Integrand size = 27, antiderivative size = 578 \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=-\frac {5^{-1-n} d e^{-\frac {5 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{32 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3^{-n} d e^{-\frac {3 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{32 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d e^{-\frac {a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )}{16 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d e^{a/b} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arccosh}(c x)}{b}\right )}{16 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3^{-n} d e^{\frac {3 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{32 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5^{-1-n} d e^{\frac {5 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{32 c^2 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:
-1/32*5^(-1-n)*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^n*GAMMA(1+n,(-5*a -5*b*arccosh(c*x))/b)/c^2/exp(5*a/b)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/((-(a+b*a rccosh(c*x))/b)^n)+1/32*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^n*GAMMA( 1+n,(-3*a-3*b*arccosh(c*x))/b)/(3^n)/c^2/exp(3*a/b)/(c*x-1)^(1/2)/(c*x+1)^ (1/2)/((-(a+b*arccosh(c*x))/b)^n)-1/16*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh (c*x))^n*GAMMA(1+n,-(a+b*arccosh(c*x))/b)/c^2/exp(a/b)/(c*x-1)^(1/2)/(c*x+ 1)^(1/2)/((-(a+b*arccosh(c*x))/b)^n)+1/16*d*exp(a/b)*(-c^2*d*x^2+d)^(1/2)* (a+b*arccosh(c*x))^n*GAMMA(1+n,(a+b*arccosh(c*x))/b)/c^2/(c*x-1)^(1/2)/(c* x+1)^(1/2)/(((a+b*arccosh(c*x))/b)^n)-1/32*d*exp(3*a/b)*(-c^2*d*x^2+d)^(1/ 2)*(a+b*arccosh(c*x))^n*GAMMA(1+n,3*(a+b*arccosh(c*x))/b)/(3^n)/c^2/(c*x-1 )^(1/2)/(c*x+1)^(1/2)/(((a+b*arccosh(c*x))/b)^n)+1/32*5^(-1-n)*d*exp(5*a/b )*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^n*GAMMA(1+n,5*(a+b*arccosh(c*x)) /b)/c^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(((a+b*arccosh(c*x))/b)^n)
Time = 1.44 (sec) , antiderivative size = 500, normalized size of antiderivative = 0.87 \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=-\frac {15^{-1-n} d^2 e^{-\frac {5 a}{b}} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) (a+b \text {arccosh}(c x))^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{-3 n} \left (2\ 15^{1+n} e^{\frac {6 a}{b}} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{2 n} \Gamma \left (1+n,\frac {a}{b}+\text {arccosh}(c x)\right )+\left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \left (-3^{1+n} \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{2 n} \Gamma \left (1+n,-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )+3\ 5^{1+n} e^{\frac {2 a}{b}} \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{2 n} \Gamma \left (1+n,-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-2\ 15^{1+n} e^{\frac {4 a}{b}} \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{2 n} \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )+5^{1+n} e^{\frac {8 a}{b}} \left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{3 n} \Gamma \left (1+n,\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-4\ 5^{1+n} e^{\frac {8 a}{b}} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{2 n} \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^n \Gamma \left (1+n,\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+3^{1+n} e^{\frac {10 a}{b}} \left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{3 n} \Gamma \left (1+n,\frac {5 (a+b \text {arccosh}(c x))}{b}\right )\right )\right )}{32 c^2 \sqrt {d-c^2 d x^2}} \] Input:
Integrate[x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n,x]
Output:
-1/32*(15^(-1 - n)*d^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcCosh [c*x])^n*(2*15^(1 + n)*E^((6*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*(-((a + b *ArcCosh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, a/b + ArcCosh[c*x]] + (a/b + Arc Cosh[c*x])^n*(-(3^(1 + n)*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, (-5*(a + b*ArcCosh[c*x]))/b]) + 3*5^(1 + n)*E^((2*a)/b)*(-((a + b*ArcCo sh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, (-3*(a + b*ArcCosh[c*x]))/b] - 2*15^(1 + n)*E^((4*a)/b)*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)] + 5^(1 + n)*E^((8*a)/b)*(a/b + ArcCosh[c*x])^n*(-((a + b*ArcCosh[c*x])/b))^(3*n)*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b] - 4* 5^(1 + n)*E^((8*a)/b)*(-((a + b*ArcCosh[c*x])/b))^(2*n)*(-((a + b*ArcCosh[ c*x])^2/b^2))^n*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b] + 3^(1 + n)*E^((1 0*a)/b)*(a/b + ArcCosh[c*x])^n*(-((a + b*ArcCosh[c*x])/b))^(3*n)*Gamma[1 + n, (5*(a + b*ArcCosh[c*x]))/b])))/(c^2*E^((5*a)/b)*Sqrt[d - c^2*d*x^2]*(- ((a + b*ArcCosh[c*x])^2/b^2))^(3*n))
Time = 0.80 (sec) , antiderivative size = 404, normalized size of antiderivative = 0.70, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.
\(\displaystyle \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx\) |
\(\Big \downarrow \) 6367 |
\(\displaystyle -\frac {d \sqrt {d-c^2 d x^2} \int (a+b \text {arccosh}(c x))^n \cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^4\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )d(a+b \text {arccosh}(c x))}{b c^2 \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle -\frac {d \sqrt {d-c^2 d x^2} \int \left (\frac {1}{16} \cosh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arccosh}(c x))}{b}\right ) (a+b \text {arccosh}(c x))^n-\frac {3}{16} \cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right ) (a+b \text {arccosh}(c x))^n+\frac {1}{8} \cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) (a+b \text {arccosh}(c x))^n\right )d(a+b \text {arccosh}(c x))}{b c^2 \sqrt {c x-1} \sqrt {c x+1}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d \sqrt {d-c^2 d x^2} \left (\frac {1}{32} b 5^{-n-1} e^{-\frac {5 a}{b}} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{32} b 3^{-n} e^{-\frac {3 a}{b}} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{16} b e^{-\frac {a}{b}} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {1}{16} b e^{a/b} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {1}{32} b 3^{-n} e^{\frac {3 a}{b}} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{32} b 5^{-n-1} e^{\frac {5 a}{b}} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {5 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b c^2 \sqrt {c x-1} \sqrt {c x+1}}\) |
Input:
Int[x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n,x]
Output:
-((d*Sqrt[d - c^2*d*x^2]*((5^(-1 - n)*b*(a + b*ArcCosh[c*x])^n*Gamma[1 + n , (-5*(a + b*ArcCosh[c*x]))/b])/(32*E^((5*a)/b)*(-((a + b*ArcCosh[c*x])/b) )^n) - (b*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-3*(a + b*ArcCosh[c*x]))/b] )/(32*3^n*E^((3*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n) + (b*(a + b*ArcCosh[c *x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)])/(16*E^(a/b)*(-((a + b*ArcC osh[c*x])/b))^n) - (b*E^(a/b)*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (a + b*A rcCosh[c*x])/b])/(16*((a + b*ArcCosh[c*x])/b)^n) + (b*E^((3*a)/b)*(a + b*A rcCosh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b])/(32*3^n*((a + b*A rcCosh[c*x])/b)^n) - (5^(-1 - n)*b*E^((5*a)/b)*(a + b*ArcCosh[c*x])^n*Gamm a[1 + n, (5*(a + b*ArcCosh[c*x]))/b])/(32*((a + b*ArcCosh[c*x])/b)^n)))/(b *c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))
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]
\[\int x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{n}d x\]
Input:
int(x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^n,x)
Output:
int(x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^n,x)
\[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} x \,d x } \] Input:
integrate(x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^n,x, algorithm="fricas ")
Output:
integral(-(c^2*d*x^3 - d*x)*sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)^n, x )
Timed out. \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\text {Timed out} \] Input:
integrate(x*(-c**2*d*x**2+d)**(3/2)*(a+b*acosh(c*x))**n,x)
Output:
Timed out
\[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} x \,d x } \] Input:
integrate(x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^n,x, algorithm="maxima ")
Output:
integrate((-c^2*d*x^2 + d)^(3/2)*(b*arccosh(c*x) + a)^n*x, x)
Exception generated. \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^n,x, algorithm="giac")
Output:
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 Value
Timed out. \[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\int x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \] Input:
int(x*(a + b*acosh(c*x))^n*(d - c^2*d*x^2)^(3/2),x)
Output:
int(x*(a + b*acosh(c*x))^n*(d - c^2*d*x^2)^(3/2), x)
\[ \int x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\sqrt {d}\, d \left (-\left (\int \left (\mathit {acosh} \left (c x \right ) b +a \right )^{n} \sqrt {-c^{2} x^{2}+1}\, x^{3}d x \right ) c^{2}+\int \left (\mathit {acosh} \left (c x \right ) b +a \right )^{n} \sqrt {-c^{2} x^{2}+1}\, x d x \right ) \] Input:
int(x*(-c^2*d*x^2+d)^(3/2)*(a+b*acosh(c*x))^n,x)
Output:
sqrt(d)*d*( - int((acosh(c*x)*b + a)**n*sqrt( - c**2*x**2 + 1)*x**3,x)*c** 2 + int((acosh(c*x)*b + a)**n*sqrt( - c**2*x**2 + 1)*x,x))