\(\int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx\) [263]

Optimal result

Integrand size = 37, antiderivative size = 544 \[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=-\frac {2 i d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^3}{3 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {8 i b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {4 b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {4 b^2 d^2 \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {4 b^2 d^2 \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {2 b^2 d^2 \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \] Output:

-2*I*d^2*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2/c/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^ 
(3/2)+2*d^2*x*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2/(d+I*c*d*x)^(3/2)/(f-I*c*f* 
x)^(3/2)+2*d^2*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))^2/c/(d+I*c*d*x)^(3/2)/ 
(f-I*c*f*x)^(3/2)-1/3*d^2*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))^3/b/c/(d+I* 
c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)+8*I*b*d^2*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c* 
x))*arctan(c*x+(c^2*x^2+1)^(1/2))/c/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)-4* 
b*d^2*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2) 
/c/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)+4*b^2*d^2*(c^2*x^2+1)^(3/2)*polylog 
(2,-I*(c*x+(c^2*x^2+1)^(1/2)))/c/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)-4*b^2 
*d^2*(c^2*x^2+1)^(3/2)*polylog(2,I*(c*x+(c^2*x^2+1)^(1/2)))/c/(d+I*c*d*x)^ 
(3/2)/(f-I*c*f*x)^(3/2)-2*b^2*d^2*(c^2*x^2+1)^(3/2)*polylog(2,-(c*x+(c^2*x 
^2+1)^(1/2))^2)/c/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)
 

Mathematica [A] (warning: unable to verify)

Time = 5.80 (sec) , antiderivative size = 530, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=\frac {\frac {6 a^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}{i+c x}-3 a^2 \sqrt {d} \sqrt {f} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )-\frac {b^2 (-i+c x) \sqrt {d+i c d x} \sqrt {f-i c f x} \left (-18 \pi \text {arcsinh}(c x)-(6-6 i) \text {arcsinh}(c x)^2+i \text {arcsinh}(c x)^3-12 (\pi -2 i \text {arcsinh}(c x)) \log \left (1+i e^{-\text {arcsinh}(c x)}\right )+24 \pi \log \left (1+e^{\text {arcsinh}(c x)}\right )+12 \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )-24 \pi \log \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-24 i \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )-\frac {12 i \text {arcsinh}(c x)^2 \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )}{\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )}\right )}{\sqrt {1+c^2 x^2} \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )^2}+\frac {3 a b \sqrt {d+i c d x} \sqrt {f-i c f x} \left (-\text {arcsinh}(c x)^2 \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+4 \text {arcsinh}(c x) \left (-i \cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+\sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+2 \left (4 \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+i \log \left (1+c^2 x^2\right )\right ) \left (i \cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+\sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )\right )}{\sqrt {1+c^2 x^2} \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}}{3 c f^2} \] Input:

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

Output:

((6*a^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/(I + c*x) - 3*a^2*Sqrt[d]*Sqr 
t[f]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] - 
(b^2*(-I + c*x)*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(-18*Pi*ArcSinh[c*x] - 
 (6 - 6*I)*ArcSinh[c*x]^2 + I*ArcSinh[c*x]^3 - 12*(Pi - (2*I)*ArcSinh[c*x] 
)*Log[1 + I/E^ArcSinh[c*x]] + 24*Pi*Log[1 + E^ArcSinh[c*x]] + 12*Pi*Log[-C 
os[(Pi + (2*I)*ArcSinh[c*x])/4]] - 24*Pi*Log[Cosh[ArcSinh[c*x]/2]] - (24*I 
)*PolyLog[2, (-I)/E^ArcSinh[c*x]] - ((12*I)*ArcSinh[c*x]^2*Sinh[ArcSinh[c* 
x]/2])/(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])))/(Sqrt[1 + c^2*x^2 
]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^2) + (3*a*b*Sqrt[d + I*c 
*d*x]*Sqrt[f - I*c*f*x]*(-(ArcSinh[c*x]^2*(Cosh[ArcSinh[c*x]/2] - I*Sinh[A 
rcSinh[c*x]/2])) + 4*ArcSinh[c*x]*((-I)*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSin 
h[c*x]/2]) + 2*(4*ArcTan[Tanh[ArcSinh[c*x]/2]] + I*Log[1 + c^2*x^2])*(I*Co 
sh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2])))/(Sqrt[1 + c^2*x^2]*(Cosh[ArcS 
inh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])))/(3*c*f^2)
 

Rubi [A] (verified)

Time = 1.19 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.45, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {6211, 27, 6259, 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.

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

\(\Big \downarrow \) 6211

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6259

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 \left (c^2 x^2+1\right )^{3/2} \left (\frac {8 i b \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{c}+\frac {2 x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}-\frac {2 i (a+b \text {arcsinh}(c x))^2}{c \sqrt {c^2 x^2+1}}-\frac {(a+b \text {arcsinh}(c x))^3}{3 b c}+\frac {2 (a+b \text {arcsinh}(c x))^2}{c}-\frac {4 b \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))}{c}+\frac {4 b^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c}-\frac {4 b^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c}-\frac {2 b^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c}\right )}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\)

Input:

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

Output:

(d^2*(1 + c^2*x^2)^(3/2)*((2*(a + b*ArcSinh[c*x])^2)/c - ((2*I)*(a + b*Arc 
Sinh[c*x])^2)/(c*Sqrt[1 + c^2*x^2]) + (2*x*(a + b*ArcSinh[c*x])^2)/Sqrt[1 
+ c^2*x^2] - (a + b*ArcSinh[c*x])^3/(3*b*c) + ((8*I)*b*(a + b*ArcSinh[c*x] 
)*ArcTan[E^ArcSinh[c*x]])/c - (4*b*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSi 
nh[c*x])])/c + (4*b^2*PolyLog[2, (-I)*E^ArcSinh[c*x]])/c - (4*b^2*PolyLog[ 
2, I*E^ArcSinh[c*x]])/c - (2*b^2*PolyLog[2, -E^(2*ArcSinh[c*x])])/c))/((d 
+ I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))
 

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 6259

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

Maple [A] (verified)

Time = 5.66 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.53

method	result	size
default	\(-\frac {\left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{3} \sqrt {-i \left (x c +i\right ) f}\, \sqrt {i \left (x c -i\right ) d}}{3 \sqrt {c^{2} x^{2}+1}\, b \,f^{2} c}+\frac {2 \left (\operatorname {arcsinh}\left (x c \right )^{2} b^{2}+2 \,\operatorname {arcsinh}\left (x c \right ) a b +a^{2}\right ) \left (x c -\sqrt {c^{2} x^{2}+1}-i\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{f^{2} c \left (c^{2} x^{2}+1\right )}+\frac {4 \left (b \operatorname {arcsinh}\left (x c \right )^{2}-2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right ) b -2 a \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right )+2 a \ln \left (x c +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (2, i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right ) b \right ) b \sqrt {-i \left (x c +i\right ) f}\, \sqrt {i \left (x c -i\right ) d}}{\sqrt {c^{2} x^{2}+1}\, f^{2} c}\)	\(286\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

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