Integrand size = 29, antiderivative size = 870 \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx=-\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^{1+n}}{128 b c^3 (1+n) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2^{-11-3 n} d^2 e^{-\frac {8 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {8 (a+b \text {arccosh}(c x))}{b}\right )}{c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2^{-7-n} 3^{-1-n} d^2 e^{-\frac {6 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2^{-2 (4+n)} d^2 e^{-\frac {4 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2^{-7-n} d^2 e^{-\frac {2 a}{b}} \sqrt {d-c^2 d x^2} (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 {-1+c x} \sqrt {1+c x}}-\frac {2^{-7-n} d^2 e^{\frac {2 a}{b}} \sqrt {d-c^2 d x^2} (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 {-1+c x} \sqrt {1+c x}}-\frac {2^{-2 (4+n)} d^2 e^{\frac {4 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2^{-7-n} 3^{-1-n} d^2 e^{\frac {6 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2^{-11-3 n} d^2 e^{\frac {8 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {8 (a+b \text {arccosh}(c x))}{b}\right )}{c^3 \sqrt {-1+c x} \sqrt {1+c x}} \]
-5/128*d^2*(a+b*arccosh(c*x))^(1+n)*(-c^2*d*x^2+d)^(1/2)/b/c^3/(1+n)/(c*x- 1)^(1/2)/(c*x+1)^(1/2)+2^(-11-3*n)*d^2*(a+b*arccosh(c*x))^n*GAMMA(1+n,-8*( a+b*arccosh(c*x))/b)*(-c^2*d*x^2+d)^(1/2)/c^3/exp(8*a/b)/(((-a-b*arccosh(c *x))/b)^n)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2^(-7-n)*3^(-1-n)*d^2*(a+b*arccosh( c*x))^n*GAMMA(1+n,-6*(a+b*arccosh(c*x))/b)*(-c^2*d*x^2+d)^(1/2)/c^3/exp(6* a/b)/(((-a-b*arccosh(c*x))/b)^n)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+d^2*(a+b*arcc osh(c*x))^n*GAMMA(1+n,-4*(a+b*arccosh(c*x))/b)*(-c^2*d*x^2+d)^(1/2)/(2^(8+ 2*n))/c^3/exp(4*a/b)/(((-a-b*arccosh(c*x))/b)^n)/(c*x-1)^(1/2)/(c*x+1)^(1/ 2)+2^(-7-n)*d^2*(a+b*arccosh(c*x))^n*GAMMA(1+n,-2*(a+b*arccosh(c*x))/b)*(- c^2*d*x^2+d)^(1/2)/c^3/exp(2*a/b)/(((-a-b*arccosh(c*x))/b)^n)/(c*x-1)^(1/2 )/(c*x+1)^(1/2)-2^(-7-n)*d^2*exp(2*a/b)*(a+b*arccosh(c*x))^n*GAMMA(1+n,2*( a+b*arccosh(c*x))/b)*(-c^2*d*x^2+d)^(1/2)/c^3/(((a+b*arccosh(c*x))/b)^n)/( c*x-1)^(1/2)/(c*x+1)^(1/2)-d^2*exp(4*a/b)*(a+b*arccosh(c*x))^n*GAMMA(1+n,4 *(a+b*arccosh(c*x))/b)*(-c^2*d*x^2+d)^(1/2)/(2^(8+2*n))/c^3/(((a+b*arccosh (c*x))/b)^n)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2^(-7-n)*3^(-1-n)*d^2*exp(6*a/b)* (a+b*arccosh(c*x))^n*GAMMA(1+n,6*(a+b*arccosh(c*x))/b)*(-c^2*d*x^2+d)^(1/2 )/c^3/(((a+b*arccosh(c*x))/b)^n)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2^(-11-3*n)*d ^2*exp(8*a/b)*(a+b*arccosh(c*x))^n*GAMMA(1+n,8*(a+b*arccosh(c*x))/b)*(-c^2 *d*x^2+d)^(1/2)/c^3/(((a+b*arccosh(c*x))/b)^n)/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 5.62 (sec) , antiderivative size = 677, normalized size of antiderivative = 0.78 \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx=\frac {2^{-11-3 n} 3^{-1-n} d^3 e^{-\frac {8 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 (-3^{1+n} b (1+n) \left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \Gamma \left (1+n,-\frac {8 (a+b \text {arccosh}(c x))}{b}\right )+4^{2+n} b e^{\frac {2 a}{b}} (1+n) \left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \Gamma \left (1+n,-\frac {6 (a+b \text {arccosh}(c x))}{b}\right )-2^{3+n} 3^{1+n} b e^{\frac {4 a}{b}} (1+n) \left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \Gamma \left (1+n,-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )-3^{1+n} 4^{2+n} b e^{\frac {6 a}{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 )+e^{\frac {8 a}{b}} \left (5\ 2^{4+3 n} 3^{1+n} a \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^n+5\ 2^{4+3 n} 3^{1+n} b \text {arccosh}(c x) \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^n+3^{1+n} 4^{2+n} b e^{\frac {2 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 )+2^{3+n} 3^{1+n} b e^{\frac {4 a}{b}} (1+n) \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {4 (a+b \text {arccosh}(c x))}{b}\right )-4^{2+n} b e^{\frac {6 a}{b}} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {6 (a+b \text {arccosh}(c x))}{b}\right )-4^{2+n} b e^{\frac {6 a}{b}} n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {6 (a+b \text {arccosh}(c x))}{b}\right )+3^{1+n} b e^{\frac {8 a}{b}} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {8 (a+b \text {arccosh}(c x))}{b}\right )+3^{1+n} b e^{\frac {8 a}{b}} n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {8 (a+b \text {arccosh}(c x))}{b}\right )\right )\right )}{b c^3 (1+n) \sqrt {d-c^2 d x^2}} \]
(2^(-11 - 3*n)*3^(-1 - n)*d^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b* ArcCosh[c*x])^n*(-(3^(1 + n)*b*(1 + n)*(a/b + ArcCosh[c*x])^n*Gamma[1 + n, (-8*(a + b*ArcCosh[c*x]))/b]) + 4^(2 + n)*b*E^((2*a)/b)*(1 + n)*(a/b + Ar cCosh[c*x])^n*Gamma[1 + n, (-6*(a + b*ArcCosh[c*x]))/b] - 2^(3 + n)*3^(1 + n)*b*E^((4*a)/b)*(1 + n)*(a/b + ArcCosh[c*x])^n*Gamma[1 + n, (-4*(a + b*A rcCosh[c*x]))/b] - 3^(1 + n)*4^(2 + n)*b*E^((6*a)/b)*(1 + n)*(a/b + ArcCos h[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]))/b] + E^((8*a)/b)*(5*2^(4 + 3*n)*3^(1 + n)*a*(-((a + b*ArcCosh[c*x])^2/b^2))^n + 5*2^(4 + 3*n)*3^(1 + n)*b*ArcCosh[c*x]*(-((a + b*ArcCosh[c*x])^2/b^2))^n + 3^(1 + n)*4^(2 + n )*b*E^((2*a)/b)*(1 + n)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b] + 2^(3 + n)*3^(1 + n)*b*E^((4*a)/b)*(1 + n)*(-((a + b *ArcCosh[c*x])/b))^n*Gamma[1 + n, (4*(a + b*ArcCosh[c*x]))/b] - 4^(2 + n)* b*E^((6*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (6*(a + b*ArcCosh [c*x]))/b] - 4^(2 + n)*b*E^((6*a)/b)*n*(-((a + b*ArcCosh[c*x])/b))^n*Gamma [1 + n, (6*(a + b*ArcCosh[c*x]))/b] + 3^(1 + n)*b*E^((8*a)/b)*(-((a + b*Ar cCosh[c*x])/b))^n*Gamma[1 + n, (8*(a + b*ArcCosh[c*x]))/b] + 3^(1 + n)*b*E ^((8*a)/b)*n*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (8*(a + b*ArcCosh[ c*x]))/b])))/(b*c^3*E^((8*a)/b)*(1 + n)*Sqrt[d - c^2*d*x^2]*(-((a + b*ArcC osh[c*x])^2/b^2))^n)
Time = 1.05 (sec) , antiderivative size = 567, normalized size of antiderivative = 0.65, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {6367, 5971, 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.
| \(\displaystyle \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx\) |
| \(\Big \downarrow \) 6367 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int (a+b \text {arccosh}(c x))^n \cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^6\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )d(a+b \text {arccosh}(c x))}{b c^3 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int \left (\frac {1}{128} \cosh \left (\frac {8 a}{b}-\frac {8 (a+b \text {arccosh}(c x))}{b}\right ) (a+b \text {arccosh}(c x))^n-\frac {1}{32} \cosh \left (\frac {6 a}{b}-\frac {6 (a+b \text {arccosh}(c x))}{b}\right ) (a+b \text {arccosh}(c x))^n+\frac {1}{32} \cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arccosh}(c x))}{b}\right ) (a+b \text {arccosh}(c x))^n+\frac {1}{32} \cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right ) (a+b \text {arccosh}(c x))^n-\frac {5}{128} (a+b \text {arccosh}(c x))^n\right )d(a+b \text {arccosh}(c x))}{b c^3 \sqrt {c x-1} \sqrt {c x+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \left (-\frac {5 (a+b \text {arccosh}(c x))^{n+1}}{128 (n+1)}+b 2^{-3 n-11} e^{-\frac {8 a}{b}} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {8 (a+b \text {arccosh}(c x))}{b}\right )-b 2^{-n-7} 3^{-n-1} e^{-\frac {6 a}{b}} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {6 (a+b \text {arccosh}(c x))}{b}\right )+b 2^{-2 (n+4)} e^{-\frac {4 a}{b}} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )+b 2^{-n-7} e^{-\frac {2 a}{b}} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-b 2^{-n-7} e^{\frac {2 a}{b}} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-b 2^{-2 (n+4)} e^{\frac {4 a}{b}} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {4 (a+b \text {arccosh}(c x))}{b}\right )+b 2^{-n-7} 3^{-n-1} e^{\frac {6 a}{b}} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {6 (a+b \text {arccosh}(c x))}{b}\right )-b 2^{-3 n-11} e^{\frac {8 a}{b}} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {8 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b c^3 \sqrt {c x-1} \sqrt {c x+1}}\) |
(d^2*Sqrt[d - c^2*d*x^2]*((-5*(a + b*ArcCosh[c*x])^(1 + n))/(128*(1 + n)) + (2^(-11 - 3*n)*b*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-8*(a + b*ArcCosh[ c*x]))/b])/(E^((8*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n) - (2^(-7 - n)*3^(-1 - n)*b*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-6*(a + b*ArcCosh[c*x]))/b])/ (E^((6*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n) + (b*(a + b*ArcCosh[c*x])^n*Ga mma[1 + n, (-4*(a + b*ArcCosh[c*x]))/b])/(2^(2*(4 + n))*E^((4*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n) + (2^(-7 - 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^(-7 - 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*E^((4*a)/b)*(a + b*A rcCosh[c*x])^n*Gamma[1 + n, (4*(a + b*ArcCosh[c*x]))/b])/(2^(2*(4 + n))*(( a + b*ArcCosh[c*x])/b)^n) + (2^(-7 - n)*3^(-1 - n)*b*E^((6*a)/b)*(a + b*Ar cCosh[c*x])^n*Gamma[1 + n, (6*(a + b*ArcCosh[c*x]))/b])/((a + b*ArcCosh[c* x])/b)^n - (2^(-11 - 3*n)*b*E^((8*a)/b)*(a + b*ArcCosh[c*x])^n*Gamma[1 + n , (8*(a + b*ArcCosh[c*x]))/b])/((a + b*ArcCosh[c*x])/b)^n))/(b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
3.5.29.3.1 Defintions of rubi rules used
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
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]
\[\int x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{n}d x\]
\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} x^{2} \,d x } \]
integral((c^4*d^2*x^6 - 2*c^2*d^2*x^4 + d^2*x^2)*sqrt(-c^2*d*x^2 + d)*(b*a rccosh(c*x) + a)^n, x)
Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx=\text {Timed out} \]
\[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} x^{2} \,d x } \]
Exception generated. \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(con st gen &
Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^n \, dx=\int x^2\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]