∫ cosh(c+dx)___ (a+b tanh2(c+dx))2 dx

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

Optimal. Leaf size=101 \[ \frac{b (4 a+b) \tan ^{-1}\left (\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} d (a+b)^{5/2}}+\frac{b^2 \sinh (c+d x)}{2 a d (a+b)^2 \left ((a+b) \sinh ^2(c+d x)+a\right )}+\frac{\sinh (c+d x)}{d (a+b)^2} \]

[Out]

(b*(4*a + b)*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^(5/2)*d) + Sinh[c + d*x]/((a + b)

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

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Rubi [A] time = 0.139196, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3676, 390, 385, 205} \[ \frac{b (4 a+b) \tan ^{-1}\left (\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} d (a+b)^{5/2}}+\frac{b^2 \sinh (c+d x)}{2 a d (a+b)^2 \left ((a+b) \sinh ^2(c+d x)+a\right )}+\frac{\sinh (c+d x)}{d (a+b)^2} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(b*(4*a + b)*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^(5/2)*d) + Sinh[c + d*x]/((a + b)

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

Rule 3676

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> With[{ff = F

reeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[ExpandToSum[b*(ff*x)^n + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 -

ff^2*x^2)^((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] &&

IntegerQ[n/2] && IntegerQ[p]

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)

^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt

Q[q, 0] && GeQ[p, -q]

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

Mathematica [A] time = 0.697244, size = 84, normalized size = 0.83 \[ \frac{\frac{b (4 a+b) \tan ^{-1}\left (\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right )}{a^{3/2} (a+b)^{5/2}}+\frac{\sinh (c+d x) \left (\frac{b^2}{a \left ((a+b) \sinh ^2(c+d x)+a\right )}+2\right )}{(a+b)^2}}{2 d} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((b*(4*a + b)*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(a^(3/2)*(a + b)^(5/2)) + (Sinh[c + d*x]*(2 + b^2/(

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

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Maple [B] time = 0.096, size = 729, normalized size = 7.2 \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)/(a+b*tanh(d*x+c)^2)^2,x)

[Out]

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

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

n(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)^2/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/d/(a+b)^2/(tanh(1/2

*d*x+1/2*c)-1)

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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{2} + a b -{\left (a^{2} e^{\left (6 \, c\right )} + a b e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} -{\left (a^{2} e^{\left (4 \, c\right )} - 3 \, a b e^{\left (4 \, c\right )} + 2 \, b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} +{\left (a^{2} e^{\left (2 \, c\right )} - 3 \, a b e^{\left (2 \, c\right )} + 2 \, b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}{2 \,{\left ({\left (a^{4} d e^{\left (5 \, c\right )} + 3 \, a^{3} b d e^{\left (5 \, c\right )} + 3 \, a^{2} b^{2} d e^{\left (5 \, c\right )} + a b^{3} d e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} + 2 \,{\left (a^{4} d e^{\left (3 \, c\right )} + a^{3} b d e^{\left (3 \, c\right )} - a^{2} b^{2} d e^{\left (3 \, c\right )} - a b^{3} d e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} +{\left (a^{4} d e^{c} + 3 \, a^{3} b d e^{c} + 3 \, a^{2} b^{2} d e^{c} + a b^{3} d e^{c}\right )} e^{\left (d x\right )}\right )}} + \frac{1}{2} \, \int \frac{2 \,{\left ({\left (4 \, a b e^{\left (3 \, c\right )} + b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} +{\left (4 \, a b e^{c} + b^{2} e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3} +{\left (a^{4} e^{\left (4 \, c\right )} + 3 \, a^{3} b e^{\left (4 \, c\right )} + 3 \, a^{2} b^{2} e^{\left (4 \, c\right )} + a b^{3} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (a^{4} e^{\left (2 \, c\right )} + a^{3} b e^{\left (2 \, c\right )} - a^{2} b^{2} e^{\left (2 \, c\right )} - a b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

*d*e^(5*c) + a*b^3*d*e^(5*c))*e^(5*d*x) + 2*(a^4*d*e^(3*c) + a^3*b*d*e^(3*c) - a^2*b^2*d*e^(3*c) - a*b^3*d*e^(

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

*b*e^(3*c) + b^2*e^(3*c))*e^(3*d*x) + (4*a*b*e^c + b^2*e^c)*e^(d*x))/(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3 + (a^4

*e^(4*c) + 3*a^3*b*e^(4*c) + 3*a^2*b^2*e^(4*c) + a*b^3*e^(4*c))*e^(4*d*x) + 2*(a^4*e^(2*c) + a^3*b*e^(2*c) - a

^2*b^2*e^(2*c) - a*b^3*e^(2*c))*e^(2*d*x)), x)

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Fricas [B] time = 2.5325, size = 8197, normalized size = 81.16 \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)/(a+b*tanh(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

[1/4*(2*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^6 + 12*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^5

+ 2*(a^4 + 2*a^3*b + a^2*b^2)*sinh(d*x + c)^6 + 2*(a^4 - 2*a^3*b - a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^4 + 2*(a^

4 - 2*a^3*b - a^2*b^2 + 2*a*b^3 + 15*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 - 2*a^4 - 4*a^

3*b - 2*a^2*b^2 + 8*(5*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^3 + (a^4 - 2*a^3*b - a^2*b^2 + 2*a*b^3)*cosh(d*

x + c))*sinh(d*x + c)^3 - 2*(a^4 - 2*a^3*b - a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^2 + 2*(15*(a^4 + 2*a^3*b + a^2*b

^2)*cosh(d*x + c)^4 - a^4 + 2*a^3*b + a^2*b^2 - 2*a*b^3 + 6*(a^4 - 2*a^3*b - a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^

2)*sinh(d*x + c)^2 - ((4*a^2*b + 5*a*b^2 + b^3)*cosh(d*x + c)^5 + 5*(4*a^2*b + 5*a*b^2 + b^3)*cosh(d*x + c)*si

nh(d*x + c)^4 + (4*a^2*b + 5*a*b^2 + b^3)*sinh(d*x + c)^5 + 2*(4*a^2*b - 3*a*b^2 - b^3)*cosh(d*x + c)^3 + 2*(4

*a^2*b - 3*a*b^2 - b^3 + 5*(4*a^2*b + 5*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(5*(4*a^2*b + 5*a*b^

2 + b^3)*cosh(d*x + c)^3 + 3*(4*a^2*b - 3*a*b^2 - b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + (4*a^2*b + 5*a*b^2 + b

^3)*cosh(d*x + c) + (5*(4*a^2*b + 5*a*b^2 + b^3)*cosh(d*x + c)^4 + 4*a^2*b + 5*a*b^2 + b^3 + 6*(4*a^2*b - 3*a*

b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c))*sqrt(-a^2 - a*b)*log(((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*(3*a + b)*cosh(d*x + c)^2 + 2*(3*(a + b)*cosh(d*x + c)^2 -

3*a - b)*sinh(d*x + c)^2 + 4*((a + b)*cosh(d*x + c)^3 - (3*a + b)*cosh(d*x + c))*sinh(d*x + c) - 4*(cosh(d*x +

c)^3 + 3*cosh(d*x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3 + (3*cosh(d*x + c)^2 - 1)*sinh(d*x + c) - cosh(d*x +

c))*sqrt(-a^2 - a*b) + a + b)/((a + b)*cosh(d*x + c)^4 + 4*(a + b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a + b)*si

nh(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)) + 4*(3*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x +

c)^5 + 2*(a^4 - 2*a^3*b - a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^3 - (a^4 - 2*a^3*b - a^2*b^2 + 2*a*b^3)*cosh(d*x +

c))*sinh(d*x + c))/((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)^5 + 5*(a^6 + 4*a^5*b + 6

*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)*sinh(d*x + c)^4 + (a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2

*b^4)*d*sinh(d*x + c)^5 + 2*(a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c)^3 + 2*(5*(a^6 + 4*a^5*b + 6*

a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)^2 + (a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4)*d)*sinh(d*x + c)^3 +

(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*cosh(d*x + c) + 2*(5*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^

3 + a^2*b^4)*d*cosh(d*x + c)^3 + 3*(a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^2 + (5

*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)^4 + 6*(a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4)

*d*cosh(d*x + c)^2 + (a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d)*sinh(d*x + c)), 1/2*((a^4 + 2*a^3*b

+ a^2*b^2)*cosh(d*x + c)^6 + 6*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^5 + (a^4 + 2*a^3*b + a^2*

b^2)*sinh(d*x + c)^6 + (a^4 - 2*a^3*b - a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^4 + (a^4 - 2*a^3*b - a^2*b^2 + 2*a*b^

3 + 15*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 - a^4 - 2*a^3*b - a^2*b^2 + 4*(5*(a^4 + 2*a^

3*b + a^2*b^2)*cosh(d*x + c)^3 + (a^4 - 2*a^3*b - a^2*b^2 + 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - (a^4 - 2

*a^3*b - a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^2 + (15*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^4 - a^4 + 2*a^3*b +

a^2*b^2 - 2*a*b^3 + 6*(a^4 - 2*a^3*b - a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((4*a^2*b + 5*a*b

^2 + b^3)*cosh(d*x + c)^5 + 5*(4*a^2*b + 5*a*b^2 + b^3)*cosh(d*x + c)*sinh(d*x + c)^4 + (4*a^2*b + 5*a*b^2 + b

^3)*sinh(d*x + c)^5 + 2*(4*a^2*b - 3*a*b^2 - b^3)*cosh(d*x + c)^3 + 2*(4*a^2*b - 3*a*b^2 - b^3 + 5*(4*a^2*b +

5*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(5*(4*a^2*b + 5*a*b^2 + b^3)*cosh(d*x + c)^3 + 3*(4*a^2*b

- 3*a*b^2 - b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + (4*a^2*b + 5*a*b^2 + b^3)*cosh(d*x + c) + (5*(4*a^2*b + 5*a*

b^2 + b^3)*cosh(d*x + c)^4 + 4*a^2*b + 5*a*b^2 + b^3 + 6*(4*a^2*b - 3*a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x +

c))*sqrt(a^2 + a*b)*arctan(1/2*((a + b)*cosh(d*x + c)^3 + 3*(a + b)*cosh(d*x + c)*sinh(d*x + c)^2 + (a + b)*s

inh(d*x + c)^3 + (3*a - b)*cosh(d*x + c) + (3*(a + b)*cosh(d*x + c)^2 + 3*a - b)*sinh(d*x + c))/sqrt(a^2 + a*b

)) + ((4*a^2*b + 5*a*b^2 + b^3)*cosh(d*x + c)^5 + 5*(4*a^2*b + 5*a*b^2 + b^3)*cosh(d*x + c)*sinh(d*x + c)^4 +

(4*a^2*b + 5*a*b^2 + b^3)*sinh(d*x + c)^5 + 2*(4*a^2*b - 3*a*b^2 - b^3)*cosh(d*x + c)^3 + 2*(4*a^2*b - 3*a*b^2

- b^3 + 5*(4*a^2*b + 5*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(5*(4*a^2*b + 5*a*b^2 + b^3)*cosh(d*

x + c)^3 + 3*(4*a^2*b - 3*a*b^2 - b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + (4*a^2*b + 5*a*b^2 + b^3)*cosh(d*x + c

) + (5*(4*a^2*b + 5*a*b^2 + b^3)*cosh(d*x + c)^4 + 4*a^2*b + 5*a*b^2 + b^3 + 6*(4*a^2*b - 3*a*b^2 - b^3)*cosh(

d*x + c)^2)*sinh(d*x + c))*sqrt(a^2 + a*b)*arctan(1/2*sqrt(a^2 + a*b)*(cosh(d*x + c) + sinh(d*x + c))/a) + 2*(

3*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^5 + 2*(a^4 - 2*a^3*b - a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^3 - (a^4 - 2

*a^3*b - a^2*b^2 + 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c))/((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d

*cosh(d*x + c)^5 + 5*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)*sinh(d*x + c)^4 + (a^6

+ 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*sinh(d*x + c)^5 + 2*(a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4)*d*cos

h(d*x + c)^3 + 2*(5*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)^2 + (a^6 + 2*a^5*b - 2*a

^3*b^3 - a^2*b^4)*d)*sinh(d*x + c)^3 + (a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*cosh(d*x + c) + 2*(

5*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)^3 + 3*(a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4

)*d*cosh(d*x + c))*sinh(d*x + c)^2 + (5*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)^4 +

6*(a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c)^2 + (a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*

d)*sinh(d*x + c))]

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [C] time = 1.88005, size = 8317, normalized size = 82.35 \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)/(a+b*tanh(d*x+c)^2)^2,x, algorithm="giac")

[Out]

1/8*(2*(3*(4*a^5*b*e^(4*c) + 13*a^4*b^2*e^(4*c) + 15*a^3*b^3*e^(4*c) + 7*a^2*b^4*e^(4*c) + a*b^5*e^(4*c))*cos(

1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3*sin(1/2

*real_part(arccos(-a/(a + b) + b/(a + b)))) - (4*a^5*b*e^(4*c) + 13*a^4*b^2*e^(4*c) + 15*a^3*b^3*e^(4*c) + 7*a

^2*b^4*e^(4*c) + a*b^5*e^(4*c))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3*sin(1/2*real_part(arccos

(-a/(a + b) + b/(a + b))))^3 - 9*(4*a^5*b*e^(4*c) + 13*a^4*b^2*e^(4*c) + 15*a^3*b^3*e^(4*c) + 7*a^2*b^4*e^(4*c

) + a*b^5*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*cosh(1/2*imag_part(arccos(-a/(a + b) +

b/(a + b))))^2*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a

+ b)))) + 3*(4*a^5*b*e^(4*c) + 13*a^4*b^2*e^(4*c) + 15*a^3*b^3*e^(4*c) + 7*a^2*b^4*e^(4*c) + a*b^5*e^(4*c))*c

osh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*sinh

(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 9*(4*a^5*b*e^(4*c) + 13*a^4*b^2*e^(4*c) + 15*a^3*b^3*e^(4*c)

+ 7*a^2*b^4*e^(4*c) + a*b^5*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*cosh(1/2*imag_part(

arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(

-a/(a + b) + b/(a + b))))^2 - 3*(4*a^5*b*e^(4*c) + 13*a^4*b^2*e^(4*c) + 15*a^3*b^3*e^(4*c) + 7*a^2*b^4*e^(4*c)

+ a*b^5*e^(4*c))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/

(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(4*a^5*b*e^(4*c) + 13*a^4*b^2*e^(4*c) +

15*a^3*b^3*e^(4*c) + 7*a^2*b^4*e^(4*c) + a*b^5*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*

sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + (4*

a^5*b*e^(4*c) + 13*a^4*b^2*e^(4*c) + 15*a^3*b^3*e^(4*c) + 7*a^2*b^4*e^(4*c) + a*b^5*e^(4*c))*sin(1/2*real_part

(arccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + (4*a^5*b*e^(4*c) +

13*a^4*b^2*e^(4*c) + 15*a^3*b^3*e^(4*c) + 7*a^2*b^4*e^(4*c) + a*b^5*e^(4*c))*cosh(1/2*imag_part(arccos(-a/(a

+ b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b)))) - (4*a^5*b*e^(4*c) + 13*a^4*b^2*e^(4*c)

+ 15*a^3*b^3*e^(4*c) + 7*a^2*b^4*e^(4*c) + a*b^5*e^(4*c))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*

sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))))*arctan((((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)/(a^4*e^(4*c)

+ 3*a^3*b*e^(4*c) + 3*a^2*b^2*e^(4*c) + a*b^3*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b))) + e^(d*x))/((

(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)/(a^4*e^(4*c) + 3*a^3*b*e^(4*c) + 3*a^2*b^2*e^(4*c) + a*b^3*e^(4*c)))^(1/4)

*sin(1/2*arccos(-(a - b)/(a + b)))))/(2*(a^3*e^(2*c) + 2*a^2*b*e^(2*c) + a*b^2*e^(2*c))^2*a*b + (a^4*e^(2*c) +

a^3*b*e^(2*c) - a^2*b^2*e^(2*c) - a*b^3*e^(2*c))*sqrt(-a*b)*abs(a^3*e^(2*c) + 2*a^2*b*e^(2*c) + a*b^2*e^(2*c)

)) + 2*(3*(4*a^5*b*e^(4*c) + 13*a^4*b^2*e^(4*c) + 15*a^3*b^3*e^(4*c) + 7*a^2*b^4*e^(4*c) + a*b^5*e^(4*c))*cos(