3.3.61 \(\int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)^2 \, dx\) [261]

3.3.61.1 Optimal result

Integrand size = 30, antiderivative size = 107 \[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)^2 \, dx=\frac {(a+b x) \sqrt {1+(a+b x)^2}}{4 b}-\frac {\text {arcsinh}(a+b x)}{4 b}-\frac {(a+b x)^2 \text {arcsinh}(a+b x)}{2 b}+\frac {(a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{2 b}+\frac {\text {arcsinh}(a+b x)^3}{6 b} \]

output

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

3.3.61.2 Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.03 \[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)^2 \, dx=\frac {3 (a+b x) \sqrt {1+a^2+2 a b x+b^2 x^2}-3 \left (1+2 a^2+4 a b x+2 b^2 x^2\right ) \text {arcsinh}(a+b x)+6 (a+b x) \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)^2+2 \text {arcsinh}(a+b x)^3}{12 b} \]

input

Integrate[Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x]^2,x]
 

output

(3*(a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] - 3*(1 + 2*a^2 + 4*a*b*x + 
2*b^2*x^2)*ArcSinh[a + b*x] + 6*(a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2 
]*ArcSinh[a + b*x]^2 + 2*ArcSinh[a + b*x]^3)/(12*b)
 

3.3.61.3 Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6275, 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.

\(\displaystyle \int \sqrt {a^2+2 a b x+b^2 x^2+1} \text {arcsinh}(a+b x)^2 \, dx\)

\(\Big \downarrow \) 6275

\(\displaystyle \frac {\int \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^2d(a+b x)}{b}\)

\(\Big \downarrow \) 6200

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 222

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

\(\Big \downarrow \) 6198

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

input

Int[Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x]^2,x]
 

output

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

3.3.61.3.1 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 6275

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((A_.) + (B_.)*(x_) + 
(C_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/d   Subst[Int[(C/d^2 + (C/d^2)*x^2 
)^p*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, A, B, C 
, n, p}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
 

3.3.61.4 Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.56

method	result	size
default	\(\frac {6 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b x -6 \,\operatorname {arcsinh}\left (b x +a \right ) b^{2} x^{2}+6 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a -12 \,\operatorname {arcsinh}\left (b x +a \right ) a b x +2 \operatorname {arcsinh}\left (b x +a \right )^{3}-6 a^{2} \operatorname {arcsinh}\left (b x +a \right )+3 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b x +3 a \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}-3 \,\operatorname {arcsinh}\left (b x +a \right )}{12 b}\)	\(167\)

input

int(arcsinh(b*x+a)^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2),x,method=_RETURNVERBOSE 
)
 

output

1/12*(6*arcsinh(b*x+a)^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*b*x-6*arcsinh(b*x+a 
)*b^2*x^2+6*arcsinh(b*x+a)^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a-12*arcsinh(b* 
x+a)*a*b*x+2*arcsinh(b*x+a)^3-6*a^2*arcsinh(b*x+a)+3*(b^2*x^2+2*a*b*x+a^2+ 
1)^(1/2)*b*x+3*a*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-3*arcsinh(b*x+a))/b
 

3.3.61.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.50 \[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)^2 \, dx=\frac {6 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b x + a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} + 2 \, \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{3} - 3 \, {\left (2 \, b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} + 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + 3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b x + a\right )}}{12 \, b} \]

input

integrate(arcsinh(b*x+a)^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2),x, algorithm="fri 
cas")
 

output

1/12*(6*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(b*x + a)*log(b*x + a + sqrt(b^2 
*x^2 + 2*a*b*x + a^2 + 1))^2 + 2*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^ 
2 + 1))^3 - 3*(2*b^2*x^2 + 4*a*b*x + 2*a^2 + 1)*log(b*x + a + sqrt(b^2*x^2 
 + 2*a*b*x + a^2 + 1)) + 3*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(b*x + a))/b
 

3.3.61.6 Sympy [F]

\[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)^2 \, dx=\int \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}\, dx \]

input

integrate(asinh(b*x+a)**2*(b**2*x**2+2*a*b*x+a**2+1)**(1/2),x)
 

output

Integral(sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)**2, x)
 

3.3.61.7 Maxima [F]

\[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)^2 \, dx=\int { \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \operatorname {arsinh}\left (b x + a\right )^{2} \,d x } \]

input

integrate(arcsinh(b*x+a)^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2),x, algorithm="max 
ima")
 

output

integrate(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*arcsinh(b*x + a)^2, x)
 

3.3.61.8 Giac [F]

\[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)^2 \, dx=\int { \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \operatorname {arsinh}\left (b x + a\right )^{2} \,d x } \]

input

integrate(arcsinh(b*x+a)^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2),x, algorithm="gia 
c")
 

output

integrate(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*arcsinh(b*x + a)^2, x)
 

3.3.61.9 Mupad [F(-1)]

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

input

int(asinh(a + b*x)^2*(a^2 + b^2*x^2 + 2*a*b*x + 1)^(1/2),x)
 

output

int(asinh(a + b*x)^2*(a^2 + b^2*x^2 + 2*a*b*x + 1)^(1/2), x)