Integrand size = 8, antiderivative size = 78 \[ \int \text {arcsinh}(a+b x)^3 \, dx=-\frac {6 \sqrt {1+(a+b x)^2}}{b}+\frac {6 (a+b x) \text {arcsinh}(a+b x)}{b}-\frac {3 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{b}+\frac {(a+b x) \text {arcsinh}(a+b x)^3}{b} \]
6*(b*x+a)*arcsinh(b*x+a)/b+(b*x+a)*arcsinh(b*x+a)^3/b-6*(1+(b*x+a)^2)^(1/2 )/b-3*arcsinh(b*x+a)^2*(1+(b*x+a)^2)^(1/2)/b
Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.90 \[ \int \text {arcsinh}(a+b x)^3 \, dx=\frac {-6 \sqrt {1+(a+b x)^2}+6 (a+b x) \text {arcsinh}(a+b x)-3 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2+(a+b x) \text {arcsinh}(a+b x)^3}{b} \]
(-6*Sqrt[1 + (a + b*x)^2] + 6*(a + b*x)*ArcSinh[a + b*x] - 3*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^2 + (a + b*x)*ArcSinh[a + b*x]^3)/b
Time = 0.37 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6273, 6187, 6213, 6187, 241}
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.
\(\displaystyle \int \text {arcsinh}(a+b x)^3 \, dx\) |
\(\Big \downarrow \) 6273 |
\(\displaystyle \frac {\int \text {arcsinh}(a+b x)^3d(a+b x)}{b}\) |
\(\Big \downarrow \) 6187 |
\(\displaystyle \frac {(a+b x) \text {arcsinh}(a+b x)^3-3 \int \frac {(a+b x) \text {arcsinh}(a+b x)^2}{\sqrt {(a+b x)^2+1}}d(a+b x)}{b}\) |
\(\Big \downarrow \) 6213 |
\(\displaystyle \frac {(a+b x) \text {arcsinh}(a+b x)^3-3 \left (\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^2-2 \int \text {arcsinh}(a+b x)d(a+b x)\right )}{b}\) |
\(\Big \downarrow \) 6187 |
\(\displaystyle \frac {(a+b x) \text {arcsinh}(a+b x)^3-3 \left (\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^2-2 \left ((a+b x) \text {arcsinh}(a+b x)-\int \frac {a+b x}{\sqrt {(a+b x)^2+1}}d(a+b x)\right )\right )}{b}\) |
\(\Big \downarrow \) 241 |
\(\displaystyle \frac {(a+b x) \text {arcsinh}(a+b x)^3-3 \left (\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^2-2 \left ((a+b x) \text {arcsinh}(a+b x)-\sqrt {(a+b x)^2+1}\right )\right )}{b}\) |
((a + b*x)*ArcSinh[a + b*x]^3 - 3*(Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^ 2 - 2*(-Sqrt[1 + (a + b*x)^2] + (a + b*x)*ArcSinh[a + b*x])))/b
3.1.77.3.1 Defintions of rubi rules used
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A rcSinh[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[ 1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d , n}, x]
Time = 0.09 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {\operatorname {arcsinh}\left (b x +a \right )^{3} \left (b x +a \right )-3 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}+6 \left (b x +a \right ) \operatorname {arcsinh}\left (b x +a \right )-6 \sqrt {1+\left (b x +a \right )^{2}}}{b}\) | \(67\) |
default | \(\frac {\operatorname {arcsinh}\left (b x +a \right )^{3} \left (b x +a \right )-3 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}+6 \left (b x +a \right ) \operatorname {arcsinh}\left (b x +a \right )-6 \sqrt {1+\left (b x +a \right )^{2}}}{b}\) | \(67\) |
1/b*(arcsinh(b*x+a)^3*(b*x+a)-3*arcsinh(b*x+a)^2*(1+(b*x+a)^2)^(1/2)+6*(b* x+a)*arcsinh(b*x+a)-6*(1+(b*x+a)^2)^(1/2))
Time = 0.29 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.78 \[ \int \text {arcsinh}(a+b x)^3 \, dx=\frac {{\left (b x + a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{3} - 3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} + 6 \, {\left (b x + a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 6 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b} \]
((b*x + a)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^3 - 3*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 + 6*(b*x + a)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 6*sqrt (b^2*x^2 + 2*a*b*x + a^2 + 1))/b
Time = 0.14 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.40 \[ \int \text {arcsinh}(a+b x)^3 \, dx=\begin {cases} \frac {a \operatorname {asinh}^{3}{\left (a + b x \right )}}{b} + \frac {6 a \operatorname {asinh}{\left (a + b x \right )}}{b} + x \operatorname {asinh}^{3}{\left (a + b x \right )} + 6 x \operatorname {asinh}{\left (a + b x \right )} - \frac {3 \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{b} - \frac {6 \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{b} & \text {for}\: b \neq 0 \\x \operatorname {asinh}^{3}{\left (a \right )} & \text {otherwise} \end {cases} \]
Piecewise((a*asinh(a + b*x)**3/b + 6*a*asinh(a + b*x)/b + x*asinh(a + b*x) **3 + 6*x*asinh(a + b*x) - 3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)**2/b - 6*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)/b, Ne(b, 0)), (x*asin h(a)**3, True))
\[ \int \text {arcsinh}(a+b x)^3 \, dx=\int { \operatorname {arsinh}\left (b x + a\right )^{3} \,d x } \]
x*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^3 - integrate(3*(b^3*x^ 3 + 2*a*b^2*x^2 + (a^2*b + b)*x + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(b^2*x ^2 + a*b*x))*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2/(b^3*x^3 + 3*a*b^2*x^2 + a^3 + (3*a^2*b + b)*x + (b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2) + a), x)
\[ \int \text {arcsinh}(a+b x)^3 \, dx=\int { \operatorname {arsinh}\left (b x + a\right )^{3} \,d x } \]
Timed out. \[ \int \text {arcsinh}(a+b x)^3 \, dx=\int {\mathrm {asinh}\left (a+b\,x\right )}^3 \,d x \]