3.6.72 \(\int \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx\) [572]

3.6.72.1 Optimal result

Integrand size = 37, antiderivative size = 244 \[ \int \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{4} b^2 x \sqrt {d+i c d x} \sqrt {f-i c f x}-\frac {b^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)}{4 c \sqrt {1+c^2 x^2}}-\frac {b c x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{2 \sqrt {1+c^2 x^2}}+\frac {1}{2} x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2+\frac {\sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {1+c^2 x^2}} \]

output

1/4*b^2*x*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)+1/2*x*(a+b*arcsinh(c*x))^2*( 
d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)-1/4*b^2*arcsinh(c*x)*(d+I*c*d*x)^(1/2)* 
(f-I*c*f*x)^(1/2)/c/(c^2*x^2+1)^(1/2)-1/2*b*c*x^2*(a+b*arcsinh(c*x))*(d+I* 
c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/(c^2*x^2+1)^(1/2)+1/6*(a+b*arcsinh(c*x))^3* 
(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/b/c/(c^2*x^2+1)^(1/2)
 

3.6.72.2 Mathematica [A] (verified)

Time = 1.88 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.44 \[ \int \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {12 a^2 c x \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+4 b^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^3-6 a b \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (2 \text {arcsinh}(c x))+12 a^2 \sqrt {d} \sqrt {f} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )+3 b^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (2 \text {arcsinh}(c x))-6 b \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x) (b \cosh (2 \text {arcsinh}(c x))-2 a \sinh (2 \text {arcsinh}(c x)))+6 b \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^2 (2 a+b \sinh (2 \text {arcsinh}(c x)))}{24 c \sqrt {1+c^2 x^2}} \]

input

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

output

(12*a^2*c*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 4*b^2* 
Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 - 6*a*b*Sqrt[d + I*c*d* 
x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] + 12*a^2*Sqrt[d]*Sqrt[f]*Sqrt[1 
+ c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f* 
x]] + 3*b^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2*ArcSinh[c*x]] - 6*b 
*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*(b*Cosh[2*ArcSinh[c*x]] 
- 2*a*Sinh[2*ArcSinh[c*x]]) + 6*b*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcS 
inh[c*x]^2*(2*a + b*Sinh[2*ArcSinh[c*x]]))/(24*c*Sqrt[1 + c^2*x^2])
 

3.6.72.3 Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.61, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {6211, 6200, 6191, 262, 222, 6198}

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.

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

\(\Big \downarrow \) 6211

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

\(\Big \downarrow \) 6200

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 222

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

\(\Big \downarrow \) 6198

\(\displaystyle \frac {\sqrt {d+i c d x} \sqrt {f-i c f x} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2-b c \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )+\frac {(a+b \text {arcsinh}(c x))^3}{6 b c}\right )}{\sqrt {c^2 x^2+1}}\)

input

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

output

(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*((x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[ 
c*x])^2)/2 + (a + b*ArcSinh[c*x])^3/(6*b*c) - b*c*((x^2*(a + b*ArcSinh[c*x 
]))/2 - (b*c*((x*Sqrt[1 + c^2*x^2])/(2*c^2) - ArcSinh[c*x]/(2*c^3)))/2)))/ 
Sqrt[1 + c^2*x^2]
 

3.6.72.3.1 Defintions of rubi rules used

rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 6191

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* 
(n/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + 
c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
 

rule 6198

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

rule 6200

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 
/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   Int[(a + b*ArcSinh[c*x])^n/Sq 
rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* 
x^2]]   Int[x*(a + b*ArcSinh[c*x])^(n - 1), 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]
 

3.6.72.4 Maple [F]

\[\int \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2} \sqrt {i c d x +d}\, \sqrt {-i c f x +f}d x\]

input

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

output

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

3.6.72.5 Fricas [F]

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

input

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

3.6.72.6 Sympy [F]

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

input

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

output

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

3.6.72.7 Maxima [F(-2)]

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

input

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

output

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

3.6.72.8 Giac [F(-2)]

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

input

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

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad 
Argument TypeDone
 

3.6.72.9 Mupad [F(-1)]

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

input

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

output

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