3.2.27 \(\int (c e+d e x)^4 (a+b \sinh ^{-1}(c+d x))^2 \, dx\) [127]
Optimal. Leaf size=197 \[ \frac {16}{75} b^2 e^4 x-\frac {8 b^2 e^4 (c+d x)^3}{225 d}+\frac {2 b^2 e^4 (c+d x)^5}{125 d}-\frac {16 b e^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{75 d}+\frac {8 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{75 d}-\frac {2 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{5 d} \]
[Out]
16/75*b^2*e^4*x-8/225*b^2*e^4*(d*x+c)^3/d+2/125*b^2*e^4*(d*x+c)^5/d+1/5*e^4*(d*x+c)^5*(a+b*arcsinh(d*x+c))^2/d
-16/75*b*e^4*(a+b*arcsinh(d*x+c))*(1+(d*x+c)^2)^(1/2)/d+8/75*b*e^4*(d*x+c)^2*(a+b*arcsinh(d*x+c))*(1+(d*x+c)^2
)^(1/2)/d-2/25*b*e^4*(d*x+c)^4*(a+b*arcsinh(d*x+c))*(1+(d*x+c)^2)^(1/2)/d
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Rubi [A]
time = 0.20, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5859, 12, 5776,
5812, 5798, 8, 30} \begin {gather*} \frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{5 d}-\frac {2 b e^4 \sqrt {(c+d x)^2+1} (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )}{25 d}+\frac {8 b e^4 \sqrt {(c+d x)^2+1} (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )}{75 d}-\frac {16 b e^4 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )}{75 d}+\frac {2 b^2 e^4 (c+d x)^5}{125 d}-\frac {8 b^2 e^4 (c+d x)^3}{225 d}+\frac {16}{75} b^2 e^4 x \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(c*e + d*e*x)^4*(a + b*ArcSinh[c + d*x])^2,x]
[Out]
(16*b^2*e^4*x)/75 - (8*b^2*e^4*(c + d*x)^3)/(225*d) + (2*b^2*e^4*(c + d*x)^5)/(125*d) - (16*b*e^4*Sqrt[1 + (c
+ d*x)^2]*(a + b*ArcSinh[c + d*x]))/(75*d) + (8*b*e^4*(c + d*x)^2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x
]))/(75*d) - (2*b*e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/(25*d) + (e^4*(c + d*x)^5*(a
+ b*ArcSinh[c + d*x])^2)/(5*d)
Rule 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 30
Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]
Rule 5776
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcS
inh[c*x])^n/(d*(m + 1))), x] - Dist[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]
Rule 5798
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] - Dist[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]
Rule 5812
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] + (-Dist[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] - Dist[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
]
Rule 5859
Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[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
]
Rubi steps
\begin {align*} \int (c e+d e x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int e^4 x^4 \left (a+b \sinh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 \text {Subst}\left (\int x^4 \left (a+b \sinh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{5 d}-\frac {\left (2 b e^4\right ) \text {Subst}\left (\int \frac {x^5 \left (a+b \sinh ^{-1}(x)\right )}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{5 d}\\ &=-\frac {2 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{5 d}+\frac {\left (8 b e^4\right ) \text {Subst}\left (\int \frac {x^3 \left (a+b \sinh ^{-1}(x)\right )}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{25 d}+\frac {\left (2 b^2 e^4\right ) \text {Subst}\left (\int x^4 \, dx,x,c+d x\right )}{25 d}\\ &=\frac {2 b^2 e^4 (c+d x)^5}{125 d}+\frac {8 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{75 d}-\frac {2 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{5 d}-\frac {\left (16 b e^4\right ) \text {Subst}\left (\int \frac {x \left (a+b \sinh ^{-1}(x)\right )}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{75 d}-\frac {\left (8 b^2 e^4\right ) \text {Subst}\left (\int x^2 \, dx,x,c+d x\right )}{75 d}\\ &=-\frac {8 b^2 e^4 (c+d x)^3}{225 d}+\frac {2 b^2 e^4 (c+d x)^5}{125 d}-\frac {16 b e^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{75 d}+\frac {8 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{75 d}-\frac {2 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{5 d}+\frac {\left (16 b^2 e^4\right ) \text {Subst}(\int 1 \, dx,x,c+d x)}{75 d}\\ &=\frac {16}{75} b^2 e^4 x-\frac {8 b^2 e^4 (c+d x)^3}{225 d}+\frac {2 b^2 e^4 (c+d x)^5}{125 d}-\frac {16 b e^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{75 d}+\frac {8 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{75 d}-\frac {2 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{5 d}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 192, normalized size = 0.97 \begin {gather*} \frac {e^4 \left (240 b^2 (c+d x)-40 b^2 (c+d x)^3+9 \left (25 a^2+2 b^2\right ) (c+d x)^5+30 a b \sqrt {1+(c+d x)^2} \left (-8+4 (c+d x)^2-3 (c+d x)^4\right )+30 b \left (15 a (c+d x)^5-8 b \sqrt {1+(c+d x)^2}+4 b (c+d x)^2 \sqrt {1+(c+d x)^2}-3 b (c+d x)^4 \sqrt {1+(c+d x)^2}\right ) \sinh ^{-1}(c+d x)+225 b^2 (c+d x)^5 \sinh ^{-1}(c+d x)^2\right )}{1125 d} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(c*e + d*e*x)^4*(a + b*ArcSinh[c + d*x])^2,x]
[Out]
(e^4*(240*b^2*(c + d*x) - 40*b^2*(c + d*x)^3 + 9*(25*a^2 + 2*b^2)*(c + d*x)^5 + 30*a*b*Sqrt[1 + (c + d*x)^2]*(
-8 + 4*(c + d*x)^2 - 3*(c + d*x)^4) + 30*b*(15*a*(c + d*x)^5 - 8*b*Sqrt[1 + (c + d*x)^2] + 4*b*(c + d*x)^2*Sqr
t[1 + (c + d*x)^2] - 3*b*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] + 225*b^2*(c + d*x)^5*ArcSinh[c +
d*x]^2))/(1125*d)
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(581\) vs.
\(2(177)=354\).
time = 3.93, size = 582, normalized size = 2.95
| | |
| method |
result |
size |
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| default |
\(\frac {e^{4} \left (d x +c \right )^{5} a^{2}}{5 d}+\frac {e^{4} b^{2} \left (18 x^{5} d^{5}+240 c -540 \arcsinh \left (d x +c \right ) \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, x^{2} c^{2} d^{2}+240 \arcsinh \left (d x +c \right ) \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, x c d -360 \arcsinh \left (d x +c \right ) \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, x^{3} c \,d^{3}+90 x \,c^{4} d -40 c^{3}-40 d^{3} x^{3}+18 c^{5}+240 d x +180 x^{2} c^{3} d^{2}-120 x^{2} c \,d^{2}-360 \arcsinh \left (d x +c \right ) \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, x \,c^{3} d -90 \arcsinh \left (d x +c \right ) \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, c^{4}+120 \arcsinh \left (d x +c \right ) \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, c^{2}+225 \arcsinh \left (d x +c \right )^{2} x^{5} d^{5}-120 x \,c^{2} d +180 x^{3} c^{2} d^{3}+90 x^{4} c \,d^{4}-90 \arcsinh \left (d x +c \right ) \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, x^{4} d^{4}+120 \arcsinh \left (d x +c \right ) \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, x^{2} d^{2}+1125 \arcsinh \left (d x +c \right )^{2} x^{4} c \,d^{4}+2250 \arcsinh \left (d x +c \right )^{2} x^{3} c^{2} d^{3}+2250 \arcsinh \left (d x +c \right )^{2} x^{2} c^{3} d^{2}+1125 \arcsinh \left (d x +c \right )^{2} x \,c^{4} d -240 \arcsinh \left (d x +c \right ) \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}+225 \arcsinh \left (d x +c \right )^{2} c^{5}\right )}{1125 d}+\frac {2 e^{4} a b \left (\frac {\left (d x +c \right )^{5} \arcsinh \left (d x +c \right )}{5}-\frac {\left (d x +c \right )^{4} \sqrt {1+\left (d x +c \right )^{2}}}{25}+\frac {4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{75}-\frac {8 \sqrt {1+\left (d x +c \right )^{2}}}{75}\right )}{d}\) |
\(582\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*e*x+c*e)^4*(a+b*arcsinh(d*x+c))^2,x,method=_RETURNVERBOSE)
[Out]
1/5*e^4*(d*x+c)^5*a^2/d+1/1125*e^4*b^2*(-240*arcsinh(d*x+c)*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)+225*arcsinh(d*x+c)^2
*c^5+18*x^5*d^5+240*c+90*x*c^4*d-40*c^3-40*d^3*x^3+18*c^5+240*d*x+180*x^2*c^3*d^2-90*arcsinh(d*x+c)*(d^2*x^2+2
*c*d*x+c^2+1)^(1/2)*c^4+120*arcsinh(d*x+c)*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*c^2+225*arcsinh(d*x+c)^2*x^5*d^5-120*
x^2*c*d^2-540*arcsinh(d*x+c)*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*x^2*c^2*d^2-360*arcsinh(d*x+c)*(d^2*x^2+2*c*d*x+c^2
+1)^(1/2)*x*c^3*d+240*arcsinh(d*x+c)*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*x*c*d-360*arcsinh(d*x+c)*(d^2*x^2+2*c*d*x+c
^2+1)^(1/2)*x^3*c*d^3-90*arcsinh(d*x+c)*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*x^4*d^4+120*arcsinh(d*x+c)*(d^2*x^2+2*c*
d*x+c^2+1)^(1/2)*x^2*d^2+1125*arcsinh(d*x+c)^2*x^4*c*d^4+2250*arcsinh(d*x+c)^2*x^3*c^2*d^3+2250*arcsinh(d*x+c)
^2*x^2*c^3*d^2+1125*arcsinh(d*x+c)^2*x*c^4*d-120*x*c^2*d+180*x^3*c^2*d^3+90*x^4*c*d^4)/d+2*e^4*a*b/d*(1/5*(d*x
+c)^5*arcsinh(d*x+c)-1/25*(d*x+c)^4*(1+(d*x+c)^2)^(1/2)+4/75*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)-8/75*(1+(d*x+c)^2)^
(1/2))
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*e*x+c*e)^4*(a+b*arcsinh(d*x+c))^2,x, algorithm="maxima")
[Out]
1/5*a^2*d^4*x^5*e^4 + a^2*c*d^3*x^4*e^4 + 2*a^2*c^2*d^2*x^3*e^4 + 2*a^2*c^3*d*x^2*e^4 + 2*(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*b*c^3*d*e^4 + 2/3*(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*b*c^2*d^2*e^4 + 1/12*(
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*arcsinh(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 - 10
5*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*b*c*d^3*e^4 + 1/300*(120*x^5*arcsinh(d*x + c) - (24*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*x^4/d
^2 - 54*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c*x^3/d^3 + 126*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c^2*x^2/d^4 - 945*
c^5*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^6 - 315*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c^
3*x/d^5 - 32*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^2 + 1)*x^2/d^4 + 1050*(c^2 + 1)*c^3*arcsinh(2*(d^2*x + c*d)/
sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^6 + 945*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c^4/d^6 + 161*sqrt(d^2*x^2 + 2
*c*d*x + c^2 + 1)*(c^2 + 1)*c*x/d^5 - 225*(c^2 + 1)^2*c*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*
d^2))/d^6 - 735*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^2 + 1)*c^2/d^6 + 64*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^
2 + 1)^2/d^6)*d)*a*b*d^4*e^4 + a^2*c^4*x*e^4 + 2*((d*x + c)*arcsinh(d*x + c) - sqrt((d*x + c)^2 + 1))*a*b*c^4*
e^4/d + 1/5*(b^2*d^4*x^5*e^4 + 5*b^2*c*d^3*x^4*e^4 + 10*b^2*c^2*d^2*x^3*e^4 + 10*b^2*c^3*d*x^2*e^4 + 5*b^2*c^4
*x*e^4)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 - integrate(2/5*(b^2*d^7*x^7*e^4 + 7*b^2*c*d^6*x^6*
e^4 + (21*b^2*c^2*d^5 + b^2*d^5)*x^5*e^4 + 5*(7*b^2*c^3*d^4 + b^2*c*d^4)*x^4*e^4 + 5*(7*b^2*c^4*d^3 + 2*b^2*c^
2*d^3)*x^3*e^4 + 10*(2*b^2*c^5*d^2 + b^2*c^3*d^2)*x^2*e^4 + 5*(b^2*c^6*d + b^2*c^4*d)*x*e^4 + (b^2*d^6*x^6*e^4
+ 6*b^2*c*d^5*x^5*e^4 + 15*b^2*c^2*d^4*x^4*e^4 + 20*b^2*c^3*d^3*x^3*e^4 + 15*b^2*c^4*d^2*x^2*e^4 + 5*b^2*c^5*
d*x*e^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))/(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)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2430 vs.
\(2 (170) = 340\).
time = 0.38, size = 2430, normalized size = 12.34 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*e*x+c*e)^4*(a+b*arcsinh(d*x+c))^2,x, algorithm="fricas")
[Out]
1/1125*((9*(25*a^2 + 2*b^2)*d^5*x^5 + 45*(25*a^2 + 2*b^2)*c*d^4*x^4 + 10*(9*(25*a^2 + 2*b^2)*c^2 - 4*b^2)*d^3*
x^3 + 30*(3*(25*a^2 + 2*b^2)*c^3 - 4*b^2*c)*d^2*x^2 + 15*(3*(25*a^2 + 2*b^2)*c^4 - 8*b^2*c^2 + 16*b^2)*d*x)*co
sh(1)^4 + 4*(9*(25*a^2 + 2*b^2)*d^5*x^5 + 45*(25*a^2 + 2*b^2)*c*d^4*x^4 + 10*(9*(25*a^2 + 2*b^2)*c^2 - 4*b^2)*
d^3*x^3 + 30*(3*(25*a^2 + 2*b^2)*c^3 - 4*b^2*c)*d^2*x^2 + 15*(3*(25*a^2 + 2*b^2)*c^4 - 8*b^2*c^2 + 16*b^2)*d*x
)*cosh(1)^3*sinh(1) + 6*(9*(25*a^2 + 2*b^2)*d^5*x^5 + 45*(25*a^2 + 2*b^2)*c*d^4*x^4 + 10*(9*(25*a^2 + 2*b^2)*c
^2 - 4*b^2)*d^3*x^3 + 30*(3*(25*a^2 + 2*b^2)*c^3 - 4*b^2*c)*d^2*x^2 + 15*(3*(25*a^2 + 2*b^2)*c^4 - 8*b^2*c^2 +
16*b^2)*d*x)*cosh(1)^2*sinh(1)^2 + 4*(9*(25*a^2 + 2*b^2)*d^5*x^5 + 45*(25*a^2 + 2*b^2)*c*d^4*x^4 + 10*(9*(25*
a^2 + 2*b^2)*c^2 - 4*b^2)*d^3*x^3 + 30*(3*(25*a^2 + 2*b^2)*c^3 - 4*b^2*c)*d^2*x^2 + 15*(3*(25*a^2 + 2*b^2)*c^4
- 8*b^2*c^2 + 16*b^2)*d*x)*cosh(1)*sinh(1)^3 + (9*(25*a^2 + 2*b^2)*d^5*x^5 + 45*(25*a^2 + 2*b^2)*c*d^4*x^4 +
10*(9*(25*a^2 + 2*b^2)*c^2 - 4*b^2)*d^3*x^3 + 30*(3*(25*a^2 + 2*b^2)*c^3 - 4*b^2*c)*d^2*x^2 + 15*(3*(25*a^2 +
2*b^2)*c^4 - 8*b^2*c^2 + 16*b^2)*d*x)*sinh(1)^4 + 225*((b^2*d^5*x^5 + 5*b^2*c*d^4*x^4 + 10*b^2*c^2*d^3*x^3 + 1
0*b^2*c^3*d^2*x^2 + 5*b^2*c^4*d*x + b^2*c^5)*cosh(1)^4 + 4*(b^2*d^5*x^5 + 5*b^2*c*d^4*x^4 + 10*b^2*c^2*d^3*x^3
+ 10*b^2*c^3*d^2*x^2 + 5*b^2*c^4*d*x + b^2*c^5)*cosh(1)^3*sinh(1) + 6*(b^2*d^5*x^5 + 5*b^2*c*d^4*x^4 + 10*b^2
*c^2*d^3*x^3 + 10*b^2*c^3*d^2*x^2 + 5*b^2*c^4*d*x + b^2*c^5)*cosh(1)^2*sinh(1)^2 + 4*(b^2*d^5*x^5 + 5*b^2*c*d^
4*x^4 + 10*b^2*c^2*d^3*x^3 + 10*b^2*c^3*d^2*x^2 + 5*b^2*c^4*d*x + b^2*c^5)*cosh(1)*sinh(1)^3 + (b^2*d^5*x^5 +
5*b^2*c*d^4*x^4 + 10*b^2*c^2*d^3*x^3 + 10*b^2*c^3*d^2*x^2 + 5*b^2*c^4*d*x + b^2*c^5)*sinh(1)^4)*log(d*x + c +
sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 + 30*(15*(a*b*d^5*x^5 + 5*a*b*c*d^4*x^4 + 10*a*b*c^2*d^3*x^3 + 10*a*b*c^3
*d^2*x^2 + 5*a*b*c^4*d*x + a*b*c^5)*cosh(1)^4 + 60*(a*b*d^5*x^5 + 5*a*b*c*d^4*x^4 + 10*a*b*c^2*d^3*x^3 + 10*a*
b*c^3*d^2*x^2 + 5*a*b*c^4*d*x + a*b*c^5)*cosh(1)^3*sinh(1) + 90*(a*b*d^5*x^5 + 5*a*b*c*d^4*x^4 + 10*a*b*c^2*d^
3*x^3 + 10*a*b*c^3*d^2*x^2 + 5*a*b*c^4*d*x + a*b*c^5)*cosh(1)^2*sinh(1)^2 + 60*(a*b*d^5*x^5 + 5*a*b*c*d^4*x^4
+ 10*a*b*c^2*d^3*x^3 + 10*a*b*c^3*d^2*x^2 + 5*a*b*c^4*d*x + a*b*c^5)*cosh(1)*sinh(1)^3 + 15*(a*b*d^5*x^5 + 5*a
*b*c*d^4*x^4 + 10*a*b*c^2*d^3*x^3 + 10*a*b*c^3*d^2*x^2 + 5*a*b*c^4*d*x + a*b*c^5)*sinh(1)^4 - ((3*b^2*d^4*x^4
+ 12*b^2*c*d^3*x^3 + 3*b^2*c^4 + 2*(9*b^2*c^2 - 2*b^2)*d^2*x^2 - 4*b^2*c^2 + 4*(3*b^2*c^3 - 2*b^2*c)*d*x + 8*b
^2)*cosh(1)^4 + 4*(3*b^2*d^4*x^4 + 12*b^2*c*d^3*x^3 + 3*b^2*c^4 + 2*(9*b^2*c^2 - 2*b^2)*d^2*x^2 - 4*b^2*c^2 +
4*(3*b^2*c^3 - 2*b^2*c)*d*x + 8*b^2)*cosh(1)^3*sinh(1) + 6*(3*b^2*d^4*x^4 + 12*b^2*c*d^3*x^3 + 3*b^2*c^4 + 2*(
9*b^2*c^2 - 2*b^2)*d^2*x^2 - 4*b^2*c^2 + 4*(3*b^2*c^3 - 2*b^2*c)*d*x + 8*b^2)*cosh(1)^2*sinh(1)^2 + 4*(3*b^2*d
^4*x^4 + 12*b^2*c*d^3*x^3 + 3*b^2*c^4 + 2*(9*b^2*c^2 - 2*b^2)*d^2*x^2 - 4*b^2*c^2 + 4*(3*b^2*c^3 - 2*b^2*c)*d*
x + 8*b^2)*cosh(1)*sinh(1)^3 + (3*b^2*d^4*x^4 + 12*b^2*c*d^3*x^3 + 3*b^2*c^4 + 2*(9*b^2*c^2 - 2*b^2)*d^2*x^2 -
4*b^2*c^2 + 4*(3*b^2*c^3 - 2*b^2*c)*d*x + 8*b^2)*sinh(1)^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)) - 30*((3*a*b*d^4*x^4 + 12*a*b*c*d^3*x^3 + 3*a*b*c^4 + 2*(9*a*b*c^2 - 2*a*b)
*d^2*x^2 - 4*a*b*c^2 + 4*(3*a*b*c^3 - 2*a*b*c)*d*x + 8*a*b)*cosh(1)^4 + 4*(3*a*b*d^4*x^4 + 12*a*b*c*d^3*x^3 +
3*a*b*c^4 + 2*(9*a*b*c^2 - 2*a*b)*d^2*x^2 - 4*a*b*c^2 + 4*(3*a*b*c^3 - 2*a*b*c)*d*x + 8*a*b)*cosh(1)^3*sinh(1)
+ 6*(3*a*b*d^4*x^4 + 12*a*b*c*d^3*x^3 + 3*a*b*c^4 + 2*(9*a*b*c^2 - 2*a*b)*d^2*x^2 - 4*a*b*c^2 + 4*(3*a*b*c^3
- 2*a*b*c)*d*x + 8*a*b)*cosh(1)^2*sinh(1)^2 + 4*(3*a*b*d^4*x^4 + 12*a*b*c*d^3*x^3 + 3*a*b*c^4 + 2*(9*a*b*c^2 -
2*a*b)*d^2*x^2 - 4*a*b*c^2 + 4*(3*a*b*c^3 - 2*a*b*c)*d*x + 8*a*b)*cosh(1)*sinh(1)^3 + (3*a*b*d^4*x^4 + 12*a*b
*c*d^3*x^3 + 3*a*b*c^4 + 2*(9*a*b*c^2 - 2*a*b)*d^2*x^2 - 4*a*b*c^2 + 4*(3*a*b*c^3 - 2*a*b*c)*d*x + 8*a*b)*sinh
(1)^4)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))/d
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1268 vs.
\(2 (184) = 368\).
time = 0.79, size = 1268, normalized size = 6.44 \begin {gather*} \begin {cases} a^{2} c^{4} e^{4} x + 2 a^{2} c^{3} d e^{4} x^{2} + 2 a^{2} c^{2} d^{2} e^{4} x^{3} + a^{2} c d^{3} e^{4} x^{4} + \frac {a^{2} d^{4} e^{4} x^{5}}{5} + \frac {2 a b c^{5} e^{4} \operatorname {asinh}{\left (c + d x \right )}}{5 d} + 2 a b c^{4} e^{4} x \operatorname {asinh}{\left (c + d x \right )} - \frac {2 a b c^{4} e^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{25 d} + 4 a b c^{3} d e^{4} x^{2} \operatorname {asinh}{\left (c + d x \right )} - \frac {8 a b c^{3} e^{4} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{25} + 4 a b c^{2} d^{2} e^{4} x^{3} \operatorname {asinh}{\left (c + d x \right )} - \frac {12 a b c^{2} d e^{4} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{25} + \frac {8 a b c^{2} e^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{75 d} + 2 a b c d^{3} e^{4} x^{4} \operatorname {asinh}{\left (c + d x \right )} - \frac {8 a b c d^{2} e^{4} x^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{25} + \frac {16 a b c e^{4} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{75} + \frac {2 a b d^{4} e^{4} x^{5} \operatorname {asinh}{\left (c + d x \right )}}{5} - \frac {2 a b d^{3} e^{4} x^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{25} + \frac {8 a b d e^{4} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{75} - \frac {16 a b e^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{75 d} + \frac {b^{2} c^{5} e^{4} \operatorname {asinh}^{2}{\left (c + d x \right )}}{5 d} + b^{2} c^{4} e^{4} x \operatorname {asinh}^{2}{\left (c + d x \right )} + \frac {2 b^{2} c^{4} e^{4} x}{25} - \frac {2 b^{2} c^{4} e^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{25 d} + 2 b^{2} c^{3} d e^{4} x^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + \frac {4 b^{2} c^{3} d e^{4} x^{2}}{25} - \frac {8 b^{2} c^{3} e^{4} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{25} + 2 b^{2} c^{2} d^{2} e^{4} x^{3} \operatorname {asinh}^{2}{\left (c + d x \right )} + \frac {4 b^{2} c^{2} d^{2} e^{4} x^{3}}{25} - \frac {12 b^{2} c^{2} d e^{4} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{25} - \frac {8 b^{2} c^{2} e^{4} x}{75} + \frac {8 b^{2} c^{2} e^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{75 d} + b^{2} c d^{3} e^{4} x^{4} \operatorname {asinh}^{2}{\left (c + d x \right )} + \frac {2 b^{2} c d^{3} e^{4} x^{4}}{25} - \frac {8 b^{2} c d^{2} e^{4} x^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{25} - \frac {8 b^{2} c d e^{4} x^{2}}{75} + \frac {16 b^{2} c e^{4} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{75} + \frac {b^{2} d^{4} e^{4} x^{5} \operatorname {asinh}^{2}{\left (c + d x \right )}}{5} + \frac {2 b^{2} d^{4} e^{4} x^{5}}{125} - \frac {2 b^{2} d^{3} e^{4} x^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{25} - \frac {8 b^{2} d^{2} e^{4} x^{3}}{225} + \frac {8 b^{2} d e^{4} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{75} + \frac {16 b^{2} e^{4} x}{75} - \frac {16 b^{2} e^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{75 d} & \text {for}\: d \neq 0 \\c^{4} e^{4} x \left (a + b \operatorname {asinh}{\left (c \right )}\right )^{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*e*x+c*e)**4*(a+b*asinh(d*x+c))**2,x)
[Out]
Piecewise((a**2*c**4*e**4*x + 2*a**2*c**3*d*e**4*x**2 + 2*a**2*c**2*d**2*e**4*x**3 + a**2*c*d**3*e**4*x**4 + a
**2*d**4*e**4*x**5/5 + 2*a*b*c**5*e**4*asinh(c + d*x)/(5*d) + 2*a*b*c**4*e**4*x*asinh(c + d*x) - 2*a*b*c**4*e*
*4*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(25*d) + 4*a*b*c**3*d*e**4*x**2*asinh(c + d*x) - 8*a*b*c**3*e**4*x*sqr
t(c**2 + 2*c*d*x + d**2*x**2 + 1)/25 + 4*a*b*c**2*d**2*e**4*x**3*asinh(c + d*x) - 12*a*b*c**2*d*e**4*x**2*sqrt
(c**2 + 2*c*d*x + d**2*x**2 + 1)/25 + 8*a*b*c**2*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(75*d) + 2*a*b*c*d*
*3*e**4*x**4*asinh(c + d*x) - 8*a*b*c*d**2*e**4*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/25 + 16*a*b*c*e**4*x
*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/75 + 2*a*b*d**4*e**4*x**5*asinh(c + d*x)/5 - 2*a*b*d**3*e**4*x**4*sqrt(c
**2 + 2*c*d*x + d**2*x**2 + 1)/25 + 8*a*b*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/75 - 16*a*b*e**4*sq
rt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(75*d) + b**2*c**5*e**4*asinh(c + d*x)**2/(5*d) + b**2*c**4*e**4*x*asinh(c
+ d*x)**2 + 2*b**2*c**4*e**4*x/25 - 2*b**2*c**4*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/(25*d
) + 2*b**2*c**3*d*e**4*x**2*asinh(c + d*x)**2 + 4*b**2*c**3*d*e**4*x**2/25 - 8*b**2*c**3*e**4*x*sqrt(c**2 + 2*
c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/25 + 2*b**2*c**2*d**2*e**4*x**3*asinh(c + d*x)**2 + 4*b**2*c**2*d**2*e**
4*x**3/25 - 12*b**2*c**2*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/25 - 8*b**2*c**2*e**4
*x/75 + 8*b**2*c**2*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/(75*d) + b**2*c*d**3*e**4*x**4*as
inh(c + d*x)**2 + 2*b**2*c*d**3*e**4*x**4/25 - 8*b**2*c*d**2*e**4*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*as
inh(c + d*x)/25 - 8*b**2*c*d*e**4*x**2/75 + 16*b**2*c*e**4*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*
x)/75 + b**2*d**4*e**4*x**5*asinh(c + d*x)**2/5 + 2*b**2*d**4*e**4*x**5/125 - 2*b**2*d**3*e**4*x**4*sqrt(c**2
+ 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/25 - 8*b**2*d**2*e**4*x**3/225 + 8*b**2*d*e**4*x**2*sqrt(c**2 + 2*c*
d*x + d**2*x**2 + 1)*asinh(c + d*x)/75 + 16*b**2*e**4*x/75 - 16*b**2*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)
*asinh(c + d*x)/(75*d), Ne(d, 0)), (c**4*e**4*x*(a + b*asinh(c))**2, True))
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*e*x+c*e)^4*(a+b*arcsinh(d*x+c))^2,x, algorithm="giac")
[Out]
integrate((d*e*x + c*e)^4*(b*arcsinh(d*x + c) + a)^2, x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (c\,e+d\,e\,x\right )}^4\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c*e + d*e*x)^4*(a + b*asinh(c + d*x))^2,x)
[Out]
int((c*e + d*e*x)^4*(a + b*asinh(c + d*x))^2, x)
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