \right ) b^{2} c^{2} x^{2}+3 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{\sqrt {c i x +1}\, \sqrt {-c i x +1}\, c^{4} x^{4}+2 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, c^{2} x^{2}+\sqrt {c i x +1}\, \sqrt {-c i x +1}}d x \right ) b^{2}+2 a^{2} c^{2} x^{3}+3 a^{2} x}{3 \sqrt {f}\, \sqrt {d}\, \sqrt {c i x +1}\, \sqrt {-c i x +1}\, d^{2} f^{2} \left (c^{2} x^{2}+1\right )} \] Input:

int((a+b*asinh(c*x))^2/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(5/2),x)
 

Output:

(6*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*int(asinh(c*x)/(sqrt(c*i*x + 1)*sqrt 
( - c*i*x + 1)*c**4*x**4 + 2*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*c**2*x**2 
+ sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)),x)*a*b*c**2*x**2 + 6*sqrt(c*i*x + 1) 
*sqrt( - c*i*x + 1)*int(asinh(c*x)/(sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*c** 
4*x**4 + 2*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*c**2*x**2 + sqrt(c*i*x + 1)* 
sqrt( - c*i*x + 1)),x)*a*b + 3*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*int(asin 
h(c*x)**2/(sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*c**4*x**4 + 2*sqrt(c*i*x + 1 
)*sqrt( - c*i*x + 1)*c**2*x**2 + sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)),x)*b* 
*2*c**2*x**2 + 3*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*int(asinh(c*x)**2/(sqr 
t(c*i*x + 1)*sqrt( - c*i*x + 1)*c**4*x**4 + 2*sqrt(c*i*x + 1)*sqrt( - c*i* 
x + 1)*c**2*x**2 + sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)),x)*b**2 + 2*a**2*c* 
*2*x**3 + 3*a**2*x)/(3*sqrt(f)*sqrt(d)*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)* 
d**2*f**2*(c**2*x**2 + 1))