3.1.19 \(\int \frac {c+d x}{\sqrt {b \tanh (e+f x)}} \, dx\) [19]

3.1.19.1 Optimal result

Integrand size = 18, antiderivative size = 1280 \[ \int \frac {c+d x}{\sqrt {b \tanh (e+f x)}} \, dx =\text {Too large to display} \]

output

-(d*x+c)*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))/f/(-b)^(1/2)-1/2*d*arct 
anh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))^2/f^2/(-b)^(1/2)+d*arctanh((b*tanh(f 
*x+e))^(1/2)/(-b)^(1/2))*ln(2/(1-(b*tanh(f*x+e))^(1/2)/(-b)^(1/2)))/f^2/(- 
b)^(1/2)-1/2*d*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))*ln(2*(b^(1/2)-(b* 
tanh(f*x+e))^(1/2))/((-b)^(1/2)+b^(1/2))/(1-(b*tanh(f*x+e))^(1/2)/(-b)^(1/ 
2)))/f^2/(-b)^(1/2)-1/2*d*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))*ln(-2* 
(b^(1/2)+(b*tanh(f*x+e))^(1/2))/((-b)^(1/2)-b^(1/2))/(1-(b*tanh(f*x+e))^(1 
/2)/(-b)^(1/2)))/f^2/(-b)^(1/2)-d*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2) 
)*ln(2/(1+(b*tanh(f*x+e))^(1/2)/(-b)^(1/2)))/f^2/(-b)^(1/2)+1/2*d*polylog( 
2,1-2/(1-(b*tanh(f*x+e))^(1/2)/(-b)^(1/2)))/f^2/(-b)^(1/2)-1/4*d*polylog(2 
,1-2*(b^(1/2)-(b*tanh(f*x+e))^(1/2))/((-b)^(1/2)+b^(1/2))/(1-(b*tanh(f*x+e 
))^(1/2)/(-b)^(1/2)))/f^2/(-b)^(1/2)-1/4*d*polylog(2,1+2*(b^(1/2)+(b*tanh( 
f*x+e))^(1/2))/((-b)^(1/2)-b^(1/2))/(1-(b*tanh(f*x+e))^(1/2)/(-b)^(1/2)))/ 
f^2/(-b)^(1/2)+1/2*d*polylog(2,1-2/(1+(b*tanh(f*x+e))^(1/2)/(-b)^(1/2)))/f 
^2/(-b)^(1/2)+(d*x+c)*arctanh((b*tanh(f*x+e))^(1/2)/b^(1/2))/f/b^(1/2)+1/2 
*d*arctanh((b*tanh(f*x+e))^(1/2)/b^(1/2))^2/f^2/b^(1/2)-d*arctanh((b*tanh( 
f*x+e))^(1/2)/b^(1/2))*ln(2*b^(1/2)/(b^(1/2)-(b*tanh(f*x+e))^(1/2)))/f^2/b 
^(1/2)+d*arctanh((b*tanh(f*x+e))^(1/2)/b^(1/2))*ln(2*b^(1/2)/(b^(1/2)+(b*t 
anh(f*x+e))^(1/2)))/f^2/b^(1/2)-1/2*d*arctanh((b*tanh(f*x+e))^(1/2)/b^(1/2 
))*ln(2*b^(1/2)*((-b)^(1/2)-(b*tanh(f*x+e))^(1/2))/((-b)^(1/2)-b^(1/2))...
 

3.1.19.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.38 (sec) , antiderivative size = 556, normalized size of antiderivative = 0.43 \[ \int \frac {c+d x}{\sqrt {b \tanh (e+f x)}} \, dx=\frac {\left (4 f (c+d x) \left (2 \arctan \left (\sqrt {\tanh (e+f x)}\right )-\log \left (1-\sqrt {\tanh (e+f x)}\right )+\log \left (1+\sqrt {\tanh (e+f x)}\right )\right )+d \left (-4 i \arctan \left (\sqrt {\tanh (e+f x)}\right )^2+4 \arctan \left (\sqrt {\tanh (e+f x)}\right ) \log \left (1+e^{4 i \arctan \left (\sqrt {\tanh (e+f x)}\right )}\right )-\log ^2\left (1-\sqrt {\tanh (e+f x)}\right )+2 \log \left (1-\sqrt {\tanh (e+f x)}\right ) \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i+\sqrt {\tanh (e+f x)}\right )\right )+2 \log \left (1-\sqrt {\tanh (e+f x)}\right ) \log \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \left (i+\sqrt {\tanh (e+f x)}\right )\right )-2 \log \left (1-\sqrt {\tanh (e+f x)}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {\tanh (e+f x)}\right )\right )-2 \log \left (1-\left (\frac {1}{2}-\frac {i}{2}\right ) \left (1+\sqrt {\tanh (e+f x)}\right )\right ) \log \left (1+\sqrt {\tanh (e+f x)}\right )+2 \log \left (\frac {1}{2} \left (1-\sqrt {\tanh (e+f x)}\right )\right ) \log \left (1+\sqrt {\tanh (e+f x)}\right )-2 \log \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (i+\sqrt {\tanh (e+f x)}\right )\right ) \log \left (1+\sqrt {\tanh (e+f x)}\right )+\log ^2\left (1+\sqrt {\tanh (e+f x)}\right )-i \operatorname {PolyLog}\left (2,-e^{4 i \arctan \left (\sqrt {\tanh (e+f x)}\right )}\right )-2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1-\sqrt {\tanh (e+f x)}\right )\right )+2 \operatorname {PolyLog}\left (2,\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (-1+\sqrt {\tanh (e+f x)}\right )\right )+2 \operatorname {PolyLog}\left (2,\left (-\frac {1}{2}+\frac {i}{2}\right ) \left (-1+\sqrt {\tanh (e+f x)}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\sqrt {\tanh (e+f x)}\right )\right )-2 \operatorname {PolyLog}\left (2,\left (\frac {1}{2}-\frac {i}{2}\right ) \left (1+\sqrt {\tanh (e+f x)}\right )\right )-2 \operatorname {PolyLog}\left (2,\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+\sqrt {\tanh (e+f x)}\right )\right )\right )\right ) \sqrt {\tanh (e+f x)}}{8 f^2 \sqrt {b \tanh (e+f x)}} \]

input

Integrate[(c + d*x)/Sqrt[b*Tanh[e + f*x]],x]
 

output

((4*f*(c + d*x)*(2*ArcTan[Sqrt[Tanh[e + f*x]]] - Log[1 - Sqrt[Tanh[e + f*x 
]]] + Log[1 + Sqrt[Tanh[e + f*x]]]) + d*((-4*I)*ArcTan[Sqrt[Tanh[e + f*x]] 
]^2 + 4*ArcTan[Sqrt[Tanh[e + f*x]]]*Log[1 + E^((4*I)*ArcTan[Sqrt[Tanh[e + 
f*x]]])] - Log[1 - Sqrt[Tanh[e + f*x]]]^2 + 2*Log[1 - Sqrt[Tanh[e + f*x]]] 
*Log[(1/2 + I/2)*(-I + Sqrt[Tanh[e + f*x]])] + 2*Log[1 - Sqrt[Tanh[e + f*x 
]]]*Log[(1/2 - I/2)*(I + Sqrt[Tanh[e + f*x]])] - 2*Log[1 - Sqrt[Tanh[e + f 
*x]]]*Log[(1 + Sqrt[Tanh[e + f*x]])/2] - 2*Log[1 - (1/2 - I/2)*(1 + Sqrt[T 
anh[e + f*x]])]*Log[1 + Sqrt[Tanh[e + f*x]]] + 2*Log[(1 - Sqrt[Tanh[e + f* 
x]])/2]*Log[1 + Sqrt[Tanh[e + f*x]]] - 2*Log[(-1/2 - I/2)*(I + Sqrt[Tanh[e 
 + f*x]])]*Log[1 + Sqrt[Tanh[e + f*x]]] + Log[1 + Sqrt[Tanh[e + f*x]]]^2 - 
 I*PolyLog[2, -E^((4*I)*ArcTan[Sqrt[Tanh[e + f*x]]])] - 2*PolyLog[2, (1 - 
Sqrt[Tanh[e + f*x]])/2] + 2*PolyLog[2, (-1/2 - I/2)*(-1 + Sqrt[Tanh[e + f* 
x]])] + 2*PolyLog[2, (-1/2 + I/2)*(-1 + Sqrt[Tanh[e + f*x]])] + 2*PolyLog[ 
2, (1 + Sqrt[Tanh[e + f*x]])/2] - 2*PolyLog[2, (1/2 - I/2)*(1 + Sqrt[Tanh[ 
e + f*x]])] - 2*PolyLog[2, (1/2 + I/2)*(1 + Sqrt[Tanh[e + f*x]])]))*Sqrt[T 
anh[e + f*x]])/(8*f^2*Sqrt[b*Tanh[e + f*x]])
 

3.1.19.3 Rubi [A] (verified)

Time = 1.68 (sec) , antiderivative size = 1187, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3042, 4221, 4853, 7267, 27, 7276, 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.

\(\displaystyle \int \frac {c+d x}{\sqrt {b \tanh (e+f x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {c+d x}{\sqrt {-i b \tan (i e+i f x)}}dx\)

\(\Big \downarrow \) 4221

\(\displaystyle \frac {d \int \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )dx}{\sqrt {-b} f}-\frac {d \int \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )dx}{\sqrt {b} f}-\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{\sqrt {-b} f}+\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{\sqrt {b} f}\)

\(\Big \downarrow \) 4853

\(\displaystyle \frac {d \int \frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{1-\tanh ^2(e+f x)}d\tanh (e+f x)}{\sqrt {-b} f^2}-\frac {d \int \frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{1-\tanh ^2(e+f x)}d\tanh (e+f x)}{\sqrt {b} f^2}-\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{\sqrt {-b} f}+\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{\sqrt {b} f}\)

\(\Big \downarrow \) 7267

\(\displaystyle \frac {2 d \int \frac {b^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}}{\sqrt {-b} b f^2}-\frac {2 d \int \frac {b^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}}{b^{3/2} f^2}-\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{\sqrt {-b} f}+\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{\sqrt {b} f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 b d \int \frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}}{\sqrt {-b} f^2}-\frac {2 \sqrt {b} d \int \frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}}{f^2}-\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{\sqrt {-b} f}+\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{\sqrt {b} f}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {2 b d \int \left (\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \sqrt {b \tanh (e+f x)}}{2 b (\tanh (e+f x) b+b)}-\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \sqrt {b \tanh (e+f x)}}{2 b (b \tanh (e+f x)-b)}\right )d\sqrt {b \tanh (e+f x)}}{\sqrt {-b} f^2}-\frac {2 \sqrt {b} d \int \left (\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tanh (e+f x)}}{2 b (\tanh (e+f x) b+b)}-\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tanh (e+f x)}}{2 b (b \tanh (e+f x)-b)}\right )d\sqrt {b \tanh (e+f x)}}{f^2}-\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{\sqrt {-b} f}+\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{\sqrt {b} f}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{\sqrt {-b} f}+\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{\sqrt {b} f}-\frac {2 \sqrt {b} d \left (-\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )^2}{4 b}+\frac {\log \left (\frac {2 \sqrt {b}}{\sqrt {b}-\sqrt {b \tanh (e+f x)}}\right ) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{2 b}-\frac {\log \left (\frac {2 \sqrt {b}}{\sqrt {b}+\sqrt {b \tanh (e+f x)}}\right ) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{2 b}+\frac {\log \left (\frac {2 \sqrt {b} \left (\sqrt {-b}-\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}-\sqrt {b}\right ) \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}\right ) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{4 b}+\frac {\log \left (\frac {2 \sqrt {b} \left (\sqrt {-b}+\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}+\sqrt {b}\right ) \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}\right ) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{4 b}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {b}}{\sqrt {b}-\sqrt {b \tanh (e+f x)}}\right )}{4 b}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {b}}{\sqrt {b}+\sqrt {b \tanh (e+f x)}}\right )}{4 b}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {b} \left (\sqrt {-b}-\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}-\sqrt {b}\right ) \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}\right )}{8 b}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {b} \left (\sqrt {-b}+\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}+\sqrt {b}\right ) \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}\right )}{8 b}\right )}{f^2}+\frac {2 b d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )^2}{4 b}-\frac {\log \left (\frac {2}{1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right ) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{2 b}+\frac {\log \left (\frac {2}{\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}+1}\right ) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{2 b}-\frac {\log \left (-\frac {2 \left (\sqrt {b}-\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}-\sqrt {b}\right ) \left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}+1\right )}\right ) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{4 b}-\frac {\log \left (\frac {2 \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}+\sqrt {b}\right ) \left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}+1\right )}\right ) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{4 b}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{4 b}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}+1}\right )}{4 b}+\frac {\operatorname {PolyLog}\left (2,\frac {2 \left (\sqrt {b}-\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}-\sqrt {b}\right ) \left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}+1\right )}+1\right )}{8 b}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2 \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}+\sqrt {b}\right ) \left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}+1\right )}\right )}{8 b}\right )}{\sqrt {-b} f^2}\)

input

Int[(c + d*x)/Sqrt[b*Tanh[e + f*x]],x]
 

output

-(((c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]])/(Sqrt[-b]*f)) + ((c 
+ d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]])/(Sqrt[b]*f) - (2*Sqrt[b]*d* 
(-1/4*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]^2/b + (ArcTanh[Sqrt[b*Tanh[e 
+ f*x]]/Sqrt[b]]*Log[(2*Sqrt[b])/(Sqrt[b] - Sqrt[b*Tanh[e + f*x]])])/(2*b) 
 - (ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b])/(Sqrt[b] + Sqrt 
[b*Tanh[e + f*x]])])/(2*b) + (ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[( 
2*Sqrt[b]*(Sqrt[-b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(Sqrt[ 
b] + Sqrt[b*Tanh[e + f*x]]))])/(4*b) + (ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt 
[b]]*Log[(2*Sqrt[b]*(Sqrt[-b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[ 
b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(4*b) + PolyLog[2, 1 - (2*Sqrt[b] 
)/(Sqrt[b] - Sqrt[b*Tanh[e + f*x]])]/(4*b) + PolyLog[2, 1 - (2*Sqrt[b])/(S 
qrt[b] + Sqrt[b*Tanh[e + f*x]])]/(4*b) - PolyLog[2, 1 - (2*Sqrt[b]*(Sqrt[- 
b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[ 
e + f*x]]))]/(8*b) - PolyLog[2, 1 - (2*Sqrt[b]*(Sqrt[-b] + Sqrt[b*Tanh[e + 
 f*x]]))/((Sqrt[-b] + Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))]/(8*b))) 
/f^2 + (2*b*d*(ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]^2/(4*b) - (ArcTanh[ 
Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[2/(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]) 
])/(2*b) + (ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[2/(1 + Sqrt[b*Tanh 
[e + f*x]]/Sqrt[-b])])/(2*b) - (ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Lo 
g[(-2*(Sqrt[b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(1 + Sqr...
 

3.1.19.3.1 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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4221

Int[((c_.) + (d_.)*(x_))/Sqrt[(a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Sym 
bol] :> Simp[(-I)*((c + d*x)/(f*Rt[a - I*b, 2]))*ArcTanh[Sqrt[a + b*Tan[e + 
 f*x]]/Rt[a - I*b, 2]], x] + (Simp[I*((c + d*x)/(f*Rt[a + I*b, 2]))*ArcTanh 
[Sqrt[a + b*Tan[e + f*x]]/Rt[a + I*b, 2]], x] + Simp[I*(d/(f*Rt[a - I*b, 2] 
))   Int[ArcTanh[Sqrt[a + b*Tan[e + f*x]]/Rt[a - I*b, 2]], x], x] - Simp[I* 
(d/(f*Rt[a + I*b, 2]))   Int[ArcTanh[Sqrt[a + b*Tan[e + f*x]]/Rt[a + I*b, 2 
]], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 + b^2, 0]
 

rule 4853

Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, Simp[With[{d = FreeFa 
ctors[Tan[v], x]}, d/Coefficient[v, x, 1]   Subst[Int[SubstFor[1/(1 + d^2*x 
^2), Tan[v]/d, u, x], x], x, Tan[v]/d]], x] /;  !FalseQ[v] && FunctionOfQ[N 
onfreeFactors[Tan[v], x], u, x, True] && TryPureTanSubst[ActivateTrig[u], x 
]]
 

rule 7267

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

3.1.19.4 Maple [F]

\[\int \frac {d x +c}{\sqrt {b \tanh \left (f x +e \right )}}d x\]

input

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

output

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

3.1.19.5 Fricas [F(-2)]

Exception generated. \[ \int \frac {c+d x}{\sqrt {b \tanh (e+f x)}} \, dx=\text {Exception raised: TypeError} \]

input

integrate((d*x+c)/(b*tanh(f*x+e))^(1/2),x, algorithm="fricas")
 

output

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

3.1.19.6 Sympy [F]

\[ \int \frac {c+d x}{\sqrt {b \tanh (e+f x)}} \, dx=\int \frac {c + d x}{\sqrt {b \tanh {\left (e + f x \right )}}}\, dx \]

input

integrate((d*x+c)/(b*tanh(f*x+e))**(1/2),x)
 

output

Integral((c + d*x)/sqrt(b*tanh(e + f*x)), x)
 

3.1.19.7 Maxima [F]

\[ \int \frac {c+d x}{\sqrt {b \tanh (e+f x)}} \, dx=\int { \frac {d x + c}{\sqrt {b \tanh \left (f x + e\right )}} \,d x } \]

input

integrate((d*x+c)/(b*tanh(f*x+e))^(1/2),x, algorithm="maxima")
 

output

integrate((d*x + c)/sqrt(b*tanh(f*x + e)), x)
 

3.1.19.8 Giac [F(-2)]

Exception generated. \[ \int \frac {c+d x}{\sqrt {b \tanh (e+f x)}} \, dx=\text {Exception raised: RuntimeError} \]

input

integrate((d*x+c)/(b*tanh(f*x+e))^(1/2),x, algorithm="giac")
 

output

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:index.cc index_m i_lex_is_greater Error: Bad Argument V 
alue
 

3.1.19.9 Mupad [F(-1)]

Timed out. \[ \int \frac {c+d x}{\sqrt {b \tanh (e+f x)}} \, dx=\int \frac {c+d\,x}{\sqrt {b\,\mathrm {tanh}\left (e+f\,x\right )}} \,d x \]

input

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

output

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