3.1.38 \(\int \frac {\text {csch}^2(c+d x)}{(a+b \tanh ^2(c+d x))^2} \, dx\) [38]

Optimal. Leaf size=82 \[ -\frac {3 \sqrt {b} \text {ArcTan}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{5/2} d}-\frac {3 \coth (c+d x)}{2 a^2 d}+\frac {\coth (c+d x)}{2 a d \left (a+b \tanh ^2(c+d x)\right )} \]

[Out]

-3/2*coth(d*x+c)/a^2/d-3/2*arctan(b^(1/2)*tanh(d*x+c)/a^(1/2))*b^(1/2)/a^(5/2)/d+1/2*coth(d*x+c)/a/d/(a+b*tanh
(d*x+c)^2)

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Rubi [A]
time = 0.05, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3744, 296, 331, 211} \begin {gather*} -\frac {3 \sqrt {b} \text {ArcTan}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{5/2} d}-\frac {3 \coth (c+d x)}{2 a^2 d}+\frac {\coth (c+d x)}{2 a d \left (a+b \tanh ^2(c+d x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(5/2)*d) - (3*Coth[c + d*x])/(2*a^2*d) + Coth[c + d*
x]/(2*a*d*(a + b*Tanh[c + d*x]^2))

Rule 211

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

Rule 296

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(c*x)^(m + 1))*((a + b*x^n)^(p + 1)/
(a*c*n*(p + 1))), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 3744

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

Rubi steps

\begin {align*} \int \frac {\text {csch}^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\coth (c+d x)}{2 a d \left (a+b \tanh ^2(c+d x)\right )}+\frac {3 \text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{2 a d}\\ &=-\frac {3 \coth (c+d x)}{2 a^2 d}+\frac {\coth (c+d x)}{2 a d \left (a+b \tanh ^2(c+d x)\right )}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{2 a^2 d}\\ &=-\frac {3 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{5/2} d}-\frac {3 \coth (c+d x)}{2 a^2 d}+\frac {\coth (c+d x)}{2 a d \left (a+b \tanh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [A]
time = 0.35, size = 86, normalized size = 1.05 \begin {gather*} \frac {-3 \sqrt {b} \text {ArcTan}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )-2 \sqrt {a} \coth (c+d x)-\frac {\sqrt {a} b \sinh (2 (c+d x))}{a-b+(a+b) \cosh (2 (c+d x))}}{2 a^{5/2} d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]] - 2*Sqrt[a]*Coth[c + d*x] - (Sqrt[a]*b*Sinh[2*(c + d*x)])/
(a - b + (a + b)*Cosh[2*(c + d*x)]))/(2*a^(5/2)*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(265\) vs. \(2(68)=136\).
time = 2.81, size = 266, normalized size = 3.24

method	result	size
risch	\(-\frac {2 a^{2} {\mathrm e}^{4 d x +4 c}+3 a b \,{\mathrm e}^{4 d x +4 c}+3 b^{2} {\mathrm e}^{4 d x +4 c}+4 a^{2} {\mathrm e}^{2 d x +2 c}-6 b^{2} {\mathrm e}^{2 d x +2 c}+2 a^{2}+5 a b +3 b^{2}}{d \left (a +b \right ) a^{2} \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right ) \left ({\mathrm e}^{2 d x +2 c}-1\right )}+\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{4 a^{3} d}-\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a +2 \sqrt {-a b}-b}{a +b}\right )}{4 a^{3} d}\)	\(252\)
derivativedivides	\(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{2}}+\frac {4 b \left (\frac {-\frac {\left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {3 a \left (-\frac {\left (-a +\sqrt {b \left (a +b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {\left (a +\sqrt {b \left (a +b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{4}\right )}{a^{2}}-\frac {1}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(266\)
default	\(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{2}}+\frac {4 b \left (\frac {-\frac {\left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {3 a \left (-\frac {\left (-a +\sqrt {b \left (a +b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {\left (a +\sqrt {b \left (a +b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{4}\right )}{a^{2}}-\frac {1}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(266\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^2/(a+b*tanh(d*x+c)^2)^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/2/a^2*tanh(1/2*d*x+1/2*c)+4*b/a^2*((-1/4*tanh(1/2*d*x+1/2*c)^3-1/4*tanh(1/2*d*x+1/2*c))/(a*tanh(1/2*d*
x+1/2*c)^4+2*a*tanh(1/2*d*x+1/2*c)^2+4*b*tanh(1/2*d*x+1/2*c)^2+a)+3/4*a*(-1/2*(-a+(b*(a+b))^(1/2)-b)/a/(b*(a+b
))^(1/2)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)
)+1/2*(a+(b*(a+b))^(1/2)+b)/a/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)
/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))))-1/2/a^2/tanh(1/2*d*x+1/2*c))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (68) = 136\).
time = 0.60, size = 212, normalized size = 2.59 \begin {gather*} -\frac {2 \, a^{2} + 5 \, a b + 3 \, b^{2} + 2 \, {\left (2 \, a^{2} - 3 \, b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (2 \, a^{2} + 3 \, a b + 3 \, b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{{\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2} + {\left (a^{4} - 2 \, a^{3} b - 3 \, a^{2} b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - {\left (a^{4} - 2 \, a^{3} b - 3 \, a^{2} b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )} d} + \frac {3 \, b \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{2} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*a^2 + 5*a*b + 3*b^2 + 2*(2*a^2 - 3*b^2)*e^(-2*d*x - 2*c) + (2*a^2 + 3*a*b + 3*b^2)*e^(-4*d*x - 4*c))/((a^4
+ 2*a^3*b + a^2*b^2 + (a^4 - 2*a^3*b - 3*a^2*b^2)*e^(-2*d*x - 2*c) - (a^4 - 2*a^3*b - 3*a^2*b^2)*e^(-4*d*x -
4*c) - (a^4 + 2*a^3*b + a^2*b^2)*e^(-6*d*x - 6*c))*d) + 3/2*b*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sq
rt(a*b))/(sqrt(a*b)*a^2*d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1120 vs. \(2 (68) = 136\).
time = 0.41, size = 2562, normalized size = 31.24 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(4*(2*a^2 + 3*a*b + 3*b^2)*cosh(d*x + c)^4 + 16*(2*a^2 + 3*a*b + 3*b^2)*cosh(d*x + c)*sinh(d*x + c)^3 +
4*(2*a^2 + 3*a*b + 3*b^2)*sinh(d*x + c)^4 + 8*(2*a^2 - 3*b^2)*cosh(d*x + c)^2 + 8*(3*(2*a^2 + 3*a*b + 3*b^2)*c
osh(d*x + c)^2 + 2*a^2 - 3*b^2)*sinh(d*x + c)^2 - 3*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 + 6*(a^2 + 2*a*b + b^
2)*cosh(d*x + c)*sinh(d*x + c)^5 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^6 + (a^2 - 2*a*b - 3*b^2)*cosh(d*x + c)^4
+ (15*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 - 2*a*b - 3*b^2)*sinh(d*x + c)^4 + 4*(5*(a^2 + 2*a*b + b^2)*c
osh(d*x + c)^3 + (a^2 - 2*a*b - 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - (a^2 - 2*a*b - 3*b^2)*cosh(d*x + c)^2
+ (15*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 6*(a^2 - 2*a*b - 3*b^2)*cosh(d*x + c)^2 - a^2 + 2*a*b + 3*b^2)*sin
h(d*x + c)^2 - a^2 - 2*a*b - b^2 + 2*(3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^5 + 2*(a^2 - 2*a*b - 3*b^2)*cosh(d*x
+ c)^3 - (a^2 - 2*a*b - 3*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-b/a)*log(((a^2 + 2*a*b + b^2)*cosh(d*x + c
)^4 + 4*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 + 2*(a^2 - b^2
)*cosh(d*x + c)^2 + 2*(3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 - b^2)*sinh(d*x + c)^2 + a^2 - 6*a*b + b^2
+ 4*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + (a^2 - b^2)*cosh(d*x + c))*sinh(d*x + c) - 4*((a^2 + a*b)*cosh(d*x
+ c)^2 + 2*(a^2 + a*b)*cosh(d*x + c)*sinh(d*x + c) + (a^2 + a*b)*sinh(d*x + c)^2 + a^2 - a*b)*sqrt(-b/a))/((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)) + 8*a^2 + 20*a*b + 12*b^2 + 16*((2*a^2 + 3*a*b + 3*b^2)*cosh(d*x + c)^3 + (2*a^
2 - 3*b^2)*cosh(d*x + c))*sinh(d*x + c))/((a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^6 + 6*(a^4 + 2*a^3*b + a^2
*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^5 + (a^4 + 2*a^3*b + a^2*b^2)*d*sinh(d*x + c)^6 + (a^4 - 2*a^3*b - 3*a^2*b
^2)*d*cosh(d*x + c)^4 + (15*(a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^2 + (a^4 - 2*a^3*b - 3*a^2*b^2)*d)*sinh(
d*x + c)^4 - (a^4 - 2*a^3*b - 3*a^2*b^2)*d*cosh(d*x + c)^2 + 4*(5*(a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^3
+ (a^4 - 2*a^3*b - 3*a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (15*(a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)
^4 + 6*(a^4 - 2*a^3*b - 3*a^2*b^2)*d*cosh(d*x + c)^2 - (a^4 - 2*a^3*b - 3*a^2*b^2)*d)*sinh(d*x + c)^2 - (a^4 +
2*a^3*b + a^2*b^2)*d + 2*(3*(a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^5 + 2*(a^4 - 2*a^3*b - 3*a^2*b^2)*d*cos
h(d*x + c)^3 - (a^4 - 2*a^3*b - 3*a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)), -1/2*(2*(2*a^2 + 3*a*b + 3*b^2)*co
sh(d*x + c)^4 + 8*(2*a^2 + 3*a*b + 3*b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + 2*(2*a^2 + 3*a*b + 3*b^2)*sinh(d*x +
c)^4 + 4*(2*a^2 - 3*b^2)*cosh(d*x + c)^2 + 4*(3*(2*a^2 + 3*a*b + 3*b^2)*cosh(d*x + c)^2 + 2*a^2 - 3*b^2)*sinh
(d*x + c)^2 + 3*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 + 6*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^5 + (
a^2 + 2*a*b + b^2)*sinh(d*x + c)^6 + (a^2 - 2*a*b - 3*b^2)*cosh(d*x + c)^4 + (15*(a^2 + 2*a*b + b^2)*cosh(d*x
+ c)^2 + a^2 - 2*a*b - 3*b^2)*sinh(d*x + c)^4 + 4*(5*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + (a^2 - 2*a*b - 3*b^
2)*cosh(d*x + c))*sinh(d*x + c)^3 - (a^2 - 2*a*b - 3*b^2)*cosh(d*x + c)^2 + (15*(a^2 + 2*a*b + b^2)*cosh(d*x +
c)^4 + 6*(a^2 - 2*a*b - 3*b^2)*cosh(d*x + c)^2 - a^2 + 2*a*b + 3*b^2)*sinh(d*x + c)^2 - a^2 - 2*a*b - b^2 + 2
*(3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^5 + 2*(a^2 - 2*a*b - 3*b^2)*cosh(d*x + c)^3 - (a^2 - 2*a*b - 3*b^2)*cosh
(d*x + c))*sinh(d*x + c))*sqrt(b/a)*arctan(1/2*((a + b)*cosh(d*x + c)^2 + 2*(a + b)*cosh(d*x + c)*sinh(d*x + c
) + (a + b)*sinh(d*x + c)^2 + a - b)*sqrt(b/a)/b) + 4*a^2 + 10*a*b + 6*b^2 + 8*((2*a^2 + 3*a*b + 3*b^2)*cosh(d
*x + c)^3 + (2*a^2 - 3*b^2)*cosh(d*x + c))*sinh(d*x + c))/((a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^6 + 6*(a^
4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^5 + (a^4 + 2*a^3*b + a^2*b^2)*d*sinh(d*x + c)^6 + (a^4 -
2*a^3*b - 3*a^2*b^2)*d*cosh(d*x + c)^4 + (15*(a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^2 + (a^4 - 2*a^3*b - 3*
a^2*b^2)*d)*sinh(d*x + c)^4 - (a^4 - 2*a^3*b - 3*a^2*b^2)*d*cosh(d*x + c)^2 + 4*(5*(a^4 + 2*a^3*b + a^2*b^2)*d
*cosh(d*x + c)^3 + (a^4 - 2*a^3*b - 3*a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (15*(a^4 + 2*a^3*b + a^2*b^2
)*d*cosh(d*x + c)^4 + 6*(a^4 - 2*a^3*b - 3*a^2*b^2)*d*cosh(d*x + c)^2 - (a^4 - 2*a^3*b - 3*a^2*b^2)*d)*sinh(d*
x + c)^2 - (a^4 + 2*a^3*b + a^2*b^2)*d + 2*(3*(a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^5 + 2*(a^4 - 2*a^3*b -
3*a^2*b^2)*d*cosh(d*x + c)^3 - (a^4 - 2*a^3*b - 3*a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c))]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {csch}^{2}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (68) = 136\).
time = 0.72, size = 227, normalized size = 2.77 \begin {gather*} -\frac {\frac {3 \, b \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {2 \, {\left (2 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 6 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{2} + 5 \, a b + 3 \, b^{2}\right )}}{{\left (a^{3} + a^{2} b\right )} {\left (a e^{\left (6 \, d x + 6 \, c\right )} + b e^{\left (6 \, d x + 6 \, c\right )} + a e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b e^{\left (4 \, d x + 4 \, c\right )} - a e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} - a - b\right )}}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(3*b*arctan(1/2*(a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/(sqrt(a*b)*a^2) + 2*(2*a^2*e^(
4*d*x + 4*c) + 3*a*b*e^(4*d*x + 4*c) + 3*b^2*e^(4*d*x + 4*c) + 4*a^2*e^(2*d*x + 2*c) - 6*b^2*e^(2*d*x + 2*c) +
2*a^2 + 5*a*b + 3*b^2)/((a^3 + a^2*b)*(a*e^(6*d*x + 6*c) + b*e^(6*d*x + 6*c) + a*e^(4*d*x + 4*c) - 3*b*e^(4*d
*x + 4*c) - a*e^(2*d*x + 2*c) + 3*b*e^(2*d*x + 2*c) - a - b)))/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)^2*(a + b*tanh(c + d*x)^2)^2),x)

[Out]

int(1/(sinh(c + d*x)^2*(a + b*tanh(c + d*x)^2)^2), x)

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