Integrand size = 12, antiderivative size = 100 \[ \int (a+b \text {arcsinh}(c+d x))^3 \, dx=6 a b^2 x-\frac {6 b^3 \sqrt {1+(c+d x)^2}}{d}+\frac {6 b^3 (c+d x) \text {arcsinh}(c+d x)}{d}-\frac {3 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{d}+\frac {(c+d x) (a+b \text {arcsinh}(c+d x))^3}{d} \]
6*a*b^2*x+6*b^3*(d*x+c)*arcsinh(d*x+c)/d+(d*x+c)*(a+b*arcsinh(d*x+c))^3/d- 6*b^3*(1+(d*x+c)^2)^(1/2)/d-3*b*(a+b*arcsinh(d*x+c))^2*(1+(d*x+c)^2)^(1/2) /d
Time = 0.15 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.47 \[ \int (a+b \text {arcsinh}(c+d x))^3 \, dx=\frac {a \left (a^2+6 b^2\right ) (c+d x)-3 b \left (a^2+2 b^2\right ) \sqrt {1+(c+d x)^2}-3 b \left (-a^2 (c+d x)-2 b^2 (c+d x)+2 a b \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)-3 b^2 \left (-a (c+d x)+b \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)^2+b^3 (c+d x) \text {arcsinh}(c+d x)^3}{d} \]
(a*(a^2 + 6*b^2)*(c + d*x) - 3*b*(a^2 + 2*b^2)*Sqrt[1 + (c + d*x)^2] - 3*b *(-(a^2*(c + d*x)) - 2*b^2*(c + d*x) + 2*a*b*Sqrt[1 + (c + d*x)^2])*ArcSin h[c + d*x] - 3*b^2*(-(a*(c + d*x)) + b*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^2 + b^3*(c + d*x)*ArcSinh[c + d*x]^3)/d
Time = 0.37 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6273, 6187, 6213, 2009}
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.
\(\displaystyle \int (a+b \text {arcsinh}(c+d x))^3 \, dx\) |
\(\Big \downarrow \) 6273 |
\(\displaystyle \frac {\int (a+b \text {arcsinh}(c+d x))^3d(c+d x)}{d}\) |
\(\Big \downarrow \) 6187 |
\(\displaystyle \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^3-3 b \int \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^2}{\sqrt {(c+d x)^2+1}}d(c+d x)}{d}\) |
\(\Big \downarrow \) 6213 |
\(\displaystyle \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^3-3 b \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2-2 b \int (a+b \text {arcsinh}(c+d x))d(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^3-3 b \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2-2 b \left (a (c+d x)+b (c+d x) \text {arcsinh}(c+d x)-b \sqrt {(c+d x)^2+1}\right )\right )}{d}\) |
((c + d*x)*(a + b*ArcSinh[c + d*x])^3 - 3*b*(Sqrt[1 + (c + d*x)^2]*(a + b* ArcSinh[c + d*x])^2 - 2*b*(a*(c + d*x) - b*Sqrt[1 + (c + d*x)^2] + b*(c + d*x)*ArcSinh[c + d*x])))/d
3.2.41.3.1 Defintions of rubi rules used
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 = 0.11 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.60
method | result | size |
derivativedivides | \(\frac {\left (d x +c \right ) a^{3}+b^{3} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{3}-3 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+6 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )-6 \sqrt {1+\left (d x +c \right )^{2}}\right )+3 a \,b^{2} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{2}-2 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 d x +2 c \right )+3 a^{2} b \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )-\sqrt {1+\left (d x +c \right )^{2}}\right )}{d}\) | \(160\) |
default | \(\frac {\left (d x +c \right ) a^{3}+b^{3} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{3}-3 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+6 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )-6 \sqrt {1+\left (d x +c \right )^{2}}\right )+3 a \,b^{2} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{2}-2 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 d x +2 c \right )+3 a^{2} b \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )-\sqrt {1+\left (d x +c \right )^{2}}\right )}{d}\) | \(160\) |
parts | \(x \,a^{3}+\frac {b^{3} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{3}-3 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+6 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )-6 \sqrt {1+\left (d x +c \right )^{2}}\right )}{d}+\frac {3 a \,b^{2} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{2}-2 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 d x +2 c \right )}{d}+\frac {3 a^{2} b \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )-\sqrt {1+\left (d x +c \right )^{2}}\right )}{d}\) | \(161\) |
1/d*((d*x+c)*a^3+b^3*((d*x+c)*arcsinh(d*x+c)^3-3*arcsinh(d*x+c)^2*(1+(d*x+ c)^2)^(1/2)+6*(d*x+c)*arcsinh(d*x+c)-6*(1+(d*x+c)^2)^(1/2))+3*a*b^2*((d*x+ c)*arcsinh(d*x+c)^2-2*arcsinh(d*x+c)*(1+(d*x+c)^2)^(1/2)+2*d*x+2*c)+3*a^2* b*((d*x+c)*arcsinh(d*x+c)-(1+(d*x+c)^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (96) = 192\).
Time = 0.26 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.39 \[ \int (a+b \text {arcsinh}(c+d x))^3 \, dx=\frac {{\left (b^{3} d x + b^{3} c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3} + {\left (a^{3} + 6 \, a b^{2}\right )} d x + 3 \, {\left (a b^{2} d x + a b^{2} c - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} b^{3}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} - 3 \, {\left (2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} a b^{2} - {\left (a^{2} b + 2 \, b^{3}\right )} d x - {\left (a^{2} b + 2 \, b^{3}\right )} c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 3 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left (a^{2} b + 2 \, b^{3}\right )}}{d} \]
((b^3*d*x + b^3*c)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^3 + (a ^3 + 6*a*b^2)*d*x + 3*(a*b^2*d*x + a*b^2*c - sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*b^3)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 - 3*(2*sqrt(d ^2*x^2 + 2*c*d*x + c^2 + 1)*a*b^2 - (a^2*b + 2*b^3)*d*x - (a^2*b + 2*b^3)* c)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) - 3*sqrt(d^2*x^2 + 2*c *d*x + c^2 + 1)*(a^2*b + 2*b^3))/d
Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (92) = 184\).
Time = 0.17 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.82 \[ \int (a+b \text {arcsinh}(c+d x))^3 \, dx=\begin {cases} a^{3} x + \frac {3 a^{2} b c \operatorname {asinh}{\left (c + d x \right )}}{d} + 3 a^{2} b x \operatorname {asinh}{\left (c + d x \right )} - \frac {3 a^{2} b \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{d} + \frac {3 a b^{2} c \operatorname {asinh}^{2}{\left (c + d x \right )}}{d} + 3 a b^{2} x \operatorname {asinh}^{2}{\left (c + d x \right )} + 6 a b^{2} x - \frac {6 a b^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{d} + \frac {b^{3} c \operatorname {asinh}^{3}{\left (c + d x \right )}}{d} + \frac {6 b^{3} c \operatorname {asinh}{\left (c + d x \right )}}{d} + b^{3} x \operatorname {asinh}^{3}{\left (c + d x \right )} + 6 b^{3} x \operatorname {asinh}{\left (c + d x \right )} - \frac {3 b^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c + d x \right )}}{d} - \frac {6 b^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {asinh}{\left (c \right )}\right )^{3} & \text {otherwise} \end {cases} \]
Piecewise((a**3*x + 3*a**2*b*c*asinh(c + d*x)/d + 3*a**2*b*x*asinh(c + d*x ) - 3*a**2*b*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/d + 3*a*b**2*c*asinh(c + d*x)**2/d + 3*a*b**2*x*asinh(c + d*x)**2 + 6*a*b**2*x - 6*a*b**2*sqrt(c** 2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/d + b**3*c*asinh(c + d*x)**3/d + 6*b**3*c*asinh(c + d*x)/d + b**3*x*asinh(c + d*x)**3 + 6*b**3*x*asinh(c + d*x) - 3*b**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2/d - 6*b**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/d, Ne(d, 0)), (x*(a + b*asin h(c))**3, True))
\[ \int (a+b \text {arcsinh}(c+d x))^3 \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
b^3*x*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^3 + a^3*x + 3*((d*x + c)*arcsinh(d*x + c) - sqrt((d*x + c)^2 + 1))*a^2*b/d + integrate(3*((c^ 3 + c)*a*b^2 + (a*b^2*d^3 - b^3*d^3)*x^3 + (3*a*b^2*c*d^2 - 2*b^3*c*d^2)*x ^2 + ((3*c^2*d + d)*a*b^2 - (c^2*d + d)*b^3)*x + ((c^2 + 1)*a*b^2 + (a*b^2 *d^2 - b^3*d^2)*x^2 + (2*a*b^2*c*d - b^3*c*d)*x)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2/(d^3*x^3 + 3* c*d^2*x^2 + c^3 + (3*c^2*d + d)*x + (d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + c), x)
\[ \int (a+b \text {arcsinh}(c+d x))^3 \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
Timed out. \[ \int (a+b \text {arcsinh}(c+d x))^3 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3 \,d x \]