\(\int \frac {(f+g x) (a+b \text {arccosh}(c x))}{(d-c^2 d x^2)^{3/2}} \, dx\) [61]

Optimal result

Integrand size = 29, antiderivative size = 142 \[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {\left (g+c^2 f x\right ) (a+b \text {arccosh}(c x))}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {b (c f-g) \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {b f \sqrt {-1+c x} \sqrt {1+c x} \log (1-c x)}{c d \sqrt {d-c^2 d x^2}} \] Output:

(c^2*f*x+g)*(a+b*arccosh(c*x))/c^2/d/(-c^2*d*x^2+d)^(1/2)-b*(c*f-g)*(c*x-1 
)^(1/2)*(c*x+1)^(1/2)*arctanh(c*x)/c^2/d/(-c^2*d*x^2+d)^(1/2)-b*f*(c*x-1)^ 
(1/2)*(c*x+1)^(1/2)*ln(-c*x+1)/c/d/(-c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.85 \[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (-\frac {2 a \left (g+c^2 f x\right )}{-1+c^2 x^2}-\frac {2 b \left (g+c^2 f x\right ) \text {arccosh}(c x)}{-1+c^2 x^2}+\frac {b ((c f+g) \log (-1+c x)+(c f-g) \log (1+c x))}{\sqrt {-1+c x} \sqrt {1+c x}}\right )}{2 c^2 d^2} \] Input:

Integrate[((f + g*x)*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(3/2),x]
 

Output:

(Sqrt[d - c^2*d*x^2]*((-2*a*(g + c^2*f*x))/(-1 + c^2*x^2) - (2*b*(g + c^2* 
f*x)*ArcCosh[c*x])/(-1 + c^2*x^2) + (b*((c*f + g)*Log[-1 + c*x] + (c*f - g 
)*Log[1 + c*x]))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/(2*c^2*d^2)
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {6387, 6389, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 6387

\(\displaystyle -\frac {\sqrt {c x-1} \sqrt {c x+1} \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{d \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 6389

\(\displaystyle -\frac {\sqrt {c x-1} \sqrt {c x+1} \left (-b c \int \left (\frac {f}{c (1-c x)}-\frac {c f-g}{c^2 (1-c x) (c x+1)}\right )dx+\frac {(c f-g) (a+b \text {arccosh}(c x))}{c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {f \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c \sqrt {c x-1}}\right )}{d \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {c x-1} \sqrt {c x+1} \left (\frac {(c f-g) (a+b \text {arccosh}(c x))}{c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {f \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c \sqrt {c x-1}}-b c \left (-\frac {\text {arctanh}(c x) (c f-g)}{c^3}-\frac {f \log (1-c x)}{c^2}\right )\right )}{d \sqrt {d-c^2 d x^2}}\)

Input:

Int[((f + g*x)*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(3/2),x]
 

Output:

-((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(((c*f - g)*(a + b*ArcCosh[c*x]))/(c^2*Sqr 
t[-1 + c*x]*Sqrt[1 + c*x]) - (f*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(c*Sqr 
t[-1 + c*x]) - b*c*(-(((c*f - g)*ArcTanh[c*x])/c^3) - (f*Log[1 - c*x])/c^2 
)))/(d*Sqrt[d - c^2*d*x^2]))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6387

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d 
_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-d)^IntPart[p]*((d + e*x^2)^Fra 
cPart[p]/((-1 + c*x)^FracPart[p]*(1 + c*x)^FracPart[p]))   Int[(f + g*x)^m* 
(-1 + c*x)^p*(1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, 
d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && IntegerQ[m]
 

rule 6389

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + 
 (e2_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(m_.), x_Symbol] :> With[{u = IntHid 
e[(f + g*x)^m*(d1 + e1*x)^p*(d2 + e2*x)^p, x]}, Simp[(a + b*ArcCosh[c*x]) 
 u, x] - Simp[b*c   Int[1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])   u, x], x]] /; Fr 
eeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c* 
d2, 0] && IGtQ[m, 0] && ILtQ[p + 1/2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && (Gt 
Q[m, 3] || LtQ[m, -2*p - 1])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(498\) vs. \(2(128)=256\).

Time = 0.87 (sec) , antiderivative size = 499, normalized size of antiderivative = 3.51

method	result	size
default	\(a \left (\frac {f x}{d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {g}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) f}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \left (c x -1\right ) \left (c x +1\right ) g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x^{2} g}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x f}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) f}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) f}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}\)	\(499\)
parts	\(a \left (\frac {f x}{d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {g}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) f}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \left (c x -1\right ) \left (c x +1\right ) g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x^{2} g}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x f}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) f}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) f}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}\)	\(499\)

Input:

int((g*x+f)*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOS 
E)
 

Output:

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

Fricas [F]

\[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((g*x+f)*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="fr 
icas")
 

Output:

integral(sqrt(-c^2*d*x^2 + d)*(a*g*x + a*f + (b*g*x + b*f)*arccosh(c*x))/( 
c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)
 

Sympy [F]

\[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (f + g x\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((g*x+f)*(a+b*acosh(c*x))/(-c**2*d*x**2+d)**(3/2),x)
 

Output:

Integral((a + b*acosh(c*x))*(f + g*x)/(-d*(c*x - 1)*(c*x + 1))**(3/2), x)
 

Maxima [F]

\[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((g*x+f)*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="ma 
xima")
 

Output:

-1/2*b*c*f*sqrt(-1/(c^4*d))*log(x^2 - 1/c^2)/d + b*g*(((c*sqrt(d)*x + sqrt 
(c*x + 1)*sqrt(c*x - 1)*sqrt(d))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/sq 
rt(-c*x + 1) + sqrt(c*x + 1)*sqrt(c*x - 1)*sqrt(d)/sqrt(-c*x + 1))/(sqrt(c 
*x + 1)*c^3*d^2*x + (c*x + 1)*sqrt(c*x - 1)*c^2*d^2) - integrate((c^2*x^3 
+ c*x^2*e^(1/2*log(c*x + 1) + 1/2*log(c*x - 1)) - x)/(sqrt(-c*x + 1)*((c^2 
*d^(3/2)*x^2 - d^(3/2))*e^(3/2*log(c*x + 1) + log(c*x - 1)) + 2*(c^3*d^(3/ 
2)*x^3 - c*d^(3/2)*x)*e^(log(c*x + 1) + 1/2*log(c*x - 1)) + (c^4*d^(3/2)*x 
^4 - c^2*d^(3/2)*x^2)*sqrt(c*x + 1))), x)) + b*f*x*arccosh(c*x)/(sqrt(-c^2 
*d*x^2 + d)*d) + a*f*x/(sqrt(-c^2*d*x^2 + d)*d) + a*g/(sqrt(-c^2*d*x^2 + d 
)*c^2*d)
 

Giac [F]

\[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((g*x+f)*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="gi 
ac")
 

Output:

integrate((g*x + f)*(b*arccosh(c*x) + a)/(-c^2*d*x^2 + d)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (f+g\,x\right )\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \] Input:

int(((f + g*x)*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^(3/2),x)
 

Output:

int(((f + g*x)*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-\sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{2} f -\sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right ) x}{\sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{2} g +a \,c^{2} f x +a g}{\sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, c^{2} d} \] Input:

int((g*x+f)*(a+b*acosh(c*x))/(-c^2*d*x^2+d)^(3/2),x)
 

Output:

( - sqrt( - c**2*x**2 + 1)*int(acosh(c*x)/(sqrt( - c**2*x**2 + 1)*c**2*x** 
2 - sqrt( - c**2*x**2 + 1)),x)*b*c**2*f - sqrt( - c**2*x**2 + 1)*int((acos 
h(c*x)*x)/(sqrt( - c**2*x**2 + 1)*c**2*x**2 - sqrt( - c**2*x**2 + 1)),x)*b 
*c**2*g + a*c**2*f*x + a*g)/(sqrt(d)*sqrt( - c**2*x**2 + 1)*c**2*d)