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

Optimal. Leaf size=36 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} d \sqrt {a+b}} \]

[Out]

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

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Rubi [A]  time = 0.04, 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 = {3676, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} d \sqrt {a+b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 36, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} d \sqrt {a+b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.42, size = 511, normalized size = 14.19 \[ \left [-\frac {\sqrt {-a^{2} - a b} \log \left (\frac {{\left (a + b\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a + b\right )} \sinh \left (d x + c\right )^{4} - 2 \, {\left (3 \, a + b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} - 3 \, a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} - {\left (3 \, a + 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 + b}{{\left (a + b\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a + b\right )} \sinh \left (d x + c\right )^{4} + 2 \, {\left (a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + {\left (a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a + b}\right )}{2 \, {\left (a^{2} + a b\right )} d}, \frac {\sqrt {a^{2} + a b} \arctan \left (\frac {{\left (a + b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (a + b\right )} \sinh \left (d x + c\right )^{3} + {\left (3 \, a - b\right )} \cosh \left (d x + c\right ) + {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + 3 \, a - 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 \, a}\right )}{{\left (a^{2} + a b\right )} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.18, size = 480, normalized size = 13.33 \[ \frac {\frac {{\left (a^{3} - 10 \, a^{2} b + 5 \, a b^{2} + {\left (5 \, a^{2} - 10 \, a b + b^{2}\right )} \sqrt {-a b}\right )} \sqrt {a^{2} - b^{2} + 2 \, \sqrt {-a b} {\left (a + b\right )}} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |} \arctan \left (\frac {e^{\left (d x\right )}}{\sqrt {\frac {a e^{\left (2 \, c\right )} - b e^{\left (2 \, c\right )} + \sqrt {{\left (a e^{\left (2 \, c\right )} - b e^{\left (2 \, c\right )}\right )}^{2} - {\left (a e^{\left (4 \, c\right )} + b e^{\left (4 \, c\right )}\right )} {\left (a + b\right )}}}{a e^{\left (4 \, c\right )} + b e^{\left (4 \, c\right )}}}}\right ) e^{\left (-2 \, c\right )}}{a^{6} - 13 \, a^{5} b - 14 \, a^{4} b^{2} + 14 \, a^{3} b^{3} + 13 \, a^{2} b^{4} - a b^{5} + 2 \, {\left (3 \, a^{5} - 4 \, a^{4} b - 14 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + 3 \, a b^{4}\right )} \sqrt {-a b}} + \frac {{\left (a^{3} - 10 \, a^{2} b + 5 \, a b^{2} - {\left (5 \, a^{2} - 10 \, a b + b^{2}\right )} \sqrt {-a b}\right )} \sqrt {a^{2} - b^{2} - 2 \, \sqrt {-a b} {\left (a + b\right )}} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |} \arctan \left (\frac {e^{\left (d x\right )}}{\sqrt {\frac {a e^{\left (2 \, c\right )} - b e^{\left (2 \, c\right )} - \sqrt {{\left (a e^{\left (2 \, c\right )} - b e^{\left (2 \, c\right )}\right )}^{2} - {\left (a e^{\left (4 \, c\right )} + b e^{\left (4 \, c\right )}\right )} {\left (a + b\right )}}}{a e^{\left (4 \, c\right )} + b e^{\left (4 \, c\right )}}}}\right ) e^{\left (-2 \, c\right )}}{a^{6} - 13 \, a^{5} b - 14 \, a^{4} b^{2} + 14 \, a^{3} b^{3} + 13 \, a^{2} b^{4} - a b^{5} - 2 \, {\left (3 \, a^{5} - 4 \, a^{4} b - 14 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + 3 \, a b^{4}\right )} \sqrt {-a b}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a^3 - 10*a^2*b + 5*a*b^2 + (5*a^2 - 10*a*b + b^2)*sqrt(-a*b))*sqrt(a^2 - b^2 + 2*sqrt(-a*b)*(a + b))*abs(a*e
^(2*c) + b*e^(2*c))*arctan(e^(d*x)/sqrt((a*e^(2*c) - b*e^(2*c) + sqrt((a*e^(2*c) - b*e^(2*c))^2 - (a*e^(4*c) +
b*e^(4*c))*(a + b)))/(a*e^(4*c) + b*e^(4*c))))*e^(-2*c)/(a^6 - 13*a^5*b - 14*a^4*b^2 + 14*a^3*b^3 + 13*a^2*b^
4 - a*b^5 + 2*(3*a^5 - 4*a^4*b - 14*a^3*b^2 - 4*a^2*b^3 + 3*a*b^4)*sqrt(-a*b)) + (a^3 - 10*a^2*b + 5*a*b^2 - (
5*a^2 - 10*a*b + b^2)*sqrt(-a*b))*sqrt(a^2 - b^2 - 2*sqrt(-a*b)*(a + b))*abs(a*e^(2*c) + b*e^(2*c))*arctan(e^(
d*x)/sqrt((a*e^(2*c) - b*e^(2*c) - sqrt((a*e^(2*c) - b*e^(2*c))^2 - (a*e^(4*c) + b*e^(4*c))*(a + b)))/(a*e^(4*
c) + b*e^(4*c))))*e^(-2*c)/(a^6 - 13*a^5*b - 14*a^4*b^2 + 14*a^3*b^3 + 13*a^2*b^4 - a*b^5 - 2*(3*a^5 - 4*a^4*b
- 14*a^3*b^2 - 4*a^2*b^3 + 3*a*b^4)*sqrt(-a*b)))/d

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maple [B]  time = 0.32, size = 235, normalized size = 6.53 \[ -\frac {\arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{d \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {\arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right ) b}{d \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {\arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{d \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}+\frac {\arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right ) b}{d \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d/((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/
d/(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))*b+1/d/((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/(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))*b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.34, size = 147, normalized size = 4.08 \[ \frac {\mathrm {atan}\left (\frac {4\,a^2\,d^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2\,d^2+b\,a\,d^2}\,\sqrt {a\,d^2\,\left (a+b\right )}+{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\sqrt {a^2\,d^2+b\,a\,d^2}\,\sqrt {a\,d^2\,\left (a+b\right )}}{2\,a\,d\,\sqrt {a\,d^2\,\left (a+b\right )}}\right )+\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a\,d^2\,\left (a+b\right )}}{2\,a\,d}\right )}{\sqrt {a^2\,d^2+b\,a\,d^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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