Integrand size = 19, antiderivative size = 449 \[ \int \frac {x (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=-\frac {(a+b \text {arccosh}(c x))^2}{2 b e}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e} \] Output:
-1/2*(a+b*arccosh(c*x))^2/b/e+1/2*(a+b*arccosh(c*x))*ln(1-e^(1/2)*(c*x+(c* x-1)^(1/2)*(c*x+1)^(1/2))/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/e+1/2*(a+b*arcc osh(c*x))*ln(1+e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*(-d)^(1/2)-(-c ^2*d-e)^(1/2)))/e+1/2*(a+b*arccosh(c*x))*ln(1-e^(1/2)*(c*x+(c*x-1)^(1/2)*( c*x+1)^(1/2))/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/e+1/2*(a+b*arccosh(c*x))*ln (1+e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2 )))/e+1/2*b*polylog(2,-e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*(-d)^( 1/2)-(-c^2*d-e)^(1/2)))/e+1/2*b*polylog(2,e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+ 1)^(1/2))/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/e+1/2*b*polylog(2,-e^(1/2)*(c*x +(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/e+1/2*b*pol ylog(2,e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*(-d)^(1/2)+(-c^2*d-e)^ (1/2)))/e
Time = 0.10 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.00 \[ \int \frac {x (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=-\frac {b \text {arccosh}(c x)^2}{2 e}+\frac {b \text {arccosh}(c x) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \text {arccosh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \text {arccosh}(c x) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \text {arccosh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {a \log \left (d+e x^2\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e} \] Input:
Integrate[(x*(a + b*ArcCosh[c*x]))/(d + e*x^2),x]
Output:
-1/2*(b*ArcCosh[c*x]^2)/e + (b*ArcCosh[c*x]*Log[1 - (Sqrt[e]*E^ArcCosh[c*x ])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e) + (b*ArcCosh[c*x]*Log[1 + (Sq rt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e) + (b*ArcCo sh[c*x]*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]) ])/(2*e) + (b*ArcCosh[c*x]*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*e) + (a*Log[d + e*x^2])/(2*e) + (b*PolyLog[2, -(( Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])/(2*e) + (b*Po lyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e ) + (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(2*e) + (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[ -(c^2*d) - e])])/(2*e)
Time = 1.63 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6374, 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.
\(\displaystyle \int \frac {x (a+b \text {arccosh}(c x))}{d+e x^2} \, dx\) |
\(\Big \downarrow \) 6374 |
\(\displaystyle \int \left (\frac {a+b \text {arccosh}(c x)}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \text {arccosh}(c x)}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 e}+\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 e}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 e}+\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 e}-\frac {(a+b \text {arccosh}(c x))^2}{2 b e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 e}\) |
Input:
Int[(x*(a + b*ArcCosh[c*x]))/(d + e*x^2),x]
Output:
-1/2*(a + b*ArcCosh[c*x])^2/(b*e) + ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e] *E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e) + ((a + b*ArcCo sh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e] )])/(2*e) + ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt [-d] + Sqrt[-(c^2*d) - e])])/(2*e) + ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e ]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*e) + (b*PolyLog[2 , -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])/(2*e) + (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])]) /(2*e) + (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2 *d) - e]))])/(2*e) + (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*e)
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e _.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
Time = 0.40 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.86
method | result | size |
parts | \(\frac {a \ln \left (e \,x^{2}+d \right )}{2 e}-\frac {b \operatorname {arccosh}\left (c x \right )^{2}}{2 e}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {-2 c^{2} d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 \sqrt {c^{4} d^{2}+c^{2} d e}-e}{-2 c^{2} d +2 \sqrt {c^{4} d^{2}+c^{2} d e}-e}\right )}{2 e}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {2 c^{2} d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 \sqrt {c^{4} d^{2}+c^{2} d e}+e}{2 c^{2} d +2 \sqrt {c^{4} d^{2}+c^{2} d e}+e}\right )}{2 e}+\frac {b \operatorname {dilog}\left (\frac {-2 c^{2} d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 \sqrt {c^{4} d^{2}+c^{2} d e}-e}{-2 c^{2} d +2 \sqrt {c^{4} d^{2}+c^{2} d e}-e}\right )}{4 e}+\frac {b \operatorname {dilog}\left (\frac {2 c^{2} d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 \sqrt {c^{4} d^{2}+c^{2} d e}+e}{2 c^{2} d +2 \sqrt {c^{4} d^{2}+c^{2} d e}+e}\right )}{4 e}\) | \(386\) |
derivativedivides | \(\frac {\frac {a \,c^{2} \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 e}+b \,c^{2} \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2 e}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {-2 c^{2} d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 \sqrt {c^{4} d^{2}+c^{2} d e}-e}{-2 c^{2} d +2 \sqrt {c^{4} d^{2}+c^{2} d e}-e}\right )}{2 e}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {2 c^{2} d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 \sqrt {c^{4} d^{2}+c^{2} d e}+e}{2 c^{2} d +2 \sqrt {c^{4} d^{2}+c^{2} d e}+e}\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {-2 c^{2} d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 \sqrt {c^{4} d^{2}+c^{2} d e}-e}{-2 c^{2} d +2 \sqrt {c^{4} d^{2}+c^{2} d e}-e}\right )}{4 e}+\frac {\operatorname {dilog}\left (\frac {2 c^{2} d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 \sqrt {c^{4} d^{2}+c^{2} d e}+e}{2 c^{2} d +2 \sqrt {c^{4} d^{2}+c^{2} d e}+e}\right )}{4 e}\right )}{c^{2}}\) | \(401\) |
default | \(\frac {\frac {a \,c^{2} \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 e}+b \,c^{2} \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2 e}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {-2 c^{2} d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 \sqrt {c^{4} d^{2}+c^{2} d e}-e}{-2 c^{2} d +2 \sqrt {c^{4} d^{2}+c^{2} d e}-e}\right )}{2 e}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {2 c^{2} d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 \sqrt {c^{4} d^{2}+c^{2} d e}+e}{2 c^{2} d +2 \sqrt {c^{4} d^{2}+c^{2} d e}+e}\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {-2 c^{2} d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 \sqrt {c^{4} d^{2}+c^{2} d e}-e}{-2 c^{2} d +2 \sqrt {c^{4} d^{2}+c^{2} d e}-e}\right )}{4 e}+\frac {\operatorname {dilog}\left (\frac {2 c^{2} d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 \sqrt {c^{4} d^{2}+c^{2} d e}+e}{2 c^{2} d +2 \sqrt {c^{4} d^{2}+c^{2} d e}+e}\right )}{4 e}\right )}{c^{2}}\) | \(401\) |
Input:
int(x*(a+b*arccosh(c*x))/(e*x^2+d),x,method=_RETURNVERBOSE)
Output:
1/2*a/e*ln(e*x^2+d)-1/2*b*arccosh(c*x)^2/e+1/2*b/e*arccosh(c*x)*ln((-2*c^2 *d-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2+2*(c^4*d^2+c^2*d*e)^(1/2)-e)/(-2* c^2*d+2*(c^4*d^2+c^2*d*e)^(1/2)-e))+1/2*b/e*arccosh(c*x)*ln((2*c^2*d+e*(c* x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2+2*(c^4*d^2+c^2*d*e)^(1/2)+e)/(2*c^2*d+2*( c^4*d^2+c^2*d*e)^(1/2)+e))+1/4*b/e*dilog((-2*c^2*d-e*(c*x+(c*x-1)^(1/2)*(c *x+1)^(1/2))^2+2*(c^4*d^2+c^2*d*e)^(1/2)-e)/(-2*c^2*d+2*(c^4*d^2+c^2*d*e)^ (1/2)-e))+1/4*b/e*dilog((2*c^2*d+e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2+2*( c^4*d^2+c^2*d*e)^(1/2)+e)/(2*c^2*d+2*(c^4*d^2+c^2*d*e)^(1/2)+e))
\[ \int \frac {x (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x}{e x^{2} + d} \,d x } \] Input:
integrate(x*(a+b*arccosh(c*x))/(e*x^2+d),x, algorithm="fricas")
Output:
integral((b*x*arccosh(c*x) + a*x)/(e*x^2 + d), x)
\[ \int \frac {x (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=\int \frac {x \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{d + e x^{2}}\, dx \] Input:
integrate(x*(a+b*acosh(c*x))/(e*x**2+d),x)
Output:
Integral(x*(a + b*acosh(c*x))/(d + e*x**2), x)
\[ \int \frac {x (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x}{e x^{2} + d} \,d x } \] Input:
integrate(x*(a+b*arccosh(c*x))/(e*x^2+d),x, algorithm="maxima")
Output:
b*integrate(x*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(e*x^2 + d), x) + 1/2 *a*log(e*x^2 + d)/e
\[ \int \frac {x (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x}{e x^{2} + d} \,d x } \] Input:
integrate(x*(a+b*arccosh(c*x))/(e*x^2+d),x, algorithm="giac")
Output:
integrate((b*arccosh(c*x) + a)*x/(e*x^2 + d), x)
Timed out. \[ \int \frac {x (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=\int \frac {x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{e\,x^2+d} \,d x \] Input:
int((x*(a + b*acosh(c*x)))/(d + e*x^2),x)
Output:
int((x*(a + b*acosh(c*x)))/(d + e*x^2), x)
\[ \int \frac {x (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=\frac {2 \left (\int \frac {\mathit {acosh} \left (c x \right ) x}{e \,x^{2}+d}d x \right ) b e +\mathrm {log}\left (e \,x^{2}+d \right ) a}{2 e} \] Input:
int(x*(a+b*acosh(c*x))/(e*x^2+d),x)
Output:
(2*int((acosh(c*x)*x)/(d + e*x**2),x)*b*e + log(d + e*x**2)*a)/(2*e)