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

Optimal. Leaf size=246 \[ \frac{\sqrt{a+b} \coth (c+d x) \sqrt{\frac{b (1-\text{sech}(c+d x))}{a+b}} \sqrt{-\frac{b (\text{sech}(c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{d}-\frac{\coth (c+d x) \sqrt{a+b \text{sech}(c+d x)}}{d}+\frac{2 \coth (c+d x) \sqrt{-\frac{b (1-\text{sech}(c+d x))}{a+b \text{sech}(c+d x)}} \sqrt{\frac{b (\text{sech}(c+d x)+1)}{a+b \text{sech}(c+d x)}} (a+b \text{sech}(c+d x)) \Pi \left (\frac{a}{a+b};\sin ^{-1}\left (\frac{\sqrt{a+b}}{\sqrt{a+b \text{sech}(c+d x)}}\right )|\frac{a-b}{a+b}\right )}{d \sqrt{a+b}} \]

[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)

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Rubi [A]  time = 0.215404, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3896, 3780, 3875, 3832} \[ -\frac{\coth (c+d x) \sqrt{a+b \text{sech}(c+d x)}}{d}+\frac{\sqrt{a+b} \coth (c+d x) \sqrt{\frac{b (1-\text{sech}(c+d x))}{a+b}} \sqrt{-\frac{b (\text{sech}(c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{d}+\frac{2 \coth (c+d x) \sqrt{-\frac{b (1-\text{sech}(c+d x))}{a+b \text{sech}(c+d x)}} \sqrt{\frac{b (\text{sech}(c+d x)+1)}{a+b \text{sech}(c+d x)}} (a+b \text{sech}(c+d x)) \Pi \left (\frac{a}{a+b};\sin ^{-1}\left (\frac{\sqrt{a+b}}{\sqrt{a+b \text{sech}(c+d x)}}\right )|\frac{a-b}{a+b}\right )}{d \sqrt{a+b}} \]

Antiderivative was successfully verified.

[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 3896

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]

Rule 3780

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

Rule 3875

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 3832

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

Rubi steps

\begin{align*} \int \coth ^2(c+d x) \sqrt{a+b \text{sech}(c+d x)} \, dx &=-\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) \Pi \left (\frac{a}{a+b};\sin ^{-1}\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) F\left (\sin ^{-1}\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) \Pi \left (\frac{a}{a+b};\sin ^{-1}\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]  time = 18.3135, size = 539, normalized size = 2.19 \[ \frac{\sqrt{a+b \text{sech}(c+d x)} \left (\frac{2 \sqrt{b} \sinh (c+d x) (a-a \cosh (c+d x))^{3/2} \sqrt{\frac{(a+b) (a \cosh (c+d x)+a)}{(a-b) (a-a \cosh (c+d x))}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a} \sqrt{a \cosh (c+d x)+b}}{\sqrt{b} \sqrt{a-a \cosh (c+d x)}}\right ),-\frac{2 b}{a-b}\right )}{a^{3/2} \sqrt{\cosh (c+d x)-1} \sqrt{\cosh (c+d x)+1} \sqrt{\text{sech}(c+d x)} \left (-\frac{a-a \cosh (c+d x)}{a}\right )^{3/2} \sqrt{\frac{a \cosh (c+d x)+a}{a}} \sqrt{-\frac{a (a+b) \cosh (c+d x)}{b (a-a \cosh (c+d x))}}}-\frac{4 b \sinh (c+d x) (a-a \cosh (c+d x)) \sqrt{-\frac{b \text{sech}(c+d x) (a \cosh (c+d x)+a)}{a (a-b)}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a} \sqrt{b+a \cosh (c+d x)}}{\sqrt{a+b} \sqrt{a \cosh (c+d x)}}\right )|\frac{a+b}{a-b}\right )}{\sqrt{a} \sqrt{a+b} \sqrt{\cosh (c+d x)-1} \sqrt{\cosh (c+d x)+1} \sqrt{\text{sech}(c+d x)} \sqrt{a \cosh (c+d x)} \sqrt{-\frac{a-a \cosh (c+d x)}{a}} \sqrt{\frac{a \cosh (c+d x)+a}{a}} \sqrt{-\frac{b \text{sech}(c+d x) (a-a \cosh (c+d x))}{a (a+b)}}}\right )}{2 d \sqrt{\text{sech}(c+d x)} \sqrt{a \cosh (c+d x)+b}}-\frac{\coth (c+d x) \sqrt{a+b \text{sech}(c+d x)}}{d} \]

Antiderivative was successfully verified.

[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]])

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Maple [F]  time = 0.196, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm coth} \left (dx+c\right ) \right ) ^{2}\sqrt{a+b{\rm sech} \left (dx+c\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \operatorname{sech}\left (d x + c\right ) + a} \coth \left (d x + c\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{b \operatorname{sech}\left (d x + c\right ) + a} \coth \left (d x + c\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \operatorname{sech}{\left (c + d x \right )}} \coth ^{2}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \operatorname{sech}\left (d x + c\right ) + a} \coth \left (d x + c\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)