\(\int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \, dx\) [229]

Optimal result

Integrand size = 35, antiderivative size = 103 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \, dx=\frac {x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b \left (1+c^2 x^2\right )^{3/2} \log \left (1+c^2 x^2\right )}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.98 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.15 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \, dx=\frac {i \sqrt {f-i c f x} \left (2 a c x+2 b c x \text {arcsinh}(c x)-b \sqrt {1+c^2 x^2} \log (d (-1+i c x))-b \sqrt {1+c^2 x^2} \log (d+i c d x)\right )}{2 c d f^2 (i+c x) \sqrt {d+i c d x}} \] Input:

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

Output:

((I/2)*Sqrt[f - I*c*f*x]*(2*a*c*x + 2*b*c*x*ArcSinh[c*x] - b*Sqrt[1 + c^2* 
x^2]*Log[d*(-1 + I*c*x)] - b*Sqrt[1 + c^2*x^2]*Log[d + I*c*d*x]))/(c*d*f^2 
*(I + c*x)*Sqrt[d + I*c*d*x])
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.80, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {6211, 6202, 240}

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.

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

\(\Big \downarrow \) 6211

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

\(\Big \downarrow \) 6202

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

\(\Big \downarrow \) 240

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

rule 6202

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), 
x_Symbol] :> Simp[x*((a + b*ArcSinh[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Simp 
[b*c*(n/d)*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]   Int[x*((a + b*ArcSinh[ 
c*x])^(n - 1)/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, 
 c^2*d] && GtQ[n, 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]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (87 ) = 174\).

Time = 6.29 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.13

method	result	size
default	\(a \left (\frac {i}{c d f \sqrt {i c d x +d}\, \sqrt {-i c f x +f}}-\frac {i \sqrt {i c d x +d}}{c f \,d^{2} \sqrt {-i c f x +f}}\right )+\frac {b \left (-\ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}+\sqrt {c^{2} x^{2}+1}\, \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x c +\operatorname {arcsinh}\left (x c \right )-\ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )\right ) \left (x c +\sqrt {c^{2} x^{2}+1}\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{c \,f^{2} d^{2} \left (c^{2} x^{2}+1\right )}\)	\(219\)
parts	\(a \left (\frac {i}{c d f \sqrt {i c d x +d}\, \sqrt {-i c f x +f}}-\frac {i \sqrt {i c d x +d}}{c f \,d^{2} \sqrt {-i c f x +f}}\right )+\frac {b \left (-\ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}+\sqrt {c^{2} x^{2}+1}\, \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x c +\operatorname {arcsinh}\left (x c \right )-\ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )\right ) \left (x c +\sqrt {c^{2} x^{2}+1}\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{c \,f^{2} d^{2} \left (c^{2} x^{2}+1\right )}\)	\(219\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

1/4*(4*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*b*x*log(c*x + sqrt(c^2*x^2 + 1 
)) + 4*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*a*x + (c^2*d^2*f^2*x^2 + d^2*f 
^2)*sqrt(b^2/(c^2*d^3*f^3))*log((b*c^2*x^4 + sqrt(c^2*x^2 + 1)*sqrt(I*c*d* 
x + d)*sqrt(-I*c*f*x + f)*c*d*f*x^2*sqrt(b^2/(c^2*d^3*f^3)) + b*x^2)/(b*c^ 
4*x^4 + 2*b*c^2*x^2 + b)) - (c^2*d^2*f^2*x^2 + d^2*f^2)*sqrt(b^2/(c^2*d^3* 
f^3))*log((b*c^2*x^4 - sqrt(c^2*x^2 + 1)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + 
 f)*c*d*f*x^2*sqrt(b^2/(c^2*d^3*f^3)) + b*x^2)/(b*c^4*x^4 + 2*b*c^2*x^2 + 
b)) - 2*(c^2*d^2*f^2*x^2 + d^2*f^2)*sqrt(b^2/(c^2*d^3*f^3))*log((b*c^2*x^3 
 + sqrt(c^2*x^2 + 1)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*c*d*f*x*sqrt(b^2 
/(c^2*d^3*f^3)) + b*x)/(b*c^2*x^2 + b)) + 2*(c^2*d^2*f^2*x^2 + d^2*f^2)*sq 
rt(b^2/(c^2*d^3*f^3))*log((b*c^2*x^3 - sqrt(c^2*x^2 + 1)*sqrt(I*c*d*x + d) 
*sqrt(-I*c*f*x + f)*c*d*f*x*sqrt(b^2/(c^2*d^3*f^3)) + b*x)/(b*c^2*x^2 + b) 
) + 4*(c^2*d^2*f^2*x^2 + d^2*f^2)*integral(-sqrt(c^2*x^2 + 1)*sqrt(I*c*d*x 
 + d)*sqrt(-I*c*f*x + f)*b*c*x/(c^4*d^2*f^2*x^4 + 2*c^2*d^2*f^2*x^2 + d^2* 
f^2), x))/(c^2*d^2*f^2*x^2 + d^2*f^2)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.80 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \, dx=\frac {b x \operatorname {arsinh}\left (c x\right )}{\sqrt {c^{2} d f x^{2} + d f} d f} + \frac {a x}{\sqrt {c^{2} d f x^{2} + d f} d f} - \frac {b \sqrt {\frac {1}{d f}} \log \left (x^{2} + \frac {1}{c^{2}}\right )}{2 \, c d f} \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

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