Integrand size = 23, antiderivative size = 279 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^3 \, dx=\frac {45 b^3 e^3 (c+d x) \sqrt {1+(c+d x)^2}}{256 d}-\frac {3 b^3 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{128 d}-\frac {45 b^3 e^3 \text {arcsinh}(c+d x)}{256 d}-\frac {9 b^2 e^3 (c+d x)^2 (a+b \text {arcsinh}(c+d x))}{32 d}+\frac {3 b^2 e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))}{32 d}+\frac {9 b e^3 (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{32 d}-\frac {3 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{16 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^3}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^3}{4 d} \]
-45/256*b^3*e^3*arcsinh(d*x+c)/d-9/32*b^2*e^3*(d*x+c)^2*(a+b*arcsinh(d*x+c ))/d+3/32*b^2*e^3*(d*x+c)^4*(a+b*arcsinh(d*x+c))/d-3/32*e^3*(a+b*arcsinh(d *x+c))^3/d+1/4*e^3*(d*x+c)^4*(a+b*arcsinh(d*x+c))^3/d+45/256*b^3*e^3*(d*x+ c)*(1+(d*x+c)^2)^(1/2)/d-3/128*b^3*e^3*(d*x+c)^3*(1+(d*x+c)^2)^(1/2)/d+9/3 2*b*e^3*(d*x+c)*(a+b*arcsinh(d*x+c))^2*(1+(d*x+c)^2)^(1/2)/d-3/16*b*e^3*(d *x+c)^3*(a+b*arcsinh(d*x+c))^2*(1+(d*x+c)^2)^(1/2)/d
Time = 0.36 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.09 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^3 \, dx=\frac {e^3 \left (-72 a b^2 (c+d x)^2+8 a \left (8 a^2+3 b^2\right ) (c+d x)^4+3 b (c+d x) \sqrt {1+(c+d x)^2} \left (3 \left (8 a^2+5 b^2\right )-2 \left (8 a^2+b^2\right ) (c+d x)^2\right )-9 b \left (8 a^2+5 b^2\right ) \text {arcsinh}(c+d x)-24 b (c+d x) \left (3 b^2 (c+d x)-8 a^2 (c+d x)^3-b^2 (c+d x)^3-6 a b \sqrt {1+(c+d x)^2}+4 a b (c+d x)^2 \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)+24 b^2 \left (-3 a+8 a (c+d x)^4+3 b (c+d x) \sqrt {1+(c+d x)^2}-2 b (c+d x)^3 \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)^2+8 b^3 \left (-3+8 (c+d x)^4\right ) \text {arcsinh}(c+d x)^3\right )}{256 d} \]
(e^3*(-72*a*b^2*(c + d*x)^2 + 8*a*(8*a^2 + 3*b^2)*(c + d*x)^4 + 3*b*(c + d *x)*Sqrt[1 + (c + d*x)^2]*(3*(8*a^2 + 5*b^2) - 2*(8*a^2 + b^2)*(c + d*x)^2 ) - 9*b*(8*a^2 + 5*b^2)*ArcSinh[c + d*x] - 24*b*(c + d*x)*(3*b^2*(c + d*x) - 8*a^2*(c + d*x)^3 - b^2*(c + d*x)^3 - 6*a*b*Sqrt[1 + (c + d*x)^2] + 4*a *b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] + 24*b^2*(-3*a + 8* a*(c + d*x)^4 + 3*b*(c + d*x)*Sqrt[1 + (c + d*x)^2] - 2*b*(c + d*x)^3*Sqrt [1 + (c + d*x)^2])*ArcSinh[c + d*x]^2 + 8*b^3*(-3 + 8*(c + d*x)^4)*ArcSinh [c + d*x]^3))/(256*d)
Time = 1.16 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {6274, 27, 6191, 6227, 6191, 262, 262, 222, 6227, 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.
| \(\displaystyle \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int e^3 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^3d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^3 \int (c+d x)^3 (a+b \text {arcsinh}(c+d x))^3d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^3-\frac {3}{4} b \int \frac {(c+d x)^4 (a+b \text {arcsinh}(c+d x))^2}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^3-\frac {3}{4} b \left (-\frac {1}{2} b \int (c+d x)^3 (a+b \text {arcsinh}(c+d x))d(c+d x)-\frac {3}{4} \int \frac {(c+d x)^2 (a+b \text {arcsinh}(c+d x))^2}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{4} \sqrt {(c+d x)^2+1} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^3-\frac {3}{4} b \left (-\frac {1}{2} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))-\frac {1}{4} b \int \frac {(c+d x)^4}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )-\frac {3}{4} \int \frac {(c+d x)^2 (a+b \text {arcsinh}(c+d x))^2}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{4} \sqrt {(c+d x)^2+1} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^3-\frac {3}{4} b \left (-\frac {1}{2} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))-\frac {1}{4} b \left (\frac {1}{4} (c+d x)^3 \sqrt {(c+d x)^2+1}-\frac {3}{4} \int \frac {(c+d x)^2}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )\right )-\frac {3}{4} \int \frac {(c+d x)^2 (a+b \text {arcsinh}(c+d x))^2}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{4} \sqrt {(c+d x)^2+1} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^3-\frac {3}{4} b \left (-\frac {1}{2} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))-\frac {1}{4} b \left (\frac {1}{4} (c+d x)^3 \sqrt {(c+d x)^2+1}-\frac {3}{4} \left (\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1}-\frac {1}{2} \int \frac {1}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )\right )\right )-\frac {3}{4} \int \frac {(c+d x)^2 (a+b \text {arcsinh}(c+d x))^2}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{4} \sqrt {(c+d x)^2+1} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^3-\frac {3}{4} b \left (-\frac {3}{4} \int \frac {(c+d x)^2 (a+b \text {arcsinh}(c+d x))^2}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{4} \sqrt {(c+d x)^2+1} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2-\frac {1}{2} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))-\frac {1}{4} b \left (\frac {1}{4} (c+d x)^3 \sqrt {(c+d x)^2+1}-\frac {3}{4} \left (\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1}-\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^3-\frac {3}{4} b \left (-\frac {3}{4} \left (-b \int (c+d x) (a+b \text {arcsinh}(c+d x))d(c+d x)-\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c+d x))^2}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2\right )+\frac {1}{4} \sqrt {(c+d x)^2+1} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2-\frac {1}{2} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))-\frac {1}{4} b \left (\frac {1}{4} (c+d x)^3 \sqrt {(c+d x)^2+1}-\frac {3}{4} \left (\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1}-\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^3-\frac {3}{4} b \left (-\frac {3}{4} \left (-b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))-\frac {1}{2} b \int \frac {(c+d x)^2}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )-\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c+d x))^2}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2\right )+\frac {1}{4} \sqrt {(c+d x)^2+1} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2-\frac {1}{2} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))-\frac {1}{4} b \left (\frac {1}{4} (c+d x)^3 \sqrt {(c+d x)^2+1}-\frac {3}{4} \left (\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1}-\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^3-\frac {3}{4} b \left (-\frac {3}{4} \left (-b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))-\frac {1}{2} b \left (\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1}-\frac {1}{2} \int \frac {1}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )\right )-\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c+d x))^2}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2\right )+\frac {1}{4} \sqrt {(c+d x)^2+1} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2-\frac {1}{2} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))-\frac {1}{4} b \left (\frac {1}{4} (c+d x)^3 \sqrt {(c+d x)^2+1}-\frac {3}{4} \left (\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1}-\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^3-\frac {3}{4} b \left (-\frac {3}{4} \left (-\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c+d x))^2}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2-b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))-\frac {1}{2} b \left (\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1}-\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )\right )+\frac {1}{4} \sqrt {(c+d x)^2+1} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2-\frac {1}{2} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))-\frac {1}{4} b \left (\frac {1}{4} (c+d x)^3 \sqrt {(c+d x)^2+1}-\frac {3}{4} \left (\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1}-\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^3-\frac {3}{4} b \left (\frac {1}{4} \sqrt {(c+d x)^2+1} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2-\frac {1}{2} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))-\frac {1}{4} b \left (\frac {1}{4} (c+d x)^3 \sqrt {(c+d x)^2+1}-\frac {3}{4} \left (\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1}-\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )\right )-\frac {3}{4} \left (-\frac {(a+b \text {arcsinh}(c+d x))^3}{6 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2-b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))-\frac {1}{2} b \left (\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1}-\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )\right )\right )\right )}{d}\) |
(e^3*(((c + d*x)^4*(a + b*ArcSinh[c + d*x])^3)/4 - (3*b*(((c + d*x)^3*Sqrt [1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^2)/4 - (b*(-1/4*(b*(((c + d*x)^ 3*Sqrt[1 + (c + d*x)^2])/4 - (3*(((c + d*x)*Sqrt[1 + (c + d*x)^2])/2 - Arc Sinh[c + d*x]/2))/4)) + ((c + d*x)^4*(a + b*ArcSinh[c + d*x]))/4))/2 - (3* (((c + d*x)*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^2)/2 - (a + b*A rcSinh[c + d*x])^3/(6*b) - b*(-1/2*(b*(((c + d*x)*Sqrt[1 + (c + d*x)^2])/2 - ArcSinh[c + d*x]/2)) + ((c + d*x)^2*(a + b*ArcSinh[c + d*x]))/2)))/4))/ 4))/d
3.2.38.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int [(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] ) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ m, 1] && NeQ[m + 2*p + 1, 0]
Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 0.38 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.31
| method | result | size |
| derivativedivides | \(\frac {\frac {e^{3} a^{3} \left (d x +c \right )^{4}}{4}+e^{3} b^{3} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{3}}{4}-\frac {3 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {9 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{32}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{3}}{32}+\frac {3 \left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )}{32}-\frac {3 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{128}+\frac {45 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{256}+\frac {27 \,\operatorname {arcsinh}\left (d x +c \right )}{256}-\frac {9 \left (1+\left (d x +c \right )^{2}\right ) \operatorname {arcsinh}\left (d x +c \right )}{32}\right )+3 e^{3} a \,b^{2} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}+\frac {3 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{16}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2}}{32}+\frac {\left (d x +c \right )^{4}}{32}-\frac {3 \left (d x +c \right )^{2}}{32}-\frac {3}{32}\right )+3 e^{3} a^{2} b \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )}{4}-\frac {\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{32}-\frac {3 \,\operatorname {arcsinh}\left (d x +c \right )}{32}\right )}{d}\) | \(365\) |
| default | \(\frac {\frac {e^{3} a^{3} \left (d x +c \right )^{4}}{4}+e^{3} b^{3} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{3}}{4}-\frac {3 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {9 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{32}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{3}}{32}+\frac {3 \left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )}{32}-\frac {3 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{128}+\frac {45 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{256}+\frac {27 \,\operatorname {arcsinh}\left (d x +c \right )}{256}-\frac {9 \left (1+\left (d x +c \right )^{2}\right ) \operatorname {arcsinh}\left (d x +c \right )}{32}\right )+3 e^{3} a \,b^{2} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}+\frac {3 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{16}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2}}{32}+\frac {\left (d x +c \right )^{4}}{32}-\frac {3 \left (d x +c \right )^{2}}{32}-\frac {3}{32}\right )+3 e^{3} a^{2} b \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )}{4}-\frac {\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{32}-\frac {3 \,\operatorname {arcsinh}\left (d x +c \right )}{32}\right )}{d}\) | \(365\) |
| parts | \(\frac {e^{3} a^{3} \left (d x +c \right )^{4}}{4 d}+\frac {e^{3} b^{3} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{3}}{4}-\frac {3 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {9 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{32}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{3}}{32}+\frac {3 \left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )}{32}-\frac {3 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{128}+\frac {45 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{256}+\frac {27 \,\operatorname {arcsinh}\left (d x +c \right )}{256}-\frac {9 \left (1+\left (d x +c \right )^{2}\right ) \operatorname {arcsinh}\left (d x +c \right )}{32}\right )}{d}+\frac {3 e^{3} a \,b^{2} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}+\frac {3 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{16}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2}}{32}+\frac {\left (d x +c \right )^{4}}{32}-\frac {3 \left (d x +c \right )^{2}}{32}-\frac {3}{32}\right )}{d}+\frac {3 e^{3} a^{2} b \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )}{4}-\frac {\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{32}-\frac {3 \,\operatorname {arcsinh}\left (d x +c \right )}{32}\right )}{d}\) | \(373\) |
1/d*(1/4*e^3*a^3*(d*x+c)^4+e^3*b^3*(1/4*(d*x+c)^4*arcsinh(d*x+c)^3-3/16*(d *x+c)^3*arcsinh(d*x+c)^2*(1+(d*x+c)^2)^(1/2)+9/32*arcsinh(d*x+c)^2*(1+(d*x +c)^2)^(1/2)*(d*x+c)-3/32*arcsinh(d*x+c)^3+3/32*(d*x+c)^4*arcsinh(d*x+c)-3 /128*(d*x+c)^3*(1+(d*x+c)^2)^(1/2)+45/256*(d*x+c)*(1+(d*x+c)^2)^(1/2)+27/2 56*arcsinh(d*x+c)-9/32*(1+(d*x+c)^2)*arcsinh(d*x+c))+3*e^3*a*b^2*(1/4*(d*x +c)^4*arcsinh(d*x+c)^2-1/8*(d*x+c)^3*arcsinh(d*x+c)*(1+(d*x+c)^2)^(1/2)+3/ 16*(1+(d*x+c)^2)^(1/2)*(d*x+c)*arcsinh(d*x+c)-3/32*arcsinh(d*x+c)^2+1/32*( d*x+c)^4-3/32*(d*x+c)^2-3/32)+3*e^3*a^2*b*(1/4*(d*x+c)^4*arcsinh(d*x+c)-1/ 16*(d*x+c)^3*(1+(d*x+c)^2)^(1/2)+3/32*(d*x+c)*(1+(d*x+c)^2)^(1/2)-3/32*arc sinh(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 832 vs. \(2 (253) = 506\).
Time = 0.32 (sec) , antiderivative size = 832, normalized size of antiderivative = 2.98 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^3 \, dx=\frac {8 \, {\left (8 \, a^{3} + 3 \, a b^{2}\right )} d^{4} e^{3} x^{4} + 32 \, {\left (8 \, a^{3} + 3 \, a b^{2}\right )} c d^{3} e^{3} x^{3} - 24 \, {\left (3 \, a b^{2} - 2 \, {\left (8 \, a^{3} + 3 \, a b^{2}\right )} c^{2}\right )} d^{2} e^{3} x^{2} - 16 \, {\left (9 \, a b^{2} c - 2 \, {\left (8 \, a^{3} + 3 \, a b^{2}\right )} c^{3}\right )} d e^{3} x + 8 \, {\left (8 \, b^{3} d^{4} e^{3} x^{4} + 32 \, b^{3} c d^{3} e^{3} x^{3} + 48 \, b^{3} c^{2} d^{2} e^{3} x^{2} + 32 \, b^{3} c^{3} d e^{3} x + {\left (8 \, b^{3} c^{4} - 3 \, b^{3}\right )} e^{3}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3} + 24 \, {\left (8 \, a b^{2} d^{4} e^{3} x^{4} + 32 \, a b^{2} c d^{3} e^{3} x^{3} + 48 \, a b^{2} c^{2} d^{2} e^{3} x^{2} + 32 \, a b^{2} c^{3} d e^{3} x + {\left (8 \, a b^{2} c^{4} - 3 \, a b^{2}\right )} e^{3} - {\left (2 \, b^{3} d^{3} e^{3} x^{3} + 6 \, b^{3} c d^{2} e^{3} x^{2} + 3 \, {\left (2 \, b^{3} c^{2} - b^{3}\right )} d e^{3} x + {\left (2 \, b^{3} c^{3} - 3 \, b^{3} c\right )} e^{3}\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 )^{2} + 3 \, {\left (8 \, {\left (8 \, a^{2} b + b^{3}\right )} d^{4} e^{3} x^{4} + 32 \, {\left (8 \, a^{2} b + b^{3}\right )} c d^{3} e^{3} x^{3} - 24 \, {\left (b^{3} - 2 \, {\left (8 \, a^{2} b + b^{3}\right )} c^{2}\right )} d^{2} e^{3} x^{2} - 16 \, {\left (3 \, b^{3} c - 2 \, {\left (8 \, a^{2} b + b^{3}\right )} c^{3}\right )} d e^{3} x - {\left (24 \, b^{3} c^{2} - 8 \, {\left (8 \, a^{2} b + b^{3}\right )} c^{4} + 24 \, a^{2} b + 15 \, b^{3}\right )} e^{3} - 16 \, {\left (2 \, a b^{2} d^{3} e^{3} x^{3} + 6 \, a b^{2} c d^{2} e^{3} x^{2} + 3 \, {\left (2 \, a b^{2} c^{2} - a b^{2}\right )} d e^{3} x + {\left (2 \, a b^{2} c^{3} - 3 \, a b^{2} c\right )} e^{3}\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 ) - 3 \, {\left (2 \, {\left (8 \, a^{2} b + b^{3}\right )} d^{3} e^{3} x^{3} + 6 \, {\left (8 \, a^{2} b + b^{3}\right )} c d^{2} e^{3} x^{2} - 3 \, {\left (8 \, a^{2} b + 5 \, b^{3} - 2 \, {\left (8 \, a^{2} b + b^{3}\right )} c^{2}\right )} d e^{3} x + {\left (2 \, {\left (8 \, a^{2} b + b^{3}\right )} c^{3} - 3 \, {\left (8 \, a^{2} b + 5 \, b^{3}\right )} c\right )} e^{3}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{256 \, d} \]
1/256*(8*(8*a^3 + 3*a*b^2)*d^4*e^3*x^4 + 32*(8*a^3 + 3*a*b^2)*c*d^3*e^3*x^ 3 - 24*(3*a*b^2 - 2*(8*a^3 + 3*a*b^2)*c^2)*d^2*e^3*x^2 - 16*(9*a*b^2*c - 2 *(8*a^3 + 3*a*b^2)*c^3)*d*e^3*x + 8*(8*b^3*d^4*e^3*x^4 + 32*b^3*c*d^3*e^3* x^3 + 48*b^3*c^2*d^2*e^3*x^2 + 32*b^3*c^3*d*e^3*x + (8*b^3*c^4 - 3*b^3)*e^ 3)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^3 + 24*(8*a*b^2*d^4*e^ 3*x^4 + 32*a*b^2*c*d^3*e^3*x^3 + 48*a*b^2*c^2*d^2*e^3*x^2 + 32*a*b^2*c^3*d *e^3*x + (8*a*b^2*c^4 - 3*a*b^2)*e^3 - (2*b^3*d^3*e^3*x^3 + 6*b^3*c*d^2*e^ 3*x^2 + 3*(2*b^3*c^2 - b^3)*d*e^3*x + (2*b^3*c^3 - 3*b^3*c)*e^3)*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 + 3*(8*(8*a^2*b + b^3)*d^4*e^3*x^4 + 32*(8*a^2*b + b^3)*c*d^3*e^3*x^3 - 24*(b^3 - 2*(8*a^2*b + b^3)*c^2)*d^2*e^3*x^2 - 16*(3*b^3*c - 2*(8*a^2*b + b^3)*c^3)*d*e^3*x - (24*b^3*c^2 - 8*(8*a^2*b + b^3)*c^4 + 24*a^2*b + 15*b ^3)*e^3 - 16*(2*a*b^2*d^3*e^3*x^3 + 6*a*b^2*c*d^2*e^3*x^2 + 3*(2*a*b^2*c^2 - a*b^2)*d*e^3*x + (2*a*b^2*c^3 - 3*a*b^2*c)*e^3)*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*(2*(8*a^2 *b + b^3)*d^3*e^3*x^3 + 6*(8*a^2*b + b^3)*c*d^2*e^3*x^2 - 3*(8*a^2*b + 5*b ^3 - 2*(8*a^2*b + b^3)*c^2)*d*e^3*x + (2*(8*a^2*b + b^3)*c^3 - 3*(8*a^2*b + 5*b^3)*c)*e^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. 1828 vs. \(2 (260) = 520\).
Time = 0.72 (sec) , antiderivative size = 1828, normalized size of antiderivative = 6.55 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^3 \, dx=\text {Too large to display} \]
Piecewise((a**3*c**3*e**3*x + 3*a**3*c**2*d*e**3*x**2/2 + a**3*c*d**2*e**3 *x**3 + a**3*d**3*e**3*x**4/4 + 3*a**2*b*c**4*e**3*asinh(c + d*x)/(4*d) + 3*a**2*b*c**3*e**3*x*asinh(c + d*x) - 3*a**2*b*c**3*e**3*sqrt(c**2 + 2*c*d *x + d**2*x**2 + 1)/(16*d) + 9*a**2*b*c**2*d*e**3*x**2*asinh(c + d*x)/2 - 9*a**2*b*c**2*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/16 + 3*a**2*b*c* d**2*e**3*x**3*asinh(c + d*x) - 9*a**2*b*c*d*e**3*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/16 + 9*a**2*b*c*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1 )/(32*d) + 3*a**2*b*d**3*e**3*x**4*asinh(c + d*x)/4 - 3*a**2*b*d**2*e**3*x **3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/16 + 9*a**2*b*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/32 - 9*a**2*b*e**3*asinh(c + d*x)/(32*d) + 3*a*b* *2*c**4*e**3*asinh(c + d*x)**2/(4*d) + 3*a*b**2*c**3*e**3*x*asinh(c + d*x) **2 + 3*a*b**2*c**3*e**3*x/8 - 3*a*b**2*c**3*e**3*sqrt(c**2 + 2*c*d*x + d* *2*x**2 + 1)*asinh(c + d*x)/(8*d) + 9*a*b**2*c**2*d*e**3*x**2*asinh(c + d* x)**2/2 + 9*a*b**2*c**2*d*e**3*x**2/16 - 9*a*b**2*c**2*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/8 + 3*a*b**2*c*d**2*e**3*x**3*asin h(c + d*x)**2 + 3*a*b**2*c*d**2*e**3*x**3/8 - 9*a*b**2*c*d*e**3*x**2*sqrt( c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/8 - 9*a*b**2*c*e**3*x/16 + 9*a*b**2*c*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/(16*d) + 3*a*b**2*d**3*e**3*x**4*asinh(c + d*x)**2/4 + 3*a*b**2*d**3*e**3*x**4/3 2 - 3*a*b**2*d**2*e**3*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(...
\[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^3 \, dx=\int { {\left (d e x + c e\right )}^{3} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
1/4*a^3*d^3*e^3*x^4 + a^3*c*d^2*e^3*x^3 + 3/2*a^3*c^2*d*e^3*x^2 + 9/4*(2*x ^2*arcsinh(d*x + c) - d*(3*c^2*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4 *(c^2 + 1)*d^2))/d^3 + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*x/d^2 - (c^2 + 1) *arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^3 - 3*sqrt( d^2*x^2 + 2*c*d*x + c^2 + 1)*c/d^3))*a^2*b*c^2*d*e^3 + 1/2*(6*x^3*arcsinh( d*x + c) - d*(2*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*x^2/d^2 - 15*c^3*arcsinh (2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^4 - 5*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c*x/d^3 + 9*(c^2 + 1)*c*arcsinh(2*(d^2*x + c*d)/sqrt( -4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^4 + 15*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)* c^2/d^4 - 4*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^2 + 1)/d^4))*a^2*b*c*d^2* e^3 + 1/32*(24*x^4*arcsinh(d*x + c) - (6*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1) *x^3/d^2 - 14*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c*x^2/d^3 + 105*c^4*arcsin h(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^5 + 35*sqrt(d^2*x^ 2 + 2*c*d*x + c^2 + 1)*c^2*x/d^4 - 90*(c^2 + 1)*c^2*arcsinh(2*(d^2*x + c*d )/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^5 - 105*sqrt(d^2*x^2 + 2*c*d*x + c ^2 + 1)*c^3/d^5 - 9*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^2 + 1)*x/d^4 + 9* (c^2 + 1)^2*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^ 5 + 55*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^2 + 1)*c/d^5)*d)*a^2*b*d^3*e^3 + a^3*c^3*e^3*x + 3*((d*x + c)*arcsinh(d*x + c) - sqrt((d*x + c)^2 + 1))* a^2*b*c^3*e^3/d + 1/4*(b^3*d^3*e^3*x^4 + 4*b^3*c*d^2*e^3*x^3 + 6*b^3*c^...
\[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^3 \, dx=\int { {\left (d e x + c e\right )}^{3} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
Timed out. \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^3 \, dx=\int {\left (c\,e+d\,e\,x\right )}^3\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3 \,d x \]