3.2.1 \(\int \frac {a+b \cosh ^{-1}(c+d x)}{(c e+d e x)^4} \, dx\) [101]

Optimal. Leaf size=99 \[ \frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{6 d e^4 (c+d x)^2}-\frac {a+b \cosh ^{-1}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \text {ArcTan}\left (\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )}{6 d e^4} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5996, 12, 5883, 105, 94, 209} \begin {gather*} -\frac {a+b \cosh ^{-1}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \text {ArcTan}\left (\sqrt {c+d x-1} \sqrt {c+d x+1}\right )}{6 d e^4}+\frac {b \sqrt {c+d x-1} \sqrt {c+d x+1}}{6 d e^4 (c+d x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

Rule 94

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 105

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] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 5883

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)/(Sqrt
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5996

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
]

Rubi steps

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

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Mathematica [A]
time = 0.16, size = 101, normalized size = 1.02 \begin {gather*} \frac {-\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{(c+d x)^3}+\frac {b \left (\frac {(-1+c+d x) (1+c+d x)}{(c+d x)^2}+\sqrt {-1+(c+d x)^2} \text {ArcTan}\left (\sqrt {-1+(c+d x)^2}\right )\right )}{\sqrt {-1+c+d x} \sqrt {1+c+d x}}}{6 d e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 4.03, size = 112, normalized size = 1.13

method	result	size
derivativedivides	\(\frac {-\frac {a}{3 e^{4} \left (d x +c \right )^{3}}-\frac {b \,\mathrm {arccosh}\left (d x +c \right )}{3 e^{4} \left (d x +c \right )^{3}}-\frac {b \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right )}{6 e^{4} \sqrt {\left (d x +c \right )^{2}-1}}+\frac {b \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{6 e^{4} \left (d x +c \right )^{2}}}{d}\)	\(112\)
default	\(\frac {-\frac {a}{3 e^{4} \left (d x +c \right )^{3}}-\frac {b \,\mathrm {arccosh}\left (d x +c \right )}{3 e^{4} \left (d x +c \right )^{3}}-\frac {b \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right )}{6 e^{4} \sqrt {\left (d x +c \right )^{2}-1}}+\frac {b \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{6 e^{4} \left (d x +c \right )^{2}}}{d}\)	\(112\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-1/3*a/e^4/(d*x+c)^3-1/3*b/e^4/(d*x+c)^3*arccosh(d*x+c)-1/6*b/e^4*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/((d*x+c
)^2-1)^(1/2)*arctan(1/((d*x+c)^2-1)^(1/2))+1/6*b/e^4*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/(d*x+c)^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((a+b*arccosh(d*x+c))/(d*e*x+c*e)^4,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (82) = 164\).
time = 0.41, size = 445, normalized size = 4.49 \begin {gather*} -\frac {2 \, a c^{3} - 2 \, {\left (b c^{3} d^{3} x^{3} + 3 \, b c^{4} d^{2} x^{2} + 3 \, b c^{5} d x + b c^{6}\right )} \arctan \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (b c^{3} d x + b c^{4}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{6 \, {\left ({\left (c^{3} d^{4} x^{3} + 3 \, c^{4} d^{3} x^{2} + 3 \, c^{5} d^{2} x + c^{6} d\right )} \cosh \left (1\right )^{4} + 4 \, {\left (c^{3} d^{4} x^{3} + 3 \, c^{4} d^{3} x^{2} + 3 \, c^{5} d^{2} x + c^{6} d\right )} \cosh \left (1\right )^{3} \sinh \left (1\right ) + 6 \, {\left (c^{3} d^{4} x^{3} + 3 \, c^{4} d^{3} x^{2} + 3 \, c^{5} d^{2} x + c^{6} d\right )} \cosh \left (1\right )^{2} \sinh \left (1\right )^{2} + 4 \, {\left (c^{3} d^{4} x^{3} + 3 \, c^{4} d^{3} x^{2} + 3 \, c^{5} d^{2} x + c^{6} d\right )} \cosh \left (1\right ) \sinh \left (1\right )^{3} + {\left (c^{3} d^{4} x^{3} + 3 \, c^{4} d^{3} x^{2} + 3 \, c^{5} d^{2} x + c^{6} d\right )} \sinh \left (1\right )^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*a*c^3 - 2*(b*c^3*d^3*x^3 + 3*b*c^4*d^2*x^2 + 3*b*c^5*d*x + b*c^6)*arctan(-d*x - c + sqrt(d^2*x^2 + 2*c
*d*x + c^2 - 1)) - 2*(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)
) - 2*(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3)*log(-d*x - c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - (b
*c^3*d*x + b*c^4)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))/((c^3*d^4*x^3 + 3*c^4*d^3*x^2 + 3*c^5*d^2*x + c^6*d)*cosh
(1)^4 + 4*(c^3*d^4*x^3 + 3*c^4*d^3*x^2 + 3*c^5*d^2*x + c^6*d)*cosh(1)^3*sinh(1) + 6*(c^3*d^4*x^3 + 3*c^4*d^3*x
^2 + 3*c^5*d^2*x + c^6*d)*cosh(1)^2*sinh(1)^2 + 4*(c^3*d^4*x^3 + 3*c^4*d^3*x^2 + 3*c^5*d^2*x + c^6*d)*cosh(1)*
sinh(1)^3 + (c^3*d^4*x^3 + 3*c^4*d^3*x^2 + 3*c^5*d^2*x + c^6*d)*sinh(1)^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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