Integrand size = 28, antiderivative size = 61 \[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x) \, dx=-\frac {(a+b x)^2}{4 b}+\frac {(a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{2 b}+\frac {\text {arcsinh}(a+b x)^2}{4 b} \] Output:
-1/4*(b*x+a)^2/b+1/2*(b*x+a)*(1+(b*x+a)^2)^(1/2)*arcsinh(b*x+a)/b+1/4*arcs inh(b*x+a)^2/b
Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x) \, dx=\frac {-b x (2 a+b x)+2 (a+b x) \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)+\text {arcsinh}(a+b x)^2}{4 b} \] Input:
Integrate[Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x],x]
Output:
(-(b*x*(2*a + b*x)) + 2*(a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSin h[a + b*x] + ArcSinh[a + b*x]^2)/(4*b)
Time = 0.34 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6275, 6200, 15, 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.
\(\displaystyle \int \sqrt {a^2+2 a b x+b^2 x^2+1} \text {arcsinh}(a+b x) \, dx\) |
\(\Big \downarrow \) 6275 |
\(\displaystyle \frac {\int \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)d(a+b x)}{b}\) |
\(\Big \downarrow \) 6200 |
\(\displaystyle \frac {\frac {1}{2} \int \frac {\text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}d(a+b x)-\frac {1}{2} \int (a+b x)d(a+b x)+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{b}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {\frac {1}{2} \int \frac {\text {arcsinh}(a+b x)}{\sqrt {(a+b x)^2+1}}d(a+b x)+\frac {1}{2} \sqrt {(a+b x)^2+1} (a+b x) \text {arcsinh}(a+b x)-\frac {1}{4} (a+b x)^2}{b}\) |
\(\Big \downarrow \) 6198 |
\(\displaystyle \frac {\frac {1}{2} \sqrt {(a+b x)^2+1} (a+b x) \text {arcsinh}(a+b x)+\frac {1}{4} \text {arcsinh}(a+b x)^2-\frac {1}{4} (a+b x)^2}{b}\) |
Input:
Int[Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x],x]
Output:
(-1/4*(a + b*x)^2 + ((a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/2 + ArcSinh[a + b*x]^2/4)/b
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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]
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]
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]
Time = 1.08 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.49
method | result | size |
default | \(\frac {2 \sqrt {x^{2} b^{2}+2 a b x +a^{2}+1}\, \operatorname {arcsinh}\left (b x +a \right ) b x -x^{2} b^{2}+2 \sqrt {x^{2} b^{2}+2 a b x +a^{2}+1}\, \operatorname {arcsinh}\left (b x +a \right ) a -2 a b x +\operatorname {arcsinh}\left (b x +a \right )^{2}-a^{2}-1}{4 b}\) | \(91\) |
Input:
int((b^2*x^2+2*a*b*x+a^2+1)^(1/2)*arcsinh(b*x+a),x,method=_RETURNVERBOSE)
Output:
1/4*(2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*arcsinh(b*x+a)*b*x-x^2*b^2+2*(b^2*x^2 +2*a*b*x+a^2+1)^(1/2)*arcsinh(b*x+a)*a-2*a*b*x+arcsinh(b*x+a)^2-a^2-1)/b
Time = 0.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.61 \[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x) \, dx=-\frac {b^{2} x^{2} + 2 \, a b x - 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b x + a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2}}{4 \, b} \] Input:
integrate((b^2*x^2+2*a*b*x+a^2+1)^(1/2)*arcsinh(b*x+a),x, algorithm="frica s")
Output:
-1/4*(b^2*x^2 + 2*a*b*x - 2*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(b*x + a)*lo g(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - log(b*x + a + sqrt(b^2*x^ 2 + 2*a*b*x + a^2 + 1))^2)/b
\[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x) \, dx=\int \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}\, dx \] Input:
integrate((b**2*x**2+2*a*b*x+a**2+1)**(1/2)*asinh(b*x+a),x)
Output:
Integral(sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (53) = 106\).
Time = 0.05 (sec) , antiderivative size = 238, normalized size of antiderivative = 3.90 \[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x) \, dx=-\frac {1}{4} \, {\left (x^{2} + \frac {2 \, a x}{b} + \frac {2 \, \operatorname {arsinh}\left (b x + a\right ) \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b^{2}} - \frac {\operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )^{2}}{b^{2}}\right )} b - \frac {1}{2} \, {\left (\frac {a^{2} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x - \frac {{\left (a^{2} + 1\right )} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{b}\right )} \operatorname {arsinh}\left (b x + a\right ) \] Input:
integrate((b^2*x^2+2*a*b*x+a^2+1)^(1/2)*arcsinh(b*x+a),x, algorithm="maxim a")
Output:
-1/4*(x^2 + 2*a*x/b + 2*arcsinh(b*x + a)*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a ^2*b^2 + 4*(a^2 + 1)*b^2))/b^2 - arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))^2/b^2)*b - 1/2*(a^2*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2 *b^2 + 4*(a^2 + 1)*b^2))/b - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*x - (a^2 + 1)*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b - sqrt(b^ 2*x^2 + 2*a*b*x + a^2 + 1)*a/b)*arcsinh(b*x + a)
\[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x) \, dx=\int { \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \operatorname {arsinh}\left (b x + a\right ) \,d x } \] Input:
integrate((b^2*x^2+2*a*b*x+a^2+1)^(1/2)*arcsinh(b*x+a),x, algorithm="giac" )
Output:
integrate(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*arcsinh(b*x + a), x)
Timed out. \[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x) \, dx=\int \mathrm {asinh}\left (a+b\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1} \,d x \] Input:
int(asinh(a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x + 1)^(1/2),x)
Output:
int(asinh(a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x + 1)^(1/2), x)
\[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x) \, dx=\int \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \mathit {asinh} \left (b x +a \right )d x \] Input:
int((b^2*x^2+2*a*b*x+a^2+1)^(1/2)*asinh(b*x+a),x)
Output:
int(sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x),x)