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

Optimal. Leaf size=87 \[ -\frac{\sqrt{a} \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right )}{d (a+b)^2}+\frac{(a-b) \tanh ^{-1}(\cosh (c+d x))}{2 d (a+b)^2}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 d (a+b)} \]

[Out]

-((Sqrt[a]*Sqrt[b]*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/((a + b)^2*d)) + ((a - b)*ArcTanh[Cosh[c + d*x]])/
(2*(a + b)^2*d) - (Coth[c + d*x]*Csch[c + d*x])/(2*(a + b)*d)

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Rubi [A]  time = 0.11841, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {4133, 471, 522, 206, 205} \[ -\frac{\sqrt{a} \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right )}{d (a+b)^2}+\frac{(a-b) \tanh ^{-1}(\cosh (c+d x))}{2 d (a+b)^2}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 d (a+b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[a]*Sqrt[b]*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/((a + b)^2*d)) + ((a - b)*ArcTanh[Cosh[c + d*x]])/
(2*(a + b)^2*d) - (Coth[c + d*x]*Csch[c + d*x])/(2*(a + b)*d)

Rule 4133

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = F
reeFactors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[((1 - ff^2*x^2)^((m - 1)/2)*(b + a*(ff*x)^n)^p)/(ff*x)^(n*
p), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p
]

Rule 471

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e^(n -
1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(n*(b*c - a*d)*(p + 1)), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\text{csch}^3(c+d x)}{a+b \text{sech}^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{\left (1-x^2\right )^2 \left (b+a x^2\right )} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 (a+b) d}-\frac{\operatorname{Subst}\left (\int \frac{b-a x^2}{\left (1-x^2\right ) \left (b+a x^2\right )} \, dx,x,\cosh (c+d x)\right )}{2 (a+b) d}\\ &=-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 (a+b) d}+\frac{(a-b) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{2 (a+b)^2 d}-\frac{(a b) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\cosh (c+d x)\right )}{(a+b)^2 d}\\ &=-\frac{\sqrt{a} \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right )}{(a+b)^2 d}+\frac{(a-b) \tanh ^{-1}(\cosh (c+d x))}{2 (a+b)^2 d}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 (a+b) d}\\ \end{align*}

Mathematica [C]  time = 1.95443, size = 338, normalized size = 3.89 \[ -\frac{\text{sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left ((a+b) \text{csch}^2\left (\frac{1}{2} (c+d x)\right )+(a+b) \text{sech}^2\left (\frac{1}{2} (c+d x)\right )+8 \sqrt{a} \sqrt{b} \tan ^{-1}\left (\frac{\sinh (c) \tanh \left (\frac{d x}{2}\right ) \left (\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right )+\cosh (c) \left (\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left (\frac{d x}{2}\right )\right )}{\sqrt{b}}\right )+8 \sqrt{a} \sqrt{b} \tan ^{-1}\left (\frac{\sinh (c) \tanh \left (\frac{d x}{2}\right ) \left (\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right )+\cosh (c) \left (\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left (\frac{d x}{2}\right )\right )}{\sqrt{b}}\right )+4 a \log \left (\sinh \left (\frac{1}{2} (c+d x)\right )\right )-4 a \log \left (\cosh \left (\frac{1}{2} (c+d x)\right )\right )-4 b \log \left (\sinh \left (\frac{1}{2} (c+d x)\right )\right )+4 b \log \left (\cosh \left (\frac{1}{2} (c+d x)\right )\right )\right )}{16 d (a+b)^2 \left (a+b \text{sech}^2(c+d x)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + 2*b + a*Cosh[2*(c + d*x)])*(8*Sqrt[a]*Sqrt[b]*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])
^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt
[b]] + 8*Sqrt[a]*Sqrt[b]*ArcTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] +
Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] + (a + b)*Csch[(c + d*x
)/2]^2 - 4*a*Log[Cosh[(c + d*x)/2]] + 4*b*Log[Cosh[(c + d*x)/2]] + 4*a*Log[Sinh[(c + d*x)/2]] - 4*b*Log[Sinh[(
c + d*x)/2]] + (a + b)*Sech[(c + d*x)/2]^2)*Sech[c + d*x]^2)/(16*(a + b)^2*d*(a + b*Sech[c + d*x]^2))

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Maple [A]  time = 0.058, size = 134, normalized size = 1.5 \begin{align*}{\frac{1}{8\,d \left ( a+b \right ) } \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{ab}{d \left ( a+b \right ) ^{2}}\arctan \left ({\frac{1}{4} \left ( 2\, \left ( a+b \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+2\,a-2\,b \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{8\,d \left ( a+b \right ) } \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}-{\frac{a}{2\,d \left ( a+b \right ) ^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }+{\frac{b}{2\,d \left ( a+b \right ) ^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/d*tanh(1/2*d*x+1/2*c)^2/(a+b)-1/d*a*b/(a+b)^2/(a*b)^(1/2)*arctan(1/4*(2*(a+b)*tanh(1/2*d*x+1/2*c)^2+2*a-2*
b)/(a*b)^(1/2))-1/8/d/(a+b)/tanh(1/2*d*x+1/2*c)^2-1/2/d/(a+b)^2*ln(tanh(1/2*d*x+1/2*c))*a+1/2/d/(a+b)^2*ln(tan
h(1/2*d*x+1/2*c))*b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (a - b\right )} \log \left ({\left (e^{\left (d x + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{2 \,{\left (a^{2} d + 2 \, a b d + b^{2} d\right )}} - \frac{{\left (a - b\right )} \log \left ({\left (e^{\left (d x + c\right )} - 1\right )} e^{\left (-c\right )}\right )}{2 \,{\left (a^{2} d + 2 \, a b d + b^{2} d\right )}} - \frac{e^{\left (3 \, d x + 3 \, c\right )} + e^{\left (d x + c\right )}}{a d + b d +{\left (a d e^{\left (4 \, c\right )} + b d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} - 2 \,{\left (a d e^{\left (2 \, c\right )} + b d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}} - 8 \, \int \frac{a b e^{\left (3 \, d x + 3 \, c\right )} - a b e^{\left (d x + c\right )}}{4 \,{\left (a^{3} + 2 \, a^{2} b + a b^{2} +{\left (a^{3} e^{\left (4 \, c\right )} + 2 \, a^{2} b e^{\left (4 \, c\right )} + a b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (a^{3} e^{\left (2 \, c\right )} + 4 \, a^{2} b e^{\left (2 \, c\right )} + 5 \, a b^{2} e^{\left (2 \, c\right )} + 2 \, b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a - b)*log((e^(d*x + c) + 1)*e^(-c))/(a^2*d + 2*a*b*d + b^2*d) - 1/2*(a - b)*log((e^(d*x + c) - 1)*e^(-c)
)/(a^2*d + 2*a*b*d + b^2*d) - (e^(3*d*x + 3*c) + e^(d*x + c))/(a*d + b*d + (a*d*e^(4*c) + b*d*e^(4*c))*e^(4*d*
x) - 2*(a*d*e^(2*c) + b*d*e^(2*c))*e^(2*d*x)) - 8*integrate(1/4*(a*b*e^(3*d*x + 3*c) - a*b*e^(d*x + c))/(a^3 +
2*a^2*b + a*b^2 + (a^3*e^(4*c) + 2*a^2*b*e^(4*c) + a*b^2*e^(4*c))*e^(4*d*x) + 2*(a^3*e^(2*c) + 4*a^2*b*e^(2*c
) + 5*a*b^2*e^(2*c) + 2*b^3*e^(2*c))*e^(2*d*x)), x)

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Fricas [B]  time = 3.18611, size = 5080, normalized size = 58.39 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(2*(a + b)*cosh(d*x + c)^3 + 6*(a + b)*cosh(d*x + c)*sinh(d*x + c)^2 + 2*(a + b)*sinh(d*x + c)^3 - (cosh
(d*x + c)^4 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4 + 2*(3*cosh(d*x + c)^2 - 1)*sinh(d*x + c)^2 -
2*cosh(d*x + c)^2 + 4*(cosh(d*x + c)^3 - cosh(d*x + c))*sinh(d*x + c) + 1)*sqrt(-a*b)*log((a*cosh(d*x + c)^4 +
4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 + 2*(a - 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2
+ a - 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 + (a - 2*b)*cosh(d*x + c))*sinh(d*x + c) - 4*(cosh(d*x + c)^
3 + 3*cosh(d*x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3 + (3*cosh(d*x + c)^2 + 1)*sinh(d*x + c) + cosh(d*x + c))
*sqrt(-a*b) + a)/(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 + 2*(a + 2*b)*cosh
(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 + a + 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 + (a + 2*b)*cosh(d*x +
c))*sinh(d*x + c) + a)) + 2*(a + b)*cosh(d*x + c) - ((a - b)*cosh(d*x + c)^4 + 4*(a - b)*cosh(d*x + c)*sinh(d*
x + c)^3 + (a - b)*sinh(d*x + c)^4 - 2*(a - b)*cosh(d*x + c)^2 + 2*(3*(a - b)*cosh(d*x + c)^2 - a + b)*sinh(d*
x + c)^2 + 4*((a - b)*cosh(d*x + c)^3 - (a - b)*cosh(d*x + c))*sinh(d*x + c) + a - b)*log(cosh(d*x + c) + sinh
(d*x + c) + 1) + ((a - b)*cosh(d*x + c)^4 + 4*(a - b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a - b)*sinh(d*x + c)^4
- 2*(a - b)*cosh(d*x + c)^2 + 2*(3*(a - b)*cosh(d*x + c)^2 - a + b)*sinh(d*x + c)^2 + 4*((a - b)*cosh(d*x + c)
^3 - (a - b)*cosh(d*x + c))*sinh(d*x + c) + a - b)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 2*(3*(a + b)*cosh(
d*x + c)^2 + a + b)*sinh(d*x + c))/((a^2 + 2*a*b + b^2)*d*cosh(d*x + c)^4 + 4*(a^2 + 2*a*b + b^2)*d*cosh(d*x +
c)*sinh(d*x + c)^3 + (a^2 + 2*a*b + b^2)*d*sinh(d*x + c)^4 - 2*(a^2 + 2*a*b + b^2)*d*cosh(d*x + c)^2 + 2*(3*(
a^2 + 2*a*b + b^2)*d*cosh(d*x + c)^2 - (a^2 + 2*a*b + b^2)*d)*sinh(d*x + c)^2 + (a^2 + 2*a*b + b^2)*d + 4*((a^
2 + 2*a*b + b^2)*d*cosh(d*x + c)^3 - (a^2 + 2*a*b + b^2)*d*cosh(d*x + c))*sinh(d*x + c)), -1/2*(2*(a + b)*cosh
(d*x + c)^3 + 6*(a + b)*cosh(d*x + c)*sinh(d*x + c)^2 + 2*(a + b)*sinh(d*x + c)^3 + 2*(cosh(d*x + c)^4 + 4*cos
h(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4 + 2*(3*cosh(d*x + c)^2 - 1)*sinh(d*x + c)^2 - 2*cosh(d*x + c)^2 +
4*(cosh(d*x + c)^3 - cosh(d*x + c))*sinh(d*x + c) + 1)*sqrt(a*b)*arctan(1/2*sqrt(a*b)*(cosh(d*x + c) + sinh(d
*x + c))/b) - 2*(cosh(d*x + c)^4 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4 + 2*(3*cosh(d*x + c)^2 -
1)*sinh(d*x + c)^2 - 2*cosh(d*x + c)^2 + 4*(cosh(d*x + c)^3 - cosh(d*x + c))*sinh(d*x + c) + 1)*sqrt(a*b)*arct
an(1/2*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3 + (a + 4*b)*cosh(d*x + c) +
(3*a*cosh(d*x + c)^2 + a + 4*b)*sinh(d*x + c))*sqrt(a*b)/(a*b)) + 2*(a + b)*cosh(d*x + c) - ((a - b)*cosh(d*x
+ c)^4 + 4*(a - b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a - b)*sinh(d*x + c)^4 - 2*(a - b)*cosh(d*x + c)^2 + 2*(3*
(a - b)*cosh(d*x + c)^2 - a + b)*sinh(d*x + c)^2 + 4*((a - b)*cosh(d*x + c)^3 - (a - b)*cosh(d*x + c))*sinh(d*
x + c) + a - b)*log(cosh(d*x + c) + sinh(d*x + c) + 1) + ((a - b)*cosh(d*x + c)^4 + 4*(a - b)*cosh(d*x + c)*si
nh(d*x + c)^3 + (a - b)*sinh(d*x + c)^4 - 2*(a - b)*cosh(d*x + c)^2 + 2*(3*(a - b)*cosh(d*x + c)^2 - a + b)*si
nh(d*x + c)^2 + 4*((a - b)*cosh(d*x + c)^3 - (a - b)*cosh(d*x + c))*sinh(d*x + c) + a - b)*log(cosh(d*x + c) +
sinh(d*x + c) - 1) + 2*(3*(a + b)*cosh(d*x + c)^2 + a + b)*sinh(d*x + c))/((a^2 + 2*a*b + b^2)*d*cosh(d*x + c
)^4 + 4*(a^2 + 2*a*b + b^2)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + 2*a*b + b^2)*d*sinh(d*x + c)^4 - 2*(a^2 +
2*a*b + b^2)*d*cosh(d*x + c)^2 + 2*(3*(a^2 + 2*a*b + b^2)*d*cosh(d*x + c)^2 - (a^2 + 2*a*b + b^2)*d)*sinh(d*x
+ c)^2 + (a^2 + 2*a*b + b^2)*d + 4*((a^2 + 2*a*b + b^2)*d*cosh(d*x + c)^3 - (a^2 + 2*a*b + b^2)*d*cosh(d*x +
c))*sinh(d*x + c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}^{3}{\left (c + d x \right )}}{a + b \operatorname{sech}^{2}{\left (c + d x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError