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:
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-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/(a+b)^(1/2)/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)*El lipticF((a+b*sech(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(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/(a+b)^(1/2)/d+2* (a+b)^(1/2)*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))*(b*(1-sech(d*x+c))/(a+b))^(1/2)*(-b*(1+sech(d*x+ c))/(a-b))^(1/2)/a^2/d-coth(d*x+c)/d/(a+b*sech(d*x+c))^(3/2)-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)
\[ \int \frac {\coth ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int \frac {\coth ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx \] 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]
Time = 2.36 (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]])
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]
\[\int \frac {\coth \left (d x +c \right )^{2}}{\left (a +b \,\operatorname {sech}\left (d x +c \right )\right )^{\frac {3}{2}}}d x\]
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)
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
\[ \int \frac {\coth ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int \frac {\coth ^{2}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \] 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)
\[ \int \frac {\coth ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int { \frac {\coth \left (d x + c\right )^{2}}{{\left (b \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] 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)
\[ \int \frac {\coth ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int { \frac {\coth \left (d x + c\right )^{2}}{{\left (b \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] 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)
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)
\[ \int \frac {\coth ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int \frac {\sqrt {\mathrm {sech}\left (d x +c \right ) b +a}\, \coth \left (d x +c \right )^{2}}{\mathrm {sech}\left (d x +c \right )^{2} b^{2}+2 \,\mathrm {sech}\left (d x +c \right ) a b +a^{2}}d x \] Input:
int(coth(d*x+c)^2/(a+b*sech(d*x+c))^(3/2),x)
Output:
int((sqrt(sech(c + d*x)*b + a)*coth(c + d*x)**2)/(sech(c + d*x)**2*b**2 + 2*sech(c + d*x)*a*b + a**2),x)