\(\int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))}{(f-i c f x)^{3/2}} \, dx\) [226]

Optimal result

Integrand size = 35, antiderivative size = 287 \[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))}{(f-i c f x)^{3/2}} \, dx=\frac {i b d^2 x \sqrt {1+c^2 x^2}}{f \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {4 i d^2 (1+i c x) (a+b \text {arcsinh}(c x))}{c f \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {i d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{c f \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {3 d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{2 b c f \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {4 b d^2 \sqrt {1+c^2 x^2} \log (i+c x)}{c f \sqrt {d+i c d x} \sqrt {f-i c f x}} \] Output:

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

Mathematica [A] (verified)

Time = 6.31 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.79 \[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))}{(f-i c f x)^{3/2}} \, dx=\frac {\frac {2 a d (5-i c x) \sqrt {d+i c d x} \sqrt {f-i c f x}}{f^2 (i+c x)}-\frac {6 a d^{3/2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )}{f^{3/2}}+\frac {b d \sqrt {d+i c d x} \sqrt {f-i c f x} \left (-\text {arcsinh}(c x)^2 \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+4 \text {arcsinh}(c x) \left (-i \cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+\sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+2 \left (4 \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+i \log \left (1+c^2 x^2\right )\right ) \left (i \cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+\sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )\right )}{f^2 \sqrt {1+c^2 x^2} \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}+\frac {2 b d \sqrt {d+i c d x} \sqrt {f-i c f x} \left (-\text {arcsinh}(c x)^2 \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+\left (c x-4 \arctan \left (\coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+i \log \left (1+c^2 x^2\right )\right ) \left (i \cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+\sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+\text {arcsinh}(c x) \left (-i \left (2+\sqrt {1+c^2 x^2}\right ) \cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-\left (-2+\sqrt {1+c^2 x^2}\right ) \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )\right )}{f^2 \sqrt {1+c^2 x^2} \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}}{2 c} \] Input:

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

Output:

((2*a*d*(5 - I*c*x)*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/(f^2*(I + c*x)) - 
 (6*a*d^(3/2)*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c 
*f*x]])/f^(3/2) + (b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(-(ArcSinh[c*x] 
^2*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])) + 4*ArcSinh[c*x]*((-I) 
*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2]) + 2*(4*ArcTan[Tanh[ArcSinh[c 
*x]/2]] + I*Log[1 + c^2*x^2])*(I*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/ 
2])))/(f^2*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2 
])) + (2*b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(-(ArcSinh[c*x]^2*(Cosh[A 
rcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])) + (c*x - 4*ArcTan[Coth[ArcSinh[c 
*x]/2]] + I*Log[1 + c^2*x^2])*(I*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/ 
2]) + ArcSinh[c*x]*((-I)*(2 + Sqrt[1 + c^2*x^2])*Cosh[ArcSinh[c*x]/2] - (- 
2 + Sqrt[1 + c^2*x^2])*Sinh[ArcSinh[c*x]/2])))/(f^2*Sqrt[1 + c^2*x^2]*(Cos 
h[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])))/(2*c)
 

Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.52, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, 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.

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

\(\Big \downarrow \) 6211

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6259

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(d^3*(1 + c^2*x^2)^(3/2)*(I*b*x - ((4*I)*(1 + I*c*x)*(a + b*ArcSinh[c*x])) 
/(c*Sqrt[1 + c^2*x^2]) - (I*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c - (3 
*(a + b*ArcSinh[c*x])^2)/(2*b*c) - (4*b*Log[I + 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 = 10.61 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.05

method	result	size
default	\(-\frac {d \left (2 i \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, b \,c^{2} x^{2}+2 i \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}+3 \operatorname {arcsinh}\left (x c \right )^{2} b \,c^{2} x^{2}+10 i \sqrt {c^{2} x^{2}+1}\, a +6 \,\operatorname {arcsinh}\left (x c \right ) a \,c^{2} x^{2}-8 \,\operatorname {arcsinh}\left (x c \right ) b \,c^{2} x^{2}+16 \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right ) b \,c^{2} x^{2}-8 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) b c x +8 a \,c^{2} x^{2}-2 i b \,c^{3} x^{3}-2 i b x c -8 \sqrt {c^{2} x^{2}+1}\, a c x +3 b \operatorname {arcsinh}\left (x c \right )^{2}+10 i \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) b +6 \,\operatorname {arcsinh}\left (x c \right ) a -8 b \,\operatorname {arcsinh}\left (x c \right )+16 b \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right )+8 a \right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{2 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} f^{2} c}\)	\(302\)

Input:

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

Output:

-1/2*d*(2*I*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*b*c^2*x^2+2*I*(c^2*x^2+1)^(1/2) 
*a*c^2*x^2+3*arcsinh(x*c)^2*b*c^2*x^2+10*I*(c^2*x^2+1)^(1/2)*a+6*arcsinh(x 
*c)*a*c^2*x^2-8*arcsinh(x*c)*b*c^2*x^2+16*ln(x*c+(c^2*x^2+1)^(1/2)+I)*b*c^ 
2*x^2-8*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*b*c*x+8*a*c^2*x^2-2*I*b*c^3*x^3-2*I 
*b*x*c-8*(c^2*x^2+1)^(1/2)*a*c*x+3*b*arcsinh(x*c)^2+10*I*(c^2*x^2+1)^(1/2) 
*arcsinh(x*c)*b+6*arcsinh(x*c)*a-8*b*arcsinh(x*c)+16*b*ln(x*c+(c^2*x^2+1)^ 
(1/2)+I)+8*a)*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)/(c^2*x^2+1)^(3/2)/f 
^2/c
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Degree mismatch inside factorisatio 
n over extensionUnable to transpose Error: Bad Argument Valuesym2poly/r2sy 
m(const g
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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