\(\int \frac {a+b \text {sech}^{-1}(c x)}{d+e x} \, dx\) [77]

Optimal result

Integrand size = 16, antiderivative size = 211 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{d+e x} \, dx=\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c d e^{\text {sech}^{-1}(c x)}}{e-\sqrt {-c^2 d^2+e^2}}\right )}{e}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c d e^{\text {sech}^{-1}(c x)}}{e+\sqrt {-c^2 d^2+e^2}}\right )}{e}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )}{e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c d e^{\text {sech}^{-1}(c x)}}{e-\sqrt {-c^2 d^2+e^2}}\right )}{e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c d e^{\text {sech}^{-1}(c x)}}{e+\sqrt {-c^2 d^2+e^2}}\right )}{e}-\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(c x)}\right )}{2 e} \] Output:

(a+b*arcsech(c*x))*ln(1+c*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/(e-(- 
c^2*d^2+e^2)^(1/2)))/e+(a+b*arcsech(c*x))*ln(1+c*d*(1/c/x+(-1+1/c/x)^(1/2) 
*(1+1/c/x)^(1/2))/(e+(-c^2*d^2+e^2)^(1/2)))/e-(a+b*arcsech(c*x))*ln(1+(1/c 
/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2)/e+b*polylog(2,-c*d*(1/c/x+(-1+1/c/ 
x)^(1/2)*(1+1/c/x)^(1/2))/(e-(-c^2*d^2+e^2)^(1/2)))/e+b*polylog(2,-c*d*(1/ 
c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/(e+(-c^2*d^2+e^2)^(1/2)))/e-1/2*b*po 
lylog(2,-(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2)/e
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.39 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.86 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{d+e x} \, dx=\frac {a \log (d+e x)}{e}+\frac {b \left (\operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(c x)}\right )-2 \left (-4 i \arcsin \left (\frac {\sqrt {1+\frac {e}{c d}}}{\sqrt {2}}\right ) \text {arctanh}\left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \text {sech}^{-1}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )+\text {sech}^{-1}(c x) \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )-\text {sech}^{-1}(c x) \log \left (1+\frac {\left (e-\sqrt {-c^2 d^2+e^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )+2 i \arcsin \left (\frac {\sqrt {1+\frac {e}{c d}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (e-\sqrt {-c^2 d^2+e^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )-\text {sech}^{-1}(c x) \log \left (1+\frac {\left (e+\sqrt {-c^2 d^2+e^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )-2 i \arcsin \left (\frac {\sqrt {1+\frac {e}{c d}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (e+\sqrt {-c^2 d^2+e^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )+\operatorname {PolyLog}\left (2,\frac {\left (-e+\sqrt {-c^2 d^2+e^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )+\operatorname {PolyLog}\left (2,-\frac {\left (e+\sqrt {-c^2 d^2+e^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )\right )\right )}{2 e} \] Input:

Integrate[(a + b*ArcSech[c*x])/(d + e*x),x]
 

Output:

(a*Log[d + e*x])/e + (b*(PolyLog[2, -E^(-2*ArcSech[c*x])] - 2*((-4*I)*ArcS 
in[Sqrt[1 + e/(c*d)]/Sqrt[2]]*ArcTanh[((-(c*d) + e)*Tanh[ArcSech[c*x]/2])/ 
Sqrt[-(c^2*d^2) + e^2]] + ArcSech[c*x]*Log[1 + E^(-2*ArcSech[c*x])] - ArcS 
ech[c*x]*Log[1 + (e - Sqrt[-(c^2*d^2) + e^2])/(c*d*E^ArcSech[c*x])] + (2*I 
)*ArcSin[Sqrt[1 + e/(c*d)]/Sqrt[2]]*Log[1 + (e - Sqrt[-(c^2*d^2) + e^2])/( 
c*d*E^ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (e + Sqrt[-(c^2*d^2) + e^2])/( 
c*d*E^ArcSech[c*x])] - (2*I)*ArcSin[Sqrt[1 + e/(c*d)]/Sqrt[2]]*Log[1 + (e 
+ Sqrt[-(c^2*d^2) + e^2])/(c*d*E^ArcSech[c*x])] + PolyLog[2, (-e + Sqrt[-( 
c^2*d^2) + e^2])/(c*d*E^ArcSech[c*x])] + PolyLog[2, -((e + Sqrt[-(c^2*d^2) 
 + e^2])/(c*d*E^ArcSech[c*x]))])))/(2*e)
 

Rubi [A] (verified)

Time = 1.37 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.09, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6841, 2998}

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.

\(\displaystyle \int \frac {a+b \text {sech}^{-1}(c x)}{d+e x} \, dx\)

\(\Big \downarrow \) 6841

\(\displaystyle \frac {b \int \frac {\sqrt {\frac {1-c x}{c x+1}} \log \left (\frac {e^{-\text {sech}^{-1}(c x)} \left (e-\sqrt {e^2-c^2 d^2}\right )}{c d}+1\right )}{x (1-c x)}dx}{e}+\frac {b \int \frac {\sqrt {\frac {1-c x}{c x+1}} \log \left (\frac {e^{-\text {sech}^{-1}(c x)} \left (e+\sqrt {e^2-c^2 d^2}\right )}{c d}+1\right )}{x (1-c x)}dx}{e}-\frac {b \int \frac {\sqrt {\frac {1-c x}{c x+1}} \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )}{x (1-c x)}dx}{e}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {\left (e-\sqrt {e^2-c^2 d^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}+1\right )}{e}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {\left (\sqrt {e^2-c^2 d^2}+e\right ) e^{-\text {sech}^{-1}(c x)}}{c d}+1\right )}{e}-\frac {\log \left (e^{-2 \text {sech}^{-1}(c x)}+1\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{e}\)

\(\Big \downarrow \) 2998

\(\displaystyle \frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {\left (e-\sqrt {e^2-c^2 d^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}+1\right )}{e}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {\left (\sqrt {e^2-c^2 d^2}+e\right ) e^{-\text {sech}^{-1}(c x)}}{c d}+1\right )}{e}-\frac {\log \left (e^{-2 \text {sech}^{-1}(c x)}+1\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{e}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\left (e-\sqrt {e^2-c^2 d^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )}{e}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\left (e+\sqrt {e^2-c^2 d^2}\right ) e^{-\text {sech}^{-1}(c x)}}{c d}\right )}{e}+\frac {b \operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(c x)}\right )}{2 e}\)

Input:

Int[(a + b*ArcSech[c*x])/(d + e*x),x]
 

Output:

-(((a + b*ArcSech[c*x])*Log[1 + E^(-2*ArcSech[c*x])])/e) + ((a + b*ArcSech 
[c*x])*Log[1 + (e - Sqrt[-(c^2*d^2) + e^2])/(c*d*E^ArcSech[c*x])])/e + ((a 
 + b*ArcSech[c*x])*Log[1 + (e + Sqrt[-(c^2*d^2) + e^2])/(c*d*E^ArcSech[c*x 
])])/e + (b*PolyLog[2, -E^(-2*ArcSech[c*x])])/(2*e) - (b*PolyLog[2, -((e - 
 Sqrt[-(c^2*d^2) + e^2])/(c*d*E^ArcSech[c*x]))])/e - (b*PolyLog[2, -((e + 
Sqrt[-(c^2*d^2) + e^2])/(c*d*E^ArcSech[c*x]))])/e
 

Defintions of rubi rules used

rule 2998

Int[Log[v_]*(u_), x_Symbol] :> With[{w = DerivativeDivides[v, u*(1 - v), x] 
}, Simp[w*PolyLog[2, 1 - v], x] /;  !FalseQ[w]]
 

rule 6841

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))/((d_.) + (e_.)*(x_)), x_Symbol] :> 
Simp[(a + b*ArcSech[c*x])*(Log[1 + (e - Sqrt[(-c^2)*d^2 + e^2])/(c*d*E^ArcS 
ech[c*x])]/e), x] + (Simp[(a + b*ArcSech[c*x])*(Log[1 + (e + Sqrt[(-c^2)*d^ 
2 + e^2])/(c*d*E^ArcSech[c*x])]/e), x] - Simp[(a + b*ArcSech[c*x])*(Log[1 + 
 1/E^(2*ArcSech[c*x])]/e), x] + Simp[b/e   Int[(Sqrt[(1 - c*x)/(1 + c*x)]*L 
og[1 + (e - Sqrt[(-c^2)*d^2 + e^2])/(c*d*E^ArcSech[c*x])])/(x*(1 - c*x)), x 
], x] + Simp[b/e   Int[(Sqrt[(1 - c*x)/(1 + c*x)]*Log[1 + (e + Sqrt[(-c^2)* 
d^2 + e^2])/(c*d*E^ArcSech[c*x])])/(x*(1 - c*x)), x], x] - Simp[b/e   Int[( 
Sqrt[(1 - c*x)/(1 + c*x)]*Log[1 + 1/E^(2*ArcSech[c*x])])/(x*(1 - c*x)), x], 
 x]) /; FreeQ[{a, b, c, d, e}, x]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.81 (sec) , antiderivative size = 511, normalized size of antiderivative = 2.42

method	result	size
parts	\(\frac {a \ln \left (e x +d \right )}{e}+\frac {b \,\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {-c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )+\sqrt {-c^{2} d^{2}+e^{2}}-e}{-e +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {b \,\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )+\sqrt {-c^{2} d^{2}+e^{2}}+e}{e +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {b \operatorname {dilog}\left (\frac {-c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )+\sqrt {-c^{2} d^{2}+e^{2}}-e}{-e +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {b \operatorname {dilog}\left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )+\sqrt {-c^{2} d^{2}+e^{2}}+e}{e +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e}-\frac {b \,\operatorname {arcsech}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {b \,\operatorname {arcsech}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {b \operatorname {dilog}\left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {b \operatorname {dilog}\left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}\)	\(511\)
derivativedivides	\(\frac {\frac {a c \ln \left (c e x +c d \right )}{e}+b c \left (\frac {\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {-c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )+\sqrt {-c^{2} d^{2}+e^{2}}-e}{-e +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )+\sqrt {-c^{2} d^{2}+e^{2}}+e}{e +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {\operatorname {dilog}\left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )+\sqrt {-c^{2} d^{2}+e^{2}}+e}{e +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {\operatorname {dilog}\left (\frac {-c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )+\sqrt {-c^{2} d^{2}+e^{2}}-e}{-e +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e}-\frac {\operatorname {arcsech}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {\operatorname {arcsech}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {\operatorname {dilog}\left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {\operatorname {dilog}\left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}\right )}{c}\)	\(515\)
default	\(\frac {\frac {a c \ln \left (c e x +c d \right )}{e}+b c \left (\frac {\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {-c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )+\sqrt {-c^{2} d^{2}+e^{2}}-e}{-e +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )+\sqrt {-c^{2} d^{2}+e^{2}}+e}{e +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {\operatorname {dilog}\left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )+\sqrt {-c^{2} d^{2}+e^{2}}+e}{e +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {\operatorname {dilog}\left (\frac {-c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )+\sqrt {-c^{2} d^{2}+e^{2}}-e}{-e +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e}-\frac {\operatorname {arcsech}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {\operatorname {arcsech}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {\operatorname {dilog}\left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {\operatorname {dilog}\left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}\right )}{c}\)	\(515\)

Input:

int((a+b*arcsech(c*x))/(e*x+d),x,method=_RETURNVERBOSE)
 

Output:

a*ln(e*x+d)/e+b/e*arcsech(c*x)*ln((-c*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^ 
(1/2))+(-c^2*d^2+e^2)^(1/2)-e)/(-e+(-c^2*d^2+e^2)^(1/2)))+b/e*arcsech(c*x) 
*ln((c*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))+(-c^2*d^2+e^2)^(1/2)+e)/ 
(e+(-c^2*d^2+e^2)^(1/2)))+b/e*dilog((-c*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x 
)^(1/2))+(-c^2*d^2+e^2)^(1/2)-e)/(-e+(-c^2*d^2+e^2)^(1/2)))+b/e*dilog((c*d 
*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))+(-c^2*d^2+e^2)^(1/2)+e)/(e+(-c^2 
*d^2+e^2)^(1/2)))-b/e*arcsech(c*x)*ln(1+I*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x 
)^(1/2)))-b/e*arcsech(c*x)*ln(1-I*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)) 
)-b/e*dilog(1+I*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)))-b/e*dilog(1-I*(1 
/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)))
 

Fricas [F]

\[ \int \frac {a+b \text {sech}^{-1}(c x)}{d+e x} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{e x + d} \,d x } \] Input:

integrate((a+b*arcsech(c*x))/(e*x+d),x, algorithm="fricas")
 

Output:

integral((b*arcsech(c*x) + a)/(e*x + d), x)
 

Sympy [F]

\[ \int \frac {a+b \text {sech}^{-1}(c x)}{d+e x} \, dx=\int \frac {a + b \operatorname {asech}{\left (c x \right )}}{d + e x}\, dx \] Input:

integrate((a+b*asech(c*x))/(e*x+d),x)
 

Output:

Integral((a + b*asech(c*x))/(d + e*x), x)
 

Maxima [F]

\[ \int \frac {a+b \text {sech}^{-1}(c x)}{d+e x} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{e x + d} \,d x } \] Input:

integrate((a+b*arcsech(c*x))/(e*x+d),x, algorithm="maxima")
 

Output:

b*integrate(log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x))/(e*x + d), 
x) + a*log(e*x + d)/e
 

Giac [F]

\[ \int \frac {a+b \text {sech}^{-1}(c x)}{d+e x} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{e x + d} \,d x } \] Input:

integrate((a+b*arcsech(c*x))/(e*x+d),x, algorithm="giac")
 

Output:

integrate((b*arcsech(c*x) + a)/(e*x + d), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

int((a + b*acosh(1/(c*x)))/(d + e*x),x)
 

Output:

int((a + b*acosh(1/(c*x)))/(d + e*x), x)
 

Reduce [F]

\[ \int \frac {a+b \text {sech}^{-1}(c x)}{d+e x} \, dx=\frac {\left (\int \frac {\mathit {asech} \left (c x \right )}{e x +d}d x \right ) b e +\mathrm {log}\left (e x +d \right ) a}{e} \] Input:

int((a+b*asech(c*x))/(e*x+d),x)
 

Output:

(int(asech(c*x)/(d + e*x),x)*b*e + log(d + e*x)*a)/e