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

Optimal. Leaf size=52 \[ \frac {\sinh (c+d x)}{a d}-\frac {b \tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{a^{3/2} d \sqrt {a+b}} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4147, 388, 205} \[ \frac {\sinh (c+d x)}{a d}-\frac {b \tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{a^{3/2} d \sqrt {a+b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 388

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

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

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Mathematica [A]  time = 0.12, size = 52, normalized size = 1.00 \[ \frac {\sqrt {a} \sinh (c+d x)-\frac {b \tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{\sqrt {a+b}}}{a^{3/2} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.46, size = 718, normalized size = 13.81 \[ \left [\frac {{\left (a^{2} + a b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} + a b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{2} + a b\right )} \sinh \left (d x + c\right )^{2} - \sqrt {-a^{2} - a b} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \log \left (\frac {a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} - 2 \, {\left (3 \, a + 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a \cosh \left (d x + c\right )^{2} - 3 \, a - 2 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a \cosh \left (d x + c\right )^{3} - {\left (3 \, a + 2 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 4 \, {\left (\cosh \left (d x + c\right )^{3} + 3 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3} + {\left (3 \, \cosh \left (d x + c\right )^{2} - 1\right )} \sinh \left (d x + c\right ) - \cosh \left (d x + c\right )\right )} \sqrt {-a^{2} - a b} + a}{a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} + 2 \, {\left (a + 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a \cosh \left (d x + c\right )^{2} + a + 2 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a \cosh \left (d x + c\right )^{3} + {\left (a + 2 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a}\right ) - a^{2} - a b}{2 \, {\left ({\left (a^{3} + a^{2} b\right )} d \cosh \left (d x + c\right ) + {\left (a^{3} + a^{2} b\right )} d \sinh \left (d x + c\right )\right )}}, \frac {{\left (a^{2} + a b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} + a b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{2} + a b\right )} \sinh \left (d x + c\right )^{2} - 2 \, \sqrt {a^{2} + a b} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \arctan \left (\frac {a \cosh \left (d x + c\right )^{3} + 3 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{3} + {\left (3 \, a + 4 \, b\right )} \cosh \left (d x + c\right ) + {\left (3 \, a \cosh \left (d x + c\right )^{2} + 3 \, a + 4 \, b\right )} \sinh \left (d x + c\right )}{2 \, \sqrt {a^{2} + a b}}\right ) - 2 \, \sqrt {a^{2} + a b} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \arctan \left (\frac {\sqrt {a^{2} + a b} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}}{2 \, {\left (a + b\right )}}\right ) - a^{2} - a b}{2 \, {\left ({\left (a^{3} + a^{2} b\right )} d \cosh \left (d x + c\right ) + {\left (a^{3} + a^{2} b\right )} d \sinh \left (d x + c\right )\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*((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 -
sqrt(-a^2 - a*b)*(b*cosh(d*x + c) + b*sinh(d*x + c))*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*c
osh(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
)) - a^2 - a*b)/((a^3 + a^2*b)*d*cosh(d*x + c) + (a^3 + a^2*b)*d*sinh(d*x + c)), 1/2*((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 - 2*sqrt(a^2 + a*b)*(b*cosh(d*x
+ c) + b*sinh(d*x + c))*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)) - 2*sqrt(a^2 +
a*b)*(b*cosh(d*x + c) + b*sinh(d*x + c))*arctan(1/2*sqrt(a^2 + a*b)*(cosh(d*x + c) + sinh(d*x + c))/(a + b))
- a^2 - a*b)/((a^3 + a^2*b)*d*cosh(d*x + c) + (a^3 + a^2*b)*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),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]=[31,78]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]=[-13,-93]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]=[-65,-82]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]=[97,-56]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wr
ong.The choice was done assuming [a,b]=[80,44]Warning, need to choose a branch for the root of a polynomial wi
th parameters. This might be wrong.The choice was done assuming [a,b]=[22,73]Undef/Unsigned Inf encountered in
limitEvaluation time: 0.84Limit: Max order reached or unable to make series expansion Error: Bad Argument Val
ue

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maple [B]  time = 0.40, size = 128, normalized size = 2.46 \[ -\frac {1}{d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {b \arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{d \,a^{\frac {3}{2}} \sqrt {a +b}}-\frac {b \arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{d \,a^{\frac {3}{2}} \sqrt {a +b}}-\frac {1}{d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.74, size = 277, normalized size = 5.33 \[ \frac {{\mathrm {e}}^{c+d\,x}}{2\,a\,d}-\frac {{\mathrm {e}}^{-c-d\,x}}{2\,a\,d}-\frac {\left (2\,\mathrm {atan}\left (\frac {b^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^3\,d^2\,\left (a+b\right )}}{2\,a\,d\,\left (a+b\right )\,{\left (b^2\right )}^{3/2}}\right )-2\,\mathrm {atan}\left (\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {2\,b^3}{a^5\,d\,{\left (a+b\right )}^2\,{\left (b^2\right )}^{3/2}}-\frac {4\,\left (2\,a^2\,d\,{\left (b^2\right )}^{3/2}+2\,a\,b\,d\,{\left (b^2\right )}^{3/2}\right )}{a^4\,b^3\,\left (a+b\right )\,\sqrt {a^4\,d^2+b\,a^3\,d^2}\,\sqrt {a^3\,d^2\,\left (a+b\right )}}\right )-\frac {2\,b^3\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}}{a^5\,d\,{\left (a+b\right )}^2\,{\left (b^2\right )}^{3/2}}\right )\,\left (\frac {a^5\,\sqrt {a^4\,d^2+b\,a^3\,d^2}}{4}+\frac {a^4\,b\,\sqrt {a^4\,d^2+b\,a^3\,d^2}}{4}\right )\right )\right )\,\sqrt {b^2}}{2\,\sqrt {a^4\,d^2+b\,a^3\,d^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh {\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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