\(\int (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx\) [244]

Optimal result

Integrand size = 37, antiderivative size = 408 \[ \int (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {15}{64} b^2 d f x \sqrt {d+i c d x} \sqrt {f-i c f x}+\frac {1}{32} b^2 d f x \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )-\frac {9 b^2 d f \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)}{64 c \sqrt {1+c^2 x^2}}-\frac {3 b c d f x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}-\frac {b d f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{8 c}+\frac {3}{8} d f x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2+\frac {1}{4} d f x \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {d f \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^3}{8 b c \sqrt {1+c^2 x^2}} \] Output:

15/64*b^2*d*f*x*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)+1/32*b^2*d*f*x*(d+I*c* 
d*x)^(1/2)*(f-I*c*f*x)^(1/2)*(c^2*x^2+1)-9/64*b^2*d*f*(d+I*c*d*x)^(1/2)*(f 
-I*c*f*x)^(1/2)*arcsinh(c*x)/c/(c^2*x^2+1)^(1/2)-3/8*b*c*d*f*x^2*(d+I*c*d* 
x)^(1/2)*(f-I*c*f*x)^(1/2)*(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2)-1/8*b*d*f* 
(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))/c 
+3/8*d*f*x*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)*(a+b*arcsinh(c*x))^2+1/4*d* 
f*x*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2+1 
/8*d*f*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)*(a+b*arcsinh(c*x))^3/b/c/(c^2*x 
^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 2.86 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.28 \[ \int (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {160 a^2 c d f x \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+64 a^2 c^3 d f x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+32 b^2 d f \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^3-64 a b d f \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (2 \text {arcsinh}(c x))-4 a b d f \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (4 \text {arcsinh}(c x))+96 a^2 d^{3/2} f^{3/2} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )+32 b^2 d f \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (2 \text {arcsinh}(c x))+b^2 d f \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (4 \text {arcsinh}(c x))+8 b d f \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^2 (12 a+8 b \sinh (2 \text {arcsinh}(c x))+b \sinh (4 \text {arcsinh}(c x)))-4 b d f \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x) (16 b \cosh (2 \text {arcsinh}(c x))+b \cosh (4 \text {arcsinh}(c x))-4 a (8 \sinh (2 \text {arcsinh}(c x))+\sinh (4 \text {arcsinh}(c x))))}{256 c \sqrt {1+c^2 x^2}} \] Input:

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

Output:

(160*a^2*c*d*f*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 6 
4*a^2*c^3*d*f*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 
32*b^2*d*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 - 64*a*b*d*f 
*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] - 4*a*b*d*f*Sqrt 
[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[4*ArcSinh[c*x]] + 96*a^2*d^(3/2)*f^(3 
/2)*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt 
[f - I*c*f*x]] + 32*b^2*d*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2*Arc 
Sinh[c*x]] + b^2*d*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[4*ArcSinh[c* 
x]] + 8*b*d*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2*(12*a + 8 
*b*Sinh[2*ArcSinh[c*x]] + b*Sinh[4*ArcSinh[c*x]]) - 4*b*d*f*Sqrt[d + I*c*d 
*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*(16*b*Cosh[2*ArcSinh[c*x]] + b*Cosh[4*A 
rcSinh[c*x]] - 4*a*(8*Sinh[2*ArcSinh[c*x]] + Sinh[4*ArcSinh[c*x]])))/(256* 
c*Sqrt[1 + c^2*x^2])
 

Rubi [A] (verified)

Time = 1.24 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.67, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {6211, 6201, 6200, 6191, 262, 222, 6198, 6213, 211, 211, 222}

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.

\(\displaystyle \int (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx\)

\(\Big \downarrow \) 6211

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

\(\Big \downarrow \) 6201

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

\(\Big \downarrow \) 6200

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 222

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

\(\Big \downarrow \) 6198

\(\displaystyle \frac {(d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (-\frac {1}{2} b c \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2-b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )+\frac {(a+b \text {arcsinh}(c x))^3}{6 b c}\right )\right )}{\left (c^2 x^2+1\right )^{3/2}}\)

\(\Big \downarrow \) 6213

\(\displaystyle \frac {(d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (-\frac {1}{2} b c \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \int \left (c^2 x^2+1\right )^{3/2}dx}{4 c}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2-b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )+\frac {(a+b \text {arcsinh}(c x))^3}{6 b c}\right )\right )}{\left (c^2 x^2+1\right )^{3/2}}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {(d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (-\frac {1}{2} b c \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \int \sqrt {c^2 x^2+1}dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2-b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )+\frac {(a+b \text {arcsinh}(c x))^3}{6 b c}\right )\right )}{\left (c^2 x^2+1\right )^{3/2}}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {(d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (-\frac {1}{2} b c \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2-b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )+\frac {(a+b \text {arcsinh}(c x))^3}{6 b c}\right )\right )}{\left (c^2 x^2+1\right )^{3/2}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {(d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2-b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )+\frac {(a+b \text {arcsinh}(c x))^3}{6 b c}\right )\right )}{\left (c^2 x^2+1\right )^{3/2}}\)

Input:

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

Output:

((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*((x*(1 + c^2*x^2)^(3/2)*(a + b*Ar 
cSinh[c*x])^2)/4 + (3*((x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/2 + (a 
 + b*ArcSinh[c*x])^3/(6*b*c) - b*c*((x^2*(a + b*ArcSinh[c*x]))/2 - (b*c*(( 
x*Sqrt[1 + c^2*x^2])/(2*c^2) - ArcSinh[c*x]/(2*c^3)))/2)))/4 - (b*c*(((1 + 
 c^2*x^2)^2*(a + b*ArcSinh[c*x]))/(4*c^2) - (b*((x*(1 + c^2*x^2)^(3/2))/4 
+ (3*((x*Sqrt[1 + c^2*x^2])/2 + ArcSinh[c*x]/(2*c)))/4))/(4*c)))/2))/(1 + 
c^2*x^2)^(3/2)
 

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 6191

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

rule 6198

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[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 6200

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

rule 6201

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

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 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. 1215 vs. \(2 (336 ) = 672\).

Time = 6.77 (sec) , antiderivative size = 1216, normalized size of antiderivative = 2.98

method	result	size
default	\(\text {Expression too large to display}\)	\(1216\)
parts	\(\text {Expression too large to display}\)	\(1216\)

Input:

int((d+I*c*d*x)^(3/2)*(f-I*c*f*x)^(3/2)*(a+b*arcsinh(x*c))^2,x,method=_RET 
URNVERBOSE)
 

Output:

1/4*I*a^2/c/f*(d+I*c*d*x)^(3/2)*(f-I*c*f*x)^(5/2)+1/4*I*a^2*d/c/f*(d+I*c*d 
*x)^(1/2)*(f-I*c*f*x)^(5/2)-1/8*I*a^2*d/c*(f-I*c*f*x)^(3/2)*(d+I*c*d*x)^(1 
/2)-3/8*I*a^2*d*f/c*(f-I*c*f*x)^(1/2)*(d+I*c*d*x)^(1/2)+3/8*a^2*d^2*f^2*(( 
f-I*c*f*x)*(d+I*c*d*x))^(1/2)/(f-I*c*f*x)^(1/2)/(d+I*c*d*x)^(1/2)*ln(c^2*d 
*f*x/(c^2*d*f)^(1/2)+(c^2*d*f*x^2+d*f)^(1/2))/(c^2*d*f)^(1/2)+b^2*(1/8*(I* 
(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)/(c^2*x^2+1)^(1/2)/c*arcsinh(x*c)^3*d 
*f+1/512*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)*(8*x^5*c^5+8*x^4*c^4*(c^ 
2*x^2+1)^(1/2)+12*x^3*c^3+8*x^2*c^2*(c^2*x^2+1)^(1/2)+4*x*c+(c^2*x^2+1)^(1 
/2))*(8*arcsinh(x*c)^2-4*arcsinh(x*c)+1)*d*f/(c^2*x^2+1)/c+1/16*(I*(x*c-I) 
*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)*(2*x^3*c^3+2*x^2*c^2*(c^2*x^2+1)^(1/2)+2*x* 
c+(c^2*x^2+1)^(1/2))*(2*arcsinh(x*c)^2-2*arcsinh(x*c)+1)*d*f/(c^2*x^2+1)/c 
+1/16*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)*(2*x^3*c^3-2*x^2*c^2*(c^2*x 
^2+1)^(1/2)+2*x*c-(c^2*x^2+1)^(1/2))*(2*arcsinh(x*c)^2+2*arcsinh(x*c)+1)*d 
*f/(c^2*x^2+1)/c+1/512*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)*(8*x^5*c^5 
-8*x^4*c^4*(c^2*x^2+1)^(1/2)+12*x^3*c^3-8*x^2*c^2*(c^2*x^2+1)^(1/2)+4*x*c- 
(c^2*x^2+1)^(1/2))*(8*arcsinh(x*c)^2+4*arcsinh(x*c)+1)*d*f/(c^2*x^2+1)/c)+ 
2*a*b*(3/16*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)/(c^2*x^2+1)^(1/2)/c*a 
rcsinh(x*c)^2*d*f+1/256*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)*(8*x^5*c^ 
5+8*x^4*c^4*(c^2*x^2+1)^(1/2)+12*x^3*c^3+8*x^2*c^2*(c^2*x^2+1)^(1/2)+4*x*c 
+(c^2*x^2+1)^(1/2))*(-1+4*arcsinh(x*c))*d*f/(c^2*x^2+1)/c+1/16*(I*(x*c-...
 

Fricas [F]

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

integrate((d+I*c*d*x)^(3/2)*(f-I*c*f*x)^(3/2)*(a+b*arcsinh(c*x))^2,x, algo 
rithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F(-2)]

Exception generated. \[ \int (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((d+I*c*d*x)^(3/2)*(f-I*c*f*x)^(3/2)*(a+b*arcsinh(c*x))^2,x, algo 
rithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F(-2)]

Exception generated. \[ \int (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \] Input:

integrate((d+I*c*d*x)^(3/2)*(f-I*c*f*x)^(3/2)*(a+b*arcsinh(c*x))^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. \[ \int (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{3/2}\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{3/2} \,d x \] Input:

int((a + b*asinh(c*x))^2*(d + c*d*x*1i)^(3/2)*(f - c*f*x*1i)^(3/2),x)
 

Output:

int((a + b*asinh(c*x))^2*(d + c*d*x*1i)^(3/2)*(f - c*f*x*1i)^(3/2), x)
 

Reduce [F]

\[ \int (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {\sqrt {f}\, \sqrt {d}\, d f \left (6 \mathit {asin} \left (\frac {\sqrt {-c i x +1}}{\sqrt {2}}\right ) a^{2} i +2 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, a^{2} c^{3} x^{3}+5 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, a^{2} c x +16 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right ) x^{2}d x \right ) a b \,c^{3}+16 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right )d x \right ) a b c +8 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}+8 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right )^{2}d x \right ) b^{2} c \right )}{8 c} \] Input:

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

Output:

(sqrt(f)*sqrt(d)*d*f*(6*asin(sqrt( - c*i*x + 1)/sqrt(2))*a**2*i + 2*sqrt(c 
*i*x + 1)*sqrt( - c*i*x + 1)*a**2*c**3*x**3 + 5*sqrt(c*i*x + 1)*sqrt( - c* 
i*x + 1)*a**2*c*x + 16*int(sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*asinh(c*x)*x 
**2,x)*a*b*c**3 + 16*int(sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*asinh(c*x),x)* 
a*b*c + 8*int(sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*asinh(c*x)**2*x**2,x)*b** 
2*c**3 + 8*int(sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*asinh(c*x)**2,x)*b**2*c) 
)/(8*c)