3.128 \(\int \frac {(a+b \cosh ^{-1}(c+d x))^4}{(c e+d e x)^3} \, dx\)

Optimal. Leaf size=195 \[ \frac {6 b^3 \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^3}-\frac {6 b^2 \log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^3}+\frac {2 b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3 (c+d x)}-\frac {2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d e^3 (c+d x)^2}+\frac {3 b^4 \text {Li}_3\left (-e^{-2 \cosh ^{-1}(c+d x)}\right )}{d e^3} \]

[Out]

-2*b*(a+b*arccosh(d*x+c))^3/d/e^3-1/2*(a+b*arccosh(d*x+c))^4/d/e^3/(d*x+c)^2-6*b^2*(a+b*arccosh(d*x+c))^2*ln(1
+1/(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)/d/e^3+6*b^3*(a+b*arccosh(d*x+c))*polylog(2,-1/(d*x+c+(d*x+c-1)^(
1/2)*(d*x+c+1)^(1/2))^2)/d/e^3+3*b^4*polylog(3,-1/(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)/d/e^3+2*b*(a+b*ar
ccosh(d*x+c))^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d/e^3/(d*x+c)

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Rubi [A]  time = 0.43, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {5866, 12, 5662, 5724, 5660, 3718, 2190, 2531, 2282, 6589} \[ -\frac {6 b^3 \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^3}+\frac {3 b^4 \text {PolyLog}\left (3,-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e^3}-\frac {6 b^2 \log \left (e^{2 \cosh ^{-1}(c+d x)}+1\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^3}+\frac {2 b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3 (c+d x)}+\frac {2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d e^3 (c+d x)^2} \]

Warning: Unable to verify antiderivative.

[In]

Int[(a + b*ArcCosh[c + d*x])^4/(c*e + d*e*x)^3,x]

[Out]

(2*b*(a + b*ArcCosh[c + d*x])^3)/(d*e^3) + (2*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^
3)/(d*e^3*(c + d*x)) - (a + b*ArcCosh[c + d*x])^4/(2*d*e^3*(c + d*x)^2) - (6*b^2*(a + b*ArcCosh[c + d*x])^2*Lo
g[1 + E^(2*ArcCosh[c + d*x])])/(d*e^3) - (6*b^3*(a + b*ArcCosh[c + d*x])*PolyLog[2, -E^(2*ArcCosh[c + d*x])])/
(d*e^3) + (3*b^4*PolyLog[3, -E^(2*ArcCosh[c + d*x])])/(d*e^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3718

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c + d*x)^(m +
1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(1 + E^(2*(-(I*e) + f*fz*x))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 5660

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Coth[x], x], x, ArcCosh
[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5724

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x
_))^(p_.), x_Symbol] :> Simp[((f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(d
1*d2*f*(m + 1)), x] + Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(f*(m
+ 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh
[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2,
0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1] && IntegerQ[p + 1/2]

Rule 5866

Int[((a_.) + ArcCosh[(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*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{(c e+d e x)^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^4}{e^3 x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^4}{x^3} \, dx,x,c+d x\right )}{d e^3}\\ &=-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d e^3 (c+d x)^2}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^3}{\sqrt {-1+x} x^2 \sqrt {1+x}} \, dx,x,c+d x\right )}{d e^3}\\ &=\frac {2 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3 (c+d x)}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d e^3 (c+d x)^2}-\frac {\left (6 b^2\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^2}{x} \, dx,x,c+d x\right )}{d e^3}\\ &=\frac {2 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3 (c+d x)}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d e^3 (c+d x)^2}-\frac {\left (6 b^2\right ) \operatorname {Subst}\left (\int (a+b x)^2 \tanh (x) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^3}\\ &=\frac {2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3}+\frac {2 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3 (c+d x)}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d e^3 (c+d x)^2}-\frac {\left (12 b^2\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)^2}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^3}\\ &=\frac {2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3}+\frac {2 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3 (c+d x)}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d e^3 (c+d x)^2}-\frac {6 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e^3}+\frac {\left (12 b^3\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^3}\\ &=\frac {2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3}+\frac {2 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3 (c+d x)}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d e^3 (c+d x)^2}-\frac {6 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e^3}-\frac {6 b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text {Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e^3}+\frac {\left (6 b^4\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^3}\\ &=\frac {2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3}+\frac {2 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3 (c+d x)}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d e^3 (c+d x)^2}-\frac {6 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e^3}-\frac {6 b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text {Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e^3}+\frac {\left (3 b^4\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c+d x)}\right )}{d e^3}\\ &=\frac {2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3}+\frac {2 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^3 (c+d x)}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{2 d e^3 (c+d x)^2}-\frac {6 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e^3}-\frac {6 b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text {Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e^3}+\frac {3 b^4 \text {Li}_3\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e^3}\\ \end {align*}

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Mathematica [B]  time = 2.31, size = 398, normalized size = 2.04 \[ \frac {-\frac {a^4}{(c+d x)^2}+\frac {4 a^3 b \sqrt {c+d x-1} \sqrt {c+d x+1}}{c+d x}-\frac {4 a^3 b \cosh ^{-1}(c+d x)}{(c+d x)^2}+12 a^2 b^2 \left (-\log (c+d x)-\frac {\cosh ^{-1}(c+d x)^2}{2 (c+d x)^2}+\frac {\sqrt {\frac {c+d x-1}{c+d x+1}} (c+d x+1) \cosh ^{-1}(c+d x)}{c+d x}\right )+4 a b^3 \left (3 \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c+d x)}\right )-\cosh ^{-1}(c+d x) \left (\frac {\cosh ^{-1}(c+d x)^2}{(c+d x)^2}-\frac {3 \sqrt {\frac {c+d x-1}{c+d x+1}} (c+d x+1) \cosh ^{-1}(c+d x)}{c+d x}+3 \cosh ^{-1}(c+d x)+6 \log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right )\right )\right )+2 b^4 \left (6 \cosh ^{-1}(c+d x) \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c+d x)}\right )+3 \text {Li}_3\left (-e^{-2 \cosh ^{-1}(c+d x)}\right )+2 \cosh ^{-1}(c+d x)^2 \left (\frac {\sqrt {\frac {c+d x-1}{c+d x+1}} (c+d x+1) \cosh ^{-1}(c+d x)}{c+d x}-\cosh ^{-1}(c+d x)-3 \log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right )\right )\right )-\frac {b^4 \cosh ^{-1}(c+d x)^4}{(c+d x)^2}}{2 d e^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCosh[c + d*x])^4/(c*e + d*e*x)^3,x]

[Out]

(-(a^4/(c + d*x)^2) + (4*a^3*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(c + d*x) - (4*a^3*b*ArcCosh[c + d*x])/(c
+ d*x)^2 - (b^4*ArcCosh[c + d*x]^4)/(c + d*x)^2 + 12*a^2*b^2*((Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*
x)*ArcCosh[c + d*x])/(c + d*x) - ArcCosh[c + d*x]^2/(2*(c + d*x)^2) - Log[c + d*x]) + 4*a*b^3*(-(ArcCosh[c + d
*x]*(3*ArcCosh[c + d*x] - (3*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*ArcCosh[c + d*x])/(c + d*x) + Ar
cCosh[c + d*x]^2/(c + d*x)^2 + 6*Log[1 + E^(-2*ArcCosh[c + d*x])])) + 3*PolyLog[2, -E^(-2*ArcCosh[c + d*x])])
+ 2*b^4*(2*ArcCosh[c + d*x]^2*(-ArcCosh[c + d*x] + (Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*ArcCosh[c
+ d*x])/(c + d*x) - 3*Log[1 + E^(-2*ArcCosh[c + d*x])]) + 6*ArcCosh[c + d*x]*PolyLog[2, -E^(-2*ArcCosh[c + d*
x])] + 3*PolyLog[3, -E^(-2*ArcCosh[c + d*x])]))/(2*d*e^3)

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fricas [F]  time = 1.34, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{4} \operatorname {arcosh}\left (d x + c\right )^{4} + 4 \, a b^{3} \operatorname {arcosh}\left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \operatorname {arcosh}\left (d x + c\right )^{2} + 4 \, a^{3} b \operatorname {arcosh}\left (d x + c\right ) + a^{4}}{d^{3} e^{3} x^{3} + 3 \, c d^{2} e^{3} x^{2} + 3 \, c^{2} d e^{3} x + c^{3} e^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))^4/(d*e*x+c*e)^3,x, algorithm="fricas")

[Out]

integral((b^4*arccosh(d*x + c)^4 + 4*a*b^3*arccosh(d*x + c)^3 + 6*a^2*b^2*arccosh(d*x + c)^2 + 4*a^3*b*arccosh
(d*x + c) + a^4)/(d^3*e^3*x^3 + 3*c*d^2*e^3*x^2 + 3*c^2*d*e^3*x + c^3*e^3), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))^4/(d*e*x+c*e)^3,x, algorithm="giac")

[Out]

integrate((b*arccosh(d*x + c) + a)^4/(d*e*x + c*e)^3, x)

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maple [B]  time = 0.29, size = 605, normalized size = 3.10 \[ -\frac {a^{4}}{2 d \,e^{3} \left (d x +c \right )^{2}}+\frac {2 b^{4} \mathrm {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c +1}\, \sqrt {d x +c -1}}{d \,e^{3} \left (d x +c \right )}+\frac {2 b^{4} \mathrm {arccosh}\left (d x +c \right )^{3}}{d \,e^{3}}-\frac {b^{4} \mathrm {arccosh}\left (d x +c \right )^{4}}{2 d \,e^{3} \left (d x +c \right )^{2}}-\frac {6 b^{4} \mathrm {arccosh}\left (d x +c \right )^{2} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{d \,e^{3}}-\frac {6 b^{4} \mathrm {arccosh}\left (d x +c \right ) \polylog \left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{d \,e^{3}}+\frac {3 b^{4} \polylog \left (3, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{d \,e^{3}}+\frac {6 a \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c +1}\, \sqrt {d x +c -1}}{d \,e^{3} \left (d x +c \right )}+\frac {6 a \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{2}}{d \,e^{3}}-\frac {2 a \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{3}}{d \,e^{3} \left (d x +c \right )^{2}}-\frac {12 a \,b^{3} \mathrm {arccosh}\left (d x +c \right ) \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{d \,e^{3}}-\frac {6 a \,b^{3} \polylog \left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{d \,e^{3}}+\frac {6 a^{2} b^{2} \mathrm {arccosh}\left (d x +c \right )}{d \,e^{3}}+\frac {6 a^{2} b^{2} \mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c +1}\, \sqrt {d x +c -1}}{d \,e^{3} \left (d x +c \right )}-\frac {3 a^{2} b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}}{d \,e^{3} \left (d x +c \right )^{2}}-\frac {6 a^{2} b^{2} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{d \,e^{3}}-\frac {2 a^{3} b \,\mathrm {arccosh}\left (d x +c \right )}{d \,e^{3} \left (d x +c \right )^{2}}+\frac {2 a^{3} b \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{d \,e^{3} \left (d x +c \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(d*x+c))^4/(d*e*x+c*e)^3,x)

[Out]

-1/2/d*a^4/e^3/(d*x+c)^2+2/d*b^4/e^3*arccosh(d*x+c)^3/(d*x+c)*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)+2/d*b^4/e^3*arcc
osh(d*x+c)^3-1/2/d*b^4/e^3*arccosh(d*x+c)^4/(d*x+c)^2-6/d*b^4/e^3*arccosh(d*x+c)^2*ln(1+(d*x+c+(d*x+c-1)^(1/2)
*(d*x+c+1)^(1/2))^2)-6/d*b^4/e^3*arccosh(d*x+c)*polylog(2,-(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)+3/d*b^4/
e^3*polylog(3,-(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)+6/d*a*b^3/e^3*arccosh(d*x+c)^2/(d*x+c)*(d*x+c+1)^(1/
2)*(d*x+c-1)^(1/2)+6/d*a*b^3/e^3*arccosh(d*x+c)^2-2/d*a*b^3/e^3*arccosh(d*x+c)^3/(d*x+c)^2-12/d*a*b^3/e^3*arcc
osh(d*x+c)*ln(1+(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)-6/d*a*b^3/e^3*polylog(2,-(d*x+c+(d*x+c-1)^(1/2)*(d*
x+c+1)^(1/2))^2)+6/d*a^2*b^2/e^3*arccosh(d*x+c)+6/d*a^2*b^2/e^3*arccosh(d*x+c)/(d*x+c)*(d*x+c+1)^(1/2)*(d*x+c-
1)^(1/2)-3/d*a^2*b^2/e^3*arccosh(d*x+c)^2/(d*x+c)^2-6/d*a^2*b^2/e^3*ln(1+(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2
))^2)-2/d*a^3*b/e^3/(d*x+c)^2*arccosh(d*x+c)+2/d*a^3*b/e^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/(d*x+c)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {b^{4} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{4}}{2 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} + 6 \, {\left (\frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d \operatorname {arcosh}\left (d x + c\right )}{d^{3} e^{3} x + c d^{2} e^{3}} - \frac {\log \left (d x + c\right )}{d e^{3}}\right )} a^{2} b^{2} + 2 \, a^{3} b {\left (\frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d}{d^{3} e^{3} x + c d^{2} e^{3}} - \frac {\operatorname {arcosh}\left (d x + c\right )}{d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}}\right )} - \frac {3 \, a^{2} b^{2} \operatorname {arcosh}\left (d x + c\right )^{2}}{d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}} - \frac {a^{4}}{2 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} + \int \frac {2 \, {\left (2 \, {\left (c^{3} - c\right )} a b^{3} + {\left (c^{3} - c\right )} b^{4} + {\left (2 \, a b^{3} d^{3} + b^{4} d^{3}\right )} x^{3} + 3 \, {\left (2 \, a b^{3} c d^{2} + b^{4} c d^{2}\right )} x^{2} + {\left (b^{4} c^{2} + 2 \, {\left (c^{2} - 1\right )} a b^{3} + {\left (2 \, a b^{3} d^{2} + b^{4} d^{2}\right )} x^{2} + 2 \, {\left (2 \, a b^{3} c d + b^{4} c d\right )} x\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left (2 \, {\left (3 \, c^{2} d - d\right )} a b^{3} + {\left (3 \, c^{2} d - d\right )} b^{4}\right )} x\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{3}}{d^{6} e^{3} x^{6} + 6 \, c d^{5} e^{3} x^{5} + c^{6} e^{3} - c^{4} e^{3} + {\left (15 \, c^{2} d^{4} e^{3} - d^{4} e^{3}\right )} x^{4} + 4 \, {\left (5 \, c^{3} d^{3} e^{3} - c d^{3} e^{3}\right )} x^{3} + 3 \, {\left (5 \, c^{4} d^{2} e^{3} - 2 \, c^{2} d^{2} e^{3}\right )} x^{2} + {\left (d^{5} e^{3} x^{5} + 5 \, c d^{4} e^{3} x^{4} + c^{5} e^{3} - c^{3} e^{3} + {\left (10 \, c^{2} d^{3} e^{3} - d^{3} e^{3}\right )} x^{3} + {\left (10 \, c^{3} d^{2} e^{3} - 3 \, c d^{2} e^{3}\right )} x^{2} + {\left (5 \, c^{4} d e^{3} - 3 \, c^{2} d e^{3}\right )} x\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + 2 \, {\left (3 \, c^{5} d e^{3} - 2 \, c^{3} d e^{3}\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))^4/(d*e*x+c*e)^3,x, algorithm="maxima")

[Out]

-1/2*b^4*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^4/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3) + 6*(s
qrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d*arccosh(d*x + c)/(d^3*e^3*x + c*d^2*e^3) - log(d*x + c)/(d*e^3))*a^2*b^2 +
2*a^3*b*(sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d/(d^3*e^3*x + c*d^2*e^3) - arccosh(d*x + c)/(d^3*e^3*x^2 + 2*c*d^2
*e^3*x + c^2*d*e^3)) - 3*a^2*b^2*arccosh(d*x + c)^2/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3) - 1/2*a^4/(d^3*e
^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3) + integrate(2*(2*(c^3 - c)*a*b^3 + (c^3 - c)*b^4 + (2*a*b^3*d^3 + b^4*d^3)
*x^3 + 3*(2*a*b^3*c*d^2 + b^4*c*d^2)*x^2 + (b^4*c^2 + 2*(c^2 - 1)*a*b^3 + (2*a*b^3*d^2 + b^4*d^2)*x^2 + 2*(2*a
*b^3*c*d + b^4*c*d)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (2*(3*c^2*d - d)*a*b^3 + (3*c^2*d - d)*b^4)*x)*lo
g(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^3/(d^6*e^3*x^6 + 6*c*d^5*e^3*x^5 + c^6*e^3 - c^4*e^3 + (15*c^
2*d^4*e^3 - d^4*e^3)*x^4 + 4*(5*c^3*d^3*e^3 - c*d^3*e^3)*x^3 + 3*(5*c^4*d^2*e^3 - 2*c^2*d^2*e^3)*x^2 + (d^5*e^
3*x^5 + 5*c*d^4*e^3*x^4 + c^5*e^3 - c^3*e^3 + (10*c^2*d^3*e^3 - d^3*e^3)*x^3 + (10*c^3*d^2*e^3 - 3*c*d^2*e^3)*
x^2 + (5*c^4*d*e^3 - 3*c^2*d*e^3)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + 2*(3*c^5*d*e^3 - 2*c^3*d*e^3)*x), x
)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4}{{\left (c\,e+d\,e\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(c + d*x))^4/(c*e + d*e*x)^3,x)

[Out]

int((a + b*acosh(c + d*x))^4/(c*e + d*e*x)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a^{4}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {4 a^{3} b \operatorname {acosh}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(d*x+c))**4/(d*e*x+c*e)**3,x)

[Out]

(Integral(a**4/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(b**4*acosh(c + d*x)**4/(c**3 + 3
*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(4*a*b**3*acosh(c + d*x)**3/(c**3 + 3*c**2*d*x + 3*c*d**2
*x**2 + d**3*x**3), x) + Integral(6*a**2*b**2*acosh(c + d*x)**2/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3
), x) + Integral(4*a**3*b*acosh(c + d*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x))/e**3

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