Integrand size = 12, antiderivative size = 115 \[ \int (a+b \text {arcsinh}(c+d x))^4 \, dx=24 b^4 x-\frac {24 b^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{d}+\frac {12 b^2 (c+d x) (a+b \text {arcsinh}(c+d x))^2}{d}-\frac {4 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{d}+\frac {(c+d x) (a+b \text {arcsinh}(c+d x))^4}{d} \]
24*b^4*x+12*b^2*(d*x+c)*(a+b*arcsinh(d*x+c))^2/d+(d*x+c)*(a+b*arcsinh(d*x+ c))^4/d-24*b^3*(a+b*arcsinh(d*x+c))*(1+(d*x+c)^2)^(1/2)/d-4*b*(a+b*arcsinh (d*x+c))^3*(1+(d*x+c)^2)^(1/2)/d
Time = 0.23 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.97 \[ \int (a+b \text {arcsinh}(c+d x))^4 \, dx=\frac {\left (a^4+12 a^2 b^2+24 b^4\right ) (c+d x)-4 a b \left (a^2+6 b^2\right ) \sqrt {1+(c+d x)^2}-4 b \left (-a^3 (c+d x)-6 a b^2 (c+d x)+3 a^2 b \sqrt {1+(c+d x)^2}+6 b^3 \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)+6 b^2 \left (a^2 (c+d x)+2 b^2 (c+d x)-2 a b \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)^2-4 b^3 \left (-a (c+d x)+b \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)^3+b^4 (c+d x) \text {arcsinh}(c+d x)^4}{d} \]
((a^4 + 12*a^2*b^2 + 24*b^4)*(c + d*x) - 4*a*b*(a^2 + 6*b^2)*Sqrt[1 + (c + d*x)^2] - 4*b*(-(a^3*(c + d*x)) - 6*a*b^2*(c + d*x) + 3*a^2*b*Sqrt[1 + (c + d*x)^2] + 6*b^3*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] + 6*b^2*(a^2*(c + d*x) + 2*b^2*(c + d*x) - 2*a*b*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^ 2 - 4*b^3*(-(a*(c + d*x)) + b*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^3 + b^4*(c + d*x)*ArcSinh[c + d*x]^4)/d
Time = 0.52 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6273, 6187, 6213, 6187, 6213, 24}
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.
| \(\displaystyle \int (a+b \text {arcsinh}(c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 6273 |
| \(\displaystyle \frac {\int (a+b \text {arcsinh}(c+d x))^4d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6187 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^4-4 b \int \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^3}{\sqrt {(c+d x)^2+1}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^4-4 b \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3-3 b \int (a+b \text {arcsinh}(c+d x))^2d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 6187 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^4-4 b \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3-3 b \left ((c+d x) (a+b \text {arcsinh}(c+d x))^2-2 b \int \frac {(c+d x) (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )\right )}{d}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^4-4 b \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3-3 b \left ((c+d x) (a+b \text {arcsinh}(c+d x))^2-2 b \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))-b \int 1d(c+d x)\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^4-4 b \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3-3 b \left ((c+d x) (a+b \text {arcsinh}(c+d x))^2-2 b \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))-b (c+d x)\right )\right )\right )}{d}\) |
((c + d*x)*(a + b*ArcSinh[c + d*x])^4 - 4*b*(Sqrt[1 + (c + d*x)^2]*(a + b* ArcSinh[c + d*x])^3 - 3*b*((c + d*x)*(a + b*ArcSinh[c + d*x])^2 - 2*b*(-(b *(c + d*x)) + Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])))))/d
3.2.50.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(244\) vs. \(2(111)=222\).
Time = 0.17 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.13
| method | result | size |
| derivativedivides | \(\frac {\left (d x +c \right ) a^{4}+b^{4} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{4}-4 \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+12 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{2}-24 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+24 d x +24 c \right )+4 a \,b^{3} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{3}-3 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+6 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )-6 \sqrt {1+\left (d x +c \right )^{2}}\right )+6 a^{2} b^{2} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{2}-2 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 d x +2 c \right )+4 b \,a^{3} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )-\sqrt {1+\left (d x +c \right )^{2}}\right )}{d}\) | \(245\) |
| default | \(\frac {\left (d x +c \right ) a^{4}+b^{4} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{4}-4 \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+12 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{2}-24 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+24 d x +24 c \right )+4 a \,b^{3} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{3}-3 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+6 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )-6 \sqrt {1+\left (d x +c \right )^{2}}\right )+6 a^{2} b^{2} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{2}-2 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 d x +2 c \right )+4 b \,a^{3} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )-\sqrt {1+\left (d x +c \right )^{2}}\right )}{d}\) | \(245\) |
| parts | \(a^{4} x +\frac {b^{4} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{4}-4 \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+12 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{2}-24 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+24 d x +24 c \right )}{d}+\frac {4 a \,b^{3} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{3}-3 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+6 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )-6 \sqrt {1+\left (d x +c \right )^{2}}\right )}{d}+\frac {6 a^{2} b^{2} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{2}-2 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 d x +2 c \right )}{d}+\frac {4 b \,a^{3} \left (\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )-\sqrt {1+\left (d x +c \right )^{2}}\right )}{d}\) | \(249\) |
1/d*((d*x+c)*a^4+b^4*((d*x+c)*arcsinh(d*x+c)^4-4*arcsinh(d*x+c)^3*(1+(d*x+ c)^2)^(1/2)+12*(d*x+c)*arcsinh(d*x+c)^2-24*arcsinh(d*x+c)*(1+(d*x+c)^2)^(1 /2)+24*d*x+24*c)+4*a*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))+6*a^2*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)+ 4*b*a^3*((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. 344 vs. \(2 (111) = 222\).
Time = 0.28 (sec) , antiderivative size = 344, normalized size of antiderivative = 2.99 \[ \int (a+b \text {arcsinh}(c+d x))^4 \, dx=\frac {{\left (b^{4} d x + b^{4} c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{4} + 4 \, {\left (a b^{3} d x + a b^{3} c - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} b^{4}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3} + {\left (a^{4} + 12 \, a^{2} b^{2} + 24 \, b^{4}\right )} d x - 6 \, {\left (2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} a b^{3} - {\left (a^{2} b^{2} + 2 \, b^{4}\right )} d x - {\left (a^{2} b^{2} + 2 \, b^{4}\right )} c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + 4 \, {\left ({\left (a^{3} b + 6 \, a b^{3}\right )} d x + {\left (a^{3} b + 6 \, a b^{3}\right )} c - 3 \, {\left (a^{2} b^{2} + 2 \, b^{4}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 4 \, {\left (a^{3} b + 6 \, a b^{3}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{d} \]
((b^4*d*x + b^4*c)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^4 + 4* (a*b^3*d*x + a*b^3*c - sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*b^4)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^3 + (a^4 + 12*a^2*b^2 + 24*b^4)*d*x - 6*(2*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*a*b^3 - (a^2*b^2 + 2*b^4)*d*x - (a ^2*b^2 + 2*b^4)*c)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 + 4* ((a^3*b + 6*a*b^3)*d*x + (a^3*b + 6*a*b^3)*c - 3*(a^2*b^2 + 2*b^4)*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 )) - 4*(a^3*b + 6*a*b^3)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))/d
Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (105) = 210\).
Time = 0.27 (sec) , antiderivative size = 444, normalized size of antiderivative = 3.86 \[ \int (a+b \text {arcsinh}(c+d x))^4 \, dx=\begin {cases} a^{4} x + \frac {4 a^{3} b c \operatorname {asinh}{\left (c + d x \right )}}{d} + 4 a^{3} b x \operatorname {asinh}{\left (c + d x \right )} - \frac {4 a^{3} b \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{d} + \frac {6 a^{2} b^{2} c \operatorname {asinh}^{2}{\left (c + d x \right )}}{d} + 6 a^{2} b^{2} x \operatorname {asinh}^{2}{\left (c + d x \right )} + 12 a^{2} b^{2} x - \frac {12 a^{2} b^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{d} + \frac {4 a b^{3} c \operatorname {asinh}^{3}{\left (c + d x \right )}}{d} + \frac {24 a b^{3} c \operatorname {asinh}{\left (c + d x \right )}}{d} + 4 a b^{3} x \operatorname {asinh}^{3}{\left (c + d x \right )} + 24 a b^{3} x \operatorname {asinh}{\left (c + d x \right )} - \frac {12 a b^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c + d x \right )}}{d} - \frac {24 a b^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{d} + \frac {b^{4} c \operatorname {asinh}^{4}{\left (c + d x \right )}}{d} + \frac {12 b^{4} c \operatorname {asinh}^{2}{\left (c + d x \right )}}{d} + b^{4} x \operatorname {asinh}^{4}{\left (c + d x \right )} + 12 b^{4} x \operatorname {asinh}^{2}{\left (c + d x \right )} + 24 b^{4} x - \frac {4 b^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (c + d x \right )}}{d} - \frac {24 b^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {asinh}{\left (c \right )}\right )^{4} & \text {otherwise} \end {cases} \]
Piecewise((a**4*x + 4*a**3*b*c*asinh(c + d*x)/d + 4*a**3*b*x*asinh(c + d*x ) - 4*a**3*b*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/d + 6*a**2*b**2*c*asinh( c + d*x)**2/d + 6*a**2*b**2*x*asinh(c + d*x)**2 + 12*a**2*b**2*x - 12*a**2 *b**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/d + 4*a*b**3*c*a sinh(c + d*x)**3/d + 24*a*b**3*c*asinh(c + d*x)/d + 4*a*b**3*x*asinh(c + d *x)**3 + 24*a*b**3*x*asinh(c + d*x) - 12*a*b**3*sqrt(c**2 + 2*c*d*x + d**2 *x**2 + 1)*asinh(c + d*x)**2/d - 24*a*b**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/d + b**4*c*asinh(c + d*x)**4/d + 12*b**4*c*asinh(c + d*x)**2/d + b** 4*x*asinh(c + d*x)**4 + 12*b**4*x*asinh(c + d*x)**2 + 24*b**4*x - 4*b**4*s qrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**3/d - 24*b**4*sqrt(c** 2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/d, Ne(d, 0)), (x*(a + b*asinh( c))**4, True))
\[ \int (a+b \text {arcsinh}(c+d x))^4 \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4} \,d x } \]
b^4*x*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^4 + a^4*x + 4*((d*x + c)*arcsinh(d*x + c) - sqrt((d*x + c)^2 + 1))*a^3*b/d + integrate(2*(2*( (c^3 + c)*a*b^3 + (a*b^3*d^3 - b^4*d^3)*x^3 + (3*a*b^3*c*d^2 - 2*b^4*c*d^2 )*x^2 + ((3*c^2*d + d)*a*b^3 - (c^2*d + d)*b^4)*x + ((c^2 + 1)*a*b^3 + (a* b^3*d^2 - b^4*d^2)*x^2 + (2*a*b^3*c*d - b^4*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))^3 + 3*(a^2*b ^2*d^3*x^3 + 3*a^2*b^2*c*d^2*x^2 + (3*c^2*d + d)*a^2*b^2*x + (c^3 + c)*a^2 *b^2 + (a^2*b^2*d^2*x^2 + 2*a^2*b^2*c*d*x + (c^2 + 1)*a^2*b^2)*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)
\[ \int (a+b \text {arcsinh}(c+d x))^4 \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4} \,d x } \]
Timed out. \[ \int (a+b \text {arcsinh}(c+d x))^4 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4 \,d x \]