\(\int (f x)^m (d+e x^2) (a+b \text {csch}^{-1}(c x)) \, dx\) [168]

Optimal result

Integrand size = 21, antiderivative size = 220 \[ \int (f x)^m \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {b e x (f x)^{1+m} \sqrt {-1-c^2 x^2}}{c f \left (6+5 m+m^2\right ) \sqrt {-c^2 x^2}}+\frac {d (f x)^{1+m} \left (a+b \text {csch}^{-1}(c x)\right )}{f (1+m)}+\frac {e (f x)^{3+m} \left (a+b \text {csch}^{-1}(c x)\right )}{f^3 (3+m)}+\frac {b \left (e (1+m)^2-c^2 d (2+m) (3+m)\right ) x (f x)^{1+m} \sqrt {1+c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},-c^2 x^2\right )}{c f (1+m)^2 (2+m) (3+m) \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}} \] Output:

b*e*x*(f*x)^(1+m)*(-c^2*x^2-1)^(1/2)/c/f/(m^2+5*m+6)/(-c^2*x^2)^(1/2)+d*(f 
*x)^(1+m)*(a+b*arccsch(c*x))/f/(1+m)+e*(f*x)^(3+m)*(a+b*arccsch(c*x))/f^3/ 
(3+m)+b*(e*(1+m)^2-c^2*d*(2+m)*(3+m))*x*(f*x)^(1+m)*(c^2*x^2+1)^(1/2)*hype 
rgeom([1/2, 1/2+1/2*m],[3/2+1/2*m],-c^2*x^2)/c/f/(1+m)^2/(2+m)/(3+m)/(-c^2 
*x^2)^(1/2)/(-c^2*x^2-1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.76 \[ \int (f x)^m \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=x (f x)^m \left (\frac {b c d \sqrt {1+\frac {1}{c^2 x^2}} x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},-c^2 x^2\right )}{(1+m)^2 \sqrt {1+c^2 x^2}}+\frac {\frac {(3+m) \left (d (3+m)+e (1+m) x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{1+m}+\frac {b c e \sqrt {1+\frac {1}{c^2 x^2}} x^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},-c^2 x^2\right )}{\sqrt {1+c^2 x^2}}}{(3+m)^2}\right ) \] Input:

Integrate[(f*x)^m*(d + e*x^2)*(a + b*ArcCsch[c*x]),x]
 

Output:

x*(f*x)^m*((b*c*d*Sqrt[1 + 1/(c^2*x^2)]*x*Hypergeometric2F1[1/2, (1 + m)/2 
, (3 + m)/2, -(c^2*x^2)])/((1 + m)^2*Sqrt[1 + c^2*x^2]) + (((3 + m)*(d*(3 
+ m) + e*(1 + m)*x^2)*(a + b*ArcCsch[c*x]))/(1 + m) + (b*c*e*Sqrt[1 + 1/(c 
^2*x^2)]*x^3*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, -(c^2*x^2)])/Sqr 
t[1 + c^2*x^2])/(3 + m)^2)
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6856, 27, 363, 279, 278}

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.

\(\displaystyle \int \left (d+e x^2\right ) (f x)^m \left (a+b \text {csch}^{-1}(c x)\right ) \, dx\)

\(\Big \downarrow \) 6856

\(\displaystyle -\frac {b c x \int \frac {(f x)^m \left (e (m+1) x^2+d (m+3)\right )}{\left (m^2+4 m+3\right ) \sqrt {-c^2 x^2-1}}dx}{\sqrt {-c^2 x^2}}+\frac {d (f x)^{m+1} \left (a+b \text {csch}^{-1}(c x)\right )}{f (m+1)}+\frac {e (f x)^{m+3} \left (a+b \text {csch}^{-1}(c x)\right )}{f^3 (m+3)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c x \int \frac {(f x)^m \left (e (m+1) x^2+d (m+3)\right )}{\sqrt {-c^2 x^2-1}}dx}{\left (m^2+4 m+3\right ) \sqrt {-c^2 x^2}}+\frac {d (f x)^{m+1} \left (a+b \text {csch}^{-1}(c x)\right )}{f (m+1)}+\frac {e (f x)^{m+3} \left (a+b \text {csch}^{-1}(c x)\right )}{f^3 (m+3)}\)

\(\Big \downarrow \) 363

\(\displaystyle -\frac {b c x \left (-\left (\frac {e (m+1)^2}{c^2 (m+2)}-d (m+3)\right ) \int \frac {(f x)^m}{\sqrt {-c^2 x^2-1}}dx-\frac {e (m+1) \sqrt {-c^2 x^2-1} (f x)^{m+1}}{c^2 f (m+2)}\right )}{\left (m^2+4 m+3\right ) \sqrt {-c^2 x^2}}+\frac {d (f x)^{m+1} \left (a+b \text {csch}^{-1}(c x)\right )}{f (m+1)}+\frac {e (f x)^{m+3} \left (a+b \text {csch}^{-1}(c x)\right )}{f^3 (m+3)}\)

\(\Big \downarrow \) 279

\(\displaystyle -\frac {b c x \left (-\frac {\sqrt {c^2 x^2+1} \left (\frac {e (m+1)^2}{c^2 (m+2)}-d (m+3)\right ) \int \frac {(f x)^m}{\sqrt {c^2 x^2+1}}dx}{\sqrt {-c^2 x^2-1}}-\frac {e (m+1) \sqrt {-c^2 x^2-1} (f x)^{m+1}}{c^2 f (m+2)}\right )}{\left (m^2+4 m+3\right ) \sqrt {-c^2 x^2}}+\frac {d (f x)^{m+1} \left (a+b \text {csch}^{-1}(c x)\right )}{f (m+1)}+\frac {e (f x)^{m+3} \left (a+b \text {csch}^{-1}(c x)\right )}{f^3 (m+3)}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {d (f x)^{m+1} \left (a+b \text {csch}^{-1}(c x)\right )}{f (m+1)}+\frac {e (f x)^{m+3} \left (a+b \text {csch}^{-1}(c x)\right )}{f^3 (m+3)}-\frac {b c x \left (-\frac {\sqrt {c^2 x^2+1} (f x)^{m+1} \left (\frac {e (m+1)^2}{c^2 (m+2)}-d (m+3)\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},-c^2 x^2\right )}{f (m+1) \sqrt {-c^2 x^2-1}}-\frac {e (m+1) \sqrt {-c^2 x^2-1} (f x)^{m+1}}{c^2 f (m+2)}\right )}{\left (m^2+4 m+3\right ) \sqrt {-c^2 x^2}}\)

Input:

Int[(f*x)^m*(d + e*x^2)*(a + b*ArcCsch[c*x]),x]
 

Output:

(d*(f*x)^(1 + m)*(a + b*ArcCsch[c*x]))/(f*(1 + m)) + (e*(f*x)^(3 + m)*(a + 
 b*ArcCsch[c*x]))/(f^3*(3 + m)) - (b*c*x*(-((e*(1 + m)*(f*x)^(1 + m)*Sqrt[ 
-1 - c^2*x^2])/(c^2*f*(2 + m))) - (((e*(1 + m)^2)/(c^2*(2 + m)) - d*(3 + m 
))*(f*x)^(1 + m)*Sqrt[1 + c^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + 
m)/2, -(c^2*x^2)])/(f*(1 + m)*Sqrt[-1 - c^2*x^2])))/((3 + 4*m + m^2)*Sqrt[ 
-(c^2*x^2)])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP 
art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(c*x)^m* 
(1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && 
!(ILtQ[p, 0] || GtQ[a, 0])
 

rule 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

rule 6856

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*( 
x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Si 
mp[(a + b*ArcCsch[c*x])   u, x] - Simp[b*c*(x/Sqrt[(-c^2)*x^2])   Int[Simpl 
ifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), x], x], x]] /; FreeQ[{a, b, c, d, e, 
 f, m, p}, x] && ((IGtQ[p, 0] &&  !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 
0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (I 
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))
 

Maple [F]

\[\int \left (f x \right )^{m} \left (x^{2} e +d \right ) \left (a +b \,\operatorname {arccsch}\left (c x \right )\right )d x\]

Input:

int((f*x)^m*(e*x^2+d)*(a+b*arccsch(c*x)),x)
 

Output:

int((f*x)^m*(e*x^2+d)*(a+b*arccsch(c*x)),x)
 

Fricas [F]

\[ \int (f x)^m \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \] Input:

integrate((f*x)^m*(e*x^2+d)*(a+b*arccsch(c*x)),x, algorithm="fricas")
 

Output:

integral((a*e*x^2 + a*d + (b*e*x^2 + b*d)*arccsch(c*x))*(f*x)^m, x)
 

Sympy [F]

\[ \int (f x)^m \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int \left (f x\right )^{m} \left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, dx \] Input:

integrate((f*x)**m*(e*x**2+d)*(a+b*acsch(c*x)),x)
 

Output:

Integral((f*x)**m*(a + b*acsch(c*x))*(d + e*x**2), x)
 

Maxima [F]

\[ \int (f x)^m \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \] Input:

integrate((f*x)^m*(e*x^2+d)*(a+b*arccsch(c*x)),x, algorithm="maxima")
 

Output:

a*e*f^m*x^3*x^m/(m + 3) + (f*x)^(m + 1)*a*d/(f*(m + 1)) - ((b*e*f^m*(m + 1 
)*x^3 + b*d*f^m*(m + 3)*x)*x^m*log(x) - (b*e*f^m*(m + 1)*x^3 + b*d*f^m*(m 
+ 3)*x)*x^m*log(sqrt(c^2*x^2 + 1) + 1))/(m^2 + 4*m + 3) + integrate((b*c^2 
*e*f^m*(m + 1)*x^4 + b*c^2*d*f^m*(m + 3)*x^2)*x^m/((m^2 + 4*m + 3)*c^2*x^2 
 + m^2 + ((m^2 + 4*m + 3)*c^2*x^2 + m^2 + 4*m + 3)*sqrt(c^2*x^2 + 1) + 4*m 
 + 3), x) - integrate(((m^2 + 4*m + 3)*b*c^2*e*f^m*x^4*log(c) + ((m^2 + 4* 
m + 3)*c^2*d*f^m*log(c) + (m^2 + 4*m + 3)*e*f^m*log(c) - e*f^m*(m + 1))*b* 
x^2 + ((m^2 + 4*m + 3)*d*f^m*log(c) - d*f^m*(m + 3))*b)*x^m/((m^2 + 4*m + 
3)*c^2*x^2 + m^2 + 4*m + 3), x)
 

Giac [F]

\[ \int (f x)^m \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \] Input:

integrate((f*x)^m*(e*x^2+d)*(a+b*arccsch(c*x)),x, algorithm="giac")
 

Output:

integrate((e*x^2 + d)*(b*arccsch(c*x) + a)*(f*x)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (f x)^m \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int {\left (f\,x\right )}^m\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right ) \,d x \] Input:

int((f*x)^m*(d + e*x^2)*(a + b*asinh(1/(c*x))),x)
 

Output:

int((f*x)^m*(d + e*x^2)*(a + b*asinh(1/(c*x))), x)
 

Reduce [F]

\[ \int (f x)^m \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {f^{m} \left (x^{m} a d m x +3 x^{m} a d x +x^{m} a e m \,x^{3}+x^{m} a e \,x^{3}+\left (\int x^{m} \mathit {acsch} \left (c x \right ) x^{2}d x \right ) b e \,m^{2}+4 \left (\int x^{m} \mathit {acsch} \left (c x \right ) x^{2}d x \right ) b e m +3 \left (\int x^{m} \mathit {acsch} \left (c x \right ) x^{2}d x \right ) b e +\left (\int x^{m} \mathit {acsch} \left (c x \right )d x \right ) b d \,m^{2}+4 \left (\int x^{m} \mathit {acsch} \left (c x \right )d x \right ) b d m +3 \left (\int x^{m} \mathit {acsch} \left (c x \right )d x \right ) b d \right )}{m^{2}+4 m +3} \] Input:

int((f*x)^m*(e*x^2+d)*(a+b*acsch(c*x)),x)
 

Output:

(f**m*(x**m*a*d*m*x + 3*x**m*a*d*x + x**m*a*e*m*x**3 + x**m*a*e*x**3 + int 
(x**m*acsch(c*x)*x**2,x)*b*e*m**2 + 4*int(x**m*acsch(c*x)*x**2,x)*b*e*m + 
3*int(x**m*acsch(c*x)*x**2,x)*b*e + int(x**m*acsch(c*x),x)*b*d*m**2 + 4*in 
t(x**m*acsch(c*x),x)*b*d*m + 3*int(x**m*acsch(c*x),x)*b*d))/(m**2 + 4*m + 
3)