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

Optimal. Leaf size=202 \[ -\frac {a+b \cosh ^{-1}(c x)}{3 e (d+e x)^3}-\frac {b c^3 d \sqrt {c x-1} \sqrt {c x+1}}{2 (c d-e)^2 (c d+e)^2 (d+e x)}-\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c^3 \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c x+1} \sqrt {c d+e}}{\sqrt {c x-1} \sqrt {c d-e}}\right )}{3 e (c d-e)^{5/2} (c d+e)^{5/2}} \]

[Out]

1/3*(-a-b*arccosh(c*x))/e/(e*x+d)^3+1/3*b*c^3*(2*c^2*d^2+e^2)*arctanh((c*d+e)^(1/2)*(c*x+1)^(1/2)/(c*d-e)^(1/2
)/(c*x-1)^(1/2))/(c*d-e)^(5/2)/e/(c*d+e)^(5/2)-1/6*b*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*d^2-e^2)/(e*x+d)^2-1/2
*b*c^3*d*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*d^2-e^2)^2/(e*x+d)

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Rubi [A]  time = 0.16, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5802, 103, 151, 12, 93, 208} \[ -\frac {a+b \cosh ^{-1}(c x)}{3 e (d+e x)^3}-\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c^3 \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c x+1} \sqrt {c d+e}}{\sqrt {c x-1} \sqrt {c d-e}}\right )}{3 e (c d-e)^{5/2} (c d+e)^{5/2}}-\frac {b c^3 d \sqrt {c x-1} \sqrt {c x+1}}{2 (c d-e)^2 (c d+e)^2 (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*(c^2*d^2 - e^2)*(d + e*x)^2) - (b*c^3*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(
2*(c*d - e)^2*(c*d + e)^2*(d + e*x)) - (a + b*ArcCosh[c*x])/(3*e*(d + e*x)^3) + (b*c^3*(2*c^2*d^2 + e^2)*ArcTa
nh[(Sqrt[c*d + e]*Sqrt[1 + c*x])/(Sqrt[c*d - e]*Sqrt[-1 + c*x])])/(3*(c*d - e)^(5/2)*e*(c*d + e)^(5/2))

Rule 12

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 208

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

Rule 5802

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

Rubi steps

\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{(d+e x)^4} \, dx &=-\frac {a+b \cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {(b c) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)^3} \, dx}{3 e}\\ &=-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {a+b \cosh ^{-1}(c x)}{3 e (d+e x)^3}-\frac {(b c) \int \frac {-2 c^2 d+c^2 e x}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2} \, dx}{6 e \left (c^2 d^2-e^2\right )}\\ &=-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b c^3 d \sqrt {-1+c x} \sqrt {1+c x}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {a+b \cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {(b c) \int \frac {c^2 \left (2 c^2 d^2+e^2\right )}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)} \, dx}{6 e \left (c^2 d^2-e^2\right )^2}\\ &=-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b c^3 d \sqrt {-1+c x} \sqrt {1+c x}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {a+b \cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {\left (b c^3 \left (2 c^2 d^2+e^2\right )\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)} \, dx}{6 e \left (c^2 d^2-e^2\right )^2}\\ &=-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b c^3 d \sqrt {-1+c x} \sqrt {1+c x}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {a+b \cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {\left (b c^3 \left (2 c^2 d^2+e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c d-e-(c d+e) x^2} \, dx,x,\frac {\sqrt {1+c x}}{\sqrt {-1+c x}}\right )}{3 e \left (c^2 d^2-e^2\right )^2}\\ &=-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {b c^3 d \sqrt {-1+c x} \sqrt {1+c x}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {a+b \cosh ^{-1}(c x)}{3 e (d+e x)^3}+\frac {b c^3 \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c d+e} \sqrt {1+c x}}{\sqrt {c d-e} \sqrt {-1+c x}}\right )}{3 (c d-e)^{5/2} e (c d+e)^{5/2}}\\ \end {align*}

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Mathematica [C]  time = 0.98, size = 259, normalized size = 1.28 \[ -\frac {\frac {2 a+\frac {b c e \sqrt {c x-1} \sqrt {c x+1} (d+e x) \left (c^2 d (4 d+3 e x)-e^2\right )}{\left (e^2-c^2 d^2\right )^2}}{(d+e x)^3}+\frac {i b c^3 \left (2 c^2 d^2+e^2\right ) \log \left (\frac {12 e^2 (e-c d)^2 (c d+e)^2 \left (\sqrt {c x-1} \sqrt {c x+1} \sqrt {e^2-c^2 d^2}-i c^2 d x-i e\right )}{b c^3 \sqrt {e^2-c^2 d^2} \left (2 c^2 d^2+e^2\right ) (d+e x)}\right )}{(e-c d)^2 (c d+e)^2 \sqrt {e^2-c^2 d^2}}+\frac {2 b \cosh ^{-1}(c x)}{(d+e x)^3}}{6 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.55, size = 1963, normalized size = 9.72 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/6*((2*a + 3*b)*c^6*d^9 - 3*(2*a + b)*c^4*d^7*e^2 + 6*a*c^2*d^5*e^4 - 2*a*d^3*e^6 + 3*(b*c^6*d^6*e^3 - b*c^
4*d^4*e^5)*x^3 + 9*(b*c^6*d^7*e^2 - b*c^4*d^5*e^4)*x^2 - (2*b*c^5*d^8 + b*c^3*d^6*e^2 + (2*b*c^5*d^5*e^3 + b*c
^3*d^3*e^5)*x^3 + 3*(2*b*c^5*d^6*e^2 + b*c^3*d^4*e^4)*x^2 + 3*(2*b*c^5*d^7*e + b*c^3*d^5*e^3)*x)*sqrt(c^2*d^2
- e^2)*log((c^3*d^2*x + c*d*e + sqrt(c^2*d^2 - e^2)*(c^2*d*x + e) + (c^2*d^2 + sqrt(c^2*d^2 - e^2)*c*d - e^2)*
sqrt(c^2*x^2 - 1))/(e*x + d)) + 9*(b*c^6*d^8*e - b*c^4*d^6*e^3)*x - 2*((b*c^6*d^6*e^3 - 3*b*c^4*d^4*e^5 + 3*b*
c^2*d^2*e^7 - b*e^9)*x^3 + 3*(b*c^6*d^7*e^2 - 3*b*c^4*d^5*e^4 + 3*b*c^2*d^3*e^6 - b*d*e^8)*x^2 + 3*(b*c^6*d^8*
e - 3*b*c^4*d^6*e^3 + 3*b*c^2*d^4*e^5 - b*d^2*e^7)*x)*log(c*x + sqrt(c^2*x^2 - 1)) - 2*(b*c^6*d^9 - 3*b*c^4*d^
7*e^2 + 3*b*c^2*d^5*e^4 - b*d^3*e^6 + (b*c^6*d^6*e^3 - 3*b*c^4*d^4*e^5 + 3*b*c^2*d^2*e^7 - b*e^9)*x^3 + 3*(b*c
^6*d^7*e^2 - 3*b*c^4*d^5*e^4 + 3*b*c^2*d^3*e^6 - b*d*e^8)*x^2 + 3*(b*c^6*d^8*e - 3*b*c^4*d^6*e^3 + 3*b*c^2*d^4
*e^5 - b*d^2*e^7)*x)*log(-c*x + sqrt(c^2*x^2 - 1)) + (4*b*c^5*d^8*e - 5*b*c^3*d^6*e^3 + b*c*d^4*e^5 + 3*(b*c^5
*d^6*e^3 - b*c^3*d^4*e^5)*x^2 + (7*b*c^5*d^7*e^2 - 8*b*c^3*d^5*e^4 + b*c*d^3*e^6)*x)*sqrt(c^2*x^2 - 1))/(c^6*d
^12*e - 3*c^4*d^10*e^3 + 3*c^2*d^8*e^5 - d^6*e^7 + (c^6*d^9*e^4 - 3*c^4*d^7*e^6 + 3*c^2*d^5*e^8 - d^3*e^10)*x^
3 + 3*(c^6*d^10*e^3 - 3*c^4*d^8*e^5 + 3*c^2*d^6*e^7 - d^4*e^9)*x^2 + 3*(c^6*d^11*e^2 - 3*c^4*d^9*e^4 + 3*c^2*d
^7*e^6 - d^5*e^8)*x), -1/6*((2*a + 3*b)*c^6*d^9 - 3*(2*a + b)*c^4*d^7*e^2 + 6*a*c^2*d^5*e^4 - 2*a*d^3*e^6 + 3*
(b*c^6*d^6*e^3 - b*c^4*d^4*e^5)*x^3 + 9*(b*c^6*d^7*e^2 - b*c^4*d^5*e^4)*x^2 + 2*(2*b*c^5*d^8 + b*c^3*d^6*e^2 +
(2*b*c^5*d^5*e^3 + b*c^3*d^3*e^5)*x^3 + 3*(2*b*c^5*d^6*e^2 + b*c^3*d^4*e^4)*x^2 + 3*(2*b*c^5*d^7*e + b*c^3*d^
5*e^3)*x)*sqrt(-c^2*d^2 + e^2)*arctan(-(sqrt(-c^2*d^2 + e^2)*sqrt(c^2*x^2 - 1)*e - sqrt(-c^2*d^2 + e^2)*(c*e*x
+ c*d))/(c^2*d^2 - e^2)) + 9*(b*c^6*d^8*e - b*c^4*d^6*e^3)*x - 2*((b*c^6*d^6*e^3 - 3*b*c^4*d^4*e^5 + 3*b*c^2*
d^2*e^7 - b*e^9)*x^3 + 3*(b*c^6*d^7*e^2 - 3*b*c^4*d^5*e^4 + 3*b*c^2*d^3*e^6 - b*d*e^8)*x^2 + 3*(b*c^6*d^8*e -
3*b*c^4*d^6*e^3 + 3*b*c^2*d^4*e^5 - b*d^2*e^7)*x)*log(c*x + sqrt(c^2*x^2 - 1)) - 2*(b*c^6*d^9 - 3*b*c^4*d^7*e^
2 + 3*b*c^2*d^5*e^4 - b*d^3*e^6 + (b*c^6*d^6*e^3 - 3*b*c^4*d^4*e^5 + 3*b*c^2*d^2*e^7 - b*e^9)*x^3 + 3*(b*c^6*d
^7*e^2 - 3*b*c^4*d^5*e^4 + 3*b*c^2*d^3*e^6 - b*d*e^8)*x^2 + 3*(b*c^6*d^8*e - 3*b*c^4*d^6*e^3 + 3*b*c^2*d^4*e^5
- b*d^2*e^7)*x)*log(-c*x + sqrt(c^2*x^2 - 1)) + (4*b*c^5*d^8*e - 5*b*c^3*d^6*e^3 + b*c*d^4*e^5 + 3*(b*c^5*d^6
*e^3 - b*c^3*d^4*e^5)*x^2 + (7*b*c^5*d^7*e^2 - 8*b*c^3*d^5*e^4 + b*c*d^3*e^6)*x)*sqrt(c^2*x^2 - 1))/(c^6*d^12*
e - 3*c^4*d^10*e^3 + 3*c^2*d^8*e^5 - d^6*e^7 + (c^6*d^9*e^4 - 3*c^4*d^7*e^6 + 3*c^2*d^5*e^8 - d^3*e^10)*x^3 +
3*(c^6*d^10*e^3 - 3*c^4*d^8*e^5 + 3*c^2*d^6*e^7 - d^4*e^9)*x^2 + 3*(c^6*d^11*e^2 - 3*c^4*d^9*e^4 + 3*c^2*d^7*e
^6 - d^5*e^8)*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.00, size = 1137, normalized size = 5.63 \[ -\frac {c^{3} a}{3 \left (c x e +c d \right )^{3} e}-\frac {c^{3} b \,\mathrm {arccosh}\left (c x \right )}{3 \left (c x e +c d \right )^{3} e}-\frac {c^{7} b \sqrt {c x +1}\, \sqrt {c x -1}\, \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{c x e +c d}\right ) x^{2} d^{2}}{3 \sqrt {c^{2} x^{2}-1}\, \left (c d +e \right ) \left (c d -e \right ) \left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {2 c^{7} b \sqrt {c x +1}\, \sqrt {c x -1}\, \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{c x e +c d}\right ) x \,d^{3}}{3 e \sqrt {c^{2} x^{2}-1}\, \left (c d +e \right ) \left (c d -e \right ) \left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {c^{7} b \sqrt {c x +1}\, \sqrt {c x -1}\, \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{c x e +c d}\right ) d^{4}}{3 e^{2} \sqrt {c^{2} x^{2}-1}\, \left (c d +e \right ) \left (c d -e \right ) \left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {c^{5} b e \sqrt {c x +1}\, \sqrt {c x -1}\, x d}{2 \left (c d +e \right ) \left (c d -e \right ) \left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2}}-\frac {2 c^{5} b \sqrt {c x +1}\, \sqrt {c x -1}\, d^{2}}{3 \left (c d +e \right ) \left (c d -e \right ) \left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2}}-\frac {c^{5} b \,e^{2} \sqrt {c x +1}\, \sqrt {c x -1}\, \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{c x e +c d}\right ) x^{2}}{6 \sqrt {c^{2} x^{2}-1}\, \left (c d +e \right ) \left (c d -e \right ) \left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {c^{5} b e \sqrt {c x +1}\, \sqrt {c x -1}\, \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{c x e +c d}\right ) x d}{3 \sqrt {c^{2} x^{2}-1}\, \left (c d +e \right ) \left (c d -e \right ) \left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {c^{5} b \sqrt {c x +1}\, \sqrt {c x -1}\, \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{c x e +c d}\right ) d^{2}}{6 \sqrt {c^{2} x^{2}-1}\, \left (c d +e \right ) \left (c d -e \right ) \left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}+\frac {c^{3} b \,e^{2} \sqrt {c x +1}\, \sqrt {c x -1}}{6 \left (c d +e \right ) \left (c d -e \right ) \left (c^{2} d^{2}-e^{2}\right ) \left (c x e +c d \right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*c^3*a/(c*e*x+c*d)^3/e-1/3*c^3*b/(c*e*x+c*d)^3/e*arccosh(c*x)-1/3*c^7*b*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c^2*x
^2-1)^(1/2)/(c*d+e)/(c*d-e)/(c^2*d^2-e^2)/(c*e*x+c*d)^2/((c^2*d^2-e^2)/e^2)^(1/2)*ln(-2*(c^2*d*x-(c^2*x^2-1)^(
1/2)*((c^2*d^2-e^2)/e^2)^(1/2)*e+e)/(c*e*x+c*d))*x^2*d^2-2/3*c^7*b/e*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c^2*x^2-1)^(
1/2)/(c*d+e)/(c*d-e)/(c^2*d^2-e^2)/(c*e*x+c*d)^2/((c^2*d^2-e^2)/e^2)^(1/2)*ln(-2*(c^2*d*x-(c^2*x^2-1)^(1/2)*((
c^2*d^2-e^2)/e^2)^(1/2)*e+e)/(c*e*x+c*d))*x*d^3-1/3*c^7*b/e^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c^2*x^2-1)^(1/2)/(c
*d+e)/(c*d-e)/(c^2*d^2-e^2)/(c*e*x+c*d)^2/((c^2*d^2-e^2)/e^2)^(1/2)*ln(-2*(c^2*d*x-(c^2*x^2-1)^(1/2)*((c^2*d^2
-e^2)/e^2)^(1/2)*e+e)/(c*e*x+c*d))*d^4-1/2*c^5*b*e*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c*d+e)/(c*d-e)/(c^2*d^2-e^2)/(
c*e*x+c*d)^2*x*d-2/3*c^5*b*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c*d+e)/(c*d-e)/(c^2*d^2-e^2)/(c*e*x+c*d)^2*d^2-1/6*c^5
*b*e^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c^2*x^2-1)^(1/2)/(c*d+e)/(c*d-e)/(c^2*d^2-e^2)/(c*e*x+c*d)^2/((c^2*d^2-e^2
)/e^2)^(1/2)*ln(-2*(c^2*d*x-(c^2*x^2-1)^(1/2)*((c^2*d^2-e^2)/e^2)^(1/2)*e+e)/(c*e*x+c*d))*x^2-1/3*c^5*b*e*(c*x
+1)^(1/2)*(c*x-1)^(1/2)/(c^2*x^2-1)^(1/2)/(c*d+e)/(c*d-e)/(c^2*d^2-e^2)/(c*e*x+c*d)^2/((c^2*d^2-e^2)/e^2)^(1/2
)*ln(-2*(c^2*d*x-(c^2*x^2-1)^(1/2)*((c^2*d^2-e^2)/e^2)^(1/2)*e+e)/(c*e*x+c*d))*x*d-1/6*c^5*b*(c*x+1)^(1/2)*(c*
x-1)^(1/2)/(c^2*x^2-1)^(1/2)/(c*d+e)/(c*d-e)/(c^2*d^2-e^2)/(c*e*x+c*d)^2/((c^2*d^2-e^2)/e^2)^(1/2)*ln(-2*(c^2*
d*x-(c^2*x^2-1)^(1/2)*((c^2*d^2-e^2)/e^2)^(1/2)*e+e)/(c*e*x+c*d))*d^2+1/6*c^3*b*e^2*(c*x+1)^(1/2)*(c*x-1)^(1/2
)/(c*d+e)/(c*d-e)/(c^2*d^2-e^2)/(c*e*x+c*d)^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{6} \, {\left (6 \, c \int \frac {1}{3 \, {\left (c^{3} e^{4} x^{6} + 3 \, c^{3} d e^{3} x^{5} - 3 \, c d^{2} e^{2} x^{2} - c d^{3} e x + {\left (3 \, c^{3} d^{2} e^{2} - c e^{4}\right )} x^{4} + {\left (c^{3} d^{3} e - 3 \, c d e^{3}\right )} x^{3} + {\left (c^{2} e^{4} x^{5} + 3 \, c^{2} d e^{3} x^{4} - 3 \, d^{2} e^{2} x - d^{3} e + {\left (3 \, c^{2} d^{2} e^{2} - e^{4}\right )} x^{3} + {\left (c^{2} d^{3} e - 3 \, d e^{3}\right )} x^{2}\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (c x - 1\right )\right )}\right )}}\,{d x} + \frac {2 \, {\left (c^{6} d^{3} + 3 \, c^{4} d e^{2}\right )} \log \left (e x + d\right )}{c^{6} d^{6} e - 3 \, c^{4} d^{4} e^{3} + 3 \, c^{2} d^{2} e^{5} - e^{7}} - \frac {3 \, c^{6} d^{6} - 2 \, c^{4} d^{4} e^{2} - c^{2} d^{2} e^{4} + 2 \, {\left (c^{6} d^{4} e^{2} - c^{2} e^{6}\right )} x^{2} + {\left (5 \, c^{6} d^{5} e - 2 \, c^{4} d^{3} e^{3} - 3 \, c^{2} d e^{5}\right )} x - 2 \, {\left (c^{6} d^{6} - 3 \, c^{4} d^{4} e^{2} + 3 \, c^{2} d^{2} e^{4} - e^{6}\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left (c^{6} d^{6} + 3 \, c^{5} d^{5} e + 3 \, c^{4} d^{4} e^{2} + c^{3} d^{3} e^{3} + {\left (c^{6} d^{3} e^{3} + 3 \, c^{5} d^{2} e^{4} + 3 \, c^{4} d e^{5} + c^{3} e^{6}\right )} x^{3} + 3 \, {\left (c^{6} d^{4} e^{2} + 3 \, c^{5} d^{3} e^{3} + 3 \, c^{4} d^{2} e^{4} + c^{3} d e^{5}\right )} x^{2} + 3 \, {\left (c^{6} d^{5} e + 3 \, c^{5} d^{4} e^{2} + 3 \, c^{4} d^{3} e^{3} + c^{3} d^{2} e^{4}\right )} x\right )} \log \left (c x + 1\right ) + {\left (c^{6} d^{6} - 3 \, c^{5} d^{5} e + 3 \, c^{4} d^{4} e^{2} - c^{3} d^{3} e^{3} + {\left (c^{6} d^{3} e^{3} - 3 \, c^{5} d^{2} e^{4} + 3 \, c^{4} d e^{5} - c^{3} e^{6}\right )} x^{3} + 3 \, {\left (c^{6} d^{4} e^{2} - 3 \, c^{5} d^{3} e^{3} + 3 \, c^{4} d^{2} e^{4} - c^{3} d e^{5}\right )} x^{2} + 3 \, {\left (c^{6} d^{5} e - 3 \, c^{5} d^{4} e^{2} + 3 \, c^{4} d^{3} e^{3} - c^{3} d^{2} e^{4}\right )} x\right )} \log \left (c x - 1\right )}{c^{6} d^{9} e - 3 \, c^{4} d^{7} e^{3} + 3 \, c^{2} d^{5} e^{5} - d^{3} e^{7} + {\left (c^{6} d^{6} e^{4} - 3 \, c^{4} d^{4} e^{6} + 3 \, c^{2} d^{2} e^{8} - e^{10}\right )} x^{3} + 3 \, {\left (c^{6} d^{7} e^{3} - 3 \, c^{4} d^{5} e^{5} + 3 \, c^{2} d^{3} e^{7} - d e^{9}\right )} x^{2} + 3 \, {\left (c^{6} d^{8} e^{2} - 3 \, c^{4} d^{6} e^{4} + 3 \, c^{2} d^{4} e^{6} - d^{2} e^{8}\right )} x}\right )} b - \frac {a}{3 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(6*c*integrate(1/3/(c^3*e^4*x^6 + 3*c^3*d*e^3*x^5 - 3*c*d^2*e^2*x^2 - c*d^3*e*x + (3*c^3*d^2*e^2 - c*e^4)
*x^4 + (c^3*d^3*e - 3*c*d*e^3)*x^3 + (c^2*e^4*x^5 + 3*c^2*d*e^3*x^4 - 3*d^2*e^2*x - d^3*e + (3*c^2*d^2*e^2 - e
^4)*x^3 + (c^2*d^3*e - 3*d*e^3)*x^2)*e^(1/2*log(c*x + 1) + 1/2*log(c*x - 1))), x) + 2*(c^6*d^3 + 3*c^4*d*e^2)*
log(e*x + d)/(c^6*d^6*e - 3*c^4*d^4*e^3 + 3*c^2*d^2*e^5 - e^7) - (3*c^6*d^6 - 2*c^4*d^4*e^2 - c^2*d^2*e^4 + 2*
(c^6*d^4*e^2 - c^2*e^6)*x^2 + (5*c^6*d^5*e - 2*c^4*d^3*e^3 - 3*c^2*d*e^5)*x - 2*(c^6*d^6 - 3*c^4*d^4*e^2 + 3*c
^2*d^2*e^4 - e^6)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + (c^6*d^6 + 3*c^5*d^5*e + 3*c^4*d^4*e^2 + c^3*d^3*e^
3 + (c^6*d^3*e^3 + 3*c^5*d^2*e^4 + 3*c^4*d*e^5 + c^3*e^6)*x^3 + 3*(c^6*d^4*e^2 + 3*c^5*d^3*e^3 + 3*c^4*d^2*e^4
+ c^3*d*e^5)*x^2 + 3*(c^6*d^5*e + 3*c^5*d^4*e^2 + 3*c^4*d^3*e^3 + c^3*d^2*e^4)*x)*log(c*x + 1) + (c^6*d^6 - 3
*c^5*d^5*e + 3*c^4*d^4*e^2 - c^3*d^3*e^3 + (c^6*d^3*e^3 - 3*c^5*d^2*e^4 + 3*c^4*d*e^5 - c^3*e^6)*x^3 + 3*(c^6*
d^4*e^2 - 3*c^5*d^3*e^3 + 3*c^4*d^2*e^4 - c^3*d*e^5)*x^2 + 3*(c^6*d^5*e - 3*c^5*d^4*e^2 + 3*c^4*d^3*e^3 - c^3*
d^2*e^4)*x)*log(c*x - 1))/(c^6*d^9*e - 3*c^4*d^7*e^3 + 3*c^2*d^5*e^5 - d^3*e^7 + (c^6*d^6*e^4 - 3*c^4*d^4*e^6
+ 3*c^2*d^2*e^8 - e^10)*x^3 + 3*(c^6*d^7*e^3 - 3*c^4*d^5*e^5 + 3*c^2*d^3*e^7 - d*e^9)*x^2 + 3*(c^6*d^8*e^2 - 3
*c^4*d^6*e^4 + 3*c^2*d^4*e^6 - d^2*e^8)*x))*b - 1/3*a/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(c*x))/(d + e*x)^4,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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