\(\int (c e+d e x) (a+b \text {arccosh}(c+d x)) \, dx\) [96]

Optimal result

Integrand size = 19, antiderivative size = 75 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x)) \, dx=-\frac {b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{4 d}-\frac {b e \text {arccosh}(c+d x)}{4 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))}{2 d} \]

[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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5996, 12, 5883, 92, 54} \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x)) \, dx=\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))}{2 d}-\frac {b e \text {arccosh}(c+d x)}{4 d}-\frac {b e \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{4 d} \]

[In]

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

[Out]

-1/4*(b*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/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 54

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 92

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 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*} \text {integral}& = \frac {\text {Subst}(\int e x (a+b \text {arccosh}(x)) \, dx,x,c+d x)}{d} \\ & = \frac {e \text {Subst}(\int x (a+b \text {arccosh}(x)) \, dx,x,c+d x)}{d} \\ & = \frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))}{2 d}-\frac {(b e) \text {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 (a+b \text {arccosh}(c+d x))}{2 d}-\frac {(b e) \text {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 \text {arccosh}(c+d x)}{4 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))}{2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.08 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x)) \, dx=\frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arccosh}(c+d x))-\frac {1}{4} b \left (\sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}+2 \text {arctanh}\left (\sqrt {\frac {-1+c+d x}{1+c+d x}}\right )\right )\right )}{d} \]

[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

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.33

method	result	size
derivativedivides	\(\frac {\frac {e a \left (d x +c \right )^{2}}{2}+e b \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{2}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+\ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{4 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\)	\(100\)
default	\(\frac {\frac {e a \left (d x +c \right )^{2}}{2}+e b \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{2}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+\ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{4 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\)	\(100\)
parts	\(e a \left (\frac {1}{2} d \,x^{2}+c x \right )+\frac {e b \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{2}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+\ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{4 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\)	\(101\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.47 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x)) \, dx=\frac {2 \, a d^{2} e x^{2} + 4 \, a c d e x + {\left (2 \, b d^{2} e x^{2} + 4 \, b c d e x + {\left (2 \, b c^{2} - b\right )} e\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (b d e x + b c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{4 \, d} \]

[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

Sympy [F]

\[ \int (c e+d e x) (a+b \text {arccosh}(c+d x)) \, dx=e \left (\int a c\, dx + \int a d x\, dx + \int b c \operatorname {acosh}{\left (c + d x \right )}\, dx + \int b d x \operatorname {acosh}{\left (c + d x \right )}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (65) = 130\).

Time = 0.22 (sec) , antiderivative size = 203, normalized size of antiderivative = 2.71 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x)) \, dx=\frac {1}{2} \, a d e x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (d x + c\right ) - d {\left (\frac {3 \, c^{2} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{3}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} x}{d^{2}} - \frac {{\left (c^{2} - 1\right )} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{3}} - \frac {3 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c}{d^{3}}\right )}\right )} b d e + a c e x + \frac {{\left ({\left (d x + c\right )} \operatorname {arcosh}\left (d x + c\right ) - \sqrt {{\left (d x + c\right )}^{2} - 1}\right )} b c e}{d} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (65) = 130\).

Time = 0.54 (sec) , antiderivative size = 245, normalized size of antiderivative = 3.27 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x)) \, dx=\frac {1}{2} \, a d e x^{2} - {\left (d {\left (\frac {c \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d {\left | d \right |}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{d^{2}}\right )} - x \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )\right )} b c e + \frac {1}{4} \, {\left (2 \, x^{2} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (\frac {x}{d^{2}} - \frac {3 \, c}{d^{3}}\right )} - \frac {{\left (2 \, c^{2} + 1\right )} \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d^{2} {\left | d \right |}}\right )} d\right )} b d e + a c e x \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x)) \, dx=\int \left (c\,e+d\,e\,x\right )\,\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right ) \,d x \]

[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)