3.2.50 \(\int \frac {\coth ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx\) [150]

3.2.50.1 Optimal result

Integrand size = 23, antiderivative size = 665 \[ \int \frac {\coth ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\frac {4 a \coth (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{(a-b) (a+b)^{3/2} d}-\frac {2 \coth (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{a \sqrt {a+b} d}-\frac {(3 a-b) \coth (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{(a-b) (a+b)^{3/2} d}+\frac {2 \coth (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{a \sqrt {a+b} d}+\frac {2 \sqrt {a+b} \coth (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{a^2 d}-\frac {\coth (c+d x)}{d (a+b \text {sech}(c+d x))^{3/2}}-\frac {b^2 \tanh (c+d x)}{\left (a^2-b^2\right ) d (a+b \text {sech}(c+d x))^{3/2}}-\frac {4 a b^2 \tanh (c+d x)}{\left (a^2-b^2\right )^2 d \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 b^2 \tanh (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}} \]

output

-coth(d*x+c)/d/(a+b*sech(d*x+c))^(3/2)+4*a*coth(d*x+c)*EllipticE((a+b*sech 
(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(b*(1-sech(d*x+c))/(a+b))^ 
(1/2)*(-b*(1+sech(d*x+c))/(a-b))^(1/2)/(a-b)/(a+b)^(3/2)/d-(3*a-b)*coth(d* 
x+c)*EllipticF((a+b*sech(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(b 
*(1-sech(d*x+c))/(a+b))^(1/2)*(-b*(1+sech(d*x+c))/(a-b))^(1/2)/(a-b)/(a+b) 
^(3/2)/d-2*coth(d*x+c)*EllipticE((a+b*sech(d*x+c))^(1/2)/(a+b)^(1/2),((a+b 
)/(a-b))^(1/2))*(b*(1-sech(d*x+c))/(a+b))^(1/2)*(-b*(1+sech(d*x+c))/(a-b)) 
^(1/2)/a/d/(a+b)^(1/2)+2*coth(d*x+c)*EllipticF((a+b*sech(d*x+c))^(1/2)/(a+ 
b)^(1/2),((a+b)/(a-b))^(1/2))*(b*(1-sech(d*x+c))/(a+b))^(1/2)*(-b*(1+sech( 
d*x+c))/(a-b))^(1/2)/a/d/(a+b)^(1/2)+2*coth(d*x+c)*EllipticPi((a+b*sech(d* 
x+c))^(1/2)/(a+b)^(1/2),(a+b)/a,((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec 
h(d*x+c))/(a+b))^(1/2)*(-b*(1+sech(d*x+c))/(a-b))^(1/2)/a^2/d-b^2*tanh(d*x 
+c)/(a^2-b^2)/d/(a+b*sech(d*x+c))^(3/2)-4*a*b^2*tanh(d*x+c)/(a^2-b^2)^2/d/ 
(a+b*sech(d*x+c))^(1/2)+2*b^2*tanh(d*x+c)/a/(a^2-b^2)/d/(a+b*sech(d*x+c))^ 
(1/2)
 

3.2.50.2 Mathematica [F]

\[ \int \frac {\coth ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int \frac {\coth ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx \]

input

Integrate[Coth[c + d*x]^2/(a + b*Sech[c + d*x])^(3/2),x]
 

output

Integrate[Coth[c + d*x]^2/(a + b*Sech[c + d*x])^(3/2), x]
 

3.2.50.3 Rubi [A] (verified)

Time = 1.76 (sec) , antiderivative size = 665, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 25, 4384, 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.

\(\displaystyle \int \frac {\coth ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {1}{\cot \left (i c+i d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \frac {1}{\cot \left (\frac {1}{2} (2 i c+\pi )+i d x\right )^2 \left (a+b \csc \left (\frac {1}{2} (2 i c+\pi )+i d x\right )\right )^{3/2}}dx\)

\(\Big \downarrow \) 4384

\(\displaystyle -\int \left (-\frac {\text {csch}^2(c+d x)}{\left (a+b \csc \left (\frac {1}{2} (2 i c+\pi )+i d x\right )\right )^{3/2}}-\frac {1}{\left (a+b \csc \left (\frac {1}{2} (2 i c+\pi )+i d x\right )\right )^{3/2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \sqrt {a+b} \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a^2 d}+\frac {2 b^2 \tanh (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \text {sech}(c+d x)}}-\frac {4 a b^2 \tanh (c+d x)}{d \left (a^2-b^2\right )^2 \sqrt {a+b \text {sech}(c+d x)}}-\frac {b^2 \tanh (c+d x)}{d \left (a^2-b^2\right ) (a+b \text {sech}(c+d x))^{3/2}}+\frac {2 \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d \sqrt {a+b}}-\frac {(3 a-b) \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d (a-b) (a+b)^{3/2}}-\frac {2 \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{a d \sqrt {a+b}}+\frac {4 a \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{d (a-b) (a+b)^{3/2}}-\frac {\coth (c+d x)}{d (a+b \text {sech}(c+d x))^{3/2}}\)

input

Int[Coth[c + d*x]^2/(a + b*Sech[c + d*x])^(3/2),x]
 

output

(4*a*Coth[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]] 
, (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Se 
ch[c + d*x]))/(a - b))])/((a - b)*(a + b)^(3/2)*d) - (2*Coth[c + d*x]*Elli 
pticE[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt 
[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))] 
)/(a*Sqrt[a + b]*d) - ((3*a - b)*Coth[c + d*x]*EllipticF[ArcSin[Sqrt[a + b 
*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]) 
)/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/((a - b)*(a + b)^(3/2 
)*d) + (2*Coth[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a 
+ b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 
 + Sech[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) + (2*Sqrt[a + b]*Coth[c + 
d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], 
(a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech 
[c + d*x]))/(a - b))])/(a^2*d) - Coth[c + d*x]/(d*(a + b*Sech[c + d*x])^(3 
/2)) - (b^2*Tanh[c + d*x])/((a^2 - b^2)*d*(a + b*Sech[c + d*x])^(3/2)) - ( 
4*a*b^2*Tanh[c + d*x])/((a^2 - b^2)^2*d*Sqrt[a + b*Sech[c + d*x]]) + (2*b^ 
2*Tanh[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sech[c + d*x]])
 

3.2.50.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 4384

Int[cot[(c_.) + (d_.)*(x_)]^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_ 
), x_Symbol] :> Int[ExpandIntegrand[(a + b*Csc[c + d*x])^n, (-1 + Sec[c + d 
*x]^2)^(-m/2), x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0] && 
ILtQ[m/2, 0] && IntegerQ[n - 1/2] && EqQ[m, -2]
 

3.2.50.4 Maple [F]

\[\int \frac {\coth \left (d x +c \right )^{2}}{\left (a +b \,\operatorname {sech}\left (d x +c \right )\right )^{\frac {3}{2}}}d x\]

input

int(coth(d*x+c)^2/(a+b*sech(d*x+c))^(3/2),x)
 

output

int(coth(d*x+c)^2/(a+b*sech(d*x+c))^(3/2),x)
 

3.2.50.5 Fricas [F(-1)]

Timed out. \[ \int \frac {\coth ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\text {Timed out} \]

input

integrate(coth(d*x+c)^2/(a+b*sech(d*x+c))^(3/2),x, algorithm="fricas")
 

output

Timed out
 

3.2.50.6 Sympy [F]

\[ \int \frac {\coth ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int \frac {\coth ^{2}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]

input

integrate(coth(d*x+c)**2/(a+b*sech(d*x+c))**(3/2),x)
 

output

Integral(coth(c + d*x)**2/(a + b*sech(c + d*x))**(3/2), x)
 

3.2.50.7 Maxima [F]

\[ \int \frac {\coth ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int { \frac {\coth \left (d x + c\right )^{2}}{{\left (b \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

input

integrate(coth(d*x+c)^2/(a+b*sech(d*x+c))^(3/2),x, algorithm="maxima")
 

output

integrate(coth(d*x + c)^2/(b*sech(d*x + c) + a)^(3/2), x)
 

3.2.50.8 Giac [F]

\[ \int \frac {\coth ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int { \frac {\coth \left (d x + c\right )^{2}}{{\left (b \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

input

integrate(coth(d*x+c)^2/(a+b*sech(d*x+c))^(3/2),x, algorithm="giac")
 

output

integrate(coth(d*x + c)^2/(b*sech(d*x + c) + a)^(3/2), x)
 

3.2.50.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\coth ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int \frac {{\mathrm {coth}\left (c+d\,x\right )}^2}{{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{3/2}} \,d x \]

input

int(coth(c + d*x)^2/(a + b/cosh(c + d*x))^(3/2),x)
 

output

int(coth(c + d*x)^2/(a + b/cosh(c + d*x))^(3/2), x)