Integrand size = 16, antiderivative size = 306 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^3} \, dx=\frac {b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{2 d \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a+b \text {sech}^{-1}(c x)}{2 e (d+e x)^2}+\frac {b c^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 d^2 \sqrt {c^2 d^2-e^2}}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{2 d^2 e} \]
1/2*(-a-b*arcsech(c*x))/e/(e*x+d)^2+1/2*b*c^2*arctan((c^2*d*x+e)/(c^2*d^2- e^2)^(1/2)/(-c^2*x^2+1)^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)/(c^2*d^2-e^ 2)^(3/2)+1/2*b*arctanh((-c^2*x^2+1)^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2) /d^2/e+1/2*b*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*x^2+1)^(1/2))*(1 /(c*x+1))^(1/2)*(c*x+1)^(1/2)/d^2/(c^2*d^2-e^2)^(1/2)+1/2*b*e*(1/(c*x+1))^ (1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/d/(c^2*d^2-e^2)/(e*x+d)
Result contains complex when optimal does not.
Time = 0.63 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.12 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^3} \, dx=\frac {1}{2} \left (-\frac {a}{e (d+e x)^2}+\frac {b \sqrt {\frac {1-c x}{1+c x}} (e+c e x)}{d (c d-e) (c d+e) (d+e x)}-\frac {b \text {sech}^{-1}(c x)}{e (d+e x)^2}-\frac {b \log (x)}{d^2 e}+\frac {b \log \left (1+\sqrt {\frac {1-c x}{1+c x}}+c x \sqrt {\frac {1-c x}{1+c x}}\right )}{d^2 e}-\frac {i b \left (2 c^2 d^2-e^2\right ) \log \left (\frac {4 d^2 e \sqrt {c^2 d^2-e^2} \left (i e+i c^2 d x+\sqrt {c^2 d^2-e^2} \sqrt {\frac {1-c x}{1+c x}}+c \sqrt {c^2 d^2-e^2} x \sqrt {\frac {1-c x}{1+c x}}\right )}{b \left (2 c^2 d^2-e^2\right ) (d+e x)}\right )}{d^2 (c d-e) (c d+e) \sqrt {c^2 d^2-e^2}}\right ) \]
(-(a/(e*(d + e*x)^2)) + (b*Sqrt[(1 - c*x)/(1 + c*x)]*(e + c*e*x))/(d*(c*d - e)*(c*d + e)*(d + e*x)) - (b*ArcSech[c*x])/(e*(d + e*x)^2) - (b*Log[x])/ (d^2*e) + (b*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c *x)]])/(d^2*e) - (I*b*(2*c^2*d^2 - e^2)*Log[(4*d^2*e*Sqrt[c^2*d^2 - e^2]*( I*e + I*c^2*d*x + Sqrt[c^2*d^2 - e^2]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c ^2*d^2 - e^2]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/(b*(2*c^2*d^2 - e^2)*(d + e*x) )])/(d^2*(c*d - e)*(c*d + e)*Sqrt[c^2*d^2 - e^2]))/2
Time = 0.48 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.80, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6842, 617, 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.
| \(\displaystyle \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 6842 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {1}{x (d+e x)^2 \sqrt {1-c^2 x^2}}dx}{2 e}-\frac {a+b \text {sech}^{-1}(c x)}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 617 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \left (-\frac {e}{d^2 (d+e x) \sqrt {1-c^2 x^2}}-\frac {e}{d (d+e x)^2 \sqrt {1-c^2 x^2}}+\frac {1}{d^2 x \sqrt {1-c^2 x^2}}\right )dx}{2 e}-\frac {a+b \text {sech}^{-1}(c x)}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a+b \text {sech}^{-1}(c x)}{2 e (d+e x)^2}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {c^2 e \arctan \left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right )}{\left (c^2 d^2-e^2\right )^{3/2}}-\frac {e \arctan \left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right )}{d^2 \sqrt {c^2 d^2-e^2}}-\frac {\text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d^2}-\frac {e^2 \sqrt {1-c^2 x^2}}{d \left (c^2 d^2-e^2\right ) (d+e x)}\right )}{2 e}\) |
-1/2*(a + b*ArcSech[c*x])/(e*(d + e*x)^2) - (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*(-((e^2*Sqrt[1 - c^2*x^2])/(d*(c^2*d^2 - e^2)*(d + e*x))) - (c^2*e *ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(c^2*d^2 - e^2)^(3/2) - (e*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^ 2])])/(d^2*Sqrt[c^2*d^2 - e^2]) - ArcTanh[Sqrt[1 - c^2*x^2]]/d^2))/(2*e)
3.1.80.3.1 Defintions of rubi rules used
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Int[ExpandIntegrand[(a + b*x^2)^p, x^m*(c + d*x)^n, x], x] /; FreeQ[{ a, b, c, d, p}, x] && ILtQ[n, 0] && IntegerQ[m] && IntegerQ[2*p]
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbo l] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcSech[c*x])/(e*(m + 1))), x] + Simp[ b*(Sqrt[1 + c*x]/(e*(m + 1)))*Sqrt[1/(1 + c*x)] Int[(d + e*x)^(m + 1)/(x* Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(592\) vs. \(2(267)=534\).
Time = 1.88 (sec) , antiderivative size = 593, normalized size of antiderivative = 1.94
| method | result | size |
| parts | \(-\frac {a}{2 \left (e x +d \right )^{2} e}+\frac {b \left (-\frac {c^{3} \operatorname {arcsech}\left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {c^{2} \sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{3} d^{3} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{3} d^{2} e \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, x -2 \ln \left (\frac {2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-c^{2} x^{2}+1}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right ) c^{3} d^{3}-2 \ln \left (\frac {2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-c^{2} x^{2}+1}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right ) c^{3} d^{2} e x -\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c d \,e^{2} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) e^{3} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c x +c d \,e^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}+\ln \left (\frac {2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-c^{2} x^{2}+1}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right ) c d \,e^{2}+\ln \left (\frac {2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-c^{2} x^{2}+1}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right ) e^{3} c x \right )}{2 e \sqrt {-c^{2} x^{2}+1}\, d^{2} \left (c d +e \right ) \left (c d -e \right ) \left (c e x +c d \right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\) | \(593\) |
| derivativedivides | \(\frac {-\frac {a \,c^{3}}{2 \left (c e x +c d \right )^{2} e}+b \,c^{3} \left (-\frac {\operatorname {arcsech}\left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {\sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{3} d^{3} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{3} d^{2} e \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, x -2 \ln \left (\frac {2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-c^{2} x^{2}+1}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right ) c^{3} d^{3}-2 \ln \left (\frac {2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-c^{2} x^{2}+1}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right ) c^{3} d^{2} e x -\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c d \,e^{2} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) e^{3} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c x +c d \,e^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}+\ln \left (\frac {2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-c^{2} x^{2}+1}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right ) c d \,e^{2}+\ln \left (\frac {2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-c^{2} x^{2}+1}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right ) e^{3} c x \right )}{2 e c \sqrt {-c^{2} x^{2}+1}\, d^{2} \left (c d -e \right ) \left (c d +e \right ) \left (c e x +c d \right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\) | \(600\) |
| default | \(\frac {-\frac {a \,c^{3}}{2 \left (c e x +c d \right )^{2} e}+b \,c^{3} \left (-\frac {\operatorname {arcsech}\left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {\sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{3} d^{3} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{3} d^{2} e \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, x -2 \ln \left (\frac {2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-c^{2} x^{2}+1}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right ) c^{3} d^{3}-2 \ln \left (\frac {2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-c^{2} x^{2}+1}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right ) c^{3} d^{2} e x -\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c d \,e^{2} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) e^{3} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, c x +c d \,e^{2} \sqrt {-c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}+\ln \left (\frac {2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-c^{2} x^{2}+1}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right ) c d \,e^{2}+\ln \left (\frac {2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-c^{2} x^{2}+1}\, e +2 d \,c^{2} x +2 e}{c e x +c d}\right ) e^{3} c x \right )}{2 e c \sqrt {-c^{2} x^{2}+1}\, d^{2} \left (c d -e \right ) \left (c d +e \right ) \left (c e x +c d \right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\) | \(600\) |
-1/2*a/(e*x+d)^2/e+b/c*(-1/2*c^3/(c*e*x+c*d)^2/e*arcsech(c*x)+1/2*c^2/e*(- (c*x-1)/c/x)^(1/2)*x*((c*x+1)/c/x)^(1/2)*(arctanh(1/(-c^2*x^2+1)^(1/2))*c^ 3*d^3*(-(c^2*d^2-e^2)/e^2)^(1/2)+arctanh(1/(-c^2*x^2+1)^(1/2))*c^3*d^2*e*( -(c^2*d^2-e^2)/e^2)^(1/2)*x-2*ln(2*((-(c^2*d^2-e^2)/e^2)^(1/2)*(-c^2*x^2+1 )^(1/2)*e+d*c^2*x+e)/(c*e*x+c*d))*c^3*d^3-2*ln(2*((-(c^2*d^2-e^2)/e^2)^(1/ 2)*(-c^2*x^2+1)^(1/2)*e+d*c^2*x+e)/(c*e*x+c*d))*c^3*d^2*e*x-arctanh(1/(-c^ 2*x^2+1)^(1/2))*c*d*e^2*(-(c^2*d^2-e^2)/e^2)^(1/2)-arctanh(1/(-c^2*x^2+1)^ (1/2))*e^3*(-(c^2*d^2-e^2)/e^2)^(1/2)*c*x+c*d*e^2*(-c^2*x^2+1)^(1/2)*(-(c^ 2*d^2-e^2)/e^2)^(1/2)+ln(2*((-(c^2*d^2-e^2)/e^2)^(1/2)*(-c^2*x^2+1)^(1/2)* e+d*c^2*x+e)/(c*e*x+c*d))*c*d*e^2+ln(2*((-(c^2*d^2-e^2)/e^2)^(1/2)*(-c^2*x ^2+1)^(1/2)*e+d*c^2*x+e)/(c*e*x+c*d))*e^3*c*x)/(-c^2*x^2+1)^(1/2)/d^2/(c*d +e)/(c*d-e)/(c*e*x+c*d)/(-(c^2*d^2-e^2)/e^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 596 vs. \(2 (200) = 400\).
Time = 0.37 (sec) , antiderivative size = 1212, normalized size of antiderivative = 3.96 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^3} \, dx=\text {Too large to display} \]
[-1/2*(a*c^4*d^6 - (2*a + b)*c^2*d^4*e^2 + (a + b)*d^2*e^4 - (b*c^2*d^2*e^ 4 - b*e^6)*x^2 + (2*b*c^2*d^4*e - b*d^2*e^3 + (2*b*c^2*d^2*e^3 - b*e^5)*x^ 2 + 2*(2*b*c^2*d^3*e^2 - b*d*e^4)*x)*sqrt(-c^2*d^2 + e^2)*log((c^2*d*e*x - (c^3*d^2 - c*e^2)*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + e^2 - sqrt(-c^2*d^2 + e^2)*(c^2*d*x + c*e*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + e))/(e*x + d)) - 2*(b*c^2*d^3*e^3 - b*d*e^5)*x + (b*c^4*d^6 - 2*b*c^2*d^4*e^2 + b*d^2*e^4 + (b*c^4*d^4*e^2 - 2*b*c^2*d^2*e^4 + b*e^6)*x^2 + 2*(b*c^4*d^5*e - 2*b*c^2* d^3*e^3 + b*d*e^5)*x)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/x) + (b *c^4*d^6 - 2*b*c^2*d^4*e^2 + b*d^2*e^4)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2* x^2)) + 1)/(c*x)) - ((b*c^3*d^3*e^3 - b*c*d*e^5)*x^2 + (b*c^3*d^4*e^2 - b* c*d^2*e^4)*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/(c^4*d^8*e - 2*c^2*d^6*e^3 + d^4*e^5 + (c^4*d^6*e^3 - 2*c^2*d^4*e^5 + d^2*e^7)*x^2 + 2*(c^4*d^7*e^2 - 2*c^2*d^5*e^4 + d^3*e^6)*x), -1/2*(a*c^4*d^6 - (2*a + b)*c^2*d^4*e^2 + (a + b)*d^2*e^4 - (b*c^2*d^2*e^4 - b*e^6)*x^2 - 2*(2*b*c^2*d^4*e - b*d^2*e^3 + (2*b*c^2*d^2*e^3 - b*e^5)*x^2 + 2*(2*b*c^2*d^3*e^2 - b*d*e^4)*x)*sqrt(c^ 2*d^2 - e^2)*arctan(-(sqrt(c^2*d^2 - e^2)*c*d*x*sqrt(-(c^2*x^2 - 1)/(c^2*x ^2)) - sqrt(c^2*d^2 - e^2)*(e*x + d))/((c^2*d^2 - e^2)*x)) - 2*(b*c^2*d^3* e^3 - b*d*e^5)*x + (b*c^4*d^6 - 2*b*c^2*d^4*e^2 + b*d^2*e^4 + (b*c^4*d^4*e ^2 - 2*b*c^2*d^2*e^4 + b*e^6)*x^2 + 2*(b*c^4*d^5*e - 2*b*c^2*d^3*e^3 + b*d *e^5)*x)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/x) + (b*c^4*d^6 -...
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^3} \, dx=\int \frac {a + b \operatorname {asech}{\left (c x \right )}}{\left (d + e x\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^3} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^3} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^3} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^3} \,d x \]