3.2.49 \(\int (c e+d e x) (a+b \sinh ^{-1}(c+d x))^4 \, dx\) [149]

Optimal. Leaf size=195 \[ \frac {3 b^4 e (c+d x)^2}{4 d}-\frac {3 b^3 e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{2 d}+\frac {3 b^2 e \left (a+b \sinh ^{-1}(c+d x)\right )^2}{4 d}+\frac {3 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d}-\frac {b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{2 d} \]

[Out]

3/4*b^4*e*(d*x+c)^2/d+3/4*b^2*e*(a+b*arcsinh(d*x+c))^2/d+3/2*b^2*e*(d*x+c)^2*(a+b*arcsinh(d*x+c))^2/d+1/4*e*(a
+b*arcsinh(d*x+c))^4/d+1/2*e*(d*x+c)^2*(a+b*arcsinh(d*x+c))^4/d-3/2*b^3*e*(d*x+c)*(a+b*arcsinh(d*x+c))*(1+(d*x
+c)^2)^(1/2)/d-b*e*(d*x+c)*(a+b*arcsinh(d*x+c))^3*(1+(d*x+c)^2)^(1/2)/d

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Rubi [A]
time = 0.21, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5859, 12, 5776, 5812, 5783, 30} \begin {gather*} -\frac {3 b^3 e (c+d x) \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )}{2 d}+\frac {3 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d}+\frac {3 b^2 e \left (a+b \sinh ^{-1}(c+d x)\right )^2}{4 d}-\frac {b e (c+d x) \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{2 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 d}+\frac {3 b^4 e (c+d x)^2}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^4,x]

[Out]

(3*b^4*e*(c + d*x)^2)/(4*d) - (3*b^3*e*(c + d*x)*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/(2*d) + (3*b^
2*e*(a + b*ArcSinh[c + d*x])^2)/(4*d) + (3*b^2*e*(c + d*x)^2*(a + b*ArcSinh[c + d*x])^2)/(2*d) - (b*e*(c + d*x
)*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^3)/d + (e*(a + b*ArcSinh[c + d*x])^4)/(4*d) + (e*(c + d*x)^2*
(a + b*ArcSinh[c + d*x])^4)/(2*d)

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 5783

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[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]

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) \left (a+b \sinh ^{-1}(c+d x)\right )^4 \, dx &=\frac {\text {Subst}\left (\int e x \left (a+b \sinh ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {e \text {Subst}\left (\int x \left (a+b \sinh ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{2 d}-\frac {(2 b e) \text {Subst}\left (\int \frac {x^2 \left (a+b \sinh ^{-1}(x)\right )^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{2 d}+\frac {(b e) \text {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d}+\frac {\left (3 b^2 e\right ) \text {Subst}\left (\int x \left (a+b \sinh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {3 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d}-\frac {b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{2 d}-\frac {\left (3 b^3 e\right ) \text {Subst}\left (\int \frac {x^2 \left (a+b \sinh ^{-1}(x)\right )}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {3 b^3 e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{2 d}+\frac {3 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d}-\frac {b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{2 d}+\frac {\left (3 b^3 e\right ) \text {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{2 d}+\frac {\left (3 b^4 e\right ) \text {Subst}(\int x \, dx,x,c+d x)}{2 d}\\ &=\frac {3 b^4 e (c+d x)^2}{4 d}-\frac {3 b^3 e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{2 d}+\frac {3 b^2 e \left (a+b \sinh ^{-1}(c+d x)\right )^2}{4 d}+\frac {3 b^2 e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d}-\frac {b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{2 d}\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 300, normalized size = 1.54 \begin {gather*} \frac {e \left (\left (2 a^4+6 a^2 b^2+3 b^4\right ) (c+d x)^2-2 a b \left (2 a^2+3 b^2\right ) (c+d x) \sqrt {1+(c+d x)^2}+2 a b \left (2 a^2+3 b^2\right ) \sinh ^{-1}(c+d x)-2 b (c+d x) \left (-4 a^3 (c+d x)-6 a b^2 (c+d x)+6 a^2 b \sqrt {1+(c+d x)^2}+3 b^3 \sqrt {1+(c+d x)^2}\right ) \sinh ^{-1}(c+d x)+3 b^2 \left (2 a^2+b^2+4 a^2 (c+d x)^2+2 b^2 (c+d x)^2-4 a b (c+d x) \sqrt {1+(c+d x)^2}\right ) \sinh ^{-1}(c+d x)^2+4 b^3 \left (a+2 a (c+d x)^2-b (c+d x) \sqrt {1+(c+d x)^2}\right ) \sinh ^{-1}(c+d x)^3+b^4 \left (1+2 (c+d x)^2\right ) \sinh ^{-1}(c+d x)^4\right )}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^4,x]

[Out]

(e*((2*a^4 + 6*a^2*b^2 + 3*b^4)*(c + d*x)^2 - 2*a*b*(2*a^2 + 3*b^2)*(c + d*x)*Sqrt[1 + (c + d*x)^2] + 2*a*b*(2
*a^2 + 3*b^2)*ArcSinh[c + d*x] - 2*b*(c + d*x)*(-4*a^3*(c + d*x) - 6*a*b^2*(c + d*x) + 6*a^2*b*Sqrt[1 + (c + d
*x)^2] + 3*b^3*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] + 3*b^2*(2*a^2 + b^2 + 4*a^2*(c + d*x)^2 + 2*b^2*(c + d
*x)^2 - 4*a*b*(c + d*x)*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^2 + 4*b^3*(a + 2*a*(c + d*x)^2 - b*(c + d*x)*S
qrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^3 + b^4*(1 + 2*(c + d*x)^2)*ArcSinh[c + d*x]^4))/(4*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(370\) vs. \(2(179)=358\).
time = 2.56, size = 371, normalized size = 1.90

method	result	size
derivativedivides	\(\frac {\frac {e \left (d x +c \right )^{2} a^{4}}{2}+e \,b^{4} \left (\frac {\arcsinh \left (d x +c \right )^{4} \left (1+\left (d x +c \right )^{2}\right )}{2}-\arcsinh \left (d x +c \right )^{3} \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}-\frac {\arcsinh \left (d x +c \right )^{4}}{4}+\frac {3 \arcsinh \left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 \arcsinh \left (d x +c \right ) \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{2}-\frac {3 \arcsinh \left (d x +c \right )^{2}}{4}+\frac {3 \left (d x +c \right )^{2}}{4}+\frac {3}{4}\right )+4 e a \,b^{3} \left (\frac {\arcsinh \left (d x +c \right )^{3} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 \arcsinh \left (d x +c \right )^{2} \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{4}-\frac {\arcsinh \left (d x +c \right )^{3}}{4}+\frac {3 \left (1+\left (d x +c \right )^{2}\right ) \arcsinh \left (d x +c \right )}{4}-\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}-\frac {3 \arcsinh \left (d x +c \right )}{8}\right )+6 e \,a^{2} b^{2} \left (\frac {\arcsinh \left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {\arcsinh \left (d x +c \right ) \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{2}-\frac {\arcsinh \left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}+\frac {1}{4}\right )+4 e \,a^{3} b \left (\frac {\left (d x +c \right )^{2} \arcsinh \left (d x +c \right )}{2}-\frac {\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {\arcsinh \left (d x +c \right )}{4}\right )}{d}\)	\(371\)
default	\(\frac {\frac {e \left (d x +c \right )^{2} a^{4}}{2}+e \,b^{4} \left (\frac {\arcsinh \left (d x +c \right )^{4} \left (1+\left (d x +c \right )^{2}\right )}{2}-\arcsinh \left (d x +c \right )^{3} \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}-\frac {\arcsinh \left (d x +c \right )^{4}}{4}+\frac {3 \arcsinh \left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 \arcsinh \left (d x +c \right ) \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{2}-\frac {3 \arcsinh \left (d x +c \right )^{2}}{4}+\frac {3 \left (d x +c \right )^{2}}{4}+\frac {3}{4}\right )+4 e a \,b^{3} \left (\frac {\arcsinh \left (d x +c \right )^{3} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 \arcsinh \left (d x +c \right )^{2} \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{4}-\frac {\arcsinh \left (d x +c \right )^{3}}{4}+\frac {3 \left (1+\left (d x +c \right )^{2}\right ) \arcsinh \left (d x +c \right )}{4}-\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}-\frac {3 \arcsinh \left (d x +c \right )}{8}\right )+6 e \,a^{2} b^{2} \left (\frac {\arcsinh \left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {\arcsinh \left (d x +c \right ) \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{2}-\frac {\arcsinh \left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}+\frac {1}{4}\right )+4 e \,a^{3} b \left (\frac {\left (d x +c \right )^{2} \arcsinh \left (d x +c \right )}{2}-\frac {\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {\arcsinh \left (d x +c \right )}{4}\right )}{d}\)	\(371\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)*(a+b*arcsinh(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/2*e*(d*x+c)^2*a^4+e*b^4*(1/2*arcsinh(d*x+c)^4*(1+(d*x+c)^2)-arcsinh(d*x+c)^3*(d*x+c)*(1+(d*x+c)^2)^(1/2
)-1/4*arcsinh(d*x+c)^4+3/2*arcsinh(d*x+c)^2*(1+(d*x+c)^2)-3/2*arcsinh(d*x+c)*(1+(d*x+c)^2)^(1/2)*(d*x+c)-3/4*a
rcsinh(d*x+c)^2+3/4*(d*x+c)^2+3/4)+4*e*a*b^3*(1/2*arcsinh(d*x+c)^3*(1+(d*x+c)^2)-3/4*arcsinh(d*x+c)^2*(d*x+c)*
(1+(d*x+c)^2)^(1/2)-1/4*arcsinh(d*x+c)^3+3/4*(1+(d*x+c)^2)*arcsinh(d*x+c)-3/8*(d*x+c)*(1+(d*x+c)^2)^(1/2)-3/8*
arcsinh(d*x+c))+6*e*a^2*b^2*(1/2*arcsinh(d*x+c)^2*(1+(d*x+c)^2)-1/2*arcsinh(d*x+c)*(1+(d*x+c)^2)^(1/2)*(d*x+c)
-1/4*arcsinh(d*x+c)^2+1/4*(d*x+c)^2+1/4)+4*e*a^3*b*(1/2*(d*x+c)^2*arcsinh(d*x+c)-1/4*(d*x+c)*(1+(d*x+c)^2)^(1/
2)+1/4*arcsinh(d*x+c)))

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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)*(a+b*arcsinh(d*x+c))^4,x, algorithm="maxima")

[Out]

1/2*a^4*d*x^2*e + (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^3*b*d*e + a^4*c*x*e + 4*((d*x + c)*arcsinh(d*x
+ c) - sqrt((d*x + c)^2 + 1))*a^3*b*c*e/d + 1/2*(b^4*d*x^2*e + 2*b^4*c*x*e)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d
*x + c^2 + 1))^4 + integrate(2*((2*(c^4 + c^2)*a*b^3*e + (2*a*b^3*d^4 - b^4*d^4)*x^4*e + 4*(2*a*b^3*c*d^3 - b^
4*c*d^3)*x^3*e + (2*(6*c^2*d^2 + d^2)*a*b^3 - (5*c^2*d^2 + d^2)*b^4)*x^2*e + 2*(2*(2*c^3*d + c*d)*a*b^3 - (c^3
*d + c*d)*b^4)*x*e + (2*(c^3 + c)*a*b^3*e + (2*a*b^3*d^3 - b^4*d^3)*x^3*e + 3*(2*a*b^3*c*d^2 - b^4*c*d^2)*x^2*
e - 2*(b^4*c^2*d - (3*c^2*d + d)*a*b^3)*x*e)*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^4*x^4*e + 4*a^2*b^2*c*d^3*x^3*e + (6*c^2*d^2 + d^2)*a^2*b^2*x^2*e + 2*(2*c
^3*d + c*d)*a^2*b^2*x*e + (c^4 + c^2)*a^2*b^2*e + (a^2*b^2*d^3*x^3*e + 3*a^2*b^2*c*d^2*x^2*e + (3*c^2*d + d)*a
^2*b^2*x*e + (c^3 + c)*a^2*b^2*e)*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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 949 vs. \(2 (186) = 372\).
time = 0.40, size = 949, normalized size = 4.87 \begin {gather*} \frac {{\left ({\left (2 \, b^{4} d^{2} x^{2} + 4 \, b^{4} c d x + 2 \, b^{4} c^{2} + b^{4}\right )} \cosh \left (1\right ) + {\left (2 \, b^{4} d^{2} x^{2} + 4 \, b^{4} c d x + 2 \, b^{4} c^{2} + b^{4}\right )} \sinh \left (1\right )\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{4} + 4 \, {\left ({\left (2 \, a b^{3} d^{2} x^{2} + 4 \, a b^{3} c d x + 2 \, a b^{3} c^{2} + a b^{3}\right )} \cosh \left (1\right ) + {\left (2 \, a b^{3} d^{2} x^{2} + 4 \, a b^{3} c d x + 2 \, a b^{3} c^{2} + a b^{3}\right )} \sinh \left (1\right ) - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left ({\left (b^{4} d x + b^{4} c\right )} \cosh \left (1\right ) + {\left (b^{4} d x + b^{4} c\right )} \sinh \left (1\right )\right )}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3} + 3 \, {\left ({\left (2 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} d^{2} x^{2} + 2 \, a^{2} b^{2} + b^{4} + 4 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} c d x + 2 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} c^{2}\right )} \cosh \left (1\right ) + {\left (2 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} d^{2} x^{2} + 2 \, a^{2} b^{2} + b^{4} + 4 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} c d x + 2 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} c^{2}\right )} \sinh \left (1\right ) - 4 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left ({\left (a b^{3} d x + a b^{3} c\right )} \cosh \left (1\right ) + {\left (a b^{3} d x + a b^{3} c\right )} \sinh \left (1\right )\right )}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + {\left ({\left (2 \, a^{4} + 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} d^{2} x^{2} + 2 \, {\left (2 \, a^{4} + 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} c d x\right )} \cosh \left (1\right ) + 2 \, {\left ({\left (2 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} d^{2} x^{2} + 2 \, a^{3} b + 3 \, a b^{3} + 4 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c d x + 2 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c^{2}\right )} \cosh \left (1\right ) + {\left (2 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} d^{2} x^{2} + 2 \, a^{3} b + 3 \, a b^{3} + 4 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c d x + 2 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c^{2}\right )} \sinh \left (1\right ) - 3 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left ({\left ({\left (2 \, a^{2} b^{2} + b^{4}\right )} d x + {\left (2 \, a^{2} b^{2} + b^{4}\right )} c\right )} \cosh \left (1\right ) + {\left ({\left (2 \, a^{2} b^{2} + b^{4}\right )} d x + {\left (2 \, a^{2} b^{2} + b^{4}\right )} c\right )} \sinh \left (1\right )\right )}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) + {\left ({\left (2 \, a^{4} + 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} d^{2} x^{2} + 2 \, {\left (2 \, a^{4} + 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} c d x\right )} \sinh \left (1\right ) - 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left ({\left ({\left (2 \, a^{3} b + 3 \, a b^{3}\right )} d x + {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c\right )} \cosh \left (1\right ) + {\left ({\left (2 \, a^{3} b + 3 \, a b^{3}\right )} d x + {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c\right )} \sinh \left (1\right )\right )}}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)*(a+b*arcsinh(d*x+c))^4,x, algorithm="fricas")

[Out]

1/4*(((2*b^4*d^2*x^2 + 4*b^4*c*d*x + 2*b^4*c^2 + b^4)*cosh(1) + (2*b^4*d^2*x^2 + 4*b^4*c*d*x + 2*b^4*c^2 + b^4
)*sinh(1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^4 + 4*((2*a*b^3*d^2*x^2 + 4*a*b^3*c*d*x + 2*a*b^3*
c^2 + a*b^3)*cosh(1) + (2*a*b^3*d^2*x^2 + 4*a*b^3*c*d*x + 2*a*b^3*c^2 + a*b^3)*sinh(1) - sqrt(d^2*x^2 + 2*c*d*
x + c^2 + 1)*((b^4*d*x + b^4*c)*cosh(1) + (b^4*d*x + b^4*c)*sinh(1)))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c
^2 + 1))^3 + 3*((2*(2*a^2*b^2 + b^4)*d^2*x^2 + 2*a^2*b^2 + b^4 + 4*(2*a^2*b^2 + b^4)*c*d*x + 2*(2*a^2*b^2 + b^
4)*c^2)*cosh(1) + (2*(2*a^2*b^2 + b^4)*d^2*x^2 + 2*a^2*b^2 + b^4 + 4*(2*a^2*b^2 + b^4)*c*d*x + 2*(2*a^2*b^2 +
b^4)*c^2)*sinh(1) - 4*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*((a*b^3*d*x + a*b^3*c)*cosh(1) + (a*b^3*d*x + a*b^3*c)
*sinh(1)))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 + ((2*a^4 + 6*a^2*b^2 + 3*b^4)*d^2*x^2 + 2*(2*a^
4 + 6*a^2*b^2 + 3*b^4)*c*d*x)*cosh(1) + 2*((2*(2*a^3*b + 3*a*b^3)*d^2*x^2 + 2*a^3*b + 3*a*b^3 + 4*(2*a^3*b + 3
*a*b^3)*c*d*x + 2*(2*a^3*b + 3*a*b^3)*c^2)*cosh(1) + (2*(2*a^3*b + 3*a*b^3)*d^2*x^2 + 2*a^3*b + 3*a*b^3 + 4*(2
*a^3*b + 3*a*b^3)*c*d*x + 2*(2*a^3*b + 3*a*b^3)*c^2)*sinh(1) - 3*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(((2*a^2*b^
2 + b^4)*d*x + (2*a^2*b^2 + b^4)*c)*cosh(1) + ((2*a^2*b^2 + b^4)*d*x + (2*a^2*b^2 + b^4)*c)*sinh(1)))*log(d*x
+ c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) + ((2*a^4 + 6*a^2*b^2 + 3*b^4)*d^2*x^2 + 2*(2*a^4 + 6*a^2*b^2 + 3*b^4
)*c*d*x)*sinh(1) - 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(((2*a^3*b + 3*a*b^3)*d*x + (2*a^3*b + 3*a*b^3)*c)*cosh
(1) + ((2*a^3*b + 3*a*b^3)*d*x + (2*a^3*b + 3*a*b^3)*c)*sinh(1)))/d

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1027 vs. \(2 (178) = 356\).
time = 0.54, size = 1027, normalized size = 5.27 \begin {gather*} \begin {cases} a^{4} c e x + \frac {a^{4} d e x^{2}}{2} + \frac {2 a^{3} b c^{2} e \operatorname {asinh}{\left (c + d x \right )}}{d} + 4 a^{3} b c e x \operatorname {asinh}{\left (c + d x \right )} - \frac {a^{3} b c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{d} + 2 a^{3} b d e x^{2} \operatorname {asinh}{\left (c + d x \right )} - a^{3} b e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} + \frac {a^{3} b e \operatorname {asinh}{\left (c + d x \right )}}{d} + \frac {3 a^{2} b^{2} c^{2} e \operatorname {asinh}^{2}{\left (c + d x \right )}}{d} + 6 a^{2} b^{2} c e x \operatorname {asinh}^{2}{\left (c + d x \right )} + 3 a^{2} b^{2} c e x - \frac {3 a^{2} b^{2} c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{d} + 3 a^{2} b^{2} d e x^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + \frac {3 a^{2} b^{2} d e x^{2}}{2} - 3 a^{2} b^{2} e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )} + \frac {3 a^{2} b^{2} e \operatorname {asinh}^{2}{\left (c + d x \right )}}{2 d} + \frac {2 a b^{3} c^{2} e \operatorname {asinh}^{3}{\left (c + d x \right )}}{d} + \frac {3 a b^{3} c^{2} e \operatorname {asinh}{\left (c + d x \right )}}{d} + 4 a b^{3} c e x \operatorname {asinh}^{3}{\left (c + d x \right )} + 6 a b^{3} c e x \operatorname {asinh}{\left (c + d x \right )} - \frac {3 a b^{3} c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c + d x \right )}}{d} - \frac {3 a b^{3} c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{2 d} + 2 a b^{3} d e x^{2} \operatorname {asinh}^{3}{\left (c + d x \right )} + 3 a b^{3} d e x^{2} \operatorname {asinh}{\left (c + d x \right )} - 3 a b^{3} e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c + d x \right )} - \frac {3 a b^{3} e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{2} + \frac {a b^{3} e \operatorname {asinh}^{3}{\left (c + d x \right )}}{d} + \frac {3 a b^{3} e \operatorname {asinh}{\left (c + d x \right )}}{2 d} + \frac {b^{4} c^{2} e \operatorname {asinh}^{4}{\left (c + d x \right )}}{2 d} + \frac {3 b^{4} c^{2} e \operatorname {asinh}^{2}{\left (c + d x \right )}}{2 d} + b^{4} c e x \operatorname {asinh}^{4}{\left (c + d x \right )} + 3 b^{4} c e x \operatorname {asinh}^{2}{\left (c + d x \right )} + \frac {3 b^{4} c e x}{2} - \frac {b^{4} c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (c + d x \right )}}{d} - \frac {3 b^{4} c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{2 d} + \frac {b^{4} d e x^{2} \operatorname {asinh}^{4}{\left (c + d x \right )}}{2} + \frac {3 b^{4} d e x^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{2} + \frac {3 b^{4} d e x^{2}}{4} - b^{4} e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (c + d x \right )} - \frac {3 b^{4} e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{2} + \frac {b^{4} e \operatorname {asinh}^{4}{\left (c + d x \right )}}{4 d} + \frac {3 b^{4} e \operatorname {asinh}^{2}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\c e x \left (a + b \operatorname {asinh}{\left (c \right )}\right )^{4} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)*(a+b*asinh(d*x+c))**4,x)

[Out]

Piecewise((a**4*c*e*x + a**4*d*e*x**2/2 + 2*a**3*b*c**2*e*asinh(c + d*x)/d + 4*a**3*b*c*e*x*asinh(c + d*x) - a
**3*b*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/d + 2*a**3*b*d*e*x**2*asinh(c + d*x) - a**3*b*e*x*sqrt(c**2 + 2
*c*d*x + d**2*x**2 + 1) + a**3*b*e*asinh(c + d*x)/d + 3*a**2*b**2*c**2*e*asinh(c + d*x)**2/d + 6*a**2*b**2*c*e
*x*asinh(c + d*x)**2 + 3*a**2*b**2*c*e*x - 3*a**2*b**2*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)
/d + 3*a**2*b**2*d*e*x**2*asinh(c + d*x)**2 + 3*a**2*b**2*d*e*x**2/2 - 3*a**2*b**2*e*x*sqrt(c**2 + 2*c*d*x + d
**2*x**2 + 1)*asinh(c + d*x) + 3*a**2*b**2*e*asinh(c + d*x)**2/(2*d) + 2*a*b**3*c**2*e*asinh(c + d*x)**3/d + 3
*a*b**3*c**2*e*asinh(c + d*x)/d + 4*a*b**3*c*e*x*asinh(c + d*x)**3 + 6*a*b**3*c*e*x*asinh(c + d*x) - 3*a*b**3*
c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2/d - 3*a*b**3*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 +
1)/(2*d) + 2*a*b**3*d*e*x**2*asinh(c + d*x)**3 + 3*a*b**3*d*e*x**2*asinh(c + d*x) - 3*a*b**3*e*x*sqrt(c**2 + 2
*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2 - 3*a*b**3*e*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/2 + a*b**3*e*asi
nh(c + d*x)**3/d + 3*a*b**3*e*asinh(c + d*x)/(2*d) + b**4*c**2*e*asinh(c + d*x)**4/(2*d) + 3*b**4*c**2*e*asinh
(c + d*x)**2/(2*d) + b**4*c*e*x*asinh(c + d*x)**4 + 3*b**4*c*e*x*asinh(c + d*x)**2 + 3*b**4*c*e*x/2 - b**4*c*e
*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**3/d - 3*b**4*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*as
inh(c + d*x)/(2*d) + b**4*d*e*x**2*asinh(c + d*x)**4/2 + 3*b**4*d*e*x**2*asinh(c + d*x)**2/2 + 3*b**4*d*e*x**2
/4 - b**4*e*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**3 - 3*b**4*e*x*sqrt(c**2 + 2*c*d*x + d**2*x
**2 + 1)*asinh(c + d*x)/2 + b**4*e*asinh(c + d*x)**4/(4*d) + 3*b**4*e*asinh(c + d*x)**2/(4*d), Ne(d, 0)), (c*e
*x*(a + b*asinh(c))**4, 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)*(a+b*arcsinh(d*x+c))^4,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)*(b*arcsinh(d*x + c) + a)^4, 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 )\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)*(a + b*asinh(c + d*x))^4,x)

[Out]

int((c*e + d*e*x)*(a + b*asinh(c + d*x))^4, x)

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