Integrand size = 18, antiderivative size = 1363 \[ \int (c+d x) (b \tanh (e+f x))^{3/2} \, dx =\text {Too large to display} \]
-2*b^(3/2)*d*arctan((b*tanh(f*x+e))^(1/2)/b^(1/2))/f^2-(-b)^(3/2)*(d*x+c)* arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))/f-1/2*(-b)^(3/2)*d*arctanh((b*ta nh(f*x+e))^(1/2)/(-b)^(1/2))^2/f^2+2*b^(3/2)*d*arctanh((b*tanh(f*x+e))^(1/ 2)/b^(1/2))/f^2+b^(3/2)*(d*x+c)*arctanh((b*tanh(f*x+e))^(1/2)/b^(1/2))/f+1 /2*b^(3/2)*d*arctanh((b*tanh(f*x+e))^(1/2)/b^(1/2))^2/f^2-b^(3/2)*d*arctan h((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^(3/2)*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^2-1/2*b^(3/2)*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))/(b^(1/2)+(b*tanh(f*x+e))^(1/2)))/f^2-1/2*b^(3/2)*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))/(b^(1/2)+(b*tanh(f*x+e))^(1/2)))/f^2+(-b)^(3/2)*d*arcta nh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))*ln(2/(1-(b*tanh(f*x+e))^(1/2)/(-b)^(1 /2)))/f^2-1/2*(-b)^(3/2)*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^2-1/2*(-b)^(3/2)*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*tan h(f*x+e))^(1/2)/(-b)^(1/2)))/f^2-(-b)^(3/2)*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^2-1/2*b^(3/2)*d *polylog(2,1-2*b^(1/2)/(b^(1/2)-(b*tanh(f*x+e))^(1/2)))/f^2-1/2*b^(3/2)...
\[ \int (c+d x) (b \tanh (e+f x))^{3/2} \, dx=\int (c+d x) (b \tanh (e+f x))^{3/2} \, dx \]
Time = 2.19 (sec) , antiderivative size = 1269, normalized size of antiderivative = 0.93, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {3042, 4203, 3042, 3957, 25, 266, 827, 216, 219, 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.
| \(\displaystyle \int (c+d x) (b \tanh (e+f x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x) (-i b \tan (i e+i f x))^{3/2}dx\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle b^2 \int \frac {c+d x}{\sqrt {b \tanh (e+f x)}}dx+\frac {2 b d \int \sqrt {b \tanh (e+f x)}dx}{f}-\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b^2 \int \frac {c+d x}{\sqrt {-i b \tan (i e+i f x)}}dx+\frac {2 b d \int \sqrt {-i b \tan (i e+i f x)}dx}{f}-\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle b^2 \int \frac {c+d x}{\sqrt {-i b \tan (i e+i f x)}}dx-\frac {2 b^2 d \int -\frac {\sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d(b \tanh (e+f x))}{f^2}-\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle b^2 \int \frac {c+d x}{\sqrt {-i b \tan (i e+i f x)}}dx+\frac {2 b^2 d \int \frac {\sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d(b \tanh (e+f x))}{f^2}-\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle b^2 \int \frac {c+d x}{\sqrt {-i b \tan (i e+i f x)}}dx+\frac {4 b^2 d \int \frac {b^2 \tanh ^2(e+f x)}{b^2-b^4 \tanh ^4(e+f x)}d\sqrt {b \tanh (e+f x)}}{f^2}-\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle b^2 \int \frac {c+d x}{\sqrt {-i b \tan (i e+i f x)}}dx+\frac {4 b^2 d \left (\frac {1}{2} \int \frac {1}{b-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}-\frac {1}{2} \int \frac {1}{b^2 \tanh ^2(e+f x)+b}d\sqrt {b \tanh (e+f x)}\right )}{f^2}-\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {4 b^2 d \left (\frac {1}{2} \int \frac {1}{b-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}-\frac {\arctan \left (\sqrt {b} \tanh (e+f x)\right )}{2 \sqrt {b}}\right )}{f^2}+b^2 \int \frac {c+d x}{\sqrt {-i b \tan (i e+i f x)}}dx-\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle b^2 \int \frac {c+d x}{\sqrt {-i b \tan (i e+i f x)}}dx+\frac {4 b^2 d \left (\frac {\text {arctanh}\left (\sqrt {b} \tanh (e+f x)\right )}{2 \sqrt {b}}-\frac {\arctan \left (\sqrt {b} \tanh (e+f x)\right )}{2 \sqrt {b}}\right )}{f^2}-\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f}\) |
| \(\Big \downarrow \) 4221 |
| \(\displaystyle b^2 \left (\frac {d \int \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )dx}{\sqrt {-b} f}-\frac {d \int \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )dx}{\sqrt {b} f}-\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{\sqrt {-b} f}+\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{\sqrt {b} f}\right )+\frac {4 b^2 d \left (\frac {\text {arctanh}\left (\sqrt {b} \tanh (e+f x)\right )}{2 \sqrt {b}}-\frac {\arctan \left (\sqrt {b} \tanh (e+f x)\right )}{2 \sqrt {b}}\right )}{f^2}-\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f}\) |
| \(\Big \downarrow \) 4853 |
| \(\displaystyle b^2 \left (\frac {d \int \frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{1-\tanh ^2(e+f x)}d\tanh (e+f x)}{\sqrt {-b} f^2}-\frac {d \int \frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{1-\tanh ^2(e+f x)}d\tanh (e+f x)}{\sqrt {b} f^2}-\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{\sqrt {-b} f}+\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{\sqrt {b} f}\right )+\frac {4 b^2 d \left (\frac {\text {arctanh}\left (\sqrt {b} \tanh (e+f x)\right )}{2 \sqrt {b}}-\frac {\arctan \left (\sqrt {b} \tanh (e+f x)\right )}{2 \sqrt {b}}\right )}{f^2}-\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle b^2 \left (\frac {2 d \int \frac {b^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}}{\sqrt {-b} b f^2}-\frac {2 d \int \frac {b^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}}{b^{3/2} f^2}-\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{\sqrt {-b} f}+\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{\sqrt {b} f}\right )+\frac {4 b^2 d \left (\frac {\text {arctanh}\left (\sqrt {b} \tanh (e+f x)\right )}{2 \sqrt {b}}-\frac {\arctan \left (\sqrt {b} \tanh (e+f x)\right )}{2 \sqrt {b}}\right )}{f^2}-\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle b^2 \left (\frac {2 b d \int \frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}}{\sqrt {-b} f^2}-\frac {2 \sqrt {b} d \int \frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}}{f^2}-\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{\sqrt {-b} f}+\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{\sqrt {b} f}\right )+\frac {4 b^2 d \left (\frac {\text {arctanh}\left (\sqrt {b} \tanh (e+f x)\right )}{2 \sqrt {b}}-\frac {\arctan \left (\sqrt {b} \tanh (e+f x)\right )}{2 \sqrt {b}}\right )}{f^2}-\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle b^2 \left (\frac {2 b d \int \left (\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \sqrt {b \tanh (e+f x)}}{2 b (\tanh (e+f x) b+b)}-\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \sqrt {b \tanh (e+f x)}}{2 b (b \tanh (e+f x)-b)}\right )d\sqrt {b \tanh (e+f x)}}{\sqrt {-b} f^2}-\frac {2 \sqrt {b} d \int \left (\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tanh (e+f x)}}{2 b (\tanh (e+f x) b+b)}-\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tanh (e+f x)}}{2 b (b \tanh (e+f x)-b)}\right )d\sqrt {b \tanh (e+f x)}}{f^2}-\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{\sqrt {-b} f}+\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{\sqrt {b} f}\right )+\frac {4 b^2 d \left (\frac {\text {arctanh}\left (\sqrt {b} \tanh (e+f x)\right )}{2 \sqrt {b}}-\frac {\arctan \left (\sqrt {b} \tanh (e+f x)\right )}{2 \sqrt {b}}\right )}{f^2}-\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 d \left (\frac {\text {arctanh}\left (\sqrt {b} \tanh (e+f x)\right )}{2 \sqrt {b}}-\frac {\arctan \left (\sqrt {b} \tanh (e+f x)\right )}{2 \sqrt {b}}\right ) b^2}{f^2}+\left (-\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}\right ) b^2-\frac {2 (c+d x) \sqrt {b \tanh (e+f x)} b}{f}\) |
(4*b^2*d*(-1/2*ArcTan[Sqrt[b]*Tanh[e + f*x]]/Sqrt[b] + ArcTanh[Sqrt[b]*Tan h[e + f*x]]/(2*Sqrt[b])))/f^2 + b^2*(-(((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]]/S qrt[b]]^2/b + (ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b])/(Sqr t[b] - Sqrt[b*Tanh[e + f*x]])])/(2*b) - (ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqr t[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])/(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*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) - (Ar...
3.1.17.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Subst[Int [x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symb ol] :> Simp[b*(c + d*x)^m*((b*Tan[e + f*x])^(n - 1)/(f*(n - 1))), x] + (-Si mp[b*d*(m/(f*(n - 1))) Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1), x] , x] - Simp[b^2 Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; Free Q[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 0]
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]
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]
\[\int \left (d x +c \right ) \left (b \tanh \left (f x +e \right )\right )^{\frac {3}{2}}d x\]
Exception generated. \[ \int (c+d x) (b \tanh (e+f x))^{3/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int (c+d x) (b \tanh (e+f x))^{3/2} \, dx=\int \left (b \tanh {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (c + d x\right )\, dx \]
\[ \int (c+d x) (b \tanh (e+f x))^{3/2} \, dx=\int { {\left (d x + c\right )} \left (b \tanh \left (f x + e\right )\right )^{\frac {3}{2}} \,d x } \]
\[ \int (c+d x) (b \tanh (e+f x))^{3/2} \, dx=\int { {\left (d x + c\right )} \left (b \tanh \left (f x + e\right )\right )^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int (c+d x) (b \tanh (e+f x))^{3/2} \, dx=\int {\left (b\,\mathrm {tanh}\left (e+f\,x\right )\right )}^{3/2}\,\left (c+d\,x\right ) \,d x \]