3.2.90 \(\int (d+e x^2) \sinh ^{-1}(a x) \log (c x^n) \, dx\) [190]

Optimal. Leaf size=244 \[ \frac {d n \sqrt {1+a^2 x^2}}{a}+\frac {\left (3 a^2 d-e\right ) n \sqrt {1+a^2 x^2}}{3 a^3}+\frac {2 e n \left (1+a^2 x^2\right )^{3/2}}{27 a^3}-d n x \sinh ^{-1}(a x)-\frac {1}{9} e n x^3 \sinh ^{-1}(a x)-\frac {\left (3 a^2 d-e\right ) n \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )}{3 a^3}-\frac {e n \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )}{9 a^3}-\frac {\left (3 a^2 d-e\right ) \sqrt {1+a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac {e \left (1+a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \sinh ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sinh ^{-1}(a x) \log \left (c x^n\right ) \]

[Out]

2/27*e*n*(a^2*x^2+1)^(3/2)/a^3-d*n*x*arcsinh(a*x)-1/9*e*n*x^3*arcsinh(a*x)-1/3*(3*a^2*d-e)*n*arctanh((a^2*x^2+
1)^(1/2))/a^3-1/9*e*n*arctanh((a^2*x^2+1)^(1/2))/a^3-1/9*e*(a^2*x^2+1)^(3/2)*ln(c*x^n)/a^3+d*x*arcsinh(a*x)*ln
(c*x^n)+1/3*e*x^3*arcsinh(a*x)*ln(c*x^n)+d*n*(a^2*x^2+1)^(1/2)/a+1/3*(3*a^2*d-e)*n*(a^2*x^2+1)^(1/2)/a^3-1/3*(
3*a^2*d-e)*ln(c*x^n)*(a^2*x^2+1)^(1/2)/a^3

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Rubi [A]
time = 0.15, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {5792, 455, 45, 2434, 272, 52, 65, 214, 5772, 267, 5776} \begin {gather*} \frac {d n \sqrt {a^2 x^2+1}}{a}-\frac {\sqrt {a^2 x^2+1} \left (3 a^2 d-e\right ) \log \left (c x^n\right )}{3 a^3}-\frac {e \left (a^2 x^2+1\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+\frac {n \sqrt {a^2 x^2+1} \left (3 a^2 d-e\right )}{3 a^3}-\frac {n \left (3 a^2 d-e\right ) \tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right )}{3 a^3}+\frac {2 e n \left (a^2 x^2+1\right )^{3/2}}{27 a^3}-\frac {e n \tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right )}{9 a^3}+d x \sinh ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sinh ^{-1}(a x) \log \left (c x^n\right )-d n x \sinh ^{-1}(a x)-\frac {1}{9} e n x^3 \sinh ^{-1}(a x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)*ArcSinh[a*x]*Log[c*x^n],x]

[Out]

(d*n*Sqrt[1 + a^2*x^2])/a + ((3*a^2*d - e)*n*Sqrt[1 + a^2*x^2])/(3*a^3) + (2*e*n*(1 + a^2*x^2)^(3/2))/(27*a^3)
- d*n*x*ArcSinh[a*x] - (e*n*x^3*ArcSinh[a*x])/9 - ((3*a^2*d - e)*n*ArcTanh[Sqrt[1 + a^2*x^2]])/(3*a^3) - (e*n
*ArcTanh[Sqrt[1 + a^2*x^2]])/(9*a^3) - ((3*a^2*d - e)*Sqrt[1 + a^2*x^2]*Log[c*x^n])/(3*a^3) - (e*(1 + a^2*x^2)
^(3/2)*Log[c*x^n])/(9*a^3) + d*x*ArcSinh[a*x]*Log[c*x^n] + (e*x^3*ArcSinh[a*x]*Log[c*x^n])/3

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 2434

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(Px_.)*(F_)[(d_.)*((e_.) + (f_.)*(x_))]^(m_.), x_Symbol] :> With[{u
= IntHide[Px*F[d*(e + f*x)]^m, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist[1/x, u, x], x], x]] /; F
reeQ[{a, b, c, d, e, f, n}, x] && PolynomialQ[Px, x] && IGtQ[m, 0] && MemberQ[{ArcSin, ArcCos, ArcSinh, ArcCos
h}, F]

Rule 5772

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

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 5792

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /;
FreeQ[{a, b, c, d, e}, x] && NeQ[e, c^2*d] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])

Rubi steps

\begin {align*} \int \left (d+e x^2\right ) \sinh ^{-1}(a x) \log \left (c x^n\right ) \, dx &=-\frac {\left (3 a^2 d-e\right ) \sqrt {1+a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac {e \left (1+a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \sinh ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sinh ^{-1}(a x) \log \left (c x^n\right )-n \int \left (-\frac {\left (3 a^2 d-e\right ) \sqrt {1+a^2 x^2}}{3 a^3 x}-\frac {e \left (1+a^2 x^2\right )^{3/2}}{9 a^3 x}+d \sinh ^{-1}(a x)+\frac {1}{3} e x^2 \sinh ^{-1}(a x)\right ) \, dx\\ &=-\frac {\left (3 a^2 d-e\right ) \sqrt {1+a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac {e \left (1+a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \sinh ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sinh ^{-1}(a x) \log \left (c x^n\right )-(d n) \int \sinh ^{-1}(a x) \, dx+\frac {\left (\left (3 a^2 d-e\right ) n\right ) \int \frac {\sqrt {1+a^2 x^2}}{x} \, dx}{3 a^3}-\frac {1}{3} (e n) \int x^2 \sinh ^{-1}(a x) \, dx+\frac {(e n) \int \frac {\left (1+a^2 x^2\right )^{3/2}}{x} \, dx}{9 a^3}\\ &=-d n x \sinh ^{-1}(a x)-\frac {1}{9} e n x^3 \sinh ^{-1}(a x)-\frac {\left (3 a^2 d-e\right ) \sqrt {1+a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac {e \left (1+a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \sinh ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sinh ^{-1}(a x) \log \left (c x^n\right )+(a d n) \int \frac {x}{\sqrt {1+a^2 x^2}} \, dx+\frac {\left (\left (3 a^2 d-e\right ) n\right ) \text {Subst}\left (\int \frac {\sqrt {1+a^2 x}}{x} \, dx,x,x^2\right )}{6 a^3}+\frac {(e n) \text {Subst}\left (\int \frac {\left (1+a^2 x\right )^{3/2}}{x} \, dx,x,x^2\right )}{18 a^3}+\frac {1}{9} (a e n) \int \frac {x^3}{\sqrt {1+a^2 x^2}} \, dx\\ &=\frac {d n \sqrt {1+a^2 x^2}}{a}+\frac {\left (3 a^2 d-e\right ) n \sqrt {1+a^2 x^2}}{3 a^3}+\frac {e n \left (1+a^2 x^2\right )^{3/2}}{27 a^3}-d n x \sinh ^{-1}(a x)-\frac {1}{9} e n x^3 \sinh ^{-1}(a x)-\frac {\left (3 a^2 d-e\right ) \sqrt {1+a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac {e \left (1+a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \sinh ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sinh ^{-1}(a x) \log \left (c x^n\right )+\frac {\left (\left (3 a^2 d-e\right ) n\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right )}{6 a^3}+\frac {(e n) \text {Subst}\left (\int \frac {\sqrt {1+a^2 x}}{x} \, dx,x,x^2\right )}{18 a^3}+\frac {1}{18} (a e n) \text {Subst}\left (\int \frac {x}{\sqrt {1+a^2 x}} \, dx,x,x^2\right )\\ &=\frac {d n \sqrt {1+a^2 x^2}}{a}+\frac {\left (3 a^2 d-e\right ) n \sqrt {1+a^2 x^2}}{3 a^3}+\frac {e n \sqrt {1+a^2 x^2}}{9 a^3}+\frac {e n \left (1+a^2 x^2\right )^{3/2}}{27 a^3}-d n x \sinh ^{-1}(a x)-\frac {1}{9} e n x^3 \sinh ^{-1}(a x)-\frac {\left (3 a^2 d-e\right ) \sqrt {1+a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac {e \left (1+a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \sinh ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sinh ^{-1}(a x) \log \left (c x^n\right )+\frac {\left (\left (3 a^2 d-e\right ) n\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {1+a^2 x^2}\right )}{3 a^5}+\frac {(e n) \text {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right )}{18 a^3}+\frac {1}{18} (a e n) \text {Subst}\left (\int \left (-\frac {1}{a^2 \sqrt {1+a^2 x}}+\frac {\sqrt {1+a^2 x}}{a^2}\right ) \, dx,x,x^2\right )\\ &=\frac {d n \sqrt {1+a^2 x^2}}{a}+\frac {\left (3 a^2 d-e\right ) n \sqrt {1+a^2 x^2}}{3 a^3}+\frac {2 e n \left (1+a^2 x^2\right )^{3/2}}{27 a^3}-d n x \sinh ^{-1}(a x)-\frac {1}{9} e n x^3 \sinh ^{-1}(a x)-\frac {\left (3 a^2 d-e\right ) n \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )}{3 a^3}-\frac {\left (3 a^2 d-e\right ) \sqrt {1+a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac {e \left (1+a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \sinh ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sinh ^{-1}(a x) \log \left (c x^n\right )+\frac {(e n) \text {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {1+a^2 x^2}\right )}{9 a^5}\\ &=\frac {d n \sqrt {1+a^2 x^2}}{a}+\frac {\left (3 a^2 d-e\right ) n \sqrt {1+a^2 x^2}}{3 a^3}+\frac {2 e n \left (1+a^2 x^2\right )^{3/2}}{27 a^3}-d n x \sinh ^{-1}(a x)-\frac {1}{9} e n x^3 \sinh ^{-1}(a x)-\frac {\left (3 a^2 d-e\right ) n \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )}{3 a^3}-\frac {e n \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )}{9 a^3}-\frac {\left (3 a^2 d-e\right ) \sqrt {1+a^2 x^2} \log \left (c x^n\right )}{3 a^3}-\frac {e \left (1+a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \sinh ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \sinh ^{-1}(a x) \log \left (c x^n\right )\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 240, normalized size = 0.98 \begin {gather*} \frac {54 a^2 d n \sqrt {1+a^2 x^2}-7 e n \sqrt {1+a^2 x^2}+2 a^2 e n x^2 \sqrt {1+a^2 x^2}+3 \left (9 a^2 d-2 e\right ) n \log (x)-27 a^2 d \sqrt {1+a^2 x^2} \log \left (c x^n\right )+6 e \sqrt {1+a^2 x^2} \log \left (c x^n\right )-3 a^2 e x^2 \sqrt {1+a^2 x^2} \log \left (c x^n\right )-3 a^3 x \sinh ^{-1}(a x) \left (n \left (9 d+e x^2\right )-3 \left (3 d+e x^2\right ) \log \left (c x^n\right )\right )-27 a^2 d n \log \left (1+\sqrt {1+a^2 x^2}\right )+6 e n \log \left (1+\sqrt {1+a^2 x^2}\right )}{27 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2)*ArcSinh[a*x]*Log[c*x^n],x]

[Out]

(54*a^2*d*n*Sqrt[1 + a^2*x^2] - 7*e*n*Sqrt[1 + a^2*x^2] + 2*a^2*e*n*x^2*Sqrt[1 + a^2*x^2] + 3*(9*a^2*d - 2*e)*
n*Log[x] - 27*a^2*d*Sqrt[1 + a^2*x^2]*Log[c*x^n] + 6*e*Sqrt[1 + a^2*x^2]*Log[c*x^n] - 3*a^2*e*x^2*Sqrt[1 + a^2
*x^2]*Log[c*x^n] - 3*a^3*x*ArcSinh[a*x]*(n*(9*d + e*x^2) - 3*(3*d + e*x^2)*Log[c*x^n]) - 27*a^2*d*n*Log[1 + Sq
rt[1 + a^2*x^2]] + 6*e*n*Log[1 + Sqrt[1 + a^2*x^2]])/(27*a^3)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 3.94, size = 4077, normalized size = 16.71

method	result	size
default	\(\text {Expression too large to display}\)	\(4077\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)*arcsinh(a*x)*ln(c*x^n),x,method=_RETURNVERBOSE)

[Out]

-1/2*I/a*(a^2*x^2+1)^(1/2)*csgn(I/(a*x+(a^2*x^2+1)^(1/2)))*Pi*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+(a^2*x^2
+1)^(1/2))^2))^2*d*n-1/2*I/a*(a^2*x^2+1)^(1/2)*Pi*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+(a^2*x^2+1)^(1/2))^2
))^2*csgn(I*(-1+(a*x+(a^2*x^2+1)^(1/2))^2))*d*n-1/2*I/a*(a^2*x^2+1)^(1/2)*Pi*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-
1+(a*x+(a^2*x^2+1)^(1/2))^2))*csgn(I/a*(-1+(a*x+(a^2*x^2+1)^(1/2))^2)/(a*x+(a^2*x^2+1)^(1/2)))^2*d*n+1/9*I/a^3
*(a^2*x^2+1)^(1/2)*csgn(I/a)*Pi*csgn(I/a*(-1+(a*x+(a^2*x^2+1)^(1/2))^2)/(a*x+(a^2*x^2+1)^(1/2)))^2*e*n+1/9*I/a
^3*(a^2*x^2+1)^(1/2)*csgn(I/(a*x+(a^2*x^2+1)^(1/2)))*Pi*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+(a^2*x^2+1)^(1
/2))^2))^2*e*n+1/6*I*csgn(I/a)*arcsinh(a*x)*Pi*csgn(I/a*(-1+(a*x+(a^2*x^2+1)^(1/2))^2)/(a*x+(a^2*x^2+1)^(1/2))
)^2*x^3*e*n+1/6*I*arcsinh(a*x)*csgn(I/(a*x+(a^2*x^2+1)^(1/2)))*Pi*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+(a^2
*x^2+1)^(1/2))^2))^2*x^3*e*n+1/6*I*arcsinh(a*x)*Pi*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+(a^2*x^2+1)^(1/2))^
2))^2*csgn(I*(-1+(a*x+(a^2*x^2+1)^(1/2))^2))*x^3*e*n+1/6*I*arcsinh(a*x)*Pi*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-1+
(a*x+(a^2*x^2+1)^(1/2))^2))*csgn(I/a*(-1+(a*x+(a^2*x^2+1)^(1/2))^2)/(a*x+(a^2*x^2+1)^(1/2)))^2*x^3*e*n+1/2*I*c
sgn(I/a)*arcsinh(a*x)*Pi*csgn(I/a*(-1+(a*x+(a^2*x^2+1)^(1/2))^2)/(a*x+(a^2*x^2+1)^(1/2)))^2*x*d*n+1/2*I*arcsin
h(a*x)*csgn(I/(a*x+(a^2*x^2+1)^(1/2)))*Pi*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+(a^2*x^2+1)^(1/2))^2))^2*x*d
*n+1/2*I*arcsinh(a*x)*Pi*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+(a^2*x^2+1)^(1/2))^2))^2*csgn(I*(-1+(a*x+(a^2
*x^2+1)^(1/2))^2))*x*d*n+1/2*I*arcsinh(a*x)*Pi*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+(a^2*x^2+1)^(1/2))^2))*
csgn(I/a*(-1+(a*x+(a^2*x^2+1)^(1/2))^2)/(a*x+(a^2*x^2+1)^(1/2)))^2*x*d*n+1/9*I/a^3*(a^2*x^2+1)^(1/2)*Pi*csgn(I
/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+(a^2*x^2+1)^(1/2))^2))^2*csgn(I*(-1+(a*x+(a^2*x^2+1)^(1/2))^2))*e*n+1/a*ln(a
*x+(a^2*x^2+1)^(1/2)-1)*d*n-1/a*ln(1+a*x+(a^2*x^2+1)^(1/2))*d*n-2/9/a^3*ln(a*x+(a^2*x^2+1)^(1/2)-1)*e*n+2/9/a^
3*ln(1+a*x+(a^2*x^2+1)^(1/2))*e*n-7/27/a^3*(a^2*x^2+1)^(1/2)*e*n-1/9*e*n*x^3*arcsinh(a*x)-1/a*(a^2*x^2+1)^(1/2
)*d*(ln(c*x^n)-n*ln(x))+2/9/a^3*(a^2*x^2+1)^(1/2)*e*(ln(c*x^n)-n*ln(x))+1/3*arcsinh(a*x)*x^3*e*(ln(c*x^n)-n*ln
(x))+arcsinh(a*x)*x*d*(ln(c*x^n)-n*ln(x))-1/9/a^3*n*(3*arcsinh(a*x)*x^3*a^3*e-(a^2*x^2+1)^(1/2)*x^2*a^2*e+9*ar
csinh(a*x)*x*a^3*d-9*(a^2*x^2+1)^(1/2)*a^2*d+2*(a^2*x^2+1)^(1/2)*e)*ln(a*x+(a^2*x^2+1)^(1/2))+1/18*I/a*(a^2*x^
2+1)^(1/2)*csgn(I/a)*Pi*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+(a^2*x^2+1)^(1/2))^2))*csgn(I/a*(-1+(a*x+(a^2*
x^2+1)^(1/2))^2)/(a*x+(a^2*x^2+1)^(1/2)))*x^2*e*n+1/18*I/a*(a^2*x^2+1)^(1/2)*csgn(I/(a*x+(a^2*x^2+1)^(1/2)))*P
i*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+(a^2*x^2+1)^(1/2))^2))*csgn(I*(-1+(a*x+(a^2*x^2+1)^(1/2))^2))*x^2*e*
n+2/27/a*(a^2*x^2+1)^(1/2)*x^2*e*n+1/a*(a^2*x^2+1)^(1/2)*ln(2)*d*n-1/a*(a^2*x^2+1)^(1/2)*ln(-1+(a*x+(a^2*x^2+1
)^(1/2))^2)*d*n+1/a*(a^2*x^2+1)^(1/2)*ln(a)*d*n-2/9/a^3*(a^2*x^2+1)^(1/2)*ln(2)*e*n+2*d*n*(a^2*x^2+1)^(1/2)/a-
d*n*x*arcsinh(a*x)+1/9*I/a^3*(a^2*x^2+1)^(1/2)*Pi*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+(a^2*x^2+1)^(1/2))^2
))*csgn(I/a*(-1+(a*x+(a^2*x^2+1)^(1/2))^2)/(a*x+(a^2*x^2+1)^(1/2)))^2*e*n+1/18*I/a*(a^2*x^2+1)^(1/2)*Pi*csgn(I
/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+(a^2*x^2+1)^(1/2))^2))^3*x^2*e*n+1/18*I/a*(a^2*x^2+1)^(1/2)*Pi*csgn(I/a*(-1+
(a*x+(a^2*x^2+1)^(1/2))^2)/(a*x+(a^2*x^2+1)^(1/2)))^3*x^2*e*n-1/2*I/a*(a^2*x^2+1)^(1/2)*csgn(I/a)*Pi*csgn(I/a*
(-1+(a*x+(a^2*x^2+1)^(1/2))^2)/(a*x+(a^2*x^2+1)^(1/2)))^2*d*n+1/9/a*(a^2*x^2+1)^(1/2)*ln(a)*x^2*e*n+1/2*I/a*(a
^2*x^2+1)^(1/2)*Pi*csgn(I/a*(-1+(a*x+(a^2*x^2+1)^(1/2))^2)/(a*x+(a^2*x^2+1)^(1/2)))^3*d*n-1/9*I/a^3*(a^2*x^2+1
)^(1/2)*Pi*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+(a^2*x^2+1)^(1/2))^2))^3*e*n-1/9*I/a^3*(a^2*x^2+1)^(1/2)*Pi
*csgn(I/a*(-1+(a*x+(a^2*x^2+1)^(1/2))^2)/(a*x+(a^2*x^2+1)^(1/2)))^3*e*n-1/6*I*arcsinh(a*x)*Pi*csgn(I/(a*x+(a^2
*x^2+1)^(1/2))*(-1+(a*x+(a^2*x^2+1)^(1/2))^2))^3*x^3*e*n-1/6*I*arcsinh(a*x)*Pi*csgn(I/a*(-1+(a*x+(a^2*x^2+1)^(
1/2))^2)/(a*x+(a^2*x^2+1)^(1/2)))^3*x^3*e*n-1/2*I*arcsinh(a*x)*Pi*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+(a^2
*x^2+1)^(1/2))^2))^3*x*d*n-1/2*I*arcsinh(a*x)*Pi*csgn(I/a*(-1+(a*x+(a^2*x^2+1)^(1/2))^2)/(a*x+(a^2*x^2+1)^(1/2
)))^3*x*d*n-1/6*I*csgn(I/a)*arcsinh(a*x)*Pi*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+(a^2*x^2+1)^(1/2))^2))*csg
n(I/a*(-1+(a*x+(a^2*x^2+1)^(1/2))^2)/(a*x+(a^2*x^2+1)^(1/2)))*x^3*e*n-1/6*I*arcsinh(a*x)*csgn(I/(a*x+(a^2*x^2+
1)^(1/2)))*Pi*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+(a^2*x^2+1)^(1/2))^2))*csgn(I*(-1+(a*x+(a^2*x^2+1)^(1/2)
)^2))*x^3*e*n-1/2*I*csgn(I/a)*arcsinh(a*x)*Pi*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+(a^2*x^2+1)^(1/2))^2))*c
sgn(I/a*(-1+(a*x+(a^2*x^2+1)^(1/2))^2)/(a*x+(a^2*x^2+1)^(1/2)))*x*d*n-1/18*I/a*(a^2*x^2+1)^(1/2)*csgn(I/a)*Pi*
csgn(I/a*(-1+(a*x+(a^2*x^2+1)^(1/2))^2)/(a*x+(a^2*x^2+1)^(1/2)))^2*x^2*e*n-1/18*I/a*(a^2*x^2+1)^(1/2)*csgn(I/(
a*x+(a^2*x^2+1)^(1/2)))*Pi*csgn(I/(a*x+(a^2*x^2+1)^(1/2))*(-1+(a*x+(a^2*x^2+1)^(1/2))^2))^2*x^2*e*n-1/18*I/a*(
a^2*x^2+1)^(1/2)*Pi*csgn(I/(a*x+(a^2*x^2+1)^(1/...

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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((e*x^2+d)*arcsinh(a*x)*log(c*x^n),x, algorithm="maxima")

[Out]

1/2*a^2*d*n*(2*x/a^2 + I*(log(I*a*x + 1) - log(-I*a*x + 1))/a^3) + 1/54*a^2*n*(2*(a^2*x^3 - 3*x)/a^4 - 3*I*(lo
g(I*a*x + 1) - log(-I*a*x + 1))/a^5)*e - 3*a^2*n*e*integrate(1/9*x^4*log(x)/(a^2*x^2 + 1), x) - 9*a^2*d*n*inte
grate(1/9*x^2*log(x)/(a^2*x^2 + 1), x) - 1/2*a^2*d*(2*x/a^2 + I*(log(I*a*x + 1) - log(-I*a*x + 1))/a^3)*log(c)
- 1/18*a^2*(2*(a^2*x^3 - 3*x)/a^4 - 3*I*(log(I*a*x + 1) - log(-I*a*x + 1))/a^5)*e*log(c) - 1/9*((n - 3*log(c)
)*x^3*e + 9*(d*n - d*log(c))*x - 3*(x^3*e + 3*d*x)*log(x^n))*log(a*x + sqrt(a^2*x^2 + 1)) - integrate(-1/9*(a*
(n - 3*log(c))*x^3*e + 9*(d*n - d*log(c))*a*x - 3*(a*x^3*e + 3*a*d*x)*log(x^n))/(a^3*x^3 + a*x + (a^2*x^2 + 1)
^(3/2)), x)

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Fricas [A]
time = 0.55, size = 429, normalized size = 1.76 \begin {gather*} -\frac {3 \, {\left (9 \, a^{3} d n x - 9 \, a^{3} d n + {\left (a^{3} n x^{3} - a^{3} n\right )} \cosh \left (1\right ) - 3 \, {\left (3 \, a^{3} d x - 3 \, a^{3} d + {\left (a^{3} x^{3} - a^{3}\right )} \cosh \left (1\right ) + {\left (a^{3} x^{3} - a^{3}\right )} \sinh \left (1\right )\right )} \log \left (c\right ) - 3 \, {\left (a^{3} n x^{3} \cosh \left (1\right ) + a^{3} n x^{3} \sinh \left (1\right ) + 3 \, a^{3} d n x\right )} \log \left (x\right ) + {\left (a^{3} n x^{3} - a^{3} n\right )} \sinh \left (1\right )\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) + 3 \, {\left (9 \, a^{2} d n - 2 \, n \cosh \left (1\right ) - 2 \, n \sinh \left (1\right )\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) - 3 \, {\left (9 \, a^{3} d n + a^{3} n \cosh \left (1\right ) + a^{3} n \sinh \left (1\right ) - 3 \, {\left (3 \, a^{3} d + a^{3} \cosh \left (1\right ) + a^{3} \sinh \left (1\right )\right )} \log \left (c\right )\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) - 3 \, {\left (9 \, a^{2} d n - 2 \, n \cosh \left (1\right ) - 2 \, n \sinh \left (1\right )\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) - {\left (54 \, a^{2} d n + {\left (2 \, a^{2} n x^{2} - 7 \, n\right )} \cosh \left (1\right ) - 3 \, {\left (9 \, a^{2} d + {\left (a^{2} x^{2} - 2\right )} \cosh \left (1\right ) + {\left (a^{2} x^{2} - 2\right )} \sinh \left (1\right )\right )} \log \left (c\right ) - 3 \, {\left (9 \, a^{2} d n + {\left (a^{2} n x^{2} - 2 \, n\right )} \cosh \left (1\right ) + {\left (a^{2} n x^{2} - 2 \, n\right )} \sinh \left (1\right )\right )} \log \left (x\right ) + {\left (2 \, a^{2} n x^{2} - 7 \, n\right )} \sinh \left (1\right )\right )} \sqrt {a^{2} x^{2} + 1}}{27 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*arcsinh(a*x)*log(c*x^n),x, algorithm="fricas")

[Out]

-1/27*(3*(9*a^3*d*n*x - 9*a^3*d*n + (a^3*n*x^3 - a^3*n)*cosh(1) - 3*(3*a^3*d*x - 3*a^3*d + (a^3*x^3 - a^3)*cos
h(1) + (a^3*x^3 - a^3)*sinh(1))*log(c) - 3*(a^3*n*x^3*cosh(1) + a^3*n*x^3*sinh(1) + 3*a^3*d*n*x)*log(x) + (a^3
*n*x^3 - a^3*n)*sinh(1))*log(a*x + sqrt(a^2*x^2 + 1)) + 3*(9*a^2*d*n - 2*n*cosh(1) - 2*n*sinh(1))*log(-a*x + s
qrt(a^2*x^2 + 1) + 1) - 3*(9*a^3*d*n + a^3*n*cosh(1) + a^3*n*sinh(1) - 3*(3*a^3*d + a^3*cosh(1) + a^3*sinh(1))
*log(c))*log(-a*x + sqrt(a^2*x^2 + 1)) - 3*(9*a^2*d*n - 2*n*cosh(1) - 2*n*sinh(1))*log(-a*x + sqrt(a^2*x^2 + 1
) - 1) - (54*a^2*d*n + (2*a^2*n*x^2 - 7*n)*cosh(1) - 3*(9*a^2*d + (a^2*x^2 - 2)*cosh(1) + (a^2*x^2 - 2)*sinh(1
))*log(c) - 3*(9*a^2*d*n + (a^2*n*x^2 - 2*n)*cosh(1) + (a^2*n*x^2 - 2*n)*sinh(1))*log(x) + (2*a^2*n*x^2 - 7*n)
*sinh(1))*sqrt(a^2*x^2 + 1))/a^3

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d + e x^{2}\right ) \log {\left (c x^{n} \right )} \operatorname {asinh}{\left (a x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)*asinh(a*x)*ln(c*x**n),x)

[Out]

Integral((d + e*x**2)*log(c*x**n)*asinh(a*x), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*arcsinh(a*x)*log(c*x^n),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \ln \left (c\,x^n\right )\,\mathrm {asinh}\left (a\,x\right )\,\left (e\,x^2+d\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*x^n)*asinh(a*x)*(d + e*x^2),x)

[Out]

int(log(c*x^n)*asinh(a*x)*(d + e*x^2), x)

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