\(\int \coth ^2(c+d x) \sqrt {a+b \text {sech}(c+d x)} \, dx\) [132]

Optimal result

Integrand size = 23, antiderivative size = 246 \[ \int \coth ^2(c+d x) \sqrt {a+b \text {sech}(c+d x)} \, dx=\frac {\sqrt {a+b} \coth (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{d}-\frac {\coth (c+d x) \sqrt {a+b \text {sech}(c+d x)}}{d}+\frac {2 \coth (c+d x) \operatorname {EllipticPi}\left (\frac {a}{a+b},\arcsin \left (\frac {\sqrt {a+b}}{\sqrt {a+b \text {sech}(c+d x)}}\right ),\frac {a-b}{a+b}\right ) \sqrt {-\frac {b (1-\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} \sqrt {\frac {b (1+\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} (a+b \text {sech}(c+d x))}{\sqrt {a+b} d} \]

[Out]

coth(d*x+c)*EllipticF((a+b*sech(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sech(d*x+c))/
(a+b))^(1/2)*(-b*(1+sech(d*x+c))/(a-b))^(1/2)/d+2*coth(d*x+c)*EllipticPi((a+b)^(1/2)/(a+b*sech(d*x+c))^(1/2),a
/(a+b),((a-b)/(a+b))^(1/2))*(a+b*sech(d*x+c))*(-b*(1-sech(d*x+c))/(a+b*sech(d*x+c)))^(1/2)*(b*(1+sech(d*x+c))/
(a+b*sech(d*x+c)))^(1/2)/d/(a+b)^(1/2)-coth(d*x+c)*(a+b*sech(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3981, 3865, 3960, 3917} \[ \int \coth ^2(c+d x) \sqrt {a+b \text {sech}(c+d x)} \, dx=\frac {\sqrt {a+b} \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}+\frac {2 \coth (c+d x) \sqrt {-\frac {b (1-\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} \sqrt {\frac {b (\text {sech}(c+d x)+1)}{a+b \text {sech}(c+d x)}} (a+b \text {sech}(c+d x)) \operatorname {EllipticPi}\left (\frac {a}{a+b},\arcsin \left (\frac {\sqrt {a+b}}{\sqrt {a+b \text {sech}(c+d x)}}\right ),\frac {a-b}{a+b}\right )}{d \sqrt {a+b}}-\frac {\coth (c+d x) \sqrt {a+b \text {sech}(c+d x)}}{d} \]

[In]

Int[Coth[c + d*x]^2*Sqrt[a + b*Sech[c + d*x]],x]

[Out]

(Sqrt[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))])/d - (Coth[c + d*x]*Sqrt[a + b*Sech[c +
d*x]])/d + (2*Coth[c + d*x]*EllipticPi[a/(a + b), ArcSin[Sqrt[a + b]/Sqrt[a + b*Sech[c + d*x]]], (a - b)/(a +
b)]*Sqrt[-((b*(1 - Sech[c + d*x]))/(a + b*Sech[c + d*x]))]*Sqrt[(b*(1 + Sech[c + d*x]))/(a + b*Sech[c + d*x])]
*(a + b*Sech[c + d*x]))/(Sqrt[a + b]*d)

Rule 3865

Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*((a + b*Csc[c + d*x])/(d*Rt[a + b, 2]*Cot[
c + d*x]))*Sqrt[b*((1 + Csc[c + d*x])/(a + b*Csc[c + d*x]))]*Sqrt[(-b)*((1 - Csc[c + d*x])/(a + b*Csc[c + d*x]
))]*EllipticPi[a/(a + b), ArcSin[Rt[a + b, 2]/Sqrt[a + b*Csc[c + d*x]]], (a - b)/(a + b)], x] /; FreeQ[{a, b,
c, d}, x] && NeQ[a^2 - b^2, 0]

Rule 3917

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(Rt[a + b, 2]/(b*
f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin
[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 3960

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)/cos[(e_.) + (f_.)*(x_)]^2, x_Symbol] :> Simp[Tan[e + f*x]*((a
+ b*Csc[e + f*x])^m/f), x] + Dist[b*m, Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e
, f, m}, x]

Rule 3981

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]

Rubi steps \begin{align*} \text {integral}& = -\int \left (-\sqrt {a+b \text {sech}(c+d x)}-\text {csch}^2(c+d x) \sqrt {a+b \text {sech}(c+d x)}\right ) \, dx \\ & = \int \sqrt {a+b \text {sech}(c+d x)} \, dx+\int \text {csch}^2(c+d x) \sqrt {a+b \text {sech}(c+d x)} \, dx \\ & = -\frac {\coth (c+d x) \sqrt {a+b \text {sech}(c+d x)}}{d}+\frac {2 \coth (c+d x) \operatorname {EllipticPi}\left (\frac {a}{a+b},\arcsin \left (\frac {\sqrt {a+b}}{\sqrt {a+b \text {sech}(c+d x)}}\right ),\frac {a-b}{a+b}\right ) \sqrt {-\frac {b (1-\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} \sqrt {\frac {b (1+\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} (a+b \text {sech}(c+d x))}{\sqrt {a+b} d}-\frac {1}{2} b \int \frac {\text {sech}(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx \\ & = \frac {\sqrt {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}}}{d}-\frac {\coth (c+d x) \sqrt {a+b \text {sech}(c+d x)}}{d}+\frac {2 \coth (c+d x) \operatorname {EllipticPi}\left (\frac {a}{a+b},\arcsin \left (\frac {\sqrt {a+b}}{\sqrt {a+b \text {sech}(c+d x)}}\right ),\frac {a-b}{a+b}\right ) \sqrt {-\frac {b (1-\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} \sqrt {\frac {b (1+\text {sech}(c+d x))}{a+b \text {sech}(c+d x)}} (a+b \text {sech}(c+d x))}{\sqrt {a+b} d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(539\) vs. \(2(246)=492\).

Time = 18.56 (sec) , antiderivative size = 539, normalized size of antiderivative = 2.19 \[ \int \coth ^2(c+d x) \sqrt {a+b \text {sech}(c+d x)} \, dx=-\frac {\coth (c+d x) \sqrt {a+b \text {sech}(c+d x)}}{d}+\frac {\sqrt {a+b \text {sech}(c+d x)} \left (\frac {2 \sqrt {b} (a-a \cosh (c+d x))^{3/2} \sqrt {\frac {(a+b) (a+a \cosh (c+d x))}{(a-b) (a-a \cosh (c+d x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {b+a \cosh (c+d x)}}{\sqrt {b} \sqrt {a-a \cosh (c+d x)}}\right ),-\frac {2 b}{a-b}\right ) \sinh (c+d x)}{a^{3/2} \sqrt {-1+\cosh (c+d x)} \sqrt {1+\cosh (c+d x)} \sqrt {-\frac {a (a+b) \cosh (c+d x)}{b (a-a \cosh (c+d x))}} \left (-\frac {a-a \cosh (c+d x)}{a}\right )^{3/2} \sqrt {\frac {a+a \cosh (c+d x)}{a}} \sqrt {\text {sech}(c+d x)}}-\frac {4 b (a-a \cosh (c+d x)) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a} \sqrt {b+a \cosh (c+d x)}}{\sqrt {a+b} \sqrt {a \cosh (c+d x)}}\right ),\frac {a+b}{a-b}\right ) \sqrt {-\frac {b (a+a \cosh (c+d x)) \text {sech}(c+d x)}{a (a-b)}} \sinh (c+d x)}{\sqrt {a} \sqrt {a+b} \sqrt {-1+\cosh (c+d x)} \sqrt {a \cosh (c+d x)} \sqrt {1+\cosh (c+d x)} \sqrt {-\frac {a-a \cosh (c+d x)}{a}} \sqrt {\frac {a+a \cosh (c+d x)}{a}} \sqrt {\text {sech}(c+d x)} \sqrt {-\frac {b (a-a \cosh (c+d x)) \text {sech}(c+d x)}{a (a+b)}}}\right )}{2 d \sqrt {b+a \cosh (c+d x)} \sqrt {\text {sech}(c+d x)}} \]

[In]

Integrate[Coth[c + d*x]^2*Sqrt[a + b*Sech[c + d*x]],x]

[Out]

-((Coth[c + d*x]*Sqrt[a + b*Sech[c + d*x]])/d) + (Sqrt[a + b*Sech[c + d*x]]*((2*Sqrt[b]*(a - a*Cosh[c + d*x])^
(3/2)*Sqrt[((a + b)*(a + a*Cosh[c + d*x]))/((a - b)*(a - a*Cosh[c + d*x]))]*EllipticF[ArcSin[(Sqrt[a]*Sqrt[b +
a*Cosh[c + d*x]])/(Sqrt[b]*Sqrt[a - a*Cosh[c + d*x]])], (-2*b)/(a - b)]*Sinh[c + d*x])/(a^(3/2)*Sqrt[-1 + Cos
h[c + d*x]]*Sqrt[1 + Cosh[c + d*x]]*Sqrt[-((a*(a + b)*Cosh[c + d*x])/(b*(a - a*Cosh[c + d*x])))]*(-((a - a*Cos
h[c + d*x])/a))^(3/2)*Sqrt[(a + a*Cosh[c + d*x])/a]*Sqrt[Sech[c + d*x]]) - (4*b*(a - a*Cosh[c + d*x])*Elliptic
Pi[(a + b)/a, ArcSin[(Sqrt[a]*Sqrt[b + a*Cosh[c + d*x]])/(Sqrt[a + b]*Sqrt[a*Cosh[c + d*x]])], (a + b)/(a - b)
]*Sqrt[-((b*(a + a*Cosh[c + d*x])*Sech[c + d*x])/(a*(a - b)))]*Sinh[c + d*x])/(Sqrt[a]*Sqrt[a + b]*Sqrt[-1 + C
osh[c + d*x]]*Sqrt[a*Cosh[c + d*x]]*Sqrt[1 + Cosh[c + d*x]]*Sqrt[-((a - a*Cosh[c + d*x])/a)]*Sqrt[(a + a*Cosh[
c + d*x])/a]*Sqrt[Sech[c + d*x]]*Sqrt[-((b*(a - a*Cosh[c + d*x])*Sech[c + d*x])/(a*(a + b)))])))/(2*d*Sqrt[b +
a*Cosh[c + d*x]]*Sqrt[Sech[c + d*x]])

Maple [F]

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

[In]

int(coth(d*x+c)^2*(a+b*sech(d*x+c))^(1/2),x)

[Out]

int(coth(d*x+c)^2*(a+b*sech(d*x+c))^(1/2),x)

Fricas [F(-1)]

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

[In]

integrate(coth(d*x+c)^2*(a+b*sech(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

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

[In]

integrate(coth(d*x+c)**2*(a+b*sech(d*x+c))**(1/2),x)

[Out]

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

Maxima [F]

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

[In]

integrate(coth(d*x+c)^2*(a+b*sech(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate(coth(d*x+c)^2*(a+b*sech(d*x+c))^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int(coth(c + d*x)^2*(a + b/cosh(c + d*x))^(1/2),x)

[Out]

int(coth(c + d*x)^2*(a + b/cosh(c + d*x))^(1/2), x)