\(\int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{3/2} \sqrt {f-i c f x}} \, dx\) [223]

Optimal result

Integrand size = 35, antiderivative size = 111 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{3/2} \sqrt {f-i c f x}} \, dx=\frac {f (i+c x) \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b f \left (1+c^2 x^2\right )^{3/2} \log (i-c x)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.02 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{3/2} \sqrt {f-i c f x}} \, dx=\frac {\sqrt {d+i c d x} \sqrt {f-i c f x} \left (a \sqrt {1+c^2 x^2}+b \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+b (i-c x) \log (d+i c d x)\right )}{c d^2 f (-i+c x) \sqrt {1+c^2 x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.80, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {6211, 27, 6252, 27, 451, 16}

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.

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

\(\Big \downarrow \) 6211

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6252

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 451

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

\(\Big \downarrow \) 16

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

Input:

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

Output:

(f*(1 + c^2*x^2)^(3/2)*(((I + c*x)*(a + b*ArcSinh[c*x]))/(c*Sqrt[1 + c^2*x 
^2]) - (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 16

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

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 451

Int[((c_) + (d_.)*(x_))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[c^2/a   In 
t[1/(c - d*x), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 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]
 

rule 6252

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_) + (g_.)*(x_))^(m_.)*((d_) + ( 
e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[(f + g*x)^m*(d + e*x^2)^p 
, x]}, Simp[(a + b*ArcSinh[c*x])   u, x] - Simp[b*c   Int[1/Sqrt[1 + c^2*x^ 
2]   u, x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IGtQ 
[m, 0] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && (LtQ[m, -2*p - 1] || GtQ[m, 3])
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (95 ) = 190\).

Time = 6.16 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.98

method	result	size
default	\(\frac {i a \sqrt {-i c f x +f}}{c d f \sqrt {i c d x +d}}+b \left (\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{\sqrt {c^{2} x^{2}+1}\, c \,d^{2} f}+\frac {\sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (x c +i-\sqrt {c^{2} x^{2}+1}\right ) \operatorname {arcsinh}\left (x c \right )}{\left (c^{2} x^{2}+1\right ) c \,d^{2} f}-\frac {2 \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{\sqrt {c^{2} x^{2}+1}\, c \,d^{2} f}\right )\)	\(220\)
parts	\(\frac {i a \sqrt {-i c f x +f}}{c d f \sqrt {i c d x +d}}+b \left (\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{\sqrt {c^{2} x^{2}+1}\, c \,d^{2} f}+\frac {\sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (x c +i-\sqrt {c^{2} x^{2}+1}\right ) \operatorname {arcsinh}\left (x c \right )}{\left (c^{2} x^{2}+1\right ) c \,d^{2} f}-\frac {2 \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{\sqrt {c^{2} x^{2}+1}\, c \,d^{2} f}\right )\)	\(220\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 443 vs. \(2 (89) = 178\).

Time = 0.17 (sec) , antiderivative size = 443, normalized size of antiderivative = 3.99 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{3/2} \sqrt {f-i c f x}} \, dx=\frac {2 \, \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (c^{2} d^{2} f x - i \, c d^{2} f\right )} \sqrt {\frac {b^{2}}{c^{2} d^{3} f}} \log \left (-\frac {{\left (i \, b c^{6} x^{2} + 2 \, b c^{5} x - 2 i \, b c^{4}\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} - {\left (i \, c^{9} d^{2} f x^{4} + 2 \, c^{8} d^{2} f x^{3} + i \, c^{7} d^{2} f x^{2} + 2 \, c^{6} d^{2} f x\right )} \sqrt {\frac {b^{2}}{c^{2} d^{3} f}}}{8 \, {\left (b c^{3} x^{3} - i \, b c^{2} x^{2} + b c x - i \, b\right )}}\right ) - {\left (c^{2} d^{2} f x - i \, c d^{2} f\right )} \sqrt {\frac {b^{2}}{c^{2} d^{3} f}} \log \left (-\frac {{\left (i \, b c^{6} x^{2} + 2 \, b c^{5} x - 2 i \, b c^{4}\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} - {\left (-i \, c^{9} d^{2} f x^{4} - 2 \, c^{8} d^{2} f x^{3} - i \, c^{7} d^{2} f x^{2} - 2 \, c^{6} d^{2} f x\right )} \sqrt {\frac {b^{2}}{c^{2} d^{3} f}}}{8 \, {\left (b c^{3} x^{3} - i \, b c^{2} x^{2} + b c x - i \, b\right )}}\right ) + 2 \, \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} a}{2 \, {\left (c^{2} d^{2} f x - i \, c d^{2} f\right )}} \] Input:

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

Output:

1/2*(2*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*b*log(c*x + sqrt(c^2*x^2 + 1)) 
 + (c^2*d^2*f*x - I*c*d^2*f)*sqrt(b^2/(c^2*d^3*f))*log(-1/8*((I*b*c^6*x^2 
+ 2*b*c^5*x - 2*I*b*c^4)*sqrt(c^2*x^2 + 1)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x 
 + f) - (I*c^9*d^2*f*x^4 + 2*c^8*d^2*f*x^3 + I*c^7*d^2*f*x^2 + 2*c^6*d^2*f 
*x)*sqrt(b^2/(c^2*d^3*f)))/(b*c^3*x^3 - I*b*c^2*x^2 + b*c*x - I*b)) - (c^2 
*d^2*f*x - I*c*d^2*f)*sqrt(b^2/(c^2*d^3*f))*log(-1/8*((I*b*c^6*x^2 + 2*b*c 
^5*x - 2*I*b*c^4)*sqrt(c^2*x^2 + 1)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f) - 
 (-I*c^9*d^2*f*x^4 - 2*c^8*d^2*f*x^3 - I*c^7*d^2*f*x^2 - 2*c^6*d^2*f*x)*sq 
rt(b^2/(c^2*d^3*f)))/(b*c^3*x^3 - I*b*c^2*x^2 + b*c*x - I*b)) + 2*sqrt(I*c 
*d*x + d)*sqrt(-I*c*f*x + f)*a)/(c^2*d^2*f*x - I*c*d^2*f)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.88 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+i c d x)^{3/2} \sqrt {f-i c f x}} \, dx=\frac {i \, \sqrt {c^{2} d f x^{2} + d f} b \operatorname {arsinh}\left (c x\right )}{i \, c^{2} d^{2} f x + c d^{2} f} + \frac {i \, \sqrt {c^{2} d f x^{2} + d f} a}{i \, c^{2} d^{2} f x + c d^{2} f} - \frac {b \log \left (i \, c x + 1\right )}{c d^{\frac {3}{2}} \sqrt {f}} \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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