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

Optimal. Leaf size=99 \[ \frac{\sqrt{b} (3 a+b) \tan ^{-1}\left (\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right )}{2 a^{3/2} d (a+b)^2}-\frac{b \cosh (c+d x)}{2 a d (a+b) \left (a \cosh ^2(c+d x)+b\right )}-\frac{\tanh ^{-1}(\cosh (c+d x))}{d (a+b)^2} \]

[Out]

(Sqrt[b]*(3*a + b)*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(2*a^(3/2)*(a + b)^2*d) - ArcTanh[Cosh[c + d*x]]/(
(a + b)^2*d) - (b*Cosh[c + d*x])/(2*a*(a + b)*d*(b + a*Cosh[c + d*x]^2))

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Rubi [A]  time = 0.129906, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {4133, 470, 522, 206, 205} \[ \frac{\sqrt{b} (3 a+b) \tan ^{-1}\left (\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right )}{2 a^{3/2} d (a+b)^2}-\frac{b \cosh (c+d x)}{2 a d (a+b) \left (a \cosh ^2(c+d x)+b\right )}-\frac{\tanh ^{-1}(\cosh (c+d x))}{d (a+b)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b]*(3*a + b)*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(2*a^(3/2)*(a + b)^2*d) - ArcTanh[Cosh[c + d*x]]/(
(a + b)^2*d) - (b*Cosh[c + d*x])/(2*a*(a + b)*d*(b + a*Cosh[c + d*x]^2))

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 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(a*e^(2
*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*n*(b*c - a*d)*(p + 1)), x] + Dist[e^(2
*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) +
(a*d*(m - n + n*q + 1) + b*c*n*(p + 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] && GtQ[m - n + 1, n] && 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}(c+d x)}{\left (a+b \text{sech}^2(c+d x)\right )^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^4}{\left (1-x^2\right ) \left (b+a x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{b \cosh (c+d x)}{2 a (a+b) d \left (b+a \cosh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{b+(-2 a-b) x^2}{\left (1-x^2\right ) \left (b+a x^2\right )} \, dx,x,\cosh (c+d x)\right )}{2 a (a+b) d}\\ &=-\frac{b \cosh (c+d x)}{2 a (a+b) d \left (b+a \cosh ^2(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{(a+b)^2 d}+\frac{(b (3 a+b)) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\cosh (c+d x)\right )}{2 a (a+b)^2 d}\\ &=\frac{\sqrt{b} (3 a+b) \tan ^{-1}\left (\frac{\sqrt{a} \cosh (c+d x)}{\sqrt{b}}\right )}{2 a^{3/2} (a+b)^2 d}-\frac{\tanh ^{-1}(\cosh (c+d x))}{(a+b)^2 d}-\frac{b \cosh (c+d x)}{2 a (a+b) d \left (b+a \cosh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [C]  time = 1.12061, size = 377, normalized size = 3.81 \[ \frac{\text{sech}^3(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (\frac{\sqrt{b} (3 a+b) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b) \tan ^{-1}\left (\frac{\sinh (c) \tanh \left (\frac{d x}{2}\right ) \left (\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right )+\cosh (c) \left (\sqrt{a}-i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left (\frac{d x}{2}\right )\right )}{\sqrt{b}}\right )}{a^{3/2}}+\frac{\sqrt{b} (3 a+b) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b) \tan ^{-1}\left (\frac{\sinh (c) \tanh \left (\frac{d x}{2}\right ) \left (\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2}\right )+\cosh (c) \left (\sqrt{a}+i \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh \left (\frac{d x}{2}\right )\right )}{\sqrt{b}}\right )}{a^{3/2}}-2 \text{sech}(c+d x) \log \left (\cosh \left (\frac{1}{2} (c+d x)\right )\right ) (a \cosh (2 (c+d x))+a+2 b)+2 \text{sech}(c+d x) \log \left (\sinh \left (\frac{1}{2} (c+d x)\right )\right ) (a \cosh (2 (c+d x))+a+2 b)-\frac{2 b (a+b)}{a}\right )}{8 d (a+b)^2 \left (a+b \text{sech}^2(c+d x)\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^3*((-2*b*(a + b))/a + (Sqrt[b]*(3*a + b)*ArcTan[((Sqrt[a] - I*S
qrt[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 + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x])/a^(3/2) + (Sqrt[b]*(3*a
+ 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 + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d
*x])/a^(3/2) - 2*(a + 2*b + a*Cosh[2*(c + d*x)])*Log[Cosh[(c + d*x)/2]]*Sech[c + d*x] + 2*(a + 2*b + a*Cosh[2*
(c + d*x)])*Log[Sinh[(c + d*x)/2]]*Sech[c + d*x]))/(8*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^2)

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Maple [B]  time = 0.07, size = 431, normalized size = 4.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d*b/(a+b)^2/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c
)^2*b+a+b)*tanh(1/2*d*x+1/2*c)^2+1/d*b^2/(a+b)^2/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d
*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)/a*tanh(1/2*d*x+1/2*c)^2-1/d*b/(a+b)^2/(tanh(1/2*d*x+1/2*c)^4*a+b*
tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)-1/d*b^2/(a+b)^2/(tanh(1/2*d*x+1
/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)/a+3/2/d*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))+1/2/d*b^2/(a+b)^2/a/(a*b)^(1/2)*arc
tan(1/4*(2*(a+b)*tanh(1/2*d*x+1/2*c)^2+2*a-2*b)/(a*b)^(1/2))+1/d/(a+b)^2*ln(tanh(1/2*d*x+1/2*c))

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

[Out]

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

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Fricas [B]  time = 3.3364, size = 5986, normalized size = 60.46 \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)/(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

[-1/4*(4*(a*b + b^2)*cosh(d*x + c)^3 + 12*(a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^2 + 4*(a*b + b^2)*sinh(d*x +
c)^3 - ((3*a^2 + a*b)*cosh(d*x + c)^4 + 4*(3*a^2 + a*b)*cosh(d*x + c)*sinh(d*x + c)^3 + (3*a^2 + a*b)*sinh(d*
x + c)^4 + 2*(3*a^2 + 7*a*b + 2*b^2)*cosh(d*x + c)^2 + 2*(3*(3*a^2 + a*b)*cosh(d*x + c)^2 + 3*a^2 + 7*a*b + 2*
b^2)*sinh(d*x + c)^2 + 3*a^2 + a*b + 4*((3*a^2 + a*b)*cosh(d*x + c)^3 + (3*a^2 + 7*a*b + 2*b^2)*cosh(d*x + c))
*sinh(d*x + c))*sqrt(-b/a)*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*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3
+ a*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 + a)*sinh(d*x + c))*sqrt(-b/a) + 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)*s
inh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 + (a + 2*b)*cosh(d*x + c))*sinh(d*x + c) + a)) + 4*(a*b + b^2)*cosh(d*x
+ c) + 4*(a^2*cosh(d*x + c)^4 + 4*a^2*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*sinh(d*x + c)^4 + 2*(a^2 + 2*a*b)*co
sh(d*x + c)^2 + 2*(3*a^2*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 + 4*(a^2*cosh(d*x + c)^3 + (a^2
+ 2*a*b)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) - 4*(a^2*cosh(d*x + c)^4 + 4*a^2
*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*sinh(d*x + c)^4 + 2*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 2*(3*a^2*cosh(d*x + c
)^2 + a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 + 4*(a^2*cosh(d*x + c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c)
)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 4*(3*(a*b + b^2)*cosh(d*x + c)^2 + a*b + b^2)*sinh(d*x + c))/((a^4
+ 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^4 + 4*(a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 +
2*a^3*b + a^2*b^2)*d*sinh(d*x + c)^4 + 2*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*d*cosh(d*x + c)^2 + 2*(3*(a^4 +
2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^2 + (a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*d)*sinh(d*x + c)^2 + (a^4 + 2*a^
3*b + a^2*b^2)*d + 4*((a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^3 + (a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*d*co
sh(d*x + c))*sinh(d*x + c)), -1/2*(2*(a*b + b^2)*cosh(d*x + c)^3 + 6*(a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^2
+ 2*(a*b + b^2)*sinh(d*x + c)^3 + ((3*a^2 + a*b)*cosh(d*x + c)^4 + 4*(3*a^2 + a*b)*cosh(d*x + c)*sinh(d*x + c
)^3 + (3*a^2 + a*b)*sinh(d*x + c)^4 + 2*(3*a^2 + 7*a*b + 2*b^2)*cosh(d*x + c)^2 + 2*(3*(3*a^2 + a*b)*cosh(d*x
+ c)^2 + 3*a^2 + 7*a*b + 2*b^2)*sinh(d*x + c)^2 + 3*a^2 + a*b + 4*((3*a^2 + a*b)*cosh(d*x + c)^3 + (3*a^2 + 7*
a*b + 2*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/a)*arctan(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(b
/a)/b) - ((3*a^2 + a*b)*cosh(d*x + c)^4 + 4*(3*a^2 + a*b)*cosh(d*x + c)*sinh(d*x + c)^3 + (3*a^2 + a*b)*sinh(d
*x + c)^4 + 2*(3*a^2 + 7*a*b + 2*b^2)*cosh(d*x + c)^2 + 2*(3*(3*a^2 + a*b)*cosh(d*x + c)^2 + 3*a^2 + 7*a*b + 2
*b^2)*sinh(d*x + c)^2 + 3*a^2 + a*b + 4*((3*a^2 + a*b)*cosh(d*x + c)^3 + (3*a^2 + 7*a*b + 2*b^2)*cosh(d*x + c)
)*sinh(d*x + c))*sqrt(b/a)*arctan(1/2*(a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt(b/a)/b) + 2*(a*b + b^2)*cosh(d*
x + c) + 2*(a^2*cosh(d*x + c)^4 + 4*a^2*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*sinh(d*x + c)^4 + 2*(a^2 + 2*a*b)*
cosh(d*x + c)^2 + 2*(3*a^2*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 + 4*(a^2*cosh(d*x + c)^3 + (a^
2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) - 2*(a^2*cosh(d*x + c)^4 + 4*a
^2*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*sinh(d*x + c)^4 + 2*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 2*(3*a^2*cosh(d*x +
c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 + 4*(a^2*cosh(d*x + c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x +
c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 2*(3*(a*b + b^2)*cosh(d*x + c)^2 + a*b + b^2)*sinh(d*x + c))/((a^
4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^4 + 4*(a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4
+ 2*a^3*b + a^2*b^2)*d*sinh(d*x + c)^4 + 2*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*d*cosh(d*x + c)^2 + 2*(3*(a^4
+ 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^2 + (a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*d)*sinh(d*x + c)^2 + (a^4 + 2*
a^3*b + a^2*b^2)*d + 4*((a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^3 + (a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*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}{\left (c + d x \right )}}{\left (a + b \operatorname{sech}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: TypeError