3.1.24 \(\int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx\) [24]

3.1.24.1 Optimal result

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)+(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*b^...
 

3.1.24.2 Mathematica [N/A]

Not integrable

Time = 31.05 (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]
 

3.1.24.3 Rubi [N/A]

Not integrable

Time = 2.15 (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 = {3042, 4204, 3042, 4221, 4223, 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.

\(\displaystyle \int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx\)

\(\Big \downarrow \) 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
 

3.1.24.3.1 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 4204

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symb 
ol] :> Simp[(c + d*x)^m*((b*Tan[e + f*x])^(n + 1)/(b*f*(n + 1))), x] + (-Si 
mp[d*(m/(b*f*(n + 1)))   Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n + 1), x] 
, x] - Simp[1/b^2   Int[(c + d*x)^m*(b*Tan[e + f*x])^(n + 2), x], x]) /; Fr 
eeQ[{b, c, d, e, f}, x] && LtQ[n, -1] && GtQ[m, 0]
 

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 4223

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_. 
), x_Symbol] :> Unintegrable[(c + d*x)^m*(a + b*Tan[e + f*x])^n, x] /; Free 
Q[{a, b, c, d, e, f, m, n}, x]
 

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]
 

3.1.24.4 Maple [N/A] (verified)

Not integrable

Time = 0.08 (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)
 

3.1.24.5 Fricas [F(-2)]

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)
 

3.1.24.6 Sympy [N/A]

Not integrable

Time = 1.40 (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)
 

3.1.24.7 Maxima [N/A]

Not integrable

Time = 0.58 (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)
 

3.1.24.8 Giac [N/A]

Not integrable

Time = 0.49 (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)
 

3.1.24.9 Mupad [N/A]

Not integrable

Time = 2.11 (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)