Integrand size = 18, antiderivative size = 1280 \[ \int (c+d x) \sqrt {b \tanh (e+f x)} \, dx =\text {Too large to display} \]
-(d*x+c)*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))*(-b)^(1/2)/f-1/2*d*arct anh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))^2*(-b)^(1/2)/f^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)))*(-b)^( 1/2)/f^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)))*(-b)^(1/2)/f^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)))*(-b)^(1/2)/f^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)))*(-b)^(1/2)/f^2+1/2*d*polylog( 2,1-2/(1-(b*tanh(f*x+e))^(1/2)/(-b)^(1/2)))*(-b)^(1/2)/f^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)))*(-b)^(1/2)/f^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)))* (-b)^(1/2)/f^2+1/2*d*polylog(2,1-2/(1+(b*tanh(f*x+e))^(1/2)/(-b)^(1/2)))*( -b)^(1/2)/f^2+(d*x+c)*arctanh((b*tanh(f*x+e))^(1/2)/b^(1/2))*b^(1/2)/f+1/2 *d*arctanh((b*tanh(f*x+e))^(1/2)/b^(1/2))^2*b^(1/2)/f^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)/f^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)))*b^(1/2)/f^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))...
Result contains complex when optimal does not.
Time = 4.71 (sec) , antiderivative size = 556, normalized size of antiderivative = 0.43 \[ \int (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 {b \tanh (e+f x)}}{8 f^2 \sqrt {\tanh (e+f x)}} \]
((-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[b *Tanh[e + f*x]])/(8*f^2*Sqrt[Tanh[e + f*x]])
Time = 1.90 (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, 4219, 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.
| \(\displaystyle \int (c+d x) \sqrt {b \tanh (e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x) \sqrt {-i b \tan (i e+i f x)}dx\) |
| \(\Big \downarrow \) 4219 |
| \(\displaystyle \frac {\sqrt {-b} d \int \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )dx}{f}-\frac {\sqrt {b} d \int \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )dx}{f}-\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}\) |
| \(\Big \downarrow \) 4853 |
| \(\displaystyle \frac {\sqrt {-b} d \int \frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{1-\tanh ^2(e+f x)}d\tanh (e+f x)}{f^2}-\frac {\sqrt {b} d \int \frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{1-\tanh ^2(e+f x)}d\tanh (e+f x)}{f^2}-\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {2 \sqrt {-b} d \int \frac {b^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}}{b f^2}-\frac {2 d \int \frac {b^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}}{\sqrt {b} f^2}-\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {-b} b d \int \frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}}{f^2}-\frac {2 b^{3/2} d \int \frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}}{f^2}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}-\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {2 b^{3/2} d \int \left (\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tanh (e+f x)}}{2 b (\tanh (e+f x) b+b)}-\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tanh (e+f x)}}{2 b (b \tanh (e+f x)-b)}\right )d\sqrt {b \tanh (e+f x)}}{f^2}+\frac {2 \sqrt {-b} b d \int \left (\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \sqrt {b \tanh (e+f x)}}{2 b (\tanh (e+f x) b+b)}-\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \sqrt {b \tanh (e+f x)}}{2 b (b \tanh (e+f x)-b)}\right )d\sqrt {b \tanh (e+f x)}}{f^2}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}-\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 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 ) b^{3/2}}{f^2}+\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}{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 ) b}{f^2}+\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b}}{f}-\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}\) |
-((Sqrt[-b]*(c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]])/f) + (Sqrt[ b]*(c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]])/f - (2*b^(3/2)*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*T anh[e + f*x]])])/(2*b) + (ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sq rt[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])/(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]])]/(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*Sqrt[-b]*b*d*(ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]^2/(4*b) - (Arc Tanh[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 ]]*Log[(-2*(Sqrt[b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(1 ...
3.1.18.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.) + (d_.)*(x_))*Sqrt[(a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Sym bol] :> Simp[(-I)*Rt[a - I*b, 2]*((c + d*x)/f)*ArcTanh[Sqrt[a + b*Tan[e + f *x]]/Rt[a - I*b, 2]], x] + (Simp[I*Rt[a + I*b, 2]*((c + d*x)/f)*ArcTanh[Sqr t[a + b*Tan[e + f*x]]/Rt[a + I*b, 2]], x] + Simp[I*d*(Rt[a - I*b, 2]/f) I nt[ArcTanh[Sqrt[a + b*Tan[e + f*x]]/Rt[a - I*b, 2]], x], x] - Simp[I*d*(Rt[ a + I*b, 2]/f) 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]
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 ]]
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]]
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]
\[\int \left (d x +c \right ) \sqrt {b \tanh \left (f x +e \right )}d x\]
Exception generated. \[ \int (c+d x) \sqrt {b \tanh (e+f x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int (c+d x) \sqrt {b \tanh (e+f x)} \, dx=\int \sqrt {b \tanh {\left (e + f x \right )}} \left (c + d x\right )\, dx \]
\[ \int (c+d x) \sqrt {b \tanh (e+f x)} \, dx=\int { {\left (d x + c\right )} \sqrt {b \tanh \left (f x + e\right )} \,d x } \]
\[ \int (c+d x) \sqrt {b \tanh (e+f x)} \, dx=\int { {\left (d x + c\right )} \sqrt {b \tanh \left (f x + e\right )} \,d x } \]
Timed out. \[ \int (c+d x) \sqrt {b \tanh (e+f x)} \, dx=\int \sqrt {b\,\mathrm {tanh}\left (e+f\,x\right )}\,\left (c+d\,x\right ) \,d x \]