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

Optimal. Leaf size=361 \[ \frac {\sqrt {\sqrt {a}-\sqrt {a+b}} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a+b}+\sqrt {a}}-\sqrt {2} \sqrt [4]{a} \tanh (x)}{\sqrt {\sqrt {a}-\sqrt {a+b}}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {a+b}}-\frac {\sqrt {\sqrt {a}-\sqrt {a+b}} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a+b}+\sqrt {a}}+\sqrt {2} \sqrt [4]{a} \tanh (x)}{\sqrt {\sqrt {a}-\sqrt {a+b}}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {a+b}}-\frac {\sqrt {\sqrt {a+b}+\sqrt {a}} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt {\sqrt {a+b}+\sqrt {a}} \tanh (x)+\sqrt {a+b}+\sqrt {a} \tanh ^2(x)\right )}{4 \sqrt {2} a^{3/4} \sqrt {a+b}}+\frac {\sqrt {\sqrt {a+b}+\sqrt {a}} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt {\sqrt {a+b}+\sqrt {a}} \tanh (x)+\sqrt {a+b}+\sqrt {a} \tanh ^2(x)\right )}{4 \sqrt {2} a^{3/4} \sqrt {a+b}} \]

[Out]

1/4*arctanh(((a^(1/2)+(a+b)^(1/2))^(1/2)-a^(1/4)*2^(1/2)*tanh(x))/(a^(1/2)-(a+b)^(1/2))^(1/2))*(a^(1/2)-(a+b)^
(1/2))^(1/2)/a^(3/4)*2^(1/2)/(a+b)^(1/2)-1/4*arctanh(((a^(1/2)+(a+b)^(1/2))^(1/2)+a^(1/4)*2^(1/2)*tanh(x))/(a^
(1/2)-(a+b)^(1/2))^(1/2))*(a^(1/2)-(a+b)^(1/2))^(1/2)/a^(3/4)*2^(1/2)/(a+b)^(1/2)-1/8*ln((a+b)^(1/2)-a^(1/4)*2
^(1/2)*(a^(1/2)+(a+b)^(1/2))^(1/2)*tanh(x)+a^(1/2)*tanh(x)^2)*(a^(1/2)+(a+b)^(1/2))^(1/2)/a^(3/4)*2^(1/2)/(a+b
)^(1/2)+1/8*ln((a+b)^(1/2)+a^(1/4)*2^(1/2)*(a^(1/2)+(a+b)^(1/2))^(1/2)*tanh(x)+a^(1/2)*tanh(x)^2)*(a^(1/2)+(a+
b)^(1/2))^(1/2)/a^(3/4)*2^(1/2)/(a+b)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.04, antiderivative size = 485, normalized size of antiderivative = 1.34, number of steps used = 10, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3209, 1169, 634, 618, 204, 628} \[ -\frac {\left (\sqrt {a+b}+\sqrt {a}\right ) \log \left ((a+b)^{3/4} \coth ^2(x)-\sqrt {2} \sqrt [4]{a} \sqrt {\sqrt {a} \sqrt {a+b}+a+b} \coth (x)+\sqrt {a} \sqrt [4]{a+b}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}+\frac {\left (\sqrt {a+b}+\sqrt {a}\right ) \log \left ((a+b)^{3/4} \coth ^2(x)+\sqrt {2} \sqrt [4]{a} \sqrt {\sqrt {a} \sqrt {a+b}+a+b} \coth (x)+\sqrt {a} \sqrt [4]{a+b}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}+\frac {\left (\sqrt {a}-\sqrt {a+b}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}-\sqrt {2} (a+b)^{3/4} \coth (x)}{\sqrt [4]{a} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}-\frac {\left (\sqrt {a}-\sqrt {a+b}\right ) \tan ^{-1}\left (\frac {\sqrt {2} (a+b)^{3/4} \coth (x)+\sqrt [4]{a} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}{\sqrt [4]{a} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cosh[x]^4)^(-1),x]

[Out]

((Sqrt[a] - Sqrt[a + b])*ArcTan[(a^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]] - Sqrt[2]*(a + b)^(3/4)*Coth[x])/(a
^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]])])/(2*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]
) - ((Sqrt[a] - Sqrt[a + b])*ArcTan[(a^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]] + Sqrt[2]*(a + b)^(3/4)*Coth[x]
)/(a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]])])/(2*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a +
b]]) - ((Sqrt[a] + Sqrt[a + b])*Log[Sqrt[a]*(a + b)^(1/4) - Sqrt[2]*a^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]]
*Coth[x] + (a + b)^(3/4)*Coth[x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]]) + ((S
qrt[a] + Sqrt[a + b])*Log[Sqrt[a]*(a + b)^(1/4) + Sqrt[2]*a^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]]*Coth[x] +
(a + b)^(3/4)*Coth[x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]])

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 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

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

Rule 1169

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

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]

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

________________________________________________________________________________________

Mathematica [C]  time = 0.24, size = 121, normalized size = 0.34 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+i \sqrt {a} \sqrt {b}}}\right )}{2 \sqrt {a} \sqrt {a+i \sqrt {a} \sqrt {b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {-a+i \sqrt {a} \sqrt {b}}}\right )}{2 \sqrt {a} \sqrt {-a+i \sqrt {a} \sqrt {b}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cosh[x]^4)^(-1),x]

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.91, size = 771, normalized size = 2.14 \[ -\frac {1}{4} \, \sqrt {\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} + a b}} \log \left (b \cosh \relax (x)^{2} + 2 \, b \cosh \relax (x) \sinh \relax (x) + b \sinh \relax (x)^{2} + 2 \, {\left (a b + {\left (a^{4} + a^{3} b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}}\right )} \sqrt {\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} + a b}} + 2 \, {\left (a^{3} + a^{2} b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + b\right ) + \frac {1}{4} \, \sqrt {\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} + a b}} \log \left (b \cosh \relax (x)^{2} + 2 \, b \cosh \relax (x) \sinh \relax (x) + b \sinh \relax (x)^{2} - 2 \, {\left (a b + {\left (a^{4} + a^{3} b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}}\right )} \sqrt {\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} + a b}} + 2 \, {\left (a^{3} + a^{2} b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + b\right ) - \frac {1}{4} \, \sqrt {-\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} + a b}} \log \left (b \cosh \relax (x)^{2} + 2 \, b \cosh \relax (x) \sinh \relax (x) + b \sinh \relax (x)^{2} + 2 \, {\left (a b - {\left (a^{4} + a^{3} b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}}\right )} \sqrt {-\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} + a b}} - 2 \, {\left (a^{3} + a^{2} b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + b\right ) + \frac {1}{4} \, \sqrt {-\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} + a b}} \log \left (b \cosh \relax (x)^{2} + 2 \, b \cosh \relax (x) \sinh \relax (x) + b \sinh \relax (x)^{2} - 2 \, {\left (a b - {\left (a^{4} + a^{3} b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}}\right )} \sqrt {-\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} + a b}} - 2 \, {\left (a^{3} + a^{2} b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + b\right ) \]

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*sin
h(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)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 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]=[-36,9]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]=[95,47]Warning, need to choose a branch for the root of a polynomial with parameters. This migh
t be wrong.The choice was done assuming [a,b]=[56,74]Warning, need to choose a branch for the root of a polyno
mial with parameters. This might be wrong.The choice was done assuming [a,b]=[10,-88]8*(sqrt(1/1024)*sqrt(1/10
48576*(1048576*a^2-1048576*a*sqrt(-a*b))/(a^4+a^3*b))*ln(abs(60*a^4*b*exp(x)^2+6*a^4*b-24*a^4*sqrt(-a*b)*exp(x
)^2+68*a^3*b^2*exp(x)^2+2*a^3*b^2+16*a^3*b*sqrt(-a*b)*exp(x)^2+12*a^3*b*sqrt(-a*b)+48*a^3*b*sqrt(a^2+a*sqrt(-a
*b))*exp(x)^2+6*a^3*b*sqrt(a^2+a*sqrt(-a*b))-24*a^3*sqrt(-a*b)*sqrt(a^2+a*sqrt(-a*b))*exp(x)^2-16*a^2*b^3*exp(
x)^2-8*a^2*b^3+64*a^2*b^2*sqrt(-a*b)*exp(x)^2+16*a^2*b^2*sqrt(-a*b)+61*a^2*b^2*sqrt(a^2+a*sqrt(-a*b))*exp(x)^2
+5*a^2*b^2*sqrt(a^2+a*sqrt(-a*b))-5*a^2*b*sqrt(-a*b)*sqrt(a^2+a*sqrt(-a*b))*exp(x)^2+9*a^2*b*sqrt(-a*b)*sqrt(a
^2+a*sqrt(-a*b))-4*a*b^3*sqrt(a^2+a*sqrt(-a*b))*exp(x)^2-4*a*b^3*sqrt(a^2+a*sqrt(-a*b))+36*a*b^2*sqrt(-a*b)*sq
rt(a^2+a*sqrt(-a*b))*exp(x)^2+12*a*b^2*sqrt(-a*b)*sqrt(a^2+a*sqrt(-a*b))))-sqrt(1/1024)*sqrt(1/1048576*(104857
6*a^2-1048576*a*sqrt(-a*b))/(a^4+a^3*b))*ln(abs(60*a^4*b*exp(x)^2+6*a^4*b-24*a^4*sqrt(-a*b)*exp(x)^2+68*a^3*b^
2*exp(x)^2+2*a^3*b^2+16*a^3*b*sqrt(-a*b)*exp(x)^2+12*a^3*b*sqrt(-a*b)-48*a^3*b*sqrt(a^2+a*sqrt(-a*b))*exp(x)^2
-6*a^3*b*sqrt(a^2+a*sqrt(-a*b))+24*a^3*sqrt(-a*b)*sqrt(a^2+a*sqrt(-a*b))*exp(x)^2-16*a^2*b^3*exp(x)^2-8*a^2*b^
3+64*a^2*b^2*sqrt(-a*b)*exp(x)^2+16*a^2*b^2*sqrt(-a*b)-61*a^2*b^2*sqrt(a^2+a*sqrt(-a*b))*exp(x)^2-5*a^2*b^2*sq
rt(a^2+a*sqrt(-a*b))+5*a^2*b*sqrt(-a*b)*sqrt(a^2+a*sqrt(-a*b))*exp(x)^2-9*a^2*b*sqrt(-a*b)*sqrt(a^2+a*sqrt(-a*
b))+4*a*b^3*sqrt(a^2+a*sqrt(-a*b))*exp(x)^2+4*a*b^3*sqrt(a^2+a*sqrt(-a*b))-36*a*b^2*sqrt(-a*b)*sqrt(a^2+a*sqrt
(-a*b))*exp(x)^2-12*a*b^2*sqrt(-a*b)*sqrt(a^2+a*sqrt(-a*b))))+sqrt(1/1024)*sqrt(1/1048576*(1048576*a^2+1048576
*a*sqrt(-a*b))/(a^4+a^3*b))*ln(abs(60*a^4*b*exp(x)^2+6*a^4*b+24*a^4*sqrt(-a*b)*exp(x)^2+68*a^3*b^2*exp(x)^2+2*
a^3*b^2-16*a^3*b*sqrt(-a*b)*exp(x)^2-12*a^3*b*sqrt(-a*b)+48*a^3*b*sqrt(a^2-a*sqrt(-a*b))*exp(x)^2+6*a^3*b*sqrt
(a^2-a*sqrt(-a*b))+24*a^3*sqrt(-a*b)*sqrt(a^2-a*sqrt(-a*b))*exp(x)^2-16*a^2*b^3*exp(x)^2-8*a^2*b^3-64*a^2*b^2*
sqrt(-a*b)*exp(x)^2-16*a^2*b^2*sqrt(-a*b)+61*a^2*b^2*sqrt(a^2-a*sqrt(-a*b))*exp(x)^2+5*a^2*b^2*sqrt(a^2-a*sqrt
(-a*b))+5*a^2*b*sqrt(-a*b)*sqrt(a^2-a*sqrt(-a*b))*exp(x)^2-9*a^2*b*sqrt(-a*b)*sqrt(a^2-a*sqrt(-a*b))-4*a*b^3*s
qrt(a^2-a*sqrt(-a*b))*exp(x)^2-4*a*b^3*sqrt(a^2-a*sqrt(-a*b))-36*a*b^2*sqrt(-a*b)*sqrt(a^2-a*sqrt(-a*b))*exp(x
)^2-12*a*b^2*sqrt(-a*b)*sqrt(a^2-a*sqrt(-a*b))))-sqrt(1/1024)*sqrt(1/1048576*(1048576*a^2+1048576*a*sqrt(-a*b)
)/(a^4+a^3*b))*ln(abs(60*a^4*b*exp(x)^2+6*a^4*b+24*a^4*sqrt(-a*b)*exp(x)^2+68*a^3*b^2*exp(x)^2+2*a^3*b^2-16*a^
3*b*sqrt(-a*b)*exp(x)^2-12*a^3*b*sqrt(-a*b)-48*a^3*b*sqrt(a^2-a*sqrt(-a*b))*exp(x)^2-6*a^3*b*sqrt(a^2-a*sqrt(-
a*b))-24*a^3*sqrt(-a*b)*sqrt(a^2-a*sqrt(-a*b))*exp(x)^2-16*a^2*b^3*exp(x)^2-8*a^2*b^3-64*a^2*b^2*sqrt(-a*b)*ex
p(x)^2-16*a^2*b^2*sqrt(-a*b)-61*a^2*b^2*sqrt(a^2-a*sqrt(-a*b))*exp(x)^2-5*a^2*b^2*sqrt(a^2-a*sqrt(-a*b))-5*a^2
*b*sqrt(-a*b)*sqrt(a^2-a*sqrt(-a*b))*exp(x)^2+9*a^2*b*sqrt(-a*b)*sqrt(a^2-a*sqrt(-a*b))+4*a*b^3*sqrt(a^2-a*sqr
t(-a*b))*exp(x)^2+4*a*b^3*sqrt(a^2-a*sqrt(-a*b))+36*a*b^2*sqrt(-a*b)*sqrt(a^2-a*sqrt(-a*b))*exp(x)^2+12*a*b^2*
sqrt(-a*b)*sqrt(a^2-a*sqrt(-a*b)))))

________________________________________________________________________________________

maple [C]  time = 0.09, size = 121, normalized size = 0.34 \[ \frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a +b \right ) \textit {\_Z}^{8}+\left (-4 a +4 b \right ) \textit {\_Z}^{6}+\left (6 a +6 b \right ) \textit {\_Z}^{4}+\left (-4 a +4 b \right ) \textit {\_Z}^{2}+a +b \right )}{\sum }\frac {\left (-\textit {\_R}^{6}+3 \textit {\_R}^{4}-3 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a +\textit {\_R}^{7} b -3 \textit {\_R}^{5} a +3 \textit {\_R}^{5} b +3 \textit {\_R}^{3} a +3 \textit {\_R}^{3} b -\textit {\_R} a +\textit {\_R} b}\right )}{4} \]

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))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{b \cosh \relax (x)^{4} + a}\,{d x} \]

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)

________________________________________________________________________________________

mupad [B]  time = 7.55, size = 1563, normalized size = 4.33 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*cosh(x)^4),x)

[Out]

log((524288*(1024*a^3*exp(2*x) - 35*b^3*exp(2*x) + 185*a*b^2 + 464*a^2*b + 256*a^3 - 24*b^3 + 988*a*b^2*exp(2*
x) + 2048*a^2*b*exp(2*x)))/(a*b^6*(a + b)^2) - (((((4194304*(253*a*b^3 - b^4*exp(2*x) + 1184*a^3*b + 512*a^4 -
b^4 + 930*a^2*b^2 + 1392*a^2*b^2*exp(2*x) + 627*a*b^3*exp(2*x) + 768*a^3*b*exp(2*x)))/(b^6*(a + b)^2) + (8388
608*a*((a^2 + (-a^3*b)^(1/2))/(a^3*(a + b)))^(1/2)*(512*a^3*exp(2*x) - 6*b^3*exp(2*x) + 181*a*b^2 + 432*a^2*b
+ 256*a^3 - 5*b^3 + 622*a*b^2*exp(2*x) + 1152*a^2*b*exp(2*x)))/(b^6*(a + b)))*((a^2 + (-a^3*b)^(1/2))/(a^3*(a
+ b)))^(1/2))/4 - (2097152*(176*a*b + 1536*a^2*exp(2*x) - 134*b^2*exp(2*x) + 256*a^2 - 75*b^2 + 1408*a*b*exp(2
*x)))/(b^6*(a + b)))*((a^2 + (-a^3*b)^(1/2))/(a^3*(a + b)))^(1/2))/4)*((a^2 + (-a^3*b)^(1/2))/(16*(a^3*b + a^4
)))^(1/2) - log((524288*(1024*a^3*exp(2*x) - 35*b^3*exp(2*x) + 185*a*b^2 + 464*a^2*b + 256*a^3 - 24*b^3 + 988*
a*b^2*exp(2*x) + 2048*a^2*b*exp(2*x)))/(a*b^6*(a + b)^2) - (((((4194304*(253*a*b^3 - b^4*exp(2*x) + 1184*a^3*b
+ 512*a^4 - b^4 + 930*a^2*b^2 + 1392*a^2*b^2*exp(2*x) + 627*a*b^3*exp(2*x) + 768*a^3*b*exp(2*x)))/(b^6*(a + b
)^2) - (8388608*a*((a^2 + (-a^3*b)^(1/2))/(a^3*(a + b)))^(1/2)*(512*a^3*exp(2*x) - 6*b^3*exp(2*x) + 181*a*b^2
+ 432*a^2*b + 256*a^3 - 5*b^3 + 622*a*b^2*exp(2*x) + 1152*a^2*b*exp(2*x)))/(b^6*(a + b)))*((a^2 + (-a^3*b)^(1/
2))/(a^3*(a + b)))^(1/2))/4 + (2097152*(176*a*b + 1536*a^2*exp(2*x) - 134*b^2*exp(2*x) + 256*a^2 - 75*b^2 + 14
08*a*b*exp(2*x)))/(b^6*(a + b)))*((a^2 + (-a^3*b)^(1/2))/(a^3*(a + b)))^(1/2))/4)*((a^2 + (-a^3*b)^(1/2))/(16*
(a^3*b + a^4)))^(1/2) - log((524288*(1024*a^3*exp(2*x) - 35*b^3*exp(2*x) + 185*a*b^2 + 464*a^2*b + 256*a^3 - 2
4*b^3 + 988*a*b^2*exp(2*x) + 2048*a^2*b*exp(2*x)))/(a*b^6*(a + b)^2) - (((((4194304*(253*a*b^3 - b^4*exp(2*x)
+ 1184*a^3*b + 512*a^4 - b^4 + 930*a^2*b^2 + 1392*a^2*b^2*exp(2*x) + 627*a*b^3*exp(2*x) + 768*a^3*b*exp(2*x)))
/(b^6*(a + b)^2) - (8388608*a*((a^2 - (-a^3*b)^(1/2))/(a^3*(a + b)))^(1/2)*(512*a^3*exp(2*x) - 6*b^3*exp(2*x)
+ 181*a*b^2 + 432*a^2*b + 256*a^3 - 5*b^3 + 622*a*b^2*exp(2*x) + 1152*a^2*b*exp(2*x)))/(b^6*(a + b)))*((a^2 -
(-a^3*b)^(1/2))/(a^3*(a + b)))^(1/2))/4 + (2097152*(176*a*b + 1536*a^2*exp(2*x) - 134*b^2*exp(2*x) + 256*a^2 -
75*b^2 + 1408*a*b*exp(2*x)))/(b^6*(a + b)))*((a^2 - (-a^3*b)^(1/2))/(a^3*(a + b)))^(1/2))/4)*((a^2 - (-a^3*b)
^(1/2))/(16*(a^3*b + a^4)))^(1/2) + log((524288*(1024*a^3*exp(2*x) - 35*b^3*exp(2*x) + 185*a*b^2 + 464*a^2*b +
256*a^3 - 24*b^3 + 988*a*b^2*exp(2*x) + 2048*a^2*b*exp(2*x)))/(a*b^6*(a + b)^2) - (((((4194304*(253*a*b^3 - b
^4*exp(2*x) + 1184*a^3*b + 512*a^4 - b^4 + 930*a^2*b^2 + 1392*a^2*b^2*exp(2*x) + 627*a*b^3*exp(2*x) + 768*a^3*
b*exp(2*x)))/(b^6*(a + b)^2) + (8388608*a*((a^2 - (-a^3*b)^(1/2))/(a^3*(a + b)))^(1/2)*(512*a^3*exp(2*x) - 6*b
^3*exp(2*x) + 181*a*b^2 + 432*a^2*b + 256*a^3 - 5*b^3 + 622*a*b^2*exp(2*x) + 1152*a^2*b*exp(2*x)))/(b^6*(a + b
)))*((a^2 - (-a^3*b)^(1/2))/(a^3*(a + b)))^(1/2))/4 - (2097152*(176*a*b + 1536*a^2*exp(2*x) - 134*b^2*exp(2*x)
+ 256*a^2 - 75*b^2 + 1408*a*b*exp(2*x)))/(b^6*(a + b)))*((a^2 - (-a^3*b)^(1/2))/(a^3*(a + b)))^(1/2))/4)*((a^
2 - (-a^3*b)^(1/2))/(16*(a^3*b + a^4)))^(1/2)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________