3.84 \(\int \frac {\cosh ^2(c+d x)}{(a+b \text {sech}^2(c+d x))^2} \, dx\)

Optimal. Leaf size=144 \[ \frac {b^{3/2} (5 a+4 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 a^3 d (a+b)^{3/2}}+\frac {x (a-4 b)}{2 a^3}+\frac {b (a+2 b) \tanh (c+d x)}{2 a^2 d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}+\frac {\sinh (c+d x) \cosh (c+d x)}{2 a d \left (a-b \tanh ^2(c+d x)+b\right )} \]

[Out]

1/2*(a-4*b)*x/a^3+1/2*b^(3/2)*(5*a+4*b)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))/a^3/(a+b)^(3/2)/d+1/2*cosh(d*
x+c)*sinh(d*x+c)/a/d/(a+b-b*tanh(d*x+c)^2)+1/2*b*(a+2*b)*tanh(d*x+c)/a^2/(a+b)/d/(a+b-b*tanh(d*x+c)^2)

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Rubi [A]  time = 0.24, antiderivative size = 144, normalized size of antiderivative = 1.00, 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 = {4146, 414, 527, 522, 206, 208} \[ \frac {b^{3/2} (5 a+4 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 a^3 d (a+b)^{3/2}}+\frac {b (a+2 b) \tanh (c+d x)}{2 a^2 d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}+\frac {x (a-4 b)}{2 a^3}+\frac {\sinh (c+d x) \cosh (c+d x)}{2 a d \left (a-b \tanh ^2(c+d x)+b\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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]

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 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 4146

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

Rubi steps

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

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Mathematica [A]  time = 1.38, size = 103, normalized size = 0.72 \[ \frac {\frac {2 b^{3/2} (5 a+4 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{3/2}}+\sinh (2 (c+d x)) \left (\frac {2 a b^2}{(a+b) (a \cosh (2 (c+d x))+a+2 b)}+a\right )+2 (a-4 b) (c+d x)}{4 a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.49, size = 3739, normalized size = 25.97 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*((a^3 + a^2*b)*cosh(d*x + c)^8 + 8*(a^3 + a^2*b)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^3 + a^2*b)*sinh(d*x +
c)^8 + 2*(a^3 + 3*a^2*b + 2*a*b^2 + 2*(a^3 - 3*a^2*b - 4*a*b^2)*d*x)*cosh(d*x + c)^6 + 2*(a^3 + 3*a^2*b + 2*a
*b^2 + 2*(a^3 - 3*a^2*b - 4*a*b^2)*d*x + 14*(a^3 + a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(14*(a^3 + a^2*
b)*cosh(d*x + c)^3 + 3*(a^3 + 3*a^2*b + 2*a*b^2 + 2*(a^3 - 3*a^2*b - 4*a*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c
)^5 - 8*(a*b^2 + 2*b^3 - (a^3 - a^2*b - 10*a*b^2 - 8*b^3)*d*x)*cosh(d*x + c)^4 + 2*(35*(a^3 + a^2*b)*cosh(d*x
+ c)^4 - 4*a*b^2 - 8*b^3 + 4*(a^3 - a^2*b - 10*a*b^2 - 8*b^3)*d*x + 15*(a^3 + 3*a^2*b + 2*a*b^2 + 2*(a^3 - 3*a
^2*b - 4*a*b^2)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(a^3 + a^2*b)*cosh(d*x + c)^5 + 5*(a^3 + 3*a^2*b
+ 2*a*b^2 + 2*(a^3 - 3*a^2*b - 4*a*b^2)*d*x)*cosh(d*x + c)^3 - 4*(a*b^2 + 2*b^3 - (a^3 - a^2*b - 10*a*b^2 - 8*
b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 - a^3 - a^2*b - 2*(a^3 + 3*a^2*b + 6*a*b^2 - 2*(a^3 - 3*a^2*b - 4*a*b
^2)*d*x)*cosh(d*x + c)^2 + 2*(14*(a^3 + a^2*b)*cosh(d*x + c)^6 + 15*(a^3 + 3*a^2*b + 2*a*b^2 + 2*(a^3 - 3*a^2*
b - 4*a*b^2)*d*x)*cosh(d*x + c)^4 - a^3 - 3*a^2*b - 6*a*b^2 + 2*(a^3 - 3*a^2*b - 4*a*b^2)*d*x - 24*(a*b^2 + 2*
b^3 - (a^3 - a^2*b - 10*a*b^2 - 8*b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*((5*a^2*b + 4*a*b^2)*cosh(d*x
+ c)^6 + 6*(5*a^2*b + 4*a*b^2)*cosh(d*x + c)*sinh(d*x + c)^5 + (5*a^2*b + 4*a*b^2)*sinh(d*x + c)^6 + 2*(5*a^2
*b + 14*a*b^2 + 8*b^3)*cosh(d*x + c)^4 + (10*a^2*b + 28*a*b^2 + 16*b^3 + 15*(5*a^2*b + 4*a*b^2)*cosh(d*x + c)^
2)*sinh(d*x + c)^4 + 4*(5*(5*a^2*b + 4*a*b^2)*cosh(d*x + c)^3 + 2*(5*a^2*b + 14*a*b^2 + 8*b^3)*cosh(d*x + c))*
sinh(d*x + c)^3 + (5*a^2*b + 4*a*b^2)*cosh(d*x + c)^2 + (15*(5*a^2*b + 4*a*b^2)*cosh(d*x + c)^4 + 5*a^2*b + 4*
a*b^2 + 12*(5*a^2*b + 14*a*b^2 + 8*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(3*(5*a^2*b + 4*a*b^2)*cosh(d*x +
c)^5 + 4*(5*a^2*b + 14*a*b^2 + 8*b^3)*cosh(d*x + c)^3 + (5*a^2*b + 4*a*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqr
t(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*
(2*(a^3 + a^2*b)*cosh(d*x + c)^7 + 3*(a^3 + 3*a^2*b + 2*a*b^2 + 2*(a^3 - 3*a^2*b - 4*a*b^2)*d*x)*cosh(d*x + c)
^5 - 8*(a*b^2 + 2*b^3 - (a^3 - a^2*b - 10*a*b^2 - 8*b^3)*d*x)*cosh(d*x + c)^3 - (a^3 + 3*a^2*b + 6*a*b^2 - 2*(
a^3 - 3*a^2*b - 4*a*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^5 + a^4*b)*d*cosh(d*x + c)^6 + 6*(a^5 + a^4*b)
*d*cosh(d*x + c)*sinh(d*x + c)^5 + (a^5 + a^4*b)*d*sinh(d*x + c)^6 + 2*(a^5 + 3*a^4*b + 2*a^3*b^2)*d*cosh(d*x
+ c)^4 + (15*(a^5 + a^4*b)*d*cosh(d*x + c)^2 + 2*(a^5 + 3*a^4*b + 2*a^3*b^2)*d)*sinh(d*x + c)^4 + (a^5 + a^4*b
)*d*cosh(d*x + c)^2 + 4*(5*(a^5 + a^4*b)*d*cosh(d*x + c)^3 + 2*(a^5 + 3*a^4*b + 2*a^3*b^2)*d*cosh(d*x + c))*si
nh(d*x + c)^3 + (15*(a^5 + a^4*b)*d*cosh(d*x + c)^4 + 12*(a^5 + 3*a^4*b + 2*a^3*b^2)*d*cosh(d*x + c)^2 + (a^5
+ a^4*b)*d)*sinh(d*x + c)^2 + 2*(3*(a^5 + a^4*b)*d*cosh(d*x + c)^5 + 4*(a^5 + 3*a^4*b + 2*a^3*b^2)*d*cosh(d*x
+ c)^3 + (a^5 + a^4*b)*d*cosh(d*x + c))*sinh(d*x + c)), 1/8*((a^3 + a^2*b)*cosh(d*x + c)^8 + 8*(a^3 + a^2*b)*c
osh(d*x + c)*sinh(d*x + c)^7 + (a^3 + a^2*b)*sinh(d*x + c)^8 + 2*(a^3 + 3*a^2*b + 2*a*b^2 + 2*(a^3 - 3*a^2*b -
4*a*b^2)*d*x)*cosh(d*x + c)^6 + 2*(a^3 + 3*a^2*b + 2*a*b^2 + 2*(a^3 - 3*a^2*b - 4*a*b^2)*d*x + 14*(a^3 + a^2*
b)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(14*(a^3 + a^2*b)*cosh(d*x + c)^3 + 3*(a^3 + 3*a^2*b + 2*a*b^2 + 2*(a^
3 - 3*a^2*b - 4*a*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 - 8*(a*b^2 + 2*b^3 - (a^3 - a^2*b - 10*a*b^2 - 8*b^
3)*d*x)*cosh(d*x + c)^4 + 2*(35*(a^3 + a^2*b)*cosh(d*x + c)^4 - 4*a*b^2 - 8*b^3 + 4*(a^3 - a^2*b - 10*a*b^2 -
8*b^3)*d*x + 15*(a^3 + 3*a^2*b + 2*a*b^2 + 2*(a^3 - 3*a^2*b - 4*a*b^2)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 +
8*(7*(a^3 + a^2*b)*cosh(d*x + c)^5 + 5*(a^3 + 3*a^2*b + 2*a*b^2 + 2*(a^3 - 3*a^2*b - 4*a*b^2)*d*x)*cosh(d*x +
c)^3 - 4*(a*b^2 + 2*b^3 - (a^3 - a^2*b - 10*a*b^2 - 8*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 - a^3 - a^2*b
- 2*(a^3 + 3*a^2*b + 6*a*b^2 - 2*(a^3 - 3*a^2*b - 4*a*b^2)*d*x)*cosh(d*x + c)^2 + 2*(14*(a^3 + a^2*b)*cosh(d*x
+ c)^6 + 15*(a^3 + 3*a^2*b + 2*a*b^2 + 2*(a^3 - 3*a^2*b - 4*a*b^2)*d*x)*cosh(d*x + c)^4 - a^3 - 3*a^2*b - 6*a
*b^2 + 2*(a^3 - 3*a^2*b - 4*a*b^2)*d*x - 24*(a*b^2 + 2*b^3 - (a^3 - a^2*b - 10*a*b^2 - 8*b^3)*d*x)*cosh(d*x +
c)^2)*sinh(d*x + c)^2 + 4*((5*a^2*b + 4*a*b^2)*cosh(d*x + c)^6 + 6*(5*a^2*b + 4*a*b^2)*cosh(d*x + c)*sinh(d*x
+ c)^5 + (5*a^2*b + 4*a*b^2)*sinh(d*x + c)^6 + 2*(5*a^2*b + 14*a*b^2 + 8*b^3)*cosh(d*x + c)^4 + (10*a^2*b + 28
*a*b^2 + 16*b^3 + 15*(5*a^2*b + 4*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*(5*a^2*b + 4*a*b^2)*cosh(d*x
+ c)^3 + 2*(5*a^2*b + 14*a*b^2 + 8*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + (5*a^2*b + 4*a*b^2)*cosh(d*x + c)^2 +
(15*(5*a^2*b + 4*a*b^2)*cosh(d*x + c)^4 + 5*a^2*b + 4*a*b^2 + 12*(5*a^2*b + 14*a*b^2 + 8*b^3)*cosh(d*x + c)^2
)*sinh(d*x + c)^2 + 2*(3*(5*a^2*b + 4*a*b^2)*cosh(d*x + c)^5 + 4*(5*a^2*b + 14*a*b^2 + 8*b^3)*cosh(d*x + c)^3
+ (5*a^2*b + 4*a*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) + 4*(2*(a^3 + a^2*b)*cosh(d*x + c)^7
+ 3*(a^3 + 3*a^2*b + 2*a*b^2 + 2*(a^3 - 3*a^2*b - 4*a*b^2)*d*x)*cosh(d*x + c)^5 - 8*(a*b^2 + 2*b^3 - (a^3 - a
^2*b - 10*a*b^2 - 8*b^3)*d*x)*cosh(d*x + c)^3 - (a^3 + 3*a^2*b + 6*a*b^2 - 2*(a^3 - 3*a^2*b - 4*a*b^2)*d*x)*co
sh(d*x + c))*sinh(d*x + c))/((a^5 + a^4*b)*d*cosh(d*x + c)^6 + 6*(a^5 + a^4*b)*d*cosh(d*x + c)*sinh(d*x + c)^5
+ (a^5 + a^4*b)*d*sinh(d*x + c)^6 + 2*(a^5 + 3*a^4*b + 2*a^3*b^2)*d*cosh(d*x + c)^4 + (15*(a^5 + a^4*b)*d*cos
h(d*x + c)^2 + 2*(a^5 + 3*a^4*b + 2*a^3*b^2)*d)*sinh(d*x + c)^4 + (a^5 + a^4*b)*d*cosh(d*x + c)^2 + 4*(5*(a^5
+ a^4*b)*d*cosh(d*x + c)^3 + 2*(a^5 + 3*a^4*b + 2*a^3*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (15*(a^5 + a^4*b
)*d*cosh(d*x + c)^4 + 12*(a^5 + 3*a^4*b + 2*a^3*b^2)*d*cosh(d*x + c)^2 + (a^5 + a^4*b)*d)*sinh(d*x + c)^2 + 2*
(3*(a^5 + a^4*b)*d*cosh(d*x + c)^5 + 4*(a^5 + 3*a^4*b + 2*a^3*b^2)*d*cosh(d*x + c)^3 + (a^5 + a^4*b)*d*cosh(d*
x + c))*sinh(d*x + c))]

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giac [B]  time = 2.02, size = 323, normalized size = 2.24 \[ \frac {\frac {12 \, {\left (5 \, a b^{2} + 4 \, b^{3}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{{\left (a^{4} + a^{3} b\right )} \sqrt {-a b - b^{2}}} - \frac {2 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 6 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 8 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 7 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 16 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 16 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 12 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 28 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a^{3} + 3 \, a^{2} b}{{\left (a^{4} + a^{3} b\right )} {\left (a e^{\left (6 \, d x + 6 \, c\right )} + 2 \, a e^{\left (4 \, d x + 4 \, c\right )} + 4 \, b e^{\left (4 \, d x + 4 \, c\right )} + a e^{\left (2 \, d x + 2 \, c\right )}\right )}} + \frac {12 \, {\left (d x + c\right )} {\left (a - 4 \, b\right )}}{a^{3}} + \frac {3 \, e^{\left (2 \, d x + 2 \, c\right )}}{a^{2}}}{24 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(12*(5*a*b^2 + 4*b^3)*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/((a^4 + a^3*b)*sqrt(-a*b
- b^2)) - (2*a^3*e^(6*d*x + 6*c) - 6*a^2*b*e^(6*d*x + 6*c) - 8*a*b^2*e^(6*d*x + 6*c) + 7*a^3*e^(4*d*x + 4*c)
- a^2*b*e^(4*d*x + 4*c) - 16*a*b^2*e^(4*d*x + 4*c) + 16*b^3*e^(4*d*x + 4*c) + 8*a^3*e^(2*d*x + 2*c) + 12*a^2*b
*e^(2*d*x + 2*c) + 28*a*b^2*e^(2*d*x + 2*c) + 3*a^3 + 3*a^2*b)/((a^4 + a^3*b)*(a*e^(6*d*x + 6*c) + 2*a*e^(4*d*
x + 4*c) + 4*b*e^(4*d*x + 4*c) + a*e^(2*d*x + 2*c))) + 12*(d*x + c)*(a - 4*b)/a^3 + 3*e^(2*d*x + 2*c)/a^2)/d

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maple [B]  time = 0.51, size = 557, normalized size = 3.87 \[ \frac {1}{2 d \,a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 d \,a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d \,a^{2}}+\frac {2 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) b}{d \,a^{3}}-\frac {1}{2 d \,a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{2 d \,a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d \,a^{2}}-\frac {2 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b}{d \,a^{3}}+\frac {b^{2} \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right ) \left (a +b \right )}+\frac {b^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{2} \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right ) \left (a +b \right )}-\frac {5 b^{\frac {3}{2}} \ln \left (-\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\sqrt {a +b}\right )}{4 d \,a^{2} \left (a +b \right )^{\frac {3}{2}}}+\frac {5 b^{\frac {3}{2}} \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {a +b}\right )}{4 d \,a^{2} \left (a +b \right )^{\frac {3}{2}}}-\frac {b^{\frac {5}{2}} \ln \left (-\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\sqrt {a +b}\right )}{d \,a^{3} \left (a +b \right )^{\frac {3}{2}}}+\frac {b^{\frac {5}{2}} \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {a +b}\right )}{d \,a^{3} \left (a +b \right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.47, size = 696, normalized size = 4.83 \[ \frac {{\left (3 \, a^{2} b + 12 \, a b^{2} + 8 \, b^{3}\right )} \log \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{16 \, {\left (a^{4} + a^{3} b\right )} \sqrt {{\left (a + b\right )} b} d} - \frac {{\left (3 \, a^{2} b + 12 \, a b^{2} + 8 \, b^{3}\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{16 \, {\left (a^{4} + a^{3} b\right )} \sqrt {{\left (a + b\right )} b} d} + \frac {{\left (3 \, a b + 2 \, b^{2}\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{8 \, {\left (a^{3} + a^{2} b\right )} \sqrt {{\left (a + b\right )} b} d} - \frac {a^{2} b + 2 \, a b^{2} + {\left (a^{2} b + 8 \, a b^{2} + 8 \, b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{4 \, {\left (a^{5} + a^{4} b + {\left (a^{5} + a^{4} b\right )} e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} d} + \frac {a^{2} b + 2 \, a b^{2} + {\left (a^{2} b + 8 \, a b^{2} + 8 \, b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{4 \, {\left (a^{5} + a^{4} b + 2 \, {\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{5} + a^{4} b\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} - \frac {a b + {\left (a b + 2 \, b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{2 \, {\left (a^{4} + a^{3} b + 2 \, {\left (a^{4} + 3 \, a^{3} b + 2 \, a^{2} b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{4} + a^{3} b\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} + \frac {d x + c}{2 \, a^{2} d} + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{8 \, a^{2} d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, a^{2} d} - \frac {b \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (a + 2 \, b\right )} e^{\left (2 \, d x + 2 \, c\right )} + a\right )}{2 \, a^{3} d} + \frac {b \log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{2 \, a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(3*a^2*b + 12*a*b^2 + 8*b^3)*log((a*e^(2*d*x + 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(2*d*x + 2*c) + a
+ 2*b + 2*sqrt((a + b)*b)))/((a^4 + a^3*b)*sqrt((a + b)*b)*d) - 1/16*(3*a^2*b + 12*a*b^2 + 8*b^3)*log((a*e^(-
2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^4 + a^3*b)
*sqrt((a + b)*b)*d) + 1/8*(3*a*b + 2*b^2)*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x
- 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^3 + a^2*b)*sqrt((a + b)*b)*d) - 1/4*(a^2*b + 2*a*b^2 + (a^2*b + 8*a
*b^2 + 8*b^3)*e^(2*d*x + 2*c))/((a^5 + a^4*b + (a^5 + a^4*b)*e^(4*d*x + 4*c) + 2*(a^5 + 3*a^4*b + 2*a^3*b^2)*e
^(2*d*x + 2*c))*d) + 1/4*(a^2*b + 2*a*b^2 + (a^2*b + 8*a*b^2 + 8*b^3)*e^(-2*d*x - 2*c))/((a^5 + a^4*b + 2*(a^5
+ 3*a^4*b + 2*a^3*b^2)*e^(-2*d*x - 2*c) + (a^5 + a^4*b)*e^(-4*d*x - 4*c))*d) - 1/2*(a*b + (a*b + 2*b^2)*e^(-2
*d*x - 2*c))/((a^4 + a^3*b + 2*(a^4 + 3*a^3*b + 2*a^2*b^2)*e^(-2*d*x - 2*c) + (a^4 + a^3*b)*e^(-4*d*x - 4*c))*
d) + 1/2*(d*x + c)/(a^2*d) + 1/8*e^(2*d*x + 2*c)/(a^2*d) - 1/8*e^(-2*d*x - 2*c)/(a^2*d) - 1/2*b*log(a*e^(4*d*x
+ 4*c) + 2*(a + 2*b)*e^(2*d*x + 2*c) + a)/(a^3*d) + 1/2*b*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*
c) + a)/(a^3*d)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2}{{\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)**2/(a+b*sech(d*x+c)**2)**2,x)

[Out]

Timed out

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