3.6 \(\int \frac{1}{(a+b \text{csch}^2(c+d x))^2} \, dx\)
Optimal. Leaf size=100 \[ -\frac{\sqrt{b} (3 a-2 b) \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{b}}\right )}{2 a^2 d (a-b)^{3/2}}+\frac{x}{a^2}+\frac{b \coth (c+d x)}{2 a d (a-b) \left (a+b \coth ^2(c+d x)-b\right )} \]
[Out]
x/a^2 - ((3*a - 2*b)*Sqrt[b]*ArcTan[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[b]])/(2*a^2*(a - b)^(3/2)*d) + (b*Coth[c
+ d*x])/(2*a*(a - b)*d*(a - b + b*Coth[c + d*x]^2))
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Rubi [A] time = 0.126582, antiderivative size = 100, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used =
{4128, 414, 522, 206, 205} \[ -\frac{\sqrt{b} (3 a-2 b) \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{b}}\right )}{2 a^2 d (a-b)^{3/2}}+\frac{x}{a^2}+\frac{b \coth (c+d x)}{2 a d (a-b) \left (a+b \coth ^2(c+d x)-b\right )} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Csch[c + d*x]^2)^(-2),x]
[Out]
x/a^2 - ((3*a - 2*b)*Sqrt[b]*ArcTan[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[b]])/(2*a^2*(a - b)^(3/2)*d) + (b*Coth[c
+ d*x])/(2*a*(a - b)*d*(a - b + b*Coth[c + d*x]^2))
Rule 4128
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]
Rule 414
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, 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{1}{\left (a+b \text{csch}^2(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \left (a-b+b x^2\right )^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=\frac{b \coth (c+d x)}{2 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{-2 a+b+b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )} \, dx,x,\coth (c+d x)\right )}{2 a (a-b) d}\\ &=\frac{b \coth (c+d x)}{2 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\coth (c+d x)\right )}{a^2 d}+\frac{((3 a-2 b) b) \operatorname{Subst}\left (\int \frac{1}{a-b+b x^2} \, dx,x,\coth (c+d x)\right )}{2 a^2 (a-b) d}\\ &=\frac{x}{a^2}-\frac{(3 a-2 b) \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{b}}\right )}{2 a^2 (a-b)^{3/2} d}+\frac{b \coth (c+d x)}{2 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.681491, size = 199, normalized size = 1.99 \[ \frac{\text{csch}^4(c+d x) (a \cosh (2 (c+d x))-a+2 b) \left (a b \sqrt{a-b} \sinh (2 (c+d x))+(a-2 b) \left (\sqrt{b} (3 a-2 b) \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{b}}\right )-2 (a-b)^{3/2} (c+d x)\right )+a \cosh (2 (c+d x)) \left (2 (a-b)^{3/2} (c+d x)+\sqrt{b} (2 b-3 a) \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{b}}\right )\right )\right )}{8 a^2 d (a-b)^{3/2} \left (a+b \text{csch}^2(c+d x)\right )^2} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Csch[c + d*x]^2)^(-2),x]
[Out]
((-a + 2*b + a*Cosh[2*(c + d*x)])*Csch[c + d*x]^4*((a - 2*b)*(-2*(a - b)^(3/2)*(c + d*x) + (3*a - 2*b)*Sqrt[b]
*ArcTan[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[b]]) + a*(2*(a - b)^(3/2)*(c + d*x) + Sqrt[b]*(-3*a + 2*b)*ArcTan[(Sq
rt[a - b]*Tanh[c + d*x])/Sqrt[b]])*Cosh[2*(c + d*x)] + a*Sqrt[a - b]*b*Sinh[2*(c + d*x)]))/(8*a^2*(a - b)^(3/2
)*d*(a + b*Csch[c + d*x]^2)^2)
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Maple [B] time = 0.059, size = 778, normalized size = 7.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b*csch(d*x+c)^2)^2,x)
[Out]
1/d/a^2*ln(tanh(1/2*d*x+1/2*c)+1)+1/d*b/a/(b*tanh(1/2*d*x+1/2*c)^4+4*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/
2*c)^2*b+b)/(a-b)*tanh(1/2*d*x+1/2*c)^3+1/d*b/a/(b*tanh(1/2*d*x+1/2*c)^4+4*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*
d*x+1/2*c)^2*b+b)/(a-b)*tanh(1/2*d*x+1/2*c)+3/2/d*b/a/(a-b)/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2)*arctan(tanh(1/
2*d*x+1/2*c)*b/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2))+3/2/d*b/(a-b)/(a*(a-b))^(1/2)/((2*(a*(a-b))^(1/2)+2*a-b)*b
)^(1/2)*arctan(tanh(1/2*d*x+1/2*c)*b/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2))-3/2/d*b/a/(a-b)/((2*(a*(a-b))^(1/2)-
2*a+b)*b)^(1/2)*arctanh(tanh(1/2*d*x+1/2*c)*b/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2))+3/2/d*b/(a-b)/(a*(a-b))^(1/
2)/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2)*arctanh(tanh(1/2*d*x+1/2*c)*b/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2))-1/d*
b^2/a^2/(a-b)/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2)*arctan(tanh(1/2*d*x+1/2*c)*b/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(
1/2))-1/d*b^2/a/(a-b)/(a*(a-b))^(1/2)/((2*(a*(a-b))^(1/2)+2*a-b)*b)^(1/2)*arctan(tanh(1/2*d*x+1/2*c)*b/((2*(a*
(a-b))^(1/2)+2*a-b)*b)^(1/2))+1/d*b^2/a^2/(a-b)/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2)*arctanh(tanh(1/2*d*x+1/2*c
)*b/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2))-1/d*b^2/a/(a-b)/(a*(a-b))^(1/2)/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2)*a
rctanh(tanh(1/2*d*x+1/2*c)*b/((2*(a*(a-b))^(1/2)-2*a+b)*b)^(1/2))-1/d/a^2*ln(tanh(1/2*d*x+1/2*c)-1)
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*csch(d*x+c)^2)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.91876, size = 4199, normalized size = 41.99 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*csch(d*x+c)^2)^2,x, algorithm="fricas")
[Out]
[1/4*(4*(a^2 - a*b)*d*x*cosh(d*x + c)^4 + 16*(a^2 - a*b)*d*x*cosh(d*x + c)*sinh(d*x + c)^3 + 4*(a^2 - a*b)*d*x
*sinh(d*x + c)^4 + 4*(a^2 - a*b)*d*x - 4*(2*(a^2 - 3*a*b + 2*b^2)*d*x - a*b + 2*b^2)*cosh(d*x + c)^2 + 4*(6*(a
^2 - a*b)*d*x*cosh(d*x + c)^2 - 2*(a^2 - 3*a*b + 2*b^2)*d*x + a*b - 2*b^2)*sinh(d*x + c)^2 + ((3*a^2 - 2*a*b)*
cosh(d*x + c)^4 + 4*(3*a^2 - 2*a*b)*cosh(d*x + c)*sinh(d*x + c)^3 + (3*a^2 - 2*a*b)*sinh(d*x + c)^4 - 2*(3*a^2
- 8*a*b + 4*b^2)*cosh(d*x + c)^2 + 2*(3*(3*a^2 - 2*a*b)*cosh(d*x + c)^2 - 3*a^2 + 8*a*b - 4*b^2)*sinh(d*x + c
)^2 + 3*a^2 - 2*a*b + 4*((3*a^2 - 2*a*b)*cosh(d*x + c)^3 - (3*a^2 - 8*a*b + 4*b^2)*cosh(d*x + c))*sinh(d*x + c
))*sqrt(-b/(a - b))*log((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 - 8*a*b + 8*b^2 +
4*(a^2*cosh(d*x + c)^3 - (a^2 - 2*a*b)*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 + 3*a*b - 2*b^2)*sqrt(-b/(a - b)))/(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)) - 4*a*b + 8*(2*(a^2 - a*b)*d*x*cosh(d*x + c)^3 - (2*(a^2 - 3*a*b + 2*b^2)*d*x - a*b + 2*b^2)*cosh(d*x + c)
)*sinh(d*x + c))/((a^4 - a^3*b)*d*cosh(d*x + c)^4 + 4*(a^4 - a^3*b)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 - a
^3*b)*d*sinh(d*x + c)^4 - 2*(a^4 - 3*a^3*b + 2*a^2*b^2)*d*cosh(d*x + c)^2 + 2*(3*(a^4 - a^3*b)*d*cosh(d*x + c)
^2 - (a^4 - 3*a^3*b + 2*a^2*b^2)*d)*sinh(d*x + c)^2 + (a^4 - a^3*b)*d + 4*((a^4 - a^3*b)*d*cosh(d*x + c)^3 - (
a^4 - 3*a^3*b + 2*a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)), 1/2*(2*(a^2 - a*b)*d*x*cosh(d*x + c)^4 + 8*(a^2 -
a*b)*d*x*cosh(d*x + c)*sinh(d*x + c)^3 + 2*(a^2 - a*b)*d*x*sinh(d*x + c)^4 + 2*(a^2 - a*b)*d*x - 2*(2*(a^2 - 3
*a*b + 2*b^2)*d*x - a*b + 2*b^2)*cosh(d*x + c)^2 + 2*(6*(a^2 - a*b)*d*x*cosh(d*x + c)^2 - 2*(a^2 - 3*a*b + 2*b
^2)*d*x + a*b - 2*b^2)*sinh(d*x + c)^2 - ((3*a^2 - 2*a*b)*cosh(d*x + c)^4 + 4*(3*a^2 - 2*a*b)*cosh(d*x + c)*si
nh(d*x + c)^3 + (3*a^2 - 2*a*b)*sinh(d*x + c)^4 - 2*(3*a^2 - 8*a*b + 4*b^2)*cosh(d*x + c)^2 + 2*(3*(3*a^2 - 2*
a*b)*cosh(d*x + c)^2 - 3*a^2 + 8*a*b - 4*b^2)*sinh(d*x + c)^2 + 3*a^2 - 2*a*b + 4*((3*a^2 - 2*a*b)*cosh(d*x +
c)^3 - (3*a^2 - 8*a*b + 4*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/(a - b))*arctan(1/2*(a*cosh(d*x + c)^2 + 2
*a*cosh(d*x + c)*sinh(d*x + c) + a*sinh(d*x + c)^2 - a + 2*b)*sqrt(b/(a - b))/b) - 2*a*b + 4*(2*(a^2 - a*b)*d*
x*cosh(d*x + c)^3 - (2*(a^2 - 3*a*b + 2*b^2)*d*x - a*b + 2*b^2)*cosh(d*x + c))*sinh(d*x + c))/((a^4 - a^3*b)*d
*cosh(d*x + c)^4 + 4*(a^4 - a^3*b)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 - a^3*b)*d*sinh(d*x + c)^4 - 2*(a^4
- 3*a^3*b + 2*a^2*b^2)*d*cosh(d*x + c)^2 + 2*(3*(a^4 - a^3*b)*d*cosh(d*x + c)^2 - (a^4 - 3*a^3*b + 2*a^2*b^2)*
d)*sinh(d*x + c)^2 + (a^4 - a^3*b)*d + 4*((a^4 - a^3*b)*d*cosh(d*x + c)^3 - (a^4 - 3*a^3*b + 2*a^2*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{1}{\left (a + b \operatorname{csch}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*csch(d*x+c)**2)**2,x)
[Out]
Integral((a + b*csch(c + d*x)**2)**(-2), x)
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Giac [A] time = 1.3347, size = 228, normalized size = 2.28 \begin{align*} -\frac{{\left (3 \, a b - 2 \, b^{2}\right )} \arctan \left (\frac{a e^{\left (2 \, d x + 2 \, c\right )} - a + 2 \, b}{2 \, \sqrt{a b - b^{2}}}\right )}{2 \,{\left (a^{3} d - a^{2} b d\right )} \sqrt{a b - b^{2}}} + \frac{a b e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - a b}{{\left (a^{3} d - a^{2} b d\right )}{\left (a e^{\left (4 \, d x + 4 \, c\right )} - 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}} + \frac{d x + c}{a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*csch(d*x+c)^2)^2,x, algorithm="giac")
[Out]
-1/2*(3*a*b - 2*b^2)*arctan(1/2*(a*e^(2*d*x + 2*c) - a + 2*b)/sqrt(a*b - b^2))/((a^3*d - a^2*b*d)*sqrt(a*b - b
^2)) + (a*b*e^(2*d*x + 2*c) - 2*b^2*e^(2*d*x + 2*c) - a*b)/((a^3*d - a^2*b*d)*(a*e^(4*d*x + 4*c) - 2*a*e^(2*d*
x + 2*c) + 4*b*e^(2*d*x + 2*c) + a)) + (d*x + c)/(a^2*d)