3.1058 \(\int \frac {-\text {csch}^3(a+b x)+\text {sech}^3(a+b x)}{\text {csch}^3(a+b x)+\text {sech}^3(a+b x)} \, dx\)

Optimal. Leaf size=47 \[ \frac {1}{3 b (\tanh (a+b x)+1)}+\frac {4 \tan ^{-1}\left (\frac {1-2 \tanh (a+b x)}{\sqrt {3}}\right )}{3 \sqrt {3} b} \]

[Out]

4/9*arctan(1/3*(1-2*tanh(b*x+a))*3^(1/2))/b*3^(1/2)+1/3/b/(1+tanh(b*x+a))

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Rubi [A]  time = 0.41, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2074, 618, 204} \[ \frac {1}{3 b (\tanh (a+b x)+1)}+\frac {4 \tan ^{-1}\left (\frac {1-2 \tanh (a+b x)}{\sqrt {3}}\right )}{3 \sqrt {3} b} \]

Antiderivative was successfully verified.

[In]

Int[(-Csch[a + b*x]^3 + Sech[a + b*x]^3)/(Csch[a + b*x]^3 + Sech[a + b*x]^3),x]

[Out]

(4*ArcTan[(1 - 2*Tanh[a + b*x])/Sqrt[3]])/(3*Sqrt[3]*b) + 1/(3*b*(1 + Tanh[a + b*x]))

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

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

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Mathematica [A]  time = 0.33, size = 52, normalized size = 1.11 \[ \frac {-3 \sinh (2 (a+b x))+3 \cosh (2 (a+b x))-8 \sqrt {3} \tan ^{-1}\left (\frac {2 \tanh (a+b x)-1}{\sqrt {3}}\right )}{18 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(-Csch[a + b*x]^3 + Sech[a + b*x]^3)/(Csch[a + b*x]^3 + Sech[a + b*x]^3),x]

[Out]

(-8*Sqrt[3]*ArcTan[(-1 + 2*Tanh[a + b*x])/Sqrt[3]] + 3*Cosh[2*(a + b*x)] - 3*Sinh[2*(a + b*x)])/(18*b)

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fricas [B]  time = 0.41, size = 127, normalized size = 2.70 \[ \frac {8 \, {\left (\sqrt {3} \cosh \left (b x + a\right )^{2} + 2 \, \sqrt {3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sqrt {3} \sinh \left (b x + a\right )^{2}\right )} \arctan \left (-\frac {\sqrt {3} \cosh \left (b x + a\right ) + \sqrt {3} \sinh \left (b x + a\right )}{3 \, {\left (\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}}\right ) + 3}{18 \, {\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-csch(b*x+a)^3+sech(b*x+a)^3)/(csch(b*x+a)^3+sech(b*x+a)^3),x, algorithm="fricas")

[Out]

1/18*(8*(sqrt(3)*cosh(b*x + a)^2 + 2*sqrt(3)*cosh(b*x + a)*sinh(b*x + a) + sqrt(3)*sinh(b*x + a)^2)*arctan(-1/
3*(sqrt(3)*cosh(b*x + a) + sqrt(3)*sinh(b*x + a))/(cosh(b*x + a) - sinh(b*x + a))) + 3)/(b*cosh(b*x + a)^2 + 2
*b*cosh(b*x + a)*sinh(b*x + a) + b*sinh(b*x + a)^2)

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giac [A]  time = 0.22, size = 37, normalized size = 0.79 \[ -\frac {8 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} e^{\left (2 \, b x + 2 \, a\right )}\right ) - 3 \, e^{\left (-2 \, b x - 2 \, a\right )}}{18 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-csch(b*x+a)^3+sech(b*x+a)^3)/(csch(b*x+a)^3+sech(b*x+a)^3),x, algorithm="giac")

[Out]

-1/18*(8*sqrt(3)*arctan(1/3*sqrt(3)*e^(2*b*x + 2*a)) - 3*e^(-2*b*x - 2*a))/b

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maple [C]  time = 1.41, size = 120, normalized size = 2.55 \[ \frac {2 i \sqrt {3}\, \ln \left (\tanh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )+\left (i \sqrt {3}-1\right ) \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{9 b}-\frac {2 i \sqrt {3}\, \ln \left (\tanh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )+\left (-i \sqrt {3}-1\right ) \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{9 b}+\frac {2}{3 b \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )^{2}}-\frac {2}{3 b \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-csch(b*x+a)^3+sech(b*x+a)^3)/(csch(b*x+a)^3+sech(b*x+a)^3),x)

[Out]

2/9*I/b*3^(1/2)*ln(tanh(1/2*b*x+1/2*a)^2+(I*3^(1/2)-1)*tanh(1/2*b*x+1/2*a)+1)-2/9*I/b*3^(1/2)*ln(tanh(1/2*b*x+
1/2*a)^2+(-I*3^(1/2)-1)*tanh(1/2*b*x+1/2*a)+1)+2/3/b/(tanh(1/2*b*x+1/2*a)+1)^2-2/3/b/(tanh(1/2*b*x+1/2*a)+1)

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maxima [B]  time = 0.46, size = 93, normalized size = 1.98 \[ -\frac {4 \, {\left (\sqrt {3} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, \sqrt {3} e^{\left (-b x - a\right )} + 3^{\frac {1}{4}} \sqrt {2}\right )}\right ) - \sqrt {3} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, \sqrt {3} e^{\left (-b x - a\right )} - 3^{\frac {1}{4}} \sqrt {2}\right )}\right )\right )}}{9 \, b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{6 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-csch(b*x+a)^3+sech(b*x+a)^3)/(csch(b*x+a)^3+sech(b*x+a)^3),x, algorithm="maxima")

[Out]

-4/9*(sqrt(3)*arctan(1/6*3^(3/4)*sqrt(2)*(2*sqrt(3)*e^(-b*x - a) + 3^(1/4)*sqrt(2))) - sqrt(3)*arctan(1/6*3^(3
/4)*sqrt(2)*(2*sqrt(3)*e^(-b*x - a) - 3^(1/4)*sqrt(2))))/b + 1/6*e^(-2*b*x - 2*a)/b

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mupad [B]  time = 2.16, size = 48, normalized size = 1.02 \[ \frac {{\mathrm {e}}^{-2\,a-2\,b\,x}}{6\,b}-\frac {4\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}\,\sqrt {b^2}}{3\,b}\right )}{9\,\sqrt {b^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/cosh(a + b*x)^3 - 1/sinh(a + b*x)^3)/(1/cosh(a + b*x)^3 + 1/sinh(a + b*x)^3),x)

[Out]

exp(- 2*a - 2*b*x)/(6*b) - (4*3^(1/2)*atan((3^(1/2)*exp(2*a)*exp(2*b*x)*(b^2)^(1/2))/(3*b)))/(9*(b^2)^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\operatorname {csch}^{3}{\left (a + b x \right )}}{\operatorname {csch}^{3}{\left (a + b x \right )} + \operatorname {sech}^{3}{\left (a + b x \right )}}\, dx - \int \left (- \frac {\operatorname {sech}^{3}{\left (a + b x \right )}}{\operatorname {csch}^{3}{\left (a + b x \right )} + \operatorname {sech}^{3}{\left (a + b x \right )}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-csch(b*x+a)**3+sech(b*x+a)**3)/(csch(b*x+a)**3+sech(b*x+a)**3),x)

[Out]

-Integral(csch(a + b*x)**3/(csch(a + b*x)**3 + sech(a + b*x)**3), x) - Integral(-sech(a + b*x)**3/(csch(a + b*
x)**3 + sech(a + b*x)**3), x)

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