\(\int \frac {(f x)^m \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx\) [158]

Optimal result

Integrand size = 24, antiderivative size = 129 \[ \int \frac {(f x)^m \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\frac {(f x)^{1+m} \text {arccosh}(a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )}{f (1+m)}-\frac {a (f x)^{2+m} \sqrt {1-a x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};a^2 x^2\right )}{f^2 \left (2+3 m+m^2\right ) \sqrt {-1+a x}} \] Output:

(f*x)^(1+m)*arccosh(a*x)*hypergeom([1/2, 1/2+1/2*m],[3/2+1/2*m],a^2*x^2)/f 
/(1+m)-a*(f*x)^(2+m)*(-a*x+1)^(1/2)*hypergeom([1, 1+1/2*m, 1+1/2*m],[2+1/2 
*m, 3/2+1/2*m],a^2*x^2)/f^2/(m^2+3*m+2)/(a*x-1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.96 \[ \int \frac {(f x)^m \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\frac {x (f x)^m \left (\text {arccosh}(a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )+\frac {a x \sqrt {-1+a x} \sqrt {1+a x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};a^2 x^2\right )}{(2+m) \sqrt {1-a^2 x^2}}\right )}{1+m} \] Input:

Integrate[((f*x)^m*ArcCosh[a*x])/Sqrt[1 - a^2*x^2],x]
 

Output:

(x*(f*x)^m*(ArcCosh[a*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, a^2* 
x^2] + (a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*HypergeometricPFQ[{1, 1 + m/2, 1 
+ m/2}, {3/2 + m/2, 2 + m/2}, a^2*x^2])/((2 + m)*Sqrt[1 - a^2*x^2])))/(1 + 
 m)
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.99, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {6363}

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.

\(\displaystyle \int \frac {\text {arccosh}(a x) (f x)^m}{\sqrt {1-a^2 x^2}} \, dx\)

\(\Big \downarrow \) 6363

\(\displaystyle \frac {a \sqrt {a x-1} (f x)^{m+2} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;a^2 x^2\right )}{f^2 (m+1) (m+2) \sqrt {1-a x}}+\frac {\text {arccosh}(a x) (f x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},a^2 x^2\right )}{f (m+1)}\)

Input:

Int[((f*x)^m*ArcCosh[a*x])/Sqrt[1 - a^2*x^2],x]
 

Output:

((f*x)^(1 + m)*ArcCosh[a*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, a 
^2*x^2])/(f*(1 + m)) + (a*(f*x)^(2 + m)*Sqrt[-1 + a*x]*HypergeometricPFQ[{ 
1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, a^2*x^2])/(f^2*(1 + m)*(2 + m) 
*Sqrt[1 - a*x])
 

Defintions of rubi rules used

rule 6363

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_ 
.)*(x_)^2], x_Symbol] :> Simp[((f*x)^(m + 1)/(f*(m + 1)))*Simp[Sqrt[1 - c^2 
*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/ 
2, (3 + m)/2, c^2*x^2], x] + Simp[b*c*((f*x)^(m + 2)/(f^2*(m + 1)*(m + 2))) 
*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])]*HypergeometricPFQ[{1, 
 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2], x] /; FreeQ[{a, b, c, d 
, e, f, m}, x] && EqQ[c^2*d + e, 0] &&  !IntegerQ[m]
 

Maple [F]

\[\int \frac {\left (f x \right )^{m} \operatorname {arccosh}\left (a x \right )}{\sqrt {-a^{2} x^{2}+1}}d x\]

Input:

int((f*x)^m*arccosh(a*x)/(-a^2*x^2+1)^(1/2),x)
 

Output:

int((f*x)^m*arccosh(a*x)/(-a^2*x^2+1)^(1/2),x)
 

Fricas [F]

\[ \int \frac {(f x)^m \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {\left (f x\right )^{m} \operatorname {arcosh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

integrate((f*x)^m*arccosh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

integral(-sqrt(-a^2*x^2 + 1)*(f*x)^m*arccosh(a*x)/(a^2*x^2 - 1), x)
 

Sympy [F]

\[ \int \frac {(f x)^m \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {\left (f x\right )^{m} \operatorname {acosh}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \] Input:

integrate((f*x)**m*acosh(a*x)/(-a**2*x**2+1)**(1/2),x)
 

Output:

Integral((f*x)**m*acosh(a*x)/sqrt(-(a*x - 1)*(a*x + 1)), x)
 

Maxima [F]

\[ \int \frac {(f x)^m \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {\left (f x\right )^{m} \operatorname {arcosh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

integrate((f*x)^m*arccosh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

integrate((f*x)^m*arccosh(a*x)/sqrt(-a^2*x^2 + 1), x)
 

Giac [F]

\[ \int \frac {(f x)^m \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {\left (f x\right )^{m} \operatorname {arcosh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

integrate((f*x)^m*arccosh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
 

Output:

integrate((f*x)^m*arccosh(a*x)/sqrt(-a^2*x^2 + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f x)^m \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {\mathrm {acosh}\left (a\,x\right )\,{\left (f\,x\right )}^m}{\sqrt {1-a^2\,x^2}} \,d x \] Input:

int((acosh(a*x)*(f*x)^m)/(1 - a^2*x^2)^(1/2),x)
 

Output:

int((acosh(a*x)*(f*x)^m)/(1 - a^2*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {(f x)^m \text {arccosh}(a x)}{\sqrt {1-a^2 x^2}} \, dx=f^{m} \left (\int \frac {x^{m} \mathit {acosh} \left (a x \right )}{\sqrt {-a^{2} x^{2}+1}}d x \right ) \] Input:

int((f*x)^m*acosh(a*x)/(-a^2*x^2+1)^(1/2),x)
 

Output:

f**m*int((x**m*acosh(a*x))/sqrt( - a**2*x**2 + 1),x)