\(\int (d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x))^n \, dx\) [426]

Optimal result

Integrand size = 26, antiderivative size = 450 \[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=-\frac {3 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^{1+n}}{8 b c (1+n) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2^{-2 (3+n)} d e^{-\frac {4 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2^{-3-n} d e^{-\frac {2 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2^{-3-n} d e^{\frac {2 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2^{-2 (3+n)} d e^{\frac {4 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{c \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

-3/8*d*(a+b*arccosh(c*x))^(1+n)*(-c^2*d*x^2+d)^(1/2)/b/c/(1+n)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-d*(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^(6+2*n))/c/exp(4*a/b)/(((-a-b*arccosh(c*x))/b)^
n)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+2^(-3-n)*d*(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/exp(2*a/b)/(((-a-b*arccosh(c*x))/b)^n)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2^(-3-n)*d*exp(2*a/b)*(a+b*arcco
sh(c*x))^n*GAMMA(1+n,2*(a+b*arccosh(c*x))/b)*(-c^2*d*x^2+d)^(1/2)/c/(((a+b*arccosh(c*x))/b)^n)/(c*x-1)^(1/2)/(
c*x+1)^(1/2)+d*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^(6+2*
n))/c/(((a+b*arccosh(c*x))/b)^n)/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5906, 3393, 3388, 2212} \[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=-\frac {3 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^{n+1}}{8 b c (n+1) \sqrt {c x-1} \sqrt {c x+1}}-\frac {d 2^{-2 (n+3)} e^{-\frac {4 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{c \sqrt {c x-1} \sqrt {c x+1}}+\frac {d 2^{-n-3} e^{-\frac {2 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{c \sqrt {c x-1} \sqrt {c x+1}}-\frac {d 2^{-n-3} e^{\frac {2 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{c \sqrt {c x-1} \sqrt {c x+1}}+\frac {d 2^{-2 (n+3)} e^{\frac {4 a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{c \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

Int[(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n,x]

[Out]

(-3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^(1 + n))/(8*b*c*(1 + n)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d*Sqrt
[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcCosh[c*x]))/b])/(2^(2*(3 + n))*c*E^((4*a)/b)
*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) + (2^(-3 - n)*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos
h[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]))/b])/(c*E^((2*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*Arc
Cosh[c*x])/b))^n) - (2^(-3 - n)*d*E^((2*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (2*(a +
b*ArcCosh[c*x]))/b])/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) + (d*E^((4*a)/b)*Sqrt[d - c^2
*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (4*(a + b*ArcCosh[c*x]))/b])/(2^(2*(3 + n))*c*Sqrt[-1 + c*x]*Sqrt[
1 + c*x]*((a + b*ArcCosh[c*x])/b)^n)

Rule 2212

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c
+ d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d))^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m
+ 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 3388

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d
, e, f, m}, x] && IntegerQ[2*k]

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[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])
)

Rule 5906

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(1/(b*c))*Simp[(d
+ e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Subst[Int[x^n*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]],
x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int x^n \sinh ^4\left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b c \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {3 x^n}{8}+\frac {1}{8} x^n \cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )-\frac {1}{2} x^n \cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b c \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {3 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^{1+n}}{8 b c (1+n) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int x^n \cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int x^n \cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b c \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {3 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^{1+n}}{8 b c (1+n) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {4 i a}{b}-\frac {4 i x}{b}\right )} x^n \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {4 i a}{b}-\frac {4 i x}{b}\right )} x^n \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )} x^n \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )} x^n \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b c \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {3 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^{1+n}}{8 b c (1+n) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4^{-3-n} d 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 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2^{-3-n} d 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 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2^{-3-n} d 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 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4^{-3-n} d 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 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.60 (sec) , antiderivative size = 384, normalized size of antiderivative = 0.85 \[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\frac {4^{-3-n} d^2 e^{-\frac {4 a}{b}} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) (a+b \text {arccosh}(c x))^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{-2 n} \left (3\ 2^{3+2 n} e^{\frac {4 a}{b}} (a+b \text {arccosh}(c x)) \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{2 n}+b (1+n) \left (\frac {a}{b}+\text {arccosh}(c x)\right )^{2 n} \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^n \Gamma \left (1+n,-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )-2^{3+n} b e^{\frac {2 a}{b}} (1+n) \left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^n \Gamma \left (1+n,-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+2^{3+n} b e^{\frac {6 a}{b}} (1+n) \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^n \Gamma \left (1+n,\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-b e^{\frac {8 a}{b}} (1+n) \left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{2 n} \Gamma \left (1+n,\frac {4 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b c (1+n) \sqrt {d-c^2 d x^2}} \]

[In]

Integrate[(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n,x]

[Out]

(4^(-3 - n)*d^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcCosh[c*x])^n*(3*2^(3 + 2*n)*E^((4*a)/b)*(a + b*
ArcCosh[c*x])*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n) + b*(1 + n)*(a/b + ArcCosh[c*x])^(2*n)*(-((a + b*ArcCosh[c
*x])/b))^n*Gamma[1 + n, (-4*(a + b*ArcCosh[c*x]))/b] - 2^(3 + n)*b*E^((2*a)/b)*(1 + n)*(a/b + ArcCosh[c*x])^n*
(-((a + b*ArcCosh[c*x])^2/b^2))^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]))/b] + 2^(3 + n)*b*E^((6*a)/b)*(1 + n)*
(-((a + b*ArcCosh[c*x])/b))^n*(-((a + b*ArcCosh[c*x])^2/b^2))^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b] - b*E
^((8*a)/b)*(1 + n)*(a/b + ArcCosh[c*x])^n*(-((a + b*ArcCosh[c*x])/b))^(2*n)*Gamma[1 + n, (4*(a + b*ArcCosh[c*x
]))/b]))/(b*c*E^((4*a)/b)*(1 + n)*Sqrt[d - c^2*d*x^2]*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n))

Maple [F]

\[\int \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{n}d x\]

[In]

int((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^n,x)

[Out]

int((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^n,x)

Fricas [F]

\[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} \,d x } \]

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^n,x, algorithm="fricas")

[Out]

integral((-c^2*d*x^2 + d)^(3/2)*(b*arccosh(c*x) + a)^n, x)

Sympy [F(-1)]

Timed out. \[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\text {Timed out} \]

[In]

integrate((-c**2*d*x**2+d)**(3/2)*(a+b*acosh(c*x))**n,x)

[Out]

Timed out

Maxima [F]

\[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} \,d x } \]

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^n,x, algorithm="maxima")

[Out]

integrate((-c^2*d*x^2 + d)^(3/2)*(b*arccosh(c*x) + a)^n, x)

Giac [F(-2)]

Exception generated. \[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^n,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^n \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]

[In]

int((a + b*acosh(c*x))^n*(d - c^2*d*x^2)^(3/2),x)

[Out]

int((a + b*acosh(c*x))^n*(d - c^2*d*x^2)^(3/2), x)