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

Optimal. Leaf size=100 \[ -\frac {b (4 a+3 b) \tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{2 a^{5/2} d (a+b)^{3/2}}+\frac {b^2 \sinh (c+d x)}{2 a^2 d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )}+\frac {\sinh (c+d x)}{a^2 d} \]

[Out]

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

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Rubi [A]  time = 0.13, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {4147, 390, 385, 205} \[ \frac {b^2 \sinh (c+d x)}{2 a^2 d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )}-\frac {b (4 a+3 b) \tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{2 a^{5/2} d (a+b)^{3/2}}+\frac {\sinh (c+d x)}{a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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 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 4147

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = Fr
eeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[ExpandToSum[b + 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]

Rubi steps

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

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Mathematica [B]  time = 1.81, size = 234, normalized size = 2.34 \[ \frac {\text {sech}^3(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (\frac {2 \sqrt {a} b^2 \tanh (c+d x)}{a+b}+2 \sqrt {a} \sinh (c) \cosh (d x) \text {sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)+2 \sqrt {a} \cosh (c) \sinh (d x) \text {sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)+\frac {b (4 a+3 b) (\cosh (c)-\sinh (c)) \text {sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b) \tan ^{-1}\left (\frac {\sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text {csch}(c+d x)}{\sqrt {a}}\right )}{(a+b)^{3/2} \sqrt {(\cosh (c)-\sinh (c))^2}}\right )}{8 a^{5/2} d \left (a+b \text {sech}^2(c+d x)\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^3*((b*(4*a + 3*b)*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(Cosh[
c] - Sinh[c])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]*(Cosh[c] - Sinh[c
]))/((a + b)^(3/2)*Sqrt[(Cosh[c] - Sinh[c])^2]) + 2*Sqrt[a]*Cosh[d*x]*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c +
d*x]*Sinh[c] + 2*Sqrt[a]*Cosh[c]*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]*Sinh[d*x] + (2*Sqrt[a]*b^2*Tan
h[c + d*x])/(a + b)))/(8*a^(5/2)*d*(a + b*Sech[c + d*x]^2)^2)

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fricas [B]  time = 0.49, size = 3154, normalized size = 31.54 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)/(a+b*sech(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 + 6*a^3*b + 11*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^4 + 2*
(a^4 + 6*a^3*b + 11*a^2*b^2 + 6*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 + 6*a^3*b + 11*a^2*b^2 + 6*a*b^3
)*cosh(d*x + c))*sinh(d*x + c)^3 - 2*(a^4 + 6*a^3*b + 11*a^2*b^2 + 6*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 - 6*a^3*b - 11*a^2*b^2 - 6*a*b^3 + 6*(a^4 + 6*a^3*b + 11*a^2*b^2 + 6*a*b
^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((4*a^2*b + 3*a*b^2)*cosh(d*x + c)^5 + 5*(4*a^2*b + 3*a*b^2)*cosh(d*x +
c)*sinh(d*x + c)^4 + (4*a^2*b + 3*a*b^2)*sinh(d*x + c)^5 + 2*(4*a^2*b + 11*a*b^2 + 6*b^3)*cosh(d*x + c)^3 + 2
*(4*a^2*b + 11*a*b^2 + 6*b^3 + 5*(4*a^2*b + 3*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(5*(4*a^2*b + 3*a*b^
2)*cosh(d*x + c)^3 + 3*(4*a^2*b + 11*a*b^2 + 6*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + (4*a^2*b + 3*a*b^2)*cosh(
d*x + c) + (5*(4*a^2*b + 3*a*b^2)*cosh(d*x + c)^4 + 4*a^2*b + 3*a*b^2 + 6*(4*a^2*b + 11*a*b^2 + 6*b^3)*cosh(d*
x + c)^2)*sinh(d*x + c))*sqrt(-a^2 - a*b)*log((a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(
d*x + c)^4 - 2*(3*a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 - 3*a - 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d
*x + c)^3 - (3*a + 2*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)/(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*(3*(
a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^5 + 2*(a^4 + 6*a^3*b + 11*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^3 - (a^4 + 6
*a^3*b + 11*a^2*b^2 + 6*a*b^3)*cosh(d*x + c))*sinh(d*x + c))/((a^6 + 2*a^5*b + a^4*b^2)*d*cosh(d*x + c)^5 + 5*
(a^6 + 2*a^5*b + a^4*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^4 + (a^6 + 2*a^5*b + a^4*b^2)*d*sinh(d*x + c)^5 + 2*(a
^6 + 4*a^5*b + 5*a^4*b^2 + 2*a^3*b^3)*d*cosh(d*x + c)^3 + 2*(5*(a^6 + 2*a^5*b + a^4*b^2)*d*cosh(d*x + c)^2 + (
a^6 + 4*a^5*b + 5*a^4*b^2 + 2*a^3*b^3)*d)*sinh(d*x + c)^3 + (a^6 + 2*a^5*b + a^4*b^2)*d*cosh(d*x + c) + 2*(5*(
a^6 + 2*a^5*b + a^4*b^2)*d*cosh(d*x + c)^3 + 3*(a^6 + 4*a^5*b + 5*a^4*b^2 + 2*a^3*b^3)*d*cosh(d*x + c))*sinh(d
*x + c)^2 + (5*(a^6 + 2*a^5*b + a^4*b^2)*d*cosh(d*x + c)^4 + 6*(a^6 + 4*a^5*b + 5*a^4*b^2 + 2*a^3*b^3)*d*cosh(
d*x + c)^2 + (a^6 + 2*a^5*b + a^4*b^2)*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 + 6*
a^3*b + 11*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^4 + (a^4 + 6*a^3*b + 11*a^2*b^2 + 6*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 + 6*a^3*b + 11*a^2*b^2 + 6*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - (a^4 + 6*a^3*b + 11*a^2*b^2 + 6
*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 - 6*a^3*b - 11*a^2*b^2 - 6*a*b^3
+ 6*(a^4 + 6*a^3*b + 11*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((4*a^2*b + 3*a*b^2)*cosh(d*x +
c)^5 + 5*(4*a^2*b + 3*a*b^2)*cosh(d*x + c)*sinh(d*x + c)^4 + (4*a^2*b + 3*a*b^2)*sinh(d*x + c)^5 + 2*(4*a^2*b
+ 11*a*b^2 + 6*b^3)*cosh(d*x + c)^3 + 2*(4*a^2*b + 11*a*b^2 + 6*b^3 + 5*(4*a^2*b + 3*a*b^2)*cosh(d*x + c)^2)*
sinh(d*x + c)^3 + 2*(5*(4*a^2*b + 3*a*b^2)*cosh(d*x + c)^3 + 3*(4*a^2*b + 11*a*b^2 + 6*b^3)*cosh(d*x + c))*sin
h(d*x + c)^2 + (4*a^2*b + 3*a*b^2)*cosh(d*x + c) + (5*(4*a^2*b + 3*a*b^2)*cosh(d*x + c)^4 + 4*a^2*b + 3*a*b^2
+ 6*(4*a^2*b + 11*a*b^2 + 6*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))*sqrt(a^2 + a*b)*arctan(1/2*(a*cosh(d*x + c)^3
+ 3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3 + (3*a + 4*b)*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 +
3*a + 4*b)*sinh(d*x + c))/sqrt(a^2 + a*b)) - ((4*a^2*b + 3*a*b^2)*cosh(d*x + c)^5 + 5*(4*a^2*b + 3*a*b^2)*cosh
(d*x + c)*sinh(d*x + c)^4 + (4*a^2*b + 3*a*b^2)*sinh(d*x + c)^5 + 2*(4*a^2*b + 11*a*b^2 + 6*b^3)*cosh(d*x + c)
^3 + 2*(4*a^2*b + 11*a*b^2 + 6*b^3 + 5*(4*a^2*b + 3*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(5*(4*a^2*b +
3*a*b^2)*cosh(d*x + c)^3 + 3*(4*a^2*b + 11*a*b^2 + 6*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + (4*a^2*b + 3*a*b^2)
*cosh(d*x + c) + (5*(4*a^2*b + 3*a*b^2)*cosh(d*x + c)^4 + 4*a^2*b + 3*a*b^2 + 6*(4*a^2*b + 11*a*b^2 + 6*b^3)*c
osh(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 +
b)) + 2*(3*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^5 + 2*(a^4 + 6*a^3*b + 11*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)
^3 - (a^4 + 6*a^3*b + 11*a^2*b^2 + 6*a*b^3)*cosh(d*x + c))*sinh(d*x + c))/((a^6 + 2*a^5*b + a^4*b^2)*d*cosh(d*
x + c)^5 + 5*(a^6 + 2*a^5*b + a^4*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^4 + (a^6 + 2*a^5*b + a^4*b^2)*d*sinh(d*x
+ c)^5 + 2*(a^6 + 4*a^5*b + 5*a^4*b^2 + 2*a^3*b^3)*d*cosh(d*x + c)^3 + 2*(5*(a^6 + 2*a^5*b + a^4*b^2)*d*cosh(d
*x + c)^2 + (a^6 + 4*a^5*b + 5*a^4*b^2 + 2*a^3*b^3)*d)*sinh(d*x + c)^3 + (a^6 + 2*a^5*b + a^4*b^2)*d*cosh(d*x
+ c) + 2*(5*(a^6 + 2*a^5*b + a^4*b^2)*d*cosh(d*x + c)^3 + 3*(a^6 + 4*a^5*b + 5*a^4*b^2 + 2*a^3*b^3)*d*cosh(d*x
+ c))*sinh(d*x + c)^2 + (5*(a^6 + 2*a^5*b + a^4*b^2)*d*cosh(d*x + c)^4 + 6*(a^6 + 4*a^5*b + 5*a^4*b^2 + 2*a^3
*b^3)*d*cosh(d*x + c)^2 + (a^6 + 2*a^5*b + a^4*b^2)*d)*sinh(d*x + c))]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [a,b]=[6,-20]Warning, need to choose a branch for the root of a polynomial with parameters. This
might be wrong.The choice was done assuming [a,b]=[89,-63]Warning, need to choose a branch for the root of a
polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[12,-32]Warning, need to ch
oose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [
a,b]=[2,72]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.
The choice was done assuming [a,b]=[67,31]Warning, need to choose a branch for the root of a polynomial with p
arameters. This might be wrong.The choice was done assuming [a,b]=[-88,66]Undef/Unsigned Inf encountered in li
mitEvaluation time: 0.99Limit: Max order reached or unable to make series expansion Error: Bad Argument Value

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maple [B]  time = 0.51, size = 385, normalized size = 3.85 \[ -\frac {1}{d \,a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{d \,a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {b^{2} \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right ) \left (a +b \right )}+\frac {b^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{2} \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right ) \left (a +b \right )}-\frac {2 b \arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{d \,a^{\frac {3}{2}} \left (a +b \right )^{\frac {3}{2}}}-\frac {2 b \arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{d \,a^{\frac {3}{2}} \left (a +b \right )^{\frac {3}{2}}}-\frac {3 b^{2} \arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 d \,a^{\frac {5}{2}} \left (a +b \right )^{\frac {3}{2}}}-\frac {3 b^{2} \arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 d \,a^{\frac {5}{2}} \left (a +b \right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {a^{2} + a b - {\left (a^{2} e^{\left (6 \, c\right )} + a b e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} - {\left (a^{2} e^{\left (4 \, c\right )} + 5 \, a b e^{\left (4 \, c\right )} + 6 \, b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + {\left (a^{2} e^{\left (2 \, c\right )} + 5 \, a b e^{\left (2 \, c\right )} + 6 \, b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}{2 \, {\left ({\left (a^{4} d e^{\left (5 \, c\right )} + a^{3} b d e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} + 2 \, {\left (a^{4} d e^{\left (3 \, c\right )} + 3 \, a^{3} b d e^{\left (3 \, c\right )} + 2 \, a^{2} b^{2} d e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + {\left (a^{4} d e^{c} + a^{3} b d e^{c}\right )} e^{\left (d x\right )}\right )}} - \frac {1}{2} \, \int \frac {2 \, {\left ({\left (4 \, a b e^{\left (3 \, c\right )} + 3 \, b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + {\left (4 \, a b e^{c} + 3 \, b^{2} e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{4} + a^{3} b + {\left (a^{4} e^{\left (4 \, c\right )} + a^{3} b e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \, {\left (a^{4} e^{\left (2 \, c\right )} + 3 \, a^{3} b e^{\left (2 \, c\right )} + 2 \, a^{2} b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)/(a+b*sech(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) + 5*a*b*e^(4*c) + 6*b^2*e^(4*c))*e^(4*d
*x) + (a^2*e^(2*c) + 5*a*b*e^(2*c) + 6*b^2*e^(2*c))*e^(2*d*x))/((a^4*d*e^(5*c) + a^3*b*d*e^(5*c))*e^(5*d*x) +
2*(a^4*d*e^(3*c) + 3*a^3*b*d*e^(3*c) + 2*a^2*b^2*d*e^(3*c))*e^(3*d*x) + (a^4*d*e^c + a^3*b*d*e^c)*e^(d*x)) - 1
/2*integrate(2*((4*a*b*e^(3*c) + 3*b^2*e^(3*c))*e^(3*d*x) + (4*a*b*e^c + 3*b^2*e^c)*e^(d*x))/(a^4 + a^3*b + (a
^4*e^(4*c) + a^3*b*e^(4*c))*e^(4*d*x) + 2*(a^4*e^(2*c) + 3*a^3*b*e^(2*c) + 2*a^2*b^2*e^(2*c))*e^(2*d*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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