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

Optimal. Leaf size=75 \[ \frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{2 d}-\frac {b e \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{4 d}-\frac {b e \cosh ^{-1}(c+d x)}{4 d} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5866, 12, 5662, 90, 52} \[ \frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{2 d}-\frac {b e \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{4 d}-\frac {b e \cosh ^{-1}(c+d x)}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(4*d) - (b*e*ArcCosh[c + d*x])/(4*d) + (e*(c + d*x)^2*(a
+ b*ArcCosh[c + d*x]))/(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 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 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 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
]

Rubi steps

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

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Mathematica [A]  time = 0.16, size = 81, normalized size = 1.08 \[ \frac {e \left (\frac {1}{2} (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )-\frac {1}{4} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)+2 \tanh ^{-1}\left (\sqrt {\frac {c+d x-1}{c+d x+1}}\right )\right )\right )}{d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [A]  time = 0.49, size = 110, normalized size = 1.47 \[ \frac {2 \, a d^{2} e x^{2} + 4 \, a c d e x + {\left (2 \, b d^{2} e x^{2} + 4 \, b c d e x + {\left (2 \, b c^{2} - b\right )} e\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (b d e x + b c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 3.72, size = 245, normalized size = 3.27 \[ \frac {1}{4} \, {\left (2 \, a d x^{2} - 4 \, {\left (d {\left (\frac {c \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d {\left | d \right |}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{d^{2}}\right )} - x \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )\right )} b c + {\left (2 \, x^{2} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (\frac {x}{d^{2}} - \frac {3 \, c}{d^{3}}\right )} - \frac {{\left (2 \, c^{2} + 1\right )} \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d^{2} {\left | d \right |}}\right )} d\right )} b d + 4 \, a c x\right )} e \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*a*d*x^2 - 4*(d*(c*log(abs(-c*d - (x*abs(d) - sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))*abs(d)))/(d*abs(d)) + s
qrt(d^2*x^2 + 2*c*d*x + c^2 - 1)/d^2) - x*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)))*b*c + (2*x^2*log(d
*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - (sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(x/d^2 - 3*c/d^3) - (2*c^2 +
1)*log(abs(-c*d - (x*abs(d) - sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))*abs(d)))/(d^2*abs(d)))*d)*b*d + 4*a*c*x)*e

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maple [B]  time = 0.01, size = 162, normalized size = 2.16 \[ \frac {a d e \,x^{2}}{2}+x a c e +\frac {a \,c^{2} e}{2 d}+\frac {d \,\mathrm {arccosh}\left (d x +c \right ) x^{2} b e}{2}+\mathrm {arccosh}\left (d x +c \right ) x b c e +\frac {\mathrm {arccosh}\left (d x +c \right ) b \,c^{2} e}{2 d}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, x b e}{4}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, b c e}{4 d}-\frac {e b \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )}{4 d \sqrt {\left (d x +c \right )^{2}-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.31, size = 203, normalized size = 2.71 \[ \frac {1}{2} \, a d e x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (d x + c\right ) - d {\left (\frac {3 \, c^{2} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{3}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} x}{d^{2}} - \frac {{\left (c^{2} - 1\right )} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{3}} - \frac {3 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c}{d^{3}}\right )}\right )} b d e + a c e x + \frac {{\left ({\left (d x + c\right )} \operatorname {arcosh}\left (d x + c\right ) - \sqrt {{\left (d x + c\right )}^{2} - 1}\right )} b c e}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.31, size = 148, normalized size = 1.97 \[ \begin {cases} a c e x + \frac {a d e x^{2}}{2} + \frac {b c^{2} e \operatorname {acosh}{\left (c + d x \right )}}{2 d} + b c e x \operatorname {acosh}{\left (c + d x \right )} - \frac {b c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{4 d} + \frac {b d e x^{2} \operatorname {acosh}{\left (c + d x \right )}}{2} - \frac {b e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{4} - \frac {b e \operatorname {acosh}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\c e x \left (a + b \operatorname {acosh}{\relax (c )}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*c*e*x + a*d*e*x**2/2 + b*c**2*e*acosh(c + d*x)/(2*d) + b*c*e*x*acosh(c + d*x) - b*c*e*sqrt(c**2 +
2*c*d*x + d**2*x**2 - 1)/(4*d) + b*d*e*x**2*acosh(c + d*x)/2 - b*e*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/4 -
b*e*acosh(c + d*x)/(4*d), Ne(d, 0)), (c*e*x*(a + b*acosh(c)), True))

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