3.61 \(\int \frac{1}{a-b \cosh ^4(x)} \, dx\)

Optimal. Leaf size=101 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a} \tanh (x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a} \tanh (x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{\sqrt{a}+\sqrt{b}}} \]

[Out]

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

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Rubi [A]  time = 0.12548, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3209, 1166, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a} \tanh (x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a} \tanh (x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{\sqrt{a}+\sqrt{b}}} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*Cosh[x]^4)^(-1),x]

[Out]

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

Rule 3209

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dis
t[ff/f, Subst[Int[(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff], x
]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[p]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
 q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{1}{a-b \cosh ^4(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1-x^2}{a-2 a x^2+(a-b) x^4} \, dx,x,\coth (x)\right )\\ &=\frac{1}{2} \left (-1-\frac{\sqrt{b}}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{-a-\sqrt{a} \sqrt{b}+(a-b) x^2} \, dx,x,\coth (x)\right )+\frac{1}{2} \left (-1+\frac{\sqrt{b}}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{-a+\sqrt{a} \sqrt{b}+(a-b) x^2} \, dx,x,\coth (x)\right )\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a} \tanh (x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a} \tanh (x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^{3/4} \sqrt{\sqrt{a}+\sqrt{b}}}\\ \end{align*}

Mathematica [A]  time = 0.191286, size = 109, normalized size = 1.08 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right )}{2 \sqrt{a} \sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{\tan ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right )}{2 \sqrt{a} \sqrt{\sqrt{a} \sqrt{b}-a}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*Cosh[x]^4)^(-1),x]

[Out]

-ArcTan[(Sqrt[a]*Tanh[x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]]/(2*Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + ArcTanh[(Sqrt[a
]*Tanh[x])/Sqrt[a + Sqrt[a]*Sqrt[b]]]/(2*Sqrt[a]*Sqrt[a + Sqrt[a]*Sqrt[b]])

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Maple [C]  time = 0.023, size = 127, normalized size = 1.3 \begin{align*}{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ( \left ( a-b \right ){{\it \_Z}}^{8}+ \left ( -4\,a-4\,b \right ){{\it \_Z}}^{6}+ \left ( 6\,a-6\,b \right ){{\it \_Z}}^{4}+ \left ( -4\,a-4\,b \right ){{\it \_Z}}^{2}+a-b \right ) }{\frac{-{{\it \_R}}^{6}+3\,{{\it \_R}}^{4}-3\,{{\it \_R}}^{2}+1}{{{\it \_R}}^{7}a-{{\it \_R}}^{7}b-3\,{{\it \_R}}^{5}a-3\,{{\it \_R}}^{5}b+3\,{{\it \_R}}^{3}a-3\,{{\it \_R}}^{3}b-{\it \_R}\,a-{\it \_R}\,b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -{\it \_R} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a-b*cosh(x)^4),x)

[Out]

1/4*sum((-_R^6+3*_R^4-3*_R^2+1)/(_R^7*a-_R^7*b-3*_R^5*a-3*_R^5*b+3*_R^3*a-3*_R^3*b-_R*a-_R*b)*ln(tanh(1/2*x)-_
R),_R=RootOf((a-b)*_Z^8+(-4*a-4*b)*_Z^6+(6*a-6*b)*_Z^4+(-4*a-4*b)*_Z^2+a-b))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{b \cosh \left (x\right )^{4} - a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*cosh(x)^4),x, algorithm="maxima")

[Out]

-integrate(1/(b*cosh(x)^4 - a), x)

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Fricas [B]  time = 2.70433, size = 1706, normalized size = 16.89 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*cosh(x)^4),x, algorithm="fricas")

[Out]

-1/4*sqrt(((a^2 - a*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) + 1)/(a^2 - a*b))*log(b*cosh(x)^2 + 2*b*cosh(x)*sinh(
x) + b*sinh(x)^2 + 2*(a*b - (a^4 - a^3*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)))*sqrt(((a^2 - a*b)*sqrt(b/(a^5 - 2
*a^4*b + a^3*b^2)) + 1)/(a^2 - a*b)) - 2*(a^3 - a^2*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) + b) + 1/4*sqrt(((a^2
 - a*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) + 1)/(a^2 - a*b))*log(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*sinh(x)^
2 - 2*(a*b - (a^4 - a^3*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)))*sqrt(((a^2 - a*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^
2)) + 1)/(a^2 - a*b)) - 2*(a^3 - a^2*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) + b) - 1/4*sqrt(-((a^2 - a*b)*sqrt(b
/(a^5 - 2*a^4*b + a^3*b^2)) - 1)/(a^2 - a*b))*log(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*sinh(x)^2 + 2*(a*b + (
a^4 - a^3*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)))*sqrt(-((a^2 - a*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) - 1)/(a^2
 - a*b)) + 2*(a^3 - a^2*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) + b) + 1/4*sqrt(-((a^2 - a*b)*sqrt(b/(a^5 - 2*a^4
*b + a^3*b^2)) - 1)/(a^2 - a*b))*log(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*sinh(x)^2 - 2*(a*b + (a^4 - a^3*b)*
sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)))*sqrt(-((a^2 - a*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) - 1)/(a^2 - a*b)) + 2*
(a^3 - a^2*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) + b)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*cosh(x)**4),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{b \cosh \left (x\right )^{4} - a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*cosh(x)^4),x, algorithm="giac")

[Out]

integrate(-1/(b*cosh(x)^4 - a), x)