3.1.77 \(\int \frac {\text {sech}(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx\) [77]

Optimal. Leaf size=36 \[ \frac {\text {ArcTan}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{\sqrt {a} \sqrt {a+b} d} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {4232, 211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{\sqrt {a} d \sqrt {a+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]]/(Sqrt[a]*Sqrt[a + b]*d)

Rule 211

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

Rule 4232

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

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Mathematica [A]
time = 0.04, size = 36, normalized size = 1.00 \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{\sqrt {a} \sqrt {a+b} d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]]/(Sqrt[a]*Sqrt[a + b]*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(79\) vs. \(2(28)=56\).
time = 1.78, size = 80, normalized size = 2.22

method	result	size
derivativedivides	\(\frac {\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{\sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{\sqrt {a +b}\, \sqrt {a}}}{d}\)	\(80\)
default	\(\frac {\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{\sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{\sqrt {a +b}\, \sqrt {a}}}{d}\)	\(80\)
risch	\(-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, d}\)	\(106\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)/(a+b*sech(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/(a+b)^(1/2)/a^(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/(a+b)^(1/2)/a^(
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)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (28) = 56\).
time = 0.36, size = 487, normalized size = 13.53 \begin {gather*} \left [-\frac {\sqrt {-a^{2} - a b} \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 )}{2 \, {\left (a^{2} + a b\right )} d}, \frac {\sqrt {a^{2} + a b} \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 ) + \sqrt {a^{2} + a b} \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 )}{{\left (a^{2} + a b\right )} d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*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)*sin
h(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)*d), (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)*cos
h(d*x + c) + (3*a*cosh(d*x + c)^2 + 3*a + 4*b)*sinh(d*x + c))/sqrt(a^2 + a*b)) + sqrt(a^2 + a*b)*arctan(1/2*sq
rt(a^2 + a*b)*(cosh(d*x + c) + sinh(d*x + c))/(a + b)))/((a^2 + a*b)*d)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(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,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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Mupad [B]
time = 1.90, size = 108, normalized size = 3.00 \begin {gather*} -\frac {\ln \left (-\frac {4\,\left (b-b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^2\,\left (a+b\right )}-\frac {8\,b\,{\mathrm {e}}^{c+d\,x}}{{\left (-a\right )}^{5/2}\,\sqrt {a+b}}\right )-\ln \left (\frac {8\,b\,{\mathrm {e}}^{c+d\,x}}{{\left (-a\right )}^{5/2}\,\sqrt {a+b}}-\frac {4\,\left (b-b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^2\,\left (a+b\right )}\right )}{2\,\sqrt {-a}\,d\,\sqrt {a+b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(c + d*x)*(a + b/cosh(c + d*x)^2)),x)

[Out]

-(log(- (4*(b - b*exp(2*c + 2*d*x)))/(a^2*(a + b)) - (8*b*exp(c + d*x))/((-a)^(5/2)*(a + b)^(1/2))) - log((8*b
*exp(c + d*x))/((-a)^(5/2)*(a + b)^(1/2)) - (4*(b - b*exp(2*c + 2*d*x)))/(a^2*(a + b))))/(2*(-a)^(1/2)*d*(a +
b)^(1/2))

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