\(\int \sqrt {d+i c d x} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx\) [245]

Optimal result

Integrand size = 37, antiderivative size = 508 \[ \int \sqrt {d+i c d x} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {4 i b^2 f \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 c}+\frac {1}{4} b^2 f x \sqrt {d+i c d x} \sqrt {f-i c f x}-\frac {2 i b^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )}{27 c}-\frac {b^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)}{4 c \sqrt {1+c^2 x^2}}+\frac {2 i b f x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{3 \sqrt {1+c^2 x^2}}-\frac {b c f x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{2 \sqrt {1+c^2 x^2}}+\frac {2 i b c^2 f x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+\frac {1}{2} f x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2-\frac {i f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c}+\frac {f \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {1+c^2 x^2}} \] Output:

-4/9*I*b^2*f*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/c+1/4*b^2*f*x*(d+I*c*d*x) 
^(1/2)*(f-I*c*f*x)^(1/2)-2/27*I*b^2*f*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)* 
(c^2*x^2+1)/c-1/4*b^2*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)+2/3*I*b*f*x*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)*(a+b*ar 
csinh(c*x))/(c^2*x^2+1)^(1/2)-1/2*b*c*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)+2/9*I*b*c^2*f*x^3*(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)+1/2*f*x*(d+I*c 
*d*x)^(1/2)*(f-I*c*f*x)^(1/2)*(a+b*arcsinh(c*x))^2-1/3*I*f*(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/c+1/6*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 = 2.83 (sec) , antiderivative size = 705, normalized size of antiderivative = 1.39 \[ \int \sqrt {d+i c d x} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {108 i a b c f x \sqrt {d+i c d x} \sqrt {f-i c f x}-72 i a^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-108 i b^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+108 a^2 c f x \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-72 i a^2 c^2 f x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+36 b^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^3-54 a b f \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (2 \text {arcsinh}(c x))-4 i b^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (3 \text {arcsinh}(c x))+108 a^2 \sqrt {d} f^{3/2} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )+27 b^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (2 \text {arcsinh}(c x))+18 b f \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^2 \left (6 a-3 i b \sqrt {1+c^2 x^2}-i b \cosh (3 \text {arcsinh}(c x))+3 b \sinh (2 \text {arcsinh}(c x))\right )+12 i a b f \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (3 \text {arcsinh}(c x))+6 b f \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x) \left (-9 b \cosh (2 \text {arcsinh}(c x))+2 \left (9 i b c x-9 i a \sqrt {1+c^2 x^2}-3 i a \cosh (3 \text {arcsinh}(c x))+9 a \sinh (2 \text {arcsinh}(c x))+i b \sinh (3 \text {arcsinh}(c x))\right )\right )}{216 c \sqrt {1+c^2 x^2}} \] Input:

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

Output:

((108*I)*a*b*c*f*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] - (72*I)*a^2*f*Sqrt 
[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - (108*I)*b^2*f*Sqrt[d + 
 I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 108*a^2*c*f*x*Sqrt[d + I*c 
*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - (72*I)*a^2*c^2*f*x^2*Sqrt[d + 
I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 36*b^2*f*Sqrt[d + I*c*d*x]* 
Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 - 54*a*b*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c 
*f*x]*Cosh[2*ArcSinh[c*x]] - (4*I)*b^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f* 
x]*Cosh[3*ArcSinh[c*x]] + 108*a^2*Sqrt[d]*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]] + 27*b^2*f*Sq 
rt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2*ArcSinh[c*x]] + 18*b*f*Sqrt[d + I 
*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2*(6*a - (3*I)*b*Sqrt[1 + c^2*x^2] 
- I*b*Cosh[3*ArcSinh[c*x]] + 3*b*Sinh[2*ArcSinh[c*x]]) + (12*I)*a*b*f*Sqrt 
[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[3*ArcSinh[c*x]] + 6*b*f*Sqrt[d + I*c* 
d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*(-9*b*Cosh[2*ArcSinh[c*x]] + 2*((9*I)* 
b*c*x - (9*I)*a*Sqrt[1 + c^2*x^2] - (3*I)*a*Cosh[3*ArcSinh[c*x]] + 9*a*Sin 
h[2*ArcSinh[c*x]] + I*b*Sinh[3*ArcSinh[c*x]])))/(216*c*Sqrt[1 + c^2*x^2])
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.51, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {6211, 27, 6253, 2009}

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.

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

\(\Big \downarrow \) 6211

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6253

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {f \sqrt {d+i c d x} \sqrt {f-i c f x} \left (\frac {2}{9} i b c^2 x^3 (a+b \text {arcsinh}(c x))+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2-\frac {i \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c}-\frac {1}{2} b c x^2 (a+b \text {arcsinh}(c x))+\frac {2}{3} i b x (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^3}{6 b c}-\frac {b^2 \text {arcsinh}(c x)}{4 c}+\frac {1}{4} b^2 x \sqrt {c^2 x^2+1}-\frac {2 i b^2 \left (c^2 x^2+1\right )^{3/2}}{27 c}-\frac {4 i b^2 \sqrt {c^2 x^2+1}}{9 c}\right )}{\sqrt {c^2 x^2+1}}\)

Input:

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

Output:

(f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*((((-4*I)/9)*b^2*Sqrt[1 + c^2*x^2]) 
/c + (b^2*x*Sqrt[1 + c^2*x^2])/4 - (((2*I)/27)*b^2*(1 + c^2*x^2)^(3/2))/c 
- (b^2*ArcSinh[c*x])/(4*c) + ((2*I)/3)*b*x*(a + b*ArcSinh[c*x]) - (b*c*x^2 
*(a + b*ArcSinh[c*x]))/2 + ((2*I)/9)*b*c^2*x^3*(a + b*ArcSinh[c*x]) + (x*S 
qrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/2 - ((I/3)*(1 + c^2*x^2)^(3/2)*(a 
 + b*ArcSinh[c*x])^2)/c + (a + b*ArcSinh[c*x])^3/(6*b*c)))/Sqrt[1 + c^2*x^ 
2]
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 6253

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d 
_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p*(a 
+ b*ArcSinh[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] 
 && EqQ[e, c^2*d] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n 
, 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] || EqQ[m, 1] || (EqQ[m, 2] 
&& LtQ[p, -2]))
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1398 vs. \(2 (413 ) = 826\).

Time = 6.88 (sec) , antiderivative size = 1399, normalized size of antiderivative = 2.75

method	result	size
default	\(\text {Expression too large to display}\)	\(1399\)
parts	\(\text {Expression too large to display}\)	\(1399\)

Input:

int((d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(3/2)*(a+b*arcsinh(x*c))^2,x,method=_RET 
URNVERBOSE)
 

Output:

1/3*I*a^2/c/f*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(5/2)-1/6*I*a^2/c*(f-I*c*f*x)^ 
(3/2)*(d+I*c*d*x)^(1/2)-1/2*I*a^2*f/c*(f-I*c*f*x)^(1/2)*(d+I*c*d*x)^(1/2)+ 
1/2*a^2*d*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*(1/6*(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*f-1/216*I*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)*(4*c^4*x 
^4+4*(c^2*x^2+1)^(1/2)*c^3*x^3+5*c^2*x^2+3*(c^2*x^2+1)^(1/2)*x*c+1)*(9*arc 
sinh(x*c)^2-6*arcsinh(x*c)+2)*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)*f/(c^2*x^2+1)/c-1/8*I*(I*(x*c- 
I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)*(c^2*x^2+(c^2*x^2+1)^(1/2)*x*c+1)*(arcsin 
h(x*c)^2-2*arcsinh(x*c)+2)*f/(c^2*x^2+1)/c-1/8*I*(I*(x*c-I)*d)^(1/2)*(-I*( 
I+x*c)*f)^(1/2)*(c^2*x^2-(c^2*x^2+1)^(1/2)*x*c+1)*(arcsinh(x*c)^2+2*arcsin 
h(x*c)+2)*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)*f/(c^2*x^2+1)/c-1/216*I*(I*(x*c-I)*d)^(1/2)*(-I*(I 
+x*c)*f)^(1/2)*(4*c^4*x^4-4*(c^2*x^2+1)^(1/2)*c^3*x^3+5*c^2*x^2-3*(c^2*x^2 
+1)^(1/2)*x*c+1)*(9*arcsinh(x*c)^2+6*arcsinh(x*c)+2)*f/(c^2*x^2+1)/c)+2*a* 
b*(1/4*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)/(c^2*x^2+1)^(1/2)/c*arcsin 
h(x*c)^2*f-1/72*I*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)*(4*c^4*x^4+4...
 

Fricas [F]

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

integrate((d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(3/2)*(a+b*arcsinh(c*x))^2,x, algo 
rithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((d+I*c*d*x)**(1/2)*(f-I*c*f*x)**(3/2)*(a+b*asinh(c*x))**2,x)
 

Output:

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

Maxima [F(-2)]

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

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

integrate((d+I*c*d*x)^(1/2)*(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:Error: Bad Argument TypeError: Bad 
Argument TypeError: Bad Argument TypeDone
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \sqrt {d+i c d x} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {\sqrt {f}\, \sqrt {d}\, f \left (6 \mathit {asin} \left (\frac {\sqrt {-c i x +1}}{\sqrt {2}}\right ) a^{2} i -2 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, a^{2} c^{2} i \,x^{2}+3 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, a^{2} c x -2 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, a^{2} i -12 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right ) x d x \right ) a b \,c^{2} i +12 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right )d x \right ) a b c -6 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right )^{2} x d x \right ) b^{2} c^{2} i +6 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right )^{2}d x \right ) b^{2} c \right )}{6 c} \] Input:

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

Output:

(sqrt(f)*sqrt(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**2*i*x**2 + 3*sqrt(c*i*x + 1)*sqrt( - c* 
i*x + 1)*a**2*c*x - 2*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*a**2*i - 12*int(s 
qrt(c*i*x + 1)*sqrt( - c*i*x + 1)*asinh(c*x)*x,x)*a*b*c**2*i + 12*int(sqrt 
(c*i*x + 1)*sqrt( - c*i*x + 1)*asinh(c*x),x)*a*b*c - 6*int(sqrt(c*i*x + 1) 
*sqrt( - c*i*x + 1)*asinh(c*x)**2*x,x)*b**2*c**2*i + 6*int(sqrt(c*i*x + 1) 
*sqrt( - c*i*x + 1)*asinh(c*x)**2,x)*b**2*c))/(6*c)