3.343 \(\int \frac{\cosh ^2(c+d x)}{(a+b \sinh ^2(c+d x))^3} \, dx\)

Optimal. Leaf size=143 \[ \frac{(4 a-3 b) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} d (a-b)^{3/2}}+\frac{(4 a-3 b) \tanh (c+d x)}{8 a^2 d (a-b) \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac{b \tanh (c+d x)}{4 a d (a-b) \left (a-(a-b) \tanh ^2(c+d x)\right )^2} \]

[Out]

((4*a - 3*b)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a - b)^(3/2)*d) - (b*Tanh[c + d*x])/(4*
a*(a - b)*d*(a - (a - b)*Tanh[c + d*x]^2)^2) + ((4*a - 3*b)*Tanh[c + d*x])/(8*a^2*(a - b)*d*(a - (a - b)*Tanh[
c + d*x]^2))

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Rubi [A]  time = 0.125523, antiderivative size = 143, normalized size of antiderivative = 1., 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 = {3191, 385, 199, 208} \[ \frac{(4 a-3 b) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} d (a-b)^{3/2}}+\frac{(4 a-3 b) \tanh (c+d x)}{8 a^2 d (a-b) \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac{b \tanh (c+d x)}{4 a d (a-b) \left (a-(a-b) \tanh ^2(c+d x)\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*a - 3*b)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a - b)^(3/2)*d) - (b*Tanh[c + d*x])/(4*
a*(a - b)*d*(a - (a - b)*Tanh[c + d*x]^2)^2) + ((4*a - 3*b)*Tanh[c + d*x])/(8*a^2*(a - b)*d*(a - (a - b)*Tanh[
c + d*x]^2))

Rule 3191

Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, T
an[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 208

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

Rubi steps

\begin{align*} \int \frac{\cosh ^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1-x^2}{\left (a-(a-b) x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{b \tanh (c+d x)}{4 a (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac{(4 a-3 b) \operatorname{Subst}\left (\int \frac{1}{\left (a+(-a+b) x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 a (a-b) d}\\ &=-\frac{b \tanh (c+d x)}{4 a (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac{(4 a-3 b) \tanh (c+d x)}{8 a^2 (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac{(4 a-3 b) \operatorname{Subst}\left (\int \frac{1}{a+(-a+b) x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^2 (a-b) d}\\ &=\frac{(4 a-3 b) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} (a-b)^{3/2} d}-\frac{b \tanh (c+d x)}{4 a (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac{(4 a-3 b) \tanh (c+d x)}{8 a^2 (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.888273, size = 124, normalized size = 0.87 \[ \frac{\frac{\sqrt{a} \sinh (2 (c+d x)) \left (8 a^2+b (2 a-3 b) \cosh (2 (c+d x))-12 a b+3 b^2\right )}{(a-b) (2 a+b \cosh (2 (c+d x))-b)^2}+\frac{(4 a-3 b) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{(a-b)^{3/2}}}{8 a^{5/2} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((4*a - 3*b)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a - b)^(3/2) + (Sqrt[a]*(8*a^2 - 12*a*b + 3*b^2 +
(2*a - 3*b)*b*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a - b)*(2*a - b + b*Cosh[2*(c + d*x)])^2))/(8*a^(5/2)*d
)

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Maple [B]  time = 0.06, size = 1322, normalized size = 9.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)^2/(a+b*sinh(d*x+c)^2)^3,x)

[Out]

1/d/(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)^2/(a-b)*tanh(1/2*d*x+1/2*c
)^7-5/4/d/(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)^2/a/(a-b)*tanh(1/2*d
*x+1/2*c)^7*b-1/d/(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)^2/(a-b)*tanh
(1/2*d*x+1/2*c)^5+13/4/d/(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)^2/a/(
a-b)*tanh(1/2*d*x+1/2*c)^5*b-3/d/(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)^2/a^2/(a-b)*tanh(1/2*d*x+1/2*c)^5*b^2-1/d/(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)^2/(a-b)*tanh(1/2*d*x+1/2*c)^3+13/4/d/(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)^2/a/(a-b)*tanh(1/2*d*x+1/2*c)^3*b-3/d/(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)^2/a^2/(a-b)*tanh(1/2*d*x+1/2*c)^3*b^2+1/d/(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)^2/(a-b)*tanh(1/2*d*x+1/2*c)-5/4/d/(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)^2/a/(a-b)*tanh(1/2*d*x+1/2*c)*b+1/2/d/a/(a-b)/((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/a/(a-b)/(-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
))*b-1/2/d/a/(a-b)/((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/(a-b)/(-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))*b-3/8/d/a^2/(a-b)*b/((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))+3/8/d/a^2/(a-b)/(-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))*b^2+3/8/d/a^2/(a-b)*b/((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))+3/8/d/a^2/(a-
b)/(-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))*b^2

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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(cosh(d*x+c)^2/(a+b*sinh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.35308, size = 11760, normalized size = 82.24 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(4*(4*a^3*b^2 - 7*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^6 + 24*(4*a^3*b^2 - 7*a^2*b^3 + 3*a*b^4)*cosh(d*x + c
)*sinh(d*x + c)^5 + 4*(4*a^3*b^2 - 7*a^2*b^3 + 3*a*b^4)*sinh(d*x + c)^6 - 8*a^3*b^2 + 20*a^2*b^3 - 12*a*b^4 -
4*(16*a^5 - 56*a^4*b + 70*a^3*b^2 - 39*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^4 - 4*(16*a^5 - 56*a^4*b + 70*a^3*b^2
- 39*a^2*b^3 + 9*a*b^4 - 15*(4*a^3*b^2 - 7*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*(5*(4*a^3*
b^2 - 7*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^3 - (16*a^5 - 56*a^4*b + 70*a^3*b^2 - 39*a^2*b^3 + 9*a*b^4)*cosh(d*x
+ c))*sinh(d*x + c)^3 - 4*(16*a^4*b - 44*a^3*b^2 + 37*a^2*b^3 - 9*a*b^4)*cosh(d*x + c)^2 - 4*(16*a^4*b - 44*a^
3*b^2 + 37*a^2*b^3 - 9*a*b^4 - 15*(4*a^3*b^2 - 7*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^4 + 6*(16*a^5 - 56*a^4*b + 7
0*a^3*b^2 - 39*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((4*a*b^3 - 3*b^4)*cosh(d*x + c)^8 + 8*(4
*a*b^3 - 3*b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (4*a*b^3 - 3*b^4)*sinh(d*x + c)^8 + 4*(8*a^2*b^2 - 10*a*b^3 +
3*b^4)*cosh(d*x + c)^6 + 4*(8*a^2*b^2 - 10*a*b^3 + 3*b^4 + 7*(4*a*b^3 - 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^
6 + 8*(7*(4*a*b^3 - 3*b^4)*cosh(d*x + c)^3 + 3*(8*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 +
2*(32*a^3*b - 56*a^2*b^2 + 36*a*b^3 - 9*b^4)*cosh(d*x + c)^4 + 2*(35*(4*a*b^3 - 3*b^4)*cosh(d*x + c)^4 + 32*a
^3*b - 56*a^2*b^2 + 36*a*b^3 - 9*b^4 + 30*(8*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*
a*b^3 - 3*b^4 + 8*(7*(4*a*b^3 - 3*b^4)*cosh(d*x + c)^5 + 10*(8*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^3 + (
32*a^3*b - 56*a^2*b^2 + 36*a*b^3 - 9*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(8*a^2*b^2 - 10*a*b^3 + 3*b^4)*co
sh(d*x + c)^2 + 4*(7*(4*a*b^3 - 3*b^4)*cosh(d*x + c)^6 + 15*(8*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^4 + 8
*a^2*b^2 - 10*a*b^3 + 3*b^4 + 3*(32*a^3*b - 56*a^2*b^2 + 36*a*b^3 - 9*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 +
8*((4*a*b^3 - 3*b^4)*cosh(d*x + c)^7 + 3*(8*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^5 + (32*a^3*b - 56*a^2*b
^2 + 36*a*b^3 - 9*b^4)*cosh(d*x + c)^3 + (8*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a^2
- a*b)*log((b^2*cosh(d*x + c)^4 + 4*b^2*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*sinh(d*x + c)^4 + 2*(2*a*b - b^2)
*cosh(d*x + c)^2 + 2*(3*b^2*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x + c)^2 + 8*a^2 - 8*a*b + b^2 + 4*(b^2*cosh
(d*x + c)^3 + (2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c) - 4*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x +
c) + b*sinh(d*x + c)^2 + 2*a - b)*sqrt(a^2 - a*b))/(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b
*sinh(d*x + c)^4 + 2*(2*a - b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 + 2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh
(d*x + c)^3 + (2*a - b)*cosh(d*x + c))*sinh(d*x + c) + b)) + 8*(3*(4*a^3*b^2 - 7*a^2*b^3 + 3*a*b^4)*cosh(d*x +
c)^5 - 2*(16*a^5 - 56*a^4*b + 70*a^3*b^2 - 39*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^3 - (16*a^4*b - 44*a^3*b^2 + 3
7*a^2*b^3 - 9*a*b^4)*cosh(d*x + c))*sinh(d*x + c))/((a^5*b^3 - 2*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^8 + 8*(a^5
*b^3 - 2*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^5*b^3 - 2*a^4*b^4 + a^3*b^5)*d*sinh(d*x + c)^
8 + 4*(2*a^6*b^2 - 5*a^5*b^3 + 4*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^6 + 4*(7*(a^5*b^3 - 2*a^4*b^4 + a^3*b^5)*d
*cosh(d*x + c)^2 + (2*a^6*b^2 - 5*a^5*b^3 + 4*a^4*b^4 - a^3*b^5)*d)*sinh(d*x + c)^6 + 2*(8*a^7*b - 24*a^6*b^2
+ 27*a^5*b^3 - 14*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^4 + 8*(7*(a^5*b^3 - 2*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c
)^3 + 3*(2*a^6*b^2 - 5*a^5*b^3 + 4*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^5*b^3 - 2*a^
4*b^4 + a^3*b^5)*d*cosh(d*x + c)^4 + 30*(2*a^6*b^2 - 5*a^5*b^3 + 4*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^2 + (8*a
^7*b - 24*a^6*b^2 + 27*a^5*b^3 - 14*a^4*b^4 + 3*a^3*b^5)*d)*sinh(d*x + c)^4 + 4*(2*a^6*b^2 - 5*a^5*b^3 + 4*a^4
*b^4 - a^3*b^5)*d*cosh(d*x + c)^2 + 8*(7*(a^5*b^3 - 2*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^5 + 10*(2*a^6*b^2 - 5
*a^5*b^3 + 4*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^3 + (8*a^7*b - 24*a^6*b^2 + 27*a^5*b^3 - 14*a^4*b^4 + 3*a^3*b^
5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^5*b^3 - 2*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^6 + 15*(2*a^6*b^2 -
5*a^5*b^3 + 4*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^4 + 3*(8*a^7*b - 24*a^6*b^2 + 27*a^5*b^3 - 14*a^4*b^4 + 3*a^
3*b^5)*d*cosh(d*x + c)^2 + (2*a^6*b^2 - 5*a^5*b^3 + 4*a^4*b^4 - a^3*b^5)*d)*sinh(d*x + c)^2 + (a^5*b^3 - 2*a^4
*b^4 + a^3*b^5)*d + 8*((a^5*b^3 - 2*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^7 + 3*(2*a^6*b^2 - 5*a^5*b^3 + 4*a^4*b^
4 - a^3*b^5)*d*cosh(d*x + c)^5 + (8*a^7*b - 24*a^6*b^2 + 27*a^5*b^3 - 14*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^
3 + (2*a^6*b^2 - 5*a^5*b^3 + 4*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)), 1/8*(2*(4*a^3*b^2 - 7*a^2*b
^3 + 3*a*b^4)*cosh(d*x + c)^6 + 12*(4*a^3*b^2 - 7*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)*sinh(d*x + c)^5 + 2*(4*a^3*
b^2 - 7*a^2*b^3 + 3*a*b^4)*sinh(d*x + c)^6 - 4*a^3*b^2 + 10*a^2*b^3 - 6*a*b^4 - 2*(16*a^5 - 56*a^4*b + 70*a^3*
b^2 - 39*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^4 - 2*(16*a^5 - 56*a^4*b + 70*a^3*b^2 - 39*a^2*b^3 + 9*a*b^4 - 15*(4
*a^3*b^2 - 7*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(5*(4*a^3*b^2 - 7*a^2*b^3 + 3*a*b^4)*cosh
(d*x + c)^3 - (16*a^5 - 56*a^4*b + 70*a^3*b^2 - 39*a^2*b^3 + 9*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 - 2*(16*a
^4*b - 44*a^3*b^2 + 37*a^2*b^3 - 9*a*b^4)*cosh(d*x + c)^2 - 2*(16*a^4*b - 44*a^3*b^2 + 37*a^2*b^3 - 9*a*b^4 -
15*(4*a^3*b^2 - 7*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^4 + 6*(16*a^5 - 56*a^4*b + 70*a^3*b^2 - 39*a^2*b^3 + 9*a*b^
4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((4*a*b^3 - 3*b^4)*cosh(d*x + c)^8 + 8*(4*a*b^3 - 3*b^4)*cosh(d*x + c)*s
inh(d*x + c)^7 + (4*a*b^3 - 3*b^4)*sinh(d*x + c)^8 + 4*(8*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^6 + 4*(8*a
^2*b^2 - 10*a*b^3 + 3*b^4 + 7*(4*a*b^3 - 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(4*a*b^3 - 3*b^4)*cosh
(d*x + c)^3 + 3*(8*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(32*a^3*b - 56*a^2*b^2 + 36*
a*b^3 - 9*b^4)*cosh(d*x + c)^4 + 2*(35*(4*a*b^3 - 3*b^4)*cosh(d*x + c)^4 + 32*a^3*b - 56*a^2*b^2 + 36*a*b^3 -
9*b^4 + 30*(8*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*a*b^3 - 3*b^4 + 8*(7*(4*a*b^3 -
3*b^4)*cosh(d*x + c)^5 + 10*(8*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^3 + (32*a^3*b - 56*a^2*b^2 + 36*a*b^
3 - 9*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(8*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^2 + 4*(7*(4*a*b^3 -
3*b^4)*cosh(d*x + c)^6 + 15*(8*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^4 + 8*a^2*b^2 - 10*a*b^3 + 3*b^4 + 3
*(32*a^3*b - 56*a^2*b^2 + 36*a*b^3 - 9*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((4*a*b^3 - 3*b^4)*cosh(d*x +
c)^7 + 3*(8*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^5 + (32*a^3*b - 56*a^2*b^2 + 36*a*b^3 - 9*b^4)*cosh(d*x
+ c)^3 + (8*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a^2 + a*b)*arctan(-1/2*(b*cosh(d*
x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + 2*a - b)*sqrt(-a^2 + a*b)/(a^2 - a*b)) + 4*(3
*(4*a^3*b^2 - 7*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^5 - 2*(16*a^5 - 56*a^4*b + 70*a^3*b^2 - 39*a^2*b^3 + 9*a*b^4)
*cosh(d*x + c)^3 - (16*a^4*b - 44*a^3*b^2 + 37*a^2*b^3 - 9*a*b^4)*cosh(d*x + c))*sinh(d*x + c))/((a^5*b^3 - 2*
a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^8 + 8*(a^5*b^3 - 2*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^
5*b^3 - 2*a^4*b^4 + a^3*b^5)*d*sinh(d*x + c)^8 + 4*(2*a^6*b^2 - 5*a^5*b^3 + 4*a^4*b^4 - a^3*b^5)*d*cosh(d*x +
c)^6 + 4*(7*(a^5*b^3 - 2*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^2 + (2*a^6*b^2 - 5*a^5*b^3 + 4*a^4*b^4 - a^3*b^5)*
d)*sinh(d*x + c)^6 + 2*(8*a^7*b - 24*a^6*b^2 + 27*a^5*b^3 - 14*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^4 + 8*(7*(
a^5*b^3 - 2*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^3 + 3*(2*a^6*b^2 - 5*a^5*b^3 + 4*a^4*b^4 - a^3*b^5)*d*cosh(d*x
+ c))*sinh(d*x + c)^5 + 2*(35*(a^5*b^3 - 2*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^4 + 30*(2*a^6*b^2 - 5*a^5*b^3 +
4*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^2 + (8*a^7*b - 24*a^6*b^2 + 27*a^5*b^3 - 14*a^4*b^4 + 3*a^3*b^5)*d)*sinh(
d*x + c)^4 + 4*(2*a^6*b^2 - 5*a^5*b^3 + 4*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^2 + 8*(7*(a^5*b^3 - 2*a^4*b^4 + a
^3*b^5)*d*cosh(d*x + c)^5 + 10*(2*a^6*b^2 - 5*a^5*b^3 + 4*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^3 + (8*a^7*b - 24
*a^6*b^2 + 27*a^5*b^3 - 14*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^5*b^3 - 2*a^4*b^4 +
a^3*b^5)*d*cosh(d*x + c)^6 + 15*(2*a^6*b^2 - 5*a^5*b^3 + 4*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^4 + 3*(8*a^7*b
- 24*a^6*b^2 + 27*a^5*b^3 - 14*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^2 + (2*a^6*b^2 - 5*a^5*b^3 + 4*a^4*b^4 - a
^3*b^5)*d)*sinh(d*x + c)^2 + (a^5*b^3 - 2*a^4*b^4 + a^3*b^5)*d + 8*((a^5*b^3 - 2*a^4*b^4 + a^3*b^5)*d*cosh(d*x
+ c)^7 + 3*(2*a^6*b^2 - 5*a^5*b^3 + 4*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^5 + (8*a^7*b - 24*a^6*b^2 + 27*a^5*b
^3 - 14*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^3 + (2*a^6*b^2 - 5*a^5*b^3 + 4*a^4*b^4 - a^3*b^5)*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(cosh(d*x+c)**2/(a+b*sinh(d*x+c)**2)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.39818, size = 365, normalized size = 2.55 \begin{align*} \frac{{\left (4 \, a - 3 \, b\right )} \arctan \left (\frac{b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt{-a^{2} + a b}}\right )}{8 \,{\left (a^{3} d - a^{2} b d\right )} \sqrt{-a^{2} + a b}} + \frac{4 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 3 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 16 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 40 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 30 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 9 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 16 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 28 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 9 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, a b^{2} + 3 \, b^{3}}{4 \,{\left (a^{3} b d - a^{2} b^{2} d\right )}{\left (b e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + b\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*a - 3*b)*arctan(1/2*(b*e^(2*d*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/((a^3*d - a^2*b*d)*sqrt(-a^2 + a*b)
) + 1/4*(4*a*b^2*e^(6*d*x + 6*c) - 3*b^3*e^(6*d*x + 6*c) - 16*a^3*e^(4*d*x + 4*c) + 40*a^2*b*e^(4*d*x + 4*c) -
30*a*b^2*e^(4*d*x + 4*c) + 9*b^3*e^(4*d*x + 4*c) - 16*a^2*b*e^(2*d*x + 2*c) + 28*a*b^2*e^(2*d*x + 2*c) - 9*b^
3*e^(2*d*x + 2*c) - 2*a*b^2 + 3*b^3)/((a^3*b*d - a^2*b^2*d)*(b*e^(4*d*x + 4*c) + 4*a*e^(2*d*x + 2*c) - 2*b*e^(
2*d*x + 2*c) + b)^2)