3.35 \(\int \frac{\sinh ^2(c+d x)}{(a+b \tanh ^2(c+d x))^2} \, dx\)

Optimal. Leaf size=132 \[ -\frac{b \tanh (c+d x)}{d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )}-\frac{\sqrt{b} (3 a-b) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{2 \sqrt{a} d (a+b)^3}+\frac{\sinh (c+d x) \cosh (c+d x)}{2 d (a+b) \left (a+b \tanh ^2(c+d x)\right )}-\frac{x (a-3 b)}{2 (a+b)^3} \]

[Out]

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

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Rubi [A]  time = 0.165343, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3663, 471, 527, 522, 206, 205} \[ -\frac{b \tanh (c+d x)}{d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )}-\frac{\sqrt{b} (3 a-b) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{2 \sqrt{a} d (a+b)^3}+\frac{\sinh (c+d x) \cosh (c+d x)}{2 d (a+b) \left (a+b \tanh ^2(c+d x)\right )}-\frac{x (a-3 b)}{2 (a+b)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3663

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]

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 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

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{\sinh ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{\left (1-x^2\right )^2 \left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\cosh (c+d x) \sinh (c+d x)}{2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{a-3 b x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{2 (a+b) d}\\ &=\frac{\cosh (c+d x) \sinh (c+d x)}{2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )}-\frac{b \tanh (c+d x)}{(a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{-2 a (a-b)+4 a b x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{4 a (a+b)^2 d}\\ &=\frac{\cosh (c+d x) \sinh (c+d x)}{2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )}-\frac{b \tanh (c+d x)}{(a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}-\frac{(a-3 b) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 (a+b)^3 d}-\frac{((3 a-b) b) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{2 (a+b)^3 d}\\ &=-\frac{(a-3 b) x}{2 (a+b)^3}-\frac{(3 a-b) \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{2 \sqrt{a} (a+b)^3 d}+\frac{\cosh (c+d x) \sinh (c+d x)}{2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )}-\frac{b \tanh (c+d x)}{(a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.649926, size = 105, normalized size = 0.8 \[ \frac{-2 (a-3 b) (c+d x)+(a+b) \sinh (2 (c+d x))+\frac{2 \sqrt{b} (b-3 a) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{2 b (a+b) \sinh (2 (c+d x))}{(a+b) \cosh (2 (c+d x))+a-b}}{4 d (a+b)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a - 3*b)*(c + d*x) + (2*Sqrt[b]*(-3*a + b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/Sqrt[a] + (a + b)*Sin
h[2*(c + d*x)] - (2*b*(a + b)*Sinh[2*(c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)]))/(4*(a + b)^3*d)

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Maple [B]  time = 0.099, size = 1128, normalized size = 8.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.67746, size = 9441, normalized size = 71.52 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^8 + 8*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2 + 2*a*b
+ b^2)*sinh(d*x + c)^8 - 2*(2*(a^2 - 2*a*b - 3*b^2)*d*x - a^2 + b^2)*cosh(d*x + c)^6 - 2*(2*(a^2 - 2*a*b - 3*
b^2)*d*x - 14*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 - a^2 + b^2)*sinh(d*x + c)^6 + 4*(14*(a^2 + 2*a*b + b^2)*cos
h(d*x + c)^3 - 3*(2*(a^2 - 2*a*b - 3*b^2)*d*x - a^2 + b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 8*((a^2 - 4*a*b +
3*b^2)*d*x - a*b + b^2)*cosh(d*x + c)^4 + 2*(35*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 - 4*(a^2 - 4*a*b + 3*b^2)*
d*x - 15*(2*(a^2 - 2*a*b - 3*b^2)*d*x - a^2 + b^2)*cosh(d*x + c)^2 + 4*a*b - 4*b^2)*sinh(d*x + c)^4 + 8*(7*(a^
2 + 2*a*b + b^2)*cosh(d*x + c)^5 - 5*(2*(a^2 - 2*a*b - 3*b^2)*d*x - a^2 + b^2)*cosh(d*x + c)^3 - 4*((a^2 - 4*a
*b + 3*b^2)*d*x - a*b + b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 2*(2*(a^2 - 2*a*b - 3*b^2)*d*x + a^2 - 4*a*b - 5
*b^2)*cosh(d*x + c)^2 + 2*(14*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 - 15*(2*(a^2 - 2*a*b - 3*b^2)*d*x - a^2 + b^
2)*cosh(d*x + c)^4 - 2*(a^2 - 2*a*b - 3*b^2)*d*x - 24*((a^2 - 4*a*b + 3*b^2)*d*x - a*b + b^2)*cosh(d*x + c)^2
- a^2 + 4*a*b + 5*b^2)*sinh(d*x + c)^2 - 2*((3*a^2 + 2*a*b - b^2)*cosh(d*x + c)^6 + 6*(3*a^2 + 2*a*b - b^2)*co
sh(d*x + c)*sinh(d*x + c)^5 + (3*a^2 + 2*a*b - b^2)*sinh(d*x + c)^6 + 2*(3*a^2 - 4*a*b + b^2)*cosh(d*x + c)^4
+ (15*(3*a^2 + 2*a*b - b^2)*cosh(d*x + c)^2 + 6*a^2 - 8*a*b + 2*b^2)*sinh(d*x + c)^4 + 4*(5*(3*a^2 + 2*a*b - b
^2)*cosh(d*x + c)^3 + 2*(3*a^2 - 4*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + (3*a^2 + 2*a*b - b^2)*cosh(d*x
+ c)^2 + (15*(3*a^2 + 2*a*b - b^2)*cosh(d*x + c)^4 + 12*(3*a^2 - 4*a*b + b^2)*cosh(d*x + c)^2 + 3*a^2 + 2*a*b
- b^2)*sinh(d*x + c)^2 + 2*(3*(3*a^2 + 2*a*b - b^2)*cosh(d*x + c)^5 + 4*(3*a^2 - 4*a*b + b^2)*cosh(d*x + c)^3
+ (3*a^2 + 2*a*b - 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)*cos
h(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))*s
inh(d*x + c) + a + b)) - a^2 - 2*a*b - b^2 + 4*(2*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^7 - 3*(2*(a^2 - 2*a*b - 3*
b^2)*d*x - a^2 + b^2)*cosh(d*x + c)^5 - 8*((a^2 - 4*a*b + 3*b^2)*d*x - a*b + b^2)*cosh(d*x + c)^3 - (2*(a^2 -
2*a*b - 3*b^2)*d*x + a^2 - 4*a*b - 5*b^2)*cosh(d*x + c))*sinh(d*x + c))/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3
+ b^4)*d*cosh(d*x + c)^6 + 6*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)*sinh(d*x + c)^5 + (a^
4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*sinh(d*x + c)^6 + 2*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*d*cosh(d*x + c)
^4 + (15*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)^2 + 2*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*d)*
sinh(d*x + c)^4 + (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)^2 + 4*(5*(a^4 + 4*a^3*b + 6*a^2*
b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)^3 + 2*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 +
(15*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)^4 + 12*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*d*cosh(
d*x + c)^2 + (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d)*sinh(d*x + c)^2 + 2*(3*(a^4 + 4*a^3*b + 6*a^2*b^2
+ 4*a*b^3 + b^4)*d*cosh(d*x + c)^5 + 4*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*d*cosh(d*x + c)^3 + (a^4 + 4*a^3*b + 6*
a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c))*sinh(d*x + c)), 1/8*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^8 + 8*(a^2 +
2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^8 - 2*(2*(a^2 - 2*a*b - 3*b^2)*
d*x - a^2 + b^2)*cosh(d*x + c)^6 - 2*(2*(a^2 - 2*a*b - 3*b^2)*d*x - 14*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 - a
^2 + b^2)*sinh(d*x + c)^6 + 4*(14*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 - 3*(2*(a^2 - 2*a*b - 3*b^2)*d*x - a^2 +
b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 8*((a^2 - 4*a*b + 3*b^2)*d*x - a*b + b^2)*cosh(d*x + c)^4 + 2*(35*(a^2
+ 2*a*b + b^2)*cosh(d*x + c)^4 - 4*(a^2 - 4*a*b + 3*b^2)*d*x - 15*(2*(a^2 - 2*a*b - 3*b^2)*d*x - a^2 + b^2)*co
sh(d*x + c)^2 + 4*a*b - 4*b^2)*sinh(d*x + c)^4 + 8*(7*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^5 - 5*(2*(a^2 - 2*a*b
- 3*b^2)*d*x - a^2 + b^2)*cosh(d*x + c)^3 - 4*((a^2 - 4*a*b + 3*b^2)*d*x - a*b + b^2)*cosh(d*x + c))*sinh(d*x
+ c)^3 - 2*(2*(a^2 - 2*a*b - 3*b^2)*d*x + a^2 - 4*a*b - 5*b^2)*cosh(d*x + c)^2 + 2*(14*(a^2 + 2*a*b + b^2)*cos
h(d*x + c)^6 - 15*(2*(a^2 - 2*a*b - 3*b^2)*d*x - a^2 + b^2)*cosh(d*x + c)^4 - 2*(a^2 - 2*a*b - 3*b^2)*d*x - 24
*((a^2 - 4*a*b + 3*b^2)*d*x - a*b + b^2)*cosh(d*x + c)^2 - a^2 + 4*a*b + 5*b^2)*sinh(d*x + c)^2 - 4*((3*a^2 +
2*a*b - b^2)*cosh(d*x + c)^6 + 6*(3*a^2 + 2*a*b - b^2)*cosh(d*x + c)*sinh(d*x + c)^5 + (3*a^2 + 2*a*b - b^2)*s
inh(d*x + c)^6 + 2*(3*a^2 - 4*a*b + b^2)*cosh(d*x + c)^4 + (15*(3*a^2 + 2*a*b - b^2)*cosh(d*x + c)^2 + 6*a^2 -
8*a*b + 2*b^2)*sinh(d*x + c)^4 + 4*(5*(3*a^2 + 2*a*b - b^2)*cosh(d*x + c)^3 + 2*(3*a^2 - 4*a*b + b^2)*cosh(d*
x + c))*sinh(d*x + c)^3 + (3*a^2 + 2*a*b - b^2)*cosh(d*x + c)^2 + (15*(3*a^2 + 2*a*b - b^2)*cosh(d*x + c)^4 +
12*(3*a^2 - 4*a*b + b^2)*cosh(d*x + c)^2 + 3*a^2 + 2*a*b - b^2)*sinh(d*x + c)^2 + 2*(3*(3*a^2 + 2*a*b - b^2)*c
osh(d*x + c)^5 + 4*(3*a^2 - 4*a*b + b^2)*cosh(d*x + c)^3 + (3*a^2 + 2*a*b - 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) - a^2 - 2*a*b - b^2 + 4*(2*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^7 - 3*(2*(a^2 - 2*a*b -
3*b^2)*d*x - a^2 + b^2)*cosh(d*x + c)^5 - 8*((a^2 - 4*a*b + 3*b^2)*d*x - a*b + b^2)*cosh(d*x + c)^3 - (2*(a^2
- 2*a*b - 3*b^2)*d*x + a^2 - 4*a*b - 5*b^2)*cosh(d*x + c))*sinh(d*x + c))/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^
3 + b^4)*d*cosh(d*x + c)^6 + 6*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)*sinh(d*x + c)^5 + (
a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*sinh(d*x + c)^6 + 2*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*d*cosh(d*x +
c)^4 + (15*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)^2 + 2*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*d
)*sinh(d*x + c)^4 + (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)^2 + 4*(5*(a^4 + 4*a^3*b + 6*a^
2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)^3 + 2*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3
+ (15*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)^4 + 12*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*d*cos
h(d*x + c)^2 + (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d)*sinh(d*x + c)^2 + 2*(3*(a^4 + 4*a^3*b + 6*a^2*b^
2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)^5 + 4*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*d*cosh(d*x + c)^3 + (a^4 + 4*a^3*b +
6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c))*sinh(d*x + c))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 2.25285, size = 540, normalized size = 4.09 \begin{align*} -\frac{\frac{12 \,{\left (a - 3 \, b\right )} d x}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac{12 \,{\left (3 \, a b e^{\left (2 \, c\right )} - b^{2} e^{\left (2 \, c\right )}\right )} \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 ) e^{\left (-2 \, c\right )}}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt{a b}} - \frac{3 \, e^{\left (2 \, d x + 12 \, c\right )}}{a^{2} e^{\left (10 \, c\right )} + 2 \, a b e^{\left (10 \, c\right )} + b^{2} e^{\left (10 \, c\right )}} - \frac{2 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} - 4 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 6 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 15 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 4 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 20 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 24 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 3 \, a^{2} - 6 \, a b - 3 \, b^{2}}{{\left (a^{3} e^{\left (2 \, c\right )} + 3 \, a^{2} b e^{\left (2 \, c\right )} + 3 \, a b^{2} e^{\left (2 \, c\right )} + b^{3} e^{\left (2 \, c\right )}\right )}{\left (a e^{\left (2 \, d x\right )} + b e^{\left (2 \, d x\right )} + a e^{\left (6 \, d x + 4 \, c\right )} + b e^{\left (6 \, d x + 4 \, c\right )} + 2 \, a e^{\left (4 \, d x + 2 \, c\right )} - 2 \, b e^{\left (4 \, d x + 2 \, c\right )}\right )}}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(12*(a - 3*b)*d*x/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) + 12*(3*a*b*e^(2*c) - b^2*e^(2*c))*arctan(1/2*(a*e^(2*
d*x + 2*c) + b*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))*e^(-2*c)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sqrt(a*b)) - 3*e^
(2*d*x + 12*c)/(a^2*e^(10*c) + 2*a*b*e^(10*c) + b^2*e^(10*c)) - (2*a^2*e^(6*d*x + 6*c) - 4*a*b*e^(6*d*x + 6*c)
- 6*b^2*e^(6*d*x + 6*c) + a^2*e^(4*d*x + 4*c) + 2*a*b*e^(4*d*x + 4*c) - 15*b^2*e^(4*d*x + 4*c) - 4*a^2*e^(2*d
*x + 2*c) + 20*a*b*e^(2*d*x + 2*c) + 24*b^2*e^(2*d*x + 2*c) - 3*a^2 - 6*a*b - 3*b^2)/((a^3*e^(2*c) + 3*a^2*b*e
^(2*c) + 3*a*b^2*e^(2*c) + b^3*e^(2*c))*(a*e^(2*d*x) + b*e^(2*d*x) + a*e^(6*d*x + 4*c) + b*e^(6*d*x + 4*c) + 2
*a*e^(4*d*x + 2*c) - 2*b*e^(4*d*x + 2*c))))/d