Integrand size = 21, antiderivative size = 542 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx=-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{2 d^2 \left (e+\frac {d}{x^2}\right )}+\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b d^2}+\frac {b \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} \text {arctanh}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{2 d^2 \sqrt {c^2 d+e} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^2} \] Output:
-1/2*e*(a+b*arcsech(c*x))/d^2/(e+d/x^2)+1/2*(a+b*arcsech(c*x))^2/b/d^2+1/2 *b*e^(1/2)*(-1+1/c^2/x^2)^(1/2)*arctanh((c^2*d+e)^(1/2)/c/e^(1/2)/(-1+1/c^ 2/x^2)^(1/2)/x)/d^2/(c^2*d+e)^(1/2)/(-1+1/c/x)^(1/2)/(1+1/c/x)^(1/2)-1/2*( a+b*arcsech(c*x))*ln(1-c*(-d)^(1/2)*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2 ))/(e^(1/2)-(c^2*d+e)^(1/2)))/d^2-1/2*(a+b*arcsech(c*x))*ln(1+c*(-d)^(1/2) *(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/(e^(1/2)-(c^2*d+e)^(1/2)))/d^2-1 /2*(a+b*arcsech(c*x))*ln(1-c*(-d)^(1/2)*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^ (1/2))/(e^(1/2)+(c^2*d+e)^(1/2)))/d^2-1/2*(a+b*arcsech(c*x))*ln(1+c*(-d)^( 1/2)*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/(e^(1/2)+(c^2*d+e)^(1/2)))/d ^2-1/2*b*polylog(2,-c*(-d)^(1/2)*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/ (e^(1/2)-(c^2*d+e)^(1/2)))/d^2-1/2*b*polylog(2,c*(-d)^(1/2)*(1/c/x+(-1+1/c /x)^(1/2)*(1+1/c/x)^(1/2))/(e^(1/2)-(c^2*d+e)^(1/2)))/d^2-1/2*b*polylog(2, -c*(-d)^(1/2)*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/(e^(1/2)+(c^2*d+e)^ (1/2)))/d^2-1/2*b*polylog(2,c*(-d)^(1/2)*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x) ^(1/2))/(e^(1/2)+(c^2*d+e)^(1/2)))/d^2
Result contains complex when optimal does not.
Time = 1.24 (sec) , antiderivative size = 1189, normalized size of antiderivative = 2.19 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*ArcSech[c*x])/(x*(d + e*x^2)^2),x]
Output:
((2*a*d)/(d + e*x^2) + (b*Sqrt[d]*ArcSech[c*x])/(Sqrt[d] - I*Sqrt[e]*x) + (b*Sqrt[d]*ArcSech[c*x])/(Sqrt[d] + I*Sqrt[e]*x) - 2*b*ArcSech[c*x]^2 - (8 *I)*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[(((-I)*c*S qrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] - (8*I)*b*ArcSin[ Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[((I*c*Sqrt[d] + Sqrt[e] )*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]] - 2*b*ArcSech[c*x]*Log[1 + (I*(Sq rt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (4*I)*b*ArcSin[Sqr t[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - 2*b*ArcSech[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (4*I)*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])) /(c*Sqrt[d]*E^ArcSech[c*x])] - 2*b*ArcSech[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt [c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (4*I)*b*ArcSin[Sqrt[1 - (I*Sqr t[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqr t[d]*E^ArcSech[c*x])] - 2*b*ArcSech[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (4*I)*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/( c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ ArcSech[c*x])] + 4*a*Log[x] + 2*b*Log[x] - 2*a*Log[d + e*x^2] - 2*b*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]] + (b*Sqrt[e]* Log[((2*I)*Sqrt[e]*(Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + (Sqrt...
Time = 1.59 (sec) , antiderivative size = 598, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6857, 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.
\(\displaystyle \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 6857 |
\(\displaystyle -\int \frac {a+b \text {arccosh}\left (\frac {1}{c x}\right )}{\left (\frac {d}{x^2}+e\right )^2 x^3}d\frac {1}{x}\) |
\(\Big \downarrow \) 6374 |
\(\displaystyle -\int \left (\frac {a+b \text {arccosh}\left (\frac {1}{c x}\right )}{d \left (\frac {d}{x^2}+e\right ) x}-\frac {e \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{d \left (\frac {d}{x^2}+e\right )^2 x}\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {c^2 d+e}}+1\right )}{2 d^2}-\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 d^2}-\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {c^2 d+e}+\sqrt {e}}+1\right )}{2 d^2}-\frac {e \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{2 d^2 \left (\frac {d}{x^2}+e\right )}+\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )^2}{2 b d^2}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 d^2}+\frac {b \sqrt {e} \sqrt {\frac {1}{c^2 x^2}-1} \text {arctanh}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} x \sqrt {\frac {1}{c^2 x^2}-1}}\right )}{2 d^2 \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1} \sqrt {c^2 d+e}}\) |
Input:
Int[(a + b*ArcSech[c*x])/(x*(d + e*x^2)^2),x]
Output:
-1/2*(e*(a + b*ArcCosh[1/(c*x)]))/(d^2*(e + d/x^2)) + (a + b*ArcCosh[1/(c* x)])^2/(2*b*d^2) + (b*Sqrt[e]*Sqrt[-1 + 1/(c^2*x^2)]*ArcTanh[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[-1 + 1/(c^2*x^2)]*x)])/(2*d^2*Sqrt[c^2*d + e]*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) - ((a + b*ArcCosh[1/(c*x)])*Log[1 - (c*Sqrt[- d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^2) - ((a + b*Arc Cosh[1/(c*x)])*Log[1 + (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2 *d + e])])/(2*d^2) - ((a + b*ArcCosh[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^ArcCo sh[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^2) - ((a + b*ArcCosh[1/(c* x)])*Log[1 + (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e])]) /(2*d^2) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt [c^2*d + e]))])/(2*d^2) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(S qrt[e] - Sqrt[c^2*d + e])])/(2*d^2) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCos h[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*d^2) - (b*PolyLog[2, (c*Sqrt [-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^2)
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]
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_ )^2)^(p_.), x_Symbol] :> -Subst[Int[(e + d*x^2)^p*((a + b*ArcCosh[x/c])^n/x ^(m + 2*(p + 1))), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0 ] && IntegersQ[m, p]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 5.51 (sec) , antiderivative size = 2226, normalized size of antiderivative = 4.11
method | result | size |
parts | \(\text {Expression too large to display}\) | \(2226\) |
derivativedivides | \(\text {Expression too large to display}\) | \(2275\) |
default | \(\text {Expression too large to display}\) | \(2275\) |
Input:
int((a+b*arcsech(c*x))/x/(e*x^2+d)^2,x,method=_RETURNVERBOSE)
Output:
-1/2*a/d^2*ln(e*x^2+d)+1/2*a/d/(e*x^2+d)+a/d^2*ln(x)+b*(1/4*(e*(c^2*d+e))^ (1/2)/d/e/(c^2*d+e)*c^2*arcsech(c*x)*ln(1-d*c^2*(1/c/x+(-1+1/c/x)^(1/2)*(1 +1/c/x)^(1/2))^2/(-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e))+(-c^2*d*(e*(c^2*d+e)) ^(1/2)+2*c^2*d*e+2*e^2-2*(e*(c^2*d+e))^(1/2)*e)/d^4*e/(c^2*d+e)/c^4*ln(1-d *c^2*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1 /2)-2*e))*arcsech(c*x)-1/2/d^2*sum((_R1^2*c^2*d+2*c^2*d+4*e)/(_R1^2*c^2*d+ c^2*d+2*e)*(arcsech(c*x)*ln((_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/_ R1)+dilog((_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/_R1)),_R1=RootOf(c^ 2*d*_Z^4+(2*c^2*d+4*e)*_Z^2+c^2*d))+1/2*arcsech(c*x)^2/d^2-(c^2*d-2*(e*(c^ 2*d+e))^(1/2)+2*e)/d^4/c^4*e*ln(1-d*c^2*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^ (1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))*arcsech(c*x)+(-c^2*d*(e*(c^2* d+e))^(1/2)+2*c^2*d*e+2*e^2-2*(e*(c^2*d+e))^(1/2)*e)/(c^2*d+e)/d^3/c^2*ln( 1-d*c^2*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e)) ^(1/2)-2*e))*arcsech(c*x)+1/2*(-c^2*d*(e*(c^2*d+e))^(1/2)+2*c^2*d*e+2*e^2- 2*(e*(c^2*d+e))^(1/2)*e)/d^4*e/(c^2*d+e)/c^4*polylog(2,d*c^2*(1/c/x+(-1+1/ c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d-2*(e*(c^2*d+e))^(1/2)-2*e))-(-c^2*d* (e*(c^2*d+e))^(1/2)+2*c^2*d*e+2*e^2-2*(e*(c^2*d+e))^(1/2)*e)/d^4*e/(c^2*d+ e)/c^4*arcsech(c*x)^2+1/8*(e*(c^2*d+e))^(1/2)/d/e/(c^2*d+e)*c^2*polylog(2, d*c^2*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2/(-c^2*d+2*(e*(c^2*d+e))^( 1/2)-2*e))-1/4*(e*(c^2*d+e))^(1/2)/d/e/(c^2*d+e)*c^2*arcsech(c*x)^2+1/4...
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x} \,d x } \] Input:
integrate((a+b*arcsech(c*x))/x/(e*x^2+d)^2,x, algorithm="fricas")
Output:
integral((b*arcsech(c*x) + a)/(e^2*x^5 + 2*d*e*x^3 + d^2*x), x)
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx=\int \frac {a + b \operatorname {asech}{\left (c x \right )}}{x \left (d + e x^{2}\right )^{2}}\, dx \] Input:
integrate((a+b*asech(c*x))/x/(e*x**2+d)**2,x)
Output:
Integral((a + b*asech(c*x))/(x*(d + e*x**2)**2), x)
Exception generated. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+b*arcsech(c*x))/x/(e*x^2+d)^2,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x} \,d x } \] Input:
integrate((a+b*arcsech(c*x))/x/(e*x^2+d)^2,x, algorithm="giac")
Output:
integrate((b*arcsech(c*x) + a)/((e*x^2 + d)^2*x), x)
Timed out. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{x\,{\left (e\,x^2+d\right )}^2} \,d x \] Input:
int((a + b*acosh(1/(c*x)))/(x*(d + e*x^2)^2),x)
Output:
int((a + b*acosh(1/(c*x)))/(x*(d + e*x^2)^2), x)
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx=\frac {2 \left (\int \frac {\mathit {asech} \left (c x \right )}{e^{2} x^{5}+2 d e \,x^{3}+d^{2} x}d x \right ) b \,d^{3}+2 \left (\int \frac {\mathit {asech} \left (c x \right )}{e^{2} x^{5}+2 d e \,x^{3}+d^{2} x}d x \right ) b \,d^{2} e \,x^{2}-\mathrm {log}\left (e \,x^{2}+d \right ) a d -\mathrm {log}\left (e \,x^{2}+d \right ) a e \,x^{2}+2 \,\mathrm {log}\left (x \right ) a d +2 \,\mathrm {log}\left (x \right ) a e \,x^{2}-a e \,x^{2}}{2 d^{2} \left (e \,x^{2}+d \right )} \] Input:
int((a+b*asech(c*x))/x/(e*x^2+d)^2,x)
Output:
(2*int(asech(c*x)/(d**2*x + 2*d*e*x**3 + e**2*x**5),x)*b*d**3 + 2*int(asec h(c*x)/(d**2*x + 2*d*e*x**3 + e**2*x**5),x)*b*d**2*e*x**2 - log(d + e*x**2 )*a*d - log(d + e*x**2)*a*e*x**2 + 2*log(x)*a*d + 2*log(x)*a*e*x**2 - a*e* x**2)/(2*d**2*(d + e*x**2))