\(\int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x} (f-i c f x)^{3/2}} \, dx\) [264]

Optimal result

Integrand size = 37, antiderivative size = 464 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x} (f-i c f x)^{3/2}} \, dx=-\frac {i d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {d x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {d \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {4 i b d \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {2 b d \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 b^2 d \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {2 b^2 d \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b^2 d \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \] Output:

-I*d*(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 
)+d*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) 
+d*(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)+4*I*b*d*(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)-2*b*d*(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)+2*b^2*d*(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*(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)-b^2*d*(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] (verified)

Time = 3.03 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {d+i c d x} (f-i c f x)^{3/2}} \, dx=\frac {\sqrt {d+i c d x} \sqrt {f-i c f x} \left ((-1-i) b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2 \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-\sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+\left (-i a^2+a^2 c x+4 i a b \sqrt {1+c^2 x^2} \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-2 i b^2 \pi \sqrt {1+c^2 x^2} \log \left (1+i e^{-\text {arcsinh}(c x)}\right )+4 i b^2 \pi \sqrt {1+c^2 x^2} \log \left (1+e^{\text {arcsinh}(c x)}\right )-a b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )+2 i b^2 \pi \sqrt {1+c^2 x^2} \log \left (-\cos \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )-4 i b^2 \pi \sqrt {1+c^2 x^2} \log \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )\right ) \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+4 b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right ) \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+b \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \left (-i \cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right ) \left (2 a+3 b \pi -4 i b \log \left (1+i e^{-\text {arcsinh}(c x)}\right )\right )+\left (2 a-3 b \pi +4 i b \log \left (1+i e^{-\text {arcsinh}(c x)}\right )\right ) \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )\right )}{c d f^2 (-i+c x) (i+c x) \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.51 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.47, 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.

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

\(\Big \downarrow \) 6211

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6253

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Time = 5.33 (sec) , antiderivative size = 401, normalized size of antiderivative = 0.86

method	result	size
default	\(-\frac {i a^{2} \sqrt {i c d x +d}}{d c f \sqrt {-i c f x +f}}+b^{2} \left (\frac {\sqrt {-i \left (x c +i\right ) f}\, \sqrt {i \left (x c -i\right ) d}\, \left (x c -\sqrt {c^{2} x^{2}+1}-i\right ) \operatorname {arcsinh}\left (x c \right )^{2}}{d \,f^{2} c \left (c^{2} x^{2}+1\right )}+\frac {2 \left (\operatorname {arcsinh}\left (x c \right )^{2}-2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )-2 \operatorname {polylog}\left (2, i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{\sqrt {c^{2} x^{2}+1}\, d \,f^{2} c}\right )+2 a b \left (\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{\sqrt {c^{2} x^{2}+1}\, f^{2} c d}+\frac {\sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (x c -\sqrt {c^{2} x^{2}+1}-i\right ) \operatorname {arcsinh}\left (x c \right )}{\left (c^{2} x^{2}+1\right ) f^{2} c d}-\frac {2 \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{\sqrt {c^{2} x^{2}+1}\, f^{2} c d}\right )\)	\(401\)
parts	\(-\frac {i a^{2} \sqrt {i c d x +d}}{d c f \sqrt {-i c f x +f}}+b^{2} \left (\frac {\sqrt {-i \left (x c +i\right ) f}\, \sqrt {i \left (x c -i\right ) d}\, \left (x c -\sqrt {c^{2} x^{2}+1}-i\right ) \operatorname {arcsinh}\left (x c \right )^{2}}{d \,f^{2} c \left (c^{2} x^{2}+1\right )}+\frac {2 \left (\operatorname {arcsinh}\left (x c \right )^{2}-2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )-2 \operatorname {polylog}\left (2, i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{\sqrt {c^{2} x^{2}+1}\, d \,f^{2} c}\right )+2 a b \left (\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{\sqrt {c^{2} x^{2}+1}\, f^{2} c d}+\frac {\sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (x c -\sqrt {c^{2} x^{2}+1}-i\right ) \operatorname {arcsinh}\left (x c \right )}{\left (c^{2} x^{2}+1\right ) f^{2} c d}-\frac {2 \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{\sqrt {c^{2} x^{2}+1}\, f^{2} c d}\right )\)	\(401\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

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