Integrand size = 20, antiderivative size = 20 \[ \int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx =\text {Too large to display} \] Output:
4*d*(d*x+c)*arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))/(-b)^(3/2)/f^2+2*d^2 *arctanh((b*tanh(f*x+e))^(1/2)/(-b)^(1/2))^2/(-b)^(3/2)/f^3+4*d*(d*x+c)*ar ctanh((b*tanh(f*x+e))^(1/2)/b^(1/2))/b^(3/2)/f^2+2*d^2*arctanh((b*tanh(f*x +e))^(1/2)/b^(1/2))^2/b^(3/2)/f^3-4*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^(3/2)/f^3+4*d^2*arcta nh((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^(3/2)/f^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)))/b^(3/2)/f^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)))/b^(3/2)/f^3-4*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)))/(-b)^(3/2)/f^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)))/(-b)^( 3/2)/f^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)))/(-b)^(3/2)/f^3+4*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)))/(-b)^(3/2)/f^3-2*d^2*polylog(2,1-2 *b^(1/2)/(b^(1/2)-(b*tanh(f*x+e))^(1/2)))/b^(3/2)/f^3-2*d^2*polylog(2,1-2* b^(1/2)/(b^(1/2)+(b*tanh(f*x+e))^(1/2)))/b^(3/2)/f^3+d^2*polylog(2,1-2*...
Not integrable
Time = 28.35 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx=\int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx \] 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 = 2.37 (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.
| \(\displaystyle \int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d x)^2}{(-i b \tan (i e+i f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 4204 |
| \(\displaystyle \frac {\int (c+d x)^2 \sqrt {b \tanh (e+f x)}dx}{b^2}+\frac {4 d \int \frac {c+d x}{\sqrt {b \tanh (e+f x)}}dx}{b f}-\frac {2 (c+d x)^2}{b f \sqrt {b \tanh (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (c+d x)^2 \sqrt {-i b \tan (i e+i f x)}dx}{b^2}+\frac {4 d \int \frac {c+d x}{\sqrt {-i b \tan (i e+i f x)}}dx}{b f}-\frac {2 (c+d x)^2}{b f \sqrt {b \tanh (e+f x)}}\) |
| \(\Big \downarrow \) 4221 |
| \(\displaystyle \frac {4 d \left (\frac {d \int \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )dx}{\sqrt {-b} f}-\frac {d \int \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )dx}{\sqrt {b} f}-\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{\sqrt {-b} f}+\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{\sqrt {b} f}\right )}{b f}+\frac {\int (c+d x)^2 \sqrt {-i b \tan (i e+i f x)}dx}{b^2}-\frac {2 (c+d x)^2}{b f \sqrt {b \tanh (e+f x)}}\) |
| \(\Big \downarrow \) 4223 |
| \(\displaystyle \frac {4 d \left (\frac {d \int \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )dx}{\sqrt {-b} f}-\frac {d \int \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )dx}{\sqrt {b} f}-\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{\sqrt {-b} f}+\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{\sqrt {b} f}\right )}{b f}+\frac {\int (c+d x)^2 \sqrt {b \tanh (e+f x)}dx}{b^2}-\frac {2 (c+d x)^2}{b f \sqrt {b \tanh (e+f x)}}\) |
| \(\Big \downarrow \) 4853 |
| \(\displaystyle \frac {4 d \left (\frac {d \int \frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{1-\tanh ^2(e+f x)}d\tanh (e+f x)}{\sqrt {-b} f^2}-\frac {d \int \frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{1-\tanh ^2(e+f x)}d\tanh (e+f x)}{\sqrt {b} f^2}-\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{\sqrt {-b} f}+\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{\sqrt {b} f}\right )}{b f}+\frac {\int (c+d x)^2 \sqrt {b \tanh (e+f x)}dx}{b^2}-\frac {2 (c+d x)^2}{b f \sqrt {b \tanh (e+f x)}}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {4 d \left (\frac {2 d \int \frac {b^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}}{\sqrt {-b} b f^2}-\frac {2 d \int \frac {b^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}}{b^{3/2} f^2}-\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{\sqrt {-b} f}+\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{\sqrt {b} f}\right )}{b f}+\frac {\int (c+d x)^2 \sqrt {b \tanh (e+f x)}dx}{b^2}-\frac {2 (c+d x)^2}{b f \sqrt {b \tanh (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 d \left (\frac {2 b d \int \frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}}{\sqrt {-b} f^2}-\frac {2 \sqrt {b} d \int \frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tanh (e+f x)}}{b^2-b^2 \tanh ^2(e+f x)}d\sqrt {b \tanh (e+f x)}}{f^2}-\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{\sqrt {-b} f}+\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{\sqrt {b} f}\right )}{b f}+\frac {\int (c+d x)^2 \sqrt {b \tanh (e+f x)}dx}{b^2}-\frac {2 (c+d x)^2}{b f \sqrt {b \tanh (e+f x)}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {4 d \left (\frac {2 b d \int \left (\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \sqrt {b \tanh (e+f x)}}{2 b (\tanh (e+f x) b+b)}-\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \sqrt {b \tanh (e+f x)}}{2 b (b \tanh (e+f x)-b)}\right )d\sqrt {b \tanh (e+f x)}}{\sqrt {-b} f^2}-\frac {2 \sqrt {b} d \int \left (\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tanh (e+f x)}}{2 b (\tanh (e+f x) b+b)}-\frac {\text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \sqrt {b \tanh (e+f x)}}{2 b (b \tanh (e+f x)-b)}\right )d\sqrt {b \tanh (e+f x)}}{f^2}-\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{\sqrt {-b} f}+\frac {(c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{\sqrt {b} f}\right )}{b f}+\frac {\int (c+d x)^2 \sqrt {b \tanh (e+f x)}dx}{b^2}-\frac {2 (c+d x)^2}{b f \sqrt {b \tanh (e+f x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 (c+d x)^2}{b f \sqrt {b \tanh (e+f x)}}+\frac {4 d \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 f}+\frac {\int (c+d x)^2 \sqrt {b \tanh (e+f x)}dx}{b^2}\) |
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
\[\int \frac {\left (d x +c \right )^{2}}{\left (b \tanh \left (f x +e \right )\right )^{\frac {3}{2}}}d x\]
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. \[ \int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] 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 = 1.43 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx=\int \frac {\left (c + d x\right )^{2}}{\left (b \tanh {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((d*x+c)**2/(b*tanh(f*x+e))**(3/2),x)
Output:
Integral((c + d*x)**2/(b*tanh(e + f*x))**(3/2), x)
Not integrable
Time = 0.46 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{\left (b \tanh \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \] 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 = 0.33 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{\left (b \tanh \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \] 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 \[ \int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^2}{{\left (b\,\mathrm {tanh}\left (e+f\,x\right )\right )}^{3/2}} \,d x \] Input:
int((c + d*x)^2/(b*tanh(e + f*x))^(3/2),x)
Output:
int((c + d*x)^2/(b*tanh(e + f*x))^(3/2), x)
Not integrable
Time = 0.25 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.85 \[ \int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx=\frac {\sqrt {b}\, \left (\left (\int \frac {\sqrt {\tanh \left (f x +e \right )}}{\tanh \left (f x +e \right )^{2}}d x \right ) c^{2}+\left (\int \frac {\sqrt {\tanh \left (f x +e \right )}\, x^{2}}{\tanh \left (f x +e \right )^{2}}d x \right ) d^{2}+2 \left (\int \frac {\sqrt {\tanh \left (f x +e \right )}\, x}{\tanh \left (f x +e \right )^{2}}d x \right ) c d \right )}{b^{2}} \] Input:
int((d*x+c)^2/(b*tanh(f*x+e))^(3/2),x)
Output:
(sqrt(b)*(int(sqrt(tanh(e + f*x))/tanh(e + f*x)**2,x)*c**2 + int((sqrt(tan h(e + f*x))*x**2)/tanh(e + f*x)**2,x)*d**2 + 2*int((sqrt(tanh(e + f*x))*x) /tanh(e + f*x)**2,x)*c*d))/b**2