Integrand size = 20, antiderivative size = 20 \[ \int (c+d x)^2 (b \tanh (e+f x))^{3/2} \, dx =\text {Too large to display} \] Output:
4*(-b)^(3/2)*d*(d*x+c)*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))/f^2+2*(-b )^(3/2)*d^2*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))^2/f^3+4*b^(3/2)*d*(d *x+c)*arctanh((b*tanh(f*x+e))^(1/2)/b^(1/2))/f^2+2*b^(3/2)*d^2*arctanh((b* tanh(f*x+e))^(1/2)/b^(1/2))^2/f^3-4*b^(3/2)*d^2*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^3+4*b^(3/2)*d ^2*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^3-2*b^(3/2)*d^2*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^3-2*b^(3/2)*d^2*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^3-4*(-b)^(3/2)*d^2*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^3+2* (-b)^(3/2)*d^2*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^3+2*(-b)^(3/2)*d^2*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))*ln((-2 *b^(1/2)-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^3+4*(-b)^(3/2)*d^2*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^3-2*b^(3/2)*d^2*polyl og(2,1-2*b^(1/2)/(b^(1/2)-(b*tanh(f*x+e))^(1/2)))/f^3-2*b^(3/2)*d^2*polylo g(2,1-2*b^(1/2)/(b^(1/2)+(b*tanh(f*x+e))^(1/2)))/f^3+b^(3/2)*d^2*polylo...
Not integrable
Time = 27.47 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (c+d x)^2 (b \tanh (e+f x))^{3/2} \, dx=\int (c+d x)^2 (b \tanh (e+f x))^{3/2} \, dx \] Input:
Integrate[(c + d*x)^2*(b*Tanh[e + f*x])^(3/2),x]
Output:
Integrate[(c + d*x)^2*(b*Tanh[e + f*x])^(3/2), x]
Not integrable
Time = 2.62 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
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.
| \(\displaystyle \int (c+d x)^2 (b \tanh (e+f x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x)^2 (-i b \tan (i e+i f x))^{3/2}dx\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle b^2 \int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}}dx+\frac {4 b d \int (c+d x) \sqrt {b \tanh (e+f x)}dx}{f}-\frac {2 b (c+d x)^2 \sqrt {b \tanh (e+f x)}}{f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b^2 \int \frac {(c+d x)^2}{\sqrt {-i b \tan (i e+i f x)}}dx+\frac {4 b d \int (c+d x) \sqrt {-i b \tan (i e+i f x)}dx}{f}-\frac {2 b (c+d x)^2 \sqrt {b \tanh (e+f x)}}{f}\) |
| \(\Big \downarrow \) 4219 |
| \(\displaystyle \frac {4 b d \left (\frac {\sqrt {-b} d \int \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )dx}{f}-\frac {\sqrt {b} d \int \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )dx}{f}-\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}\right )}{f}+b^2 \int \frac {(c+d x)^2}{\sqrt {-i b \tan (i e+i f x)}}dx-\frac {2 b (c+d x)^2 \sqrt {b \tanh (e+f x)}}{f}\) |
| \(\Big \downarrow \) 4223 |
| \(\displaystyle \frac {4 b d \left (\frac {\sqrt {-b} d \int \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )dx}{f}-\frac {\sqrt {b} d \int \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )dx}{f}-\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}\right )}{f}+b^2 \int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}}dx-\frac {2 b (c+d x)^2 \sqrt {b \tanh (e+f x)}}{f}\) |
| \(\Big \downarrow \) 4853 |
| \(\displaystyle \frac {4 b d \left (\frac {\sqrt {-b} d \int \frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{1-\tanh ^2(e+f x)}d\tanh (e+f x)}{f^2}-\frac {\sqrt {b} d \int \frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{1-\tanh ^2(e+f x)}d\tanh (e+f x)}{f^2}-\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}\right )}{f}+b^2 \int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}}dx-\frac {2 b (c+d x)^2 \sqrt {b \tanh (e+f x)}}{f}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {4 b d \left (\frac {2 \sqrt {-b} d \int \frac {b^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}}{b f^2}-\frac {2 d \int \frac {b^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}}{\sqrt {b} f^2}-\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}\right )}{f}+b^2 \int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}}dx-\frac {2 b (c+d x)^2 \sqrt {b \tanh (e+f x)}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 b d \left (\frac {2 \sqrt {-b} b d \int \frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}}{f^2}-\frac {2 b^{3/2} d \int \frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}}{f^2}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}-\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}\right )}{f}+b^2 \int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}}dx-\frac {2 b (c+d x)^2 \sqrt {b \tanh (e+f x)}}{f}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {4 b d \left (-\frac {2 b^{3/2} d \int \left (\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tanh (e+f x)}}{2 b (\tanh (e+f x) b+b)}-\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tanh (e+f x)}}{2 b (b \tanh (e+f x)-b)}\right )d\sqrt {b \tanh (e+f x)}}{f^2}+\frac {2 \sqrt {-b} b d \int \left (\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \sqrt {b \tanh (e+f x)}}{2 b (\tanh (e+f x) b+b)}-\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \sqrt {b \tanh (e+f x)}}{2 b (b \tanh (e+f x)-b)}\right )d\sqrt {b \tanh (e+f x)}}{f^2}+\frac {\sqrt {b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}-\frac {\sqrt {-b} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}\right )}{f}+b^2 \int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}}dx-\frac {2 b (c+d x)^2 \sqrt {b \tanh (e+f x)}}{f}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {(c+d x)^2}{\sqrt {b \tanh (e+f x)}}dx b^2+\frac {4 d \left (-\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}\right ) b}{f}-\frac {2 (c+d x)^2 \sqrt {b \tanh (e+f x)} b}{f}\) |
Input:
Int[(c + d*x)^2*(b*Tanh[e + f*x])^(3/2),x]
Output:
$Aborted
Not integrable
Time = 0.16 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90
\[\int \left (d x +c \right )^{2} \left (b \tanh \left (f x +e \right )\right )^{\frac {3}{2}}d x\]
Input:
int((d*x+c)^2*(b*tanh(f*x+e))^(3/2),x)
Output:
int((d*x+c)^2*(b*tanh(f*x+e))^(3/2),x)
Exception generated. \[ \int (c+d x)^2 (b \tanh (e+f x))^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((d*x+c)^2*(b*tanh(f*x+e))^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Not integrable
Time = 3.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int (c+d x)^2 (b \tanh (e+f x))^{3/2} \, dx=\int \left (b \tanh {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (c + d x\right )^{2}\, dx \] Input:
integrate((d*x+c)**2*(b*tanh(f*x+e))**(3/2),x)
Output:
Integral((b*tanh(e + f*x))**(3/2)*(c + d*x)**2, x)
Not integrable
Time = 0.48 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int (c+d x)^2 (b \tanh (e+f x))^{3/2} \, dx=\int { {\left (d x + c\right )}^{2} \left (b \tanh \left (f x + e\right )\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((d*x+c)^2*(b*tanh(f*x+e))^(3/2),x, algorithm="maxima")
Output:
integrate((d*x + c)^2*(b*tanh(f*x + e))^(3/2), x)
Not integrable
Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int (c+d x)^2 (b \tanh (e+f x))^{3/2} \, dx=\int { {\left (d x + c\right )}^{2} \left (b \tanh \left (f x + e\right )\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((d*x+c)^2*(b*tanh(f*x+e))^(3/2),x, algorithm="giac")
Output:
integrate((d*x + c)^2*(b*tanh(f*x + e))^(3/2), x)
Not integrable
Time = 2.54 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int (c+d x)^2 (b \tanh (e+f x))^{3/2} \, dx=\int {\left (b\,\mathrm {tanh}\left (e+f\,x\right )\right )}^{3/2}\,{\left (c+d\,x\right )}^2 \,d x \] Input:
int((b*tanh(e + f*x))^(3/2)*(c + d*x)^2,x)
Output:
int((b*tanh(e + f*x))^(3/2)*(c + d*x)^2, x)
Not integrable
Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 4.45 \[ \int (c+d x)^2 (b \tanh (e+f x))^{3/2} \, dx=\frac {\sqrt {b}\, b \left (-2 \sqrt {\tanh \left (f x +e \right )}\, c^{2}+\left (\int \frac {\sqrt {\tanh \left (f x +e \right )}}{\tanh \left (f x +e \right )}d x \right ) c^{2} f +\left (\int \sqrt {\tanh \left (f x +e \right )}\, \tanh \left (f x +e \right ) x^{2}d x \right ) d^{2} f +2 \left (\int \sqrt {\tanh \left (f x +e \right )}\, \tanh \left (f x +e \right ) x d x \right ) c d f \right )}{f} \] Input:
int((d*x+c)^2*(b*tanh(f*x+e))^(3/2),x)
Output:
(sqrt(b)*b*( - 2*sqrt(tanh(e + f*x))*c**2 + int(sqrt(tanh(e + f*x))/tanh(e + f*x),x)*c**2*f + int(sqrt(tanh(e + f*x))*tanh(e + f*x)*x**2,x)*d**2*f + 2*int(sqrt(tanh(e + f*x))*tanh(e + f*x)*x,x)*c*d*f))/f