Integrand size = 29, antiderivative size = 253 \[ \int \frac {x^2 (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^{1+n}}{2 b c^3 (1+n) \sqrt {d-c^2 d x^2}}+\frac {2^{-3-n} e^{-\frac {2 a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{c^3 \sqrt {d-c^2 d x^2}}-\frac {2^{-3-n} e^{\frac {2 a}{b}} \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{c^3 \sqrt {d-c^2 d x^2}} \] Output:
1/2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))^(1+n)/b/c^3/(1+n)/(-c^2 *d*x^2+d)^(1/2)+2^(-3-n)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))^n* GAMMA(1+n,(-2*a-2*b*arccosh(c*x))/b)/c^3/exp(2*a/b)/(-c^2*d*x^2+d)^(1/2)/( (-(a+b*arccosh(c*x))/b)^n)-2^(-3-n)*exp(2*a/b)*(c*x-1)^(1/2)*(c*x+1)^(1/2) *(a+b*arccosh(c*x))^n*GAMMA(1+n,2*(a+b*arccosh(c*x))/b)/c^3/(-c^2*d*x^2+d) ^(1/2)/(((a+b*arccosh(c*x))/b)^n)
Time = 0.59 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.84 \[ \int \frac {x^2 (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx=\frac {2^{-3-n} e^{-\frac {2 a}{b}} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) (a+b \text {arccosh}(c x))^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{-n} \left (2^{2+n} e^{\frac {2 a}{b}} (a+b \text {arccosh}(c x)) \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^n+b (1+n) \left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \Gamma \left (1+n,-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-b e^{\frac {4 a}{b}} (1+n) \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {2 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b c^3 (1+n) \sqrt {d-c^2 d x^2}} \] Input:
Integrate[(x^2*(a + b*ArcCosh[c*x])^n)/Sqrt[d - c^2*d*x^2],x]
Output:
(2^(-3 - n)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcCosh[c*x])^n*(2 ^(2 + n)*E^((2*a)/b)*(a + b*ArcCosh[c*x])*(-((a + b*ArcCosh[c*x])^2/b^2))^ n + b*(1 + n)*(a/b + ArcCosh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]) )/b] - b*E^((4*a)/b)*(1 + n)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (2 *(a + b*ArcCosh[c*x]))/b]))/(b*c^3*E^((2*a)/b)*(1 + n)*Sqrt[d - c^2*d*x^2] *(-((a + b*ArcCosh[c*x])^2/b^2))^n)
Time = 0.57 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.73, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {6367, 3042, 3793, 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.
\(\displaystyle \int \frac {x^2 (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx\) |
\(\Big \downarrow \) 6367 |
\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x))^n \cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )d(a+b \text {arccosh}(c x))}{b c^3 \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x))^n \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}+\frac {\pi }{2}\right )^2d(a+b \text {arccosh}(c x))}{b c^3 \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \int \left (\frac {1}{2} \cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right ) (a+b \text {arccosh}(c x))^n+\frac {1}{2} (a+b \text {arccosh}(c x))^n\right )d(a+b \text {arccosh}(c x))}{b c^3 \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \left (\frac {(a+b \text {arccosh}(c x))^{n+1}}{2 (n+1)}+b 2^{-n-3} e^{-\frac {2 a}{b}} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-b 2^{-n-3} e^{\frac {2 a}{b}} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b c^3 \sqrt {d-c^2 d x^2}}\) |
Input:
Int[(x^2*(a + b*ArcCosh[c*x])^n)/Sqrt[d - c^2*d*x^2],x]
Output:
(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])^(1 + n)/(2*(1 + n)) + (2^(-3 - n)*b*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]) )/b])/(E^((2*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n) - (2^(-3 - n)*b*E^((2*a) /b)*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b])/((a + b*ArcCosh[c*x])/b)^n))/(b*c^3*Sqrt[d - c^2*d*x^2])
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/((1 + c*x )^p*(-1 + c*x)^p)] Subst[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && Eq Q[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
\[\int \frac {x^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{n}}{\sqrt {-c^{2} d \,x^{2}+d}}d x\]
Input:
int(x^2*(a+b*arccosh(c*x))^n/(-c^2*d*x^2+d)^(1/2),x)
Output:
int(x^2*(a+b*arccosh(c*x))^n/(-c^2*d*x^2+d)^(1/2),x)
\[ \int \frac {x^2 (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} x^{2}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \] Input:
integrate(x^2*(a+b*arccosh(c*x))^n/(-c^2*d*x^2+d)^(1/2),x, algorithm="fric as")
Output:
integral(-sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)^n*x^2/(c^2*d*x^2 - d), x)
\[ \int \frac {x^2 (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^{2} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{n}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \] Input:
integrate(x**2*(a+b*acosh(c*x))**n/(-c**2*d*x**2+d)**(1/2),x)
Output:
Integral(x**2*(a + b*acosh(c*x))**n/sqrt(-d*(c*x - 1)*(c*x + 1)), x)
\[ \int \frac {x^2 (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} x^{2}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \] Input:
integrate(x^2*(a+b*arccosh(c*x))^n/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxi ma")
Output:
integrate((b*arccosh(c*x) + a)^n*x^2/sqrt(-c^2*d*x^2 + d), x)
\[ \int \frac {x^2 (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} x^{2}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \] Input:
integrate(x^2*(a+b*arccosh(c*x))^n/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac ")
Output:
integrate((b*arccosh(c*x) + a)^n*x^2/sqrt(-c^2*d*x^2 + d), x)
Timed out. \[ \int \frac {x^2 (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^2\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n}{\sqrt {d-c^2\,d\,x^2}} \,d x \] Input:
int((x^2*(a + b*acosh(c*x))^n)/(d - c^2*d*x^2)^(1/2),x)
Output:
int((x^2*(a + b*acosh(c*x))^n)/(d - c^2*d*x^2)^(1/2), x)
\[ \int \frac {x^2 (a+b \text {arccosh}(c x))^n}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\int \frac {\left (\mathit {acosh} \left (c x \right ) b +a \right )^{n} x^{2}}{\sqrt {-c^{2} x^{2}+1}}d x}{\sqrt {d}} \] Input:
int(x^2*(a+b*acosh(c*x))^n/(-c^2*d*x^2+d)^(1/2),x)
Output:
int(((acosh(c*x)*b + a)**n*x**2)/sqrt( - c**2*x**2 + 1),x)/sqrt(d)