Integrand size = 29, antiderivative size = 142 \[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {\left (g+c^2 f x\right ) (a+b \text {arccosh}(c x))}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {b (c f-g) \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {b f \sqrt {-1+c x} \sqrt {1+c x} \log (1-c x)}{c d \sqrt {d-c^2 d x^2}} \]
(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)*arctan h(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^2/d/(-c^2*d*x^2+d)^(1/2)-b*f*ln(-c*x+ 1)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/d/(-c^2*d*x^2+d)^(1/2)
Time = 0.75 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.85 \[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (-\frac {2 a \left (g+c^2 f x\right )}{-1+c^2 x^2}-\frac {2 b \left (g+c^2 f x\right ) \text {arccosh}(c x)}{-1+c^2 x^2}+\frac {b ((c f+g) \log (-1+c x)+(c f-g) \log (1+c x))}{\sqrt {-1+c x} \sqrt {1+c x}}\right )}{2 c^2 d^2} \]
(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)
Time = 0.61 (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 |
\(\displaystyle -\frac {\sqrt {c x-1} \sqrt {c x+1} \left (\frac {(c f-g) (a+b \text {arccosh}(c x))}{c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {f \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c \sqrt {c x-1}}-b c \left (-\frac {\text {arctanh}(c x) (c f-g)}{c^3}-\frac {f \log (1-c x)}{c^2}\right )\right )}{d \sqrt {d-c^2 d x^2}}\) |
-((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]))
3.1.73.3.1 Defintions of rubi rules used
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]
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])
Leaf count of result is larger than twice the leaf count of optimal. \(498\) vs. \(2(128)=256\).
Time = 2.66 (sec) , antiderivative size = 499, normalized size of antiderivative = 3.51
method | result | size |
default | \(a \left (\frac {f x}{d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {g}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, f \,\operatorname {arccosh}\left (c x \right )}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \left (c x +1\right ) \left (c x -1\right ) g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x^{2} g}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x f}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) f}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) f}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}\) | \(499\) |
parts | \(a \left (\frac {f x}{d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {g}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}\right )-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, f \,\operatorname {arccosh}\left (c x \right )}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \left (c x +1\right ) \left (c x -1\right ) g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x^{2} g}{d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x f}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) f}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) f}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) g}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}\) | \(499\) |
a*(f/d*x/(-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)*f*arccosh(c*x)+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
\[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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)
\[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (f + g x\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
-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)
\[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (f+g\,x\right )\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]