Integrand size = 26, antiderivative size = 56 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^3}{3 b c d \sqrt {-1+c x} \sqrt {1+c x}} \] Output:
-1/3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^3/b/c/d/(c*x-1)^(1/2)/(c*x+1) ^(1/2)
Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^3}{3 b c \sqrt {d-c^2 d x^2}} \] Input:
Integrate[(a + b*ArcCosh[c*x])^2/Sqrt[d - c^2*d*x^2],x]
Output:
(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^3)/(3*b*c*Sqrt[d - c^2* d*x^2])
Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.95, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {6307}
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.
\(\displaystyle \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx\) |
\(\Big \downarrow \) 6307 |
\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^3}{3 b c \sqrt {d-c^2 d x^2}}\) |
Input:
Int[(a + b*ArcCosh[c*x])^2/Sqrt[d - c^2*d*x^2],x]
Output:
(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^3)/(3*b*c*Sqrt[d - c^2* d*x^2])
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])]*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x ] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(148\) vs. \(2(48)=96\).
Time = 0.20 (sec) , antiderivative size = 149, normalized size of antiderivative = 2.66
method | result | size |
default | \(\frac {a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {b^{2} \sqrt {-d \left (c x -1\right ) \left (c x +1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{3}}{3 d \left (c^{2} x^{2}-1\right ) c}-\frac {a b \sqrt {-d \left (c x -1\right ) \left (c x +1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}}{d \left (c^{2} x^{2}-1\right ) c}\) | \(149\) |
parts | \(\frac {a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {b^{2} \sqrt {-d \left (c x -1\right ) \left (c x +1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{3}}{3 d \left (c^{2} x^{2}-1\right ) c}-\frac {a b \sqrt {-d \left (c x -1\right ) \left (c x +1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}}{d \left (c^{2} x^{2}-1\right ) c}\) | \(149\) |
Input:
int((a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)
Output:
a^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-1/3*b^2*(-d *(c*x-1)*(c*x+1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(c^2*x^2-1)/c*arccos h(c*x)^3-a*b*(-d*(c*x-1)*(c*x+1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(c^2 *x^2-1)/c*arccosh(c*x)^2
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \] Input:
integrate((a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas")
Output:
integral(-sqrt(-c^2*d*x^2 + d)*(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2)/(c^2*d*x^2 - d), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \] Input:
integrate((a+b*acosh(c*x))**2/(-c**2*d*x**2+d)**(1/2),x)
Output:
Integral((a + b*acosh(c*x))**2/sqrt(-d*(c*x - 1)*(c*x + 1)), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \] Input:
integrate((a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")
Output:
a^2*arcsin(c*x)/(c*sqrt(d)) + integrate(b^2*log(c*x + sqrt(c*x + 1)*sqrt(c *x - 1))^2/sqrt(-c^2*d*x^2 + d) + 2*a*b*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/sqrt(-c^2*d*x^2 + d), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \] Input:
integrate((a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")
Output:
integrate((b*arccosh(c*x) + a)^2/sqrt(-c^2*d*x^2 + d), x)
Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \] Input:
int((a + b*acosh(c*x))^2/(d - c^2*d*x^2)^(1/2),x)
Output:
int((a + b*acosh(c*x))^2/(d - c^2*d*x^2)^(1/2), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\mathit {asin} \left (c x \right ) a^{2}+2 \left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}}d x \right ) a b c +\left (\int \frac {\mathit {acosh} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2} c}{\sqrt {d}\, c} \] Input:
int((a+b*acosh(c*x))^2/(-c^2*d*x^2+d)^(1/2),x)
Output:
(asin(c*x)*a**2 + 2*int(acosh(c*x)/sqrt( - c**2*x**2 + 1),x)*a*b*c + int(a cosh(c*x)**2/sqrt( - c**2*x**2 + 1),x)*b**2*c)/(sqrt(d)*c)