3.58 \(\int \frac {1}{1+\cosh ^3(x)} \, dx\)

Optimal. Leaf size=91 \[ -\frac {2 \sqrt [4]{-\frac {1}{3}} \tanh ^{-1}\left ((-1)^{3/4} \sqrt [4]{3} \tanh \left (\frac {x}{2}\right )\right )}{3 \left (1+(-1)^{2/3}\right )}-\frac {2 \sqrt [4]{-\frac {1}{3}} \tan ^{-1}\left ((-1)^{3/4} \sqrt [4]{3} \tanh \left (\frac {x}{2}\right )\right )}{3 \left (1-\sqrt [3]{-1}\right )}+\frac {\sinh (x)}{3 (\cosh (x)+1)} \]

[Out]

-2/9*(-1)^(1/4)*3^(3/4)*arctan((-1)^(3/4)*3^(1/4)*tanh(1/2*x))/(1-(-1)^(1/3))-2/9*(-1)^(1/4)*3^(3/4)*arctanh((
-1)^(3/4)*3^(1/4)*tanh(1/2*x))/(1+(-1)^(2/3))+1/3*sinh(x)/(1+cosh(x))

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Rubi [A]  time = 0.13, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {3213, 2648, 2659, 205, 208} \[ -\frac {2 \sqrt [4]{-\frac {1}{3}} \tanh ^{-1}\left ((-1)^{3/4} \sqrt [4]{3} \tanh \left (\frac {x}{2}\right )\right )}{3 \left (1+(-1)^{2/3}\right )}-\frac {2 \sqrt [4]{-\frac {1}{3}} \tan ^{-1}\left ((-1)^{3/4} \sqrt [4]{3} \tanh \left (\frac {x}{2}\right )\right )}{3 \left (1-\sqrt [3]{-1}\right )}+\frac {\sinh (x)}{3 (\cosh (x)+1)} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Cosh[x]^3)^(-1),x]

[Out]

(-2*(-1/3)^(1/4)*ArcTan[(-1)^(3/4)*3^(1/4)*Tanh[x/2]])/(3*(1 - (-1)^(1/3))) - (2*(-1/3)^(1/4)*ArcTanh[(-1)^(3/
4)*3^(1/4)*Tanh[x/2]])/(3*(1 + (-1)^(2/3))) + Sinh[x]/(3*(1 + Cosh[x]))

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 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]

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 3213

Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Int[ExpandTrig[(a + b*(c*sin[e + f*
x])^n)^p, x], x] /; FreeQ[{a, b, c, e, f, n}, x] && (IGtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

Rubi steps

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

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Mathematica [C]  time = 0.99, size = 133, normalized size = 1.46 \[ \frac {1}{18} \left (6 \tanh \left (\frac {x}{2}\right )-\sqrt {6+2 i \sqrt {3}} \left (\sqrt {3}+3 i\right ) \tan ^{-1}\left (\frac {\left (3+i \sqrt {3}\right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {6-2 i \sqrt {3}}}\right )-\sqrt {6-2 i \sqrt {3}} \left (\sqrt {3}-3 i\right ) \tan ^{-1}\left (\frac {\left (3-i \sqrt {3}\right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {6+2 i \sqrt {3}}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Cosh[x]^3)^(-1),x]

[Out]

(-(Sqrt[6 + (2*I)*Sqrt[3]]*(3*I + Sqrt[3])*ArcTan[((3 + I*Sqrt[3])*Tanh[x/2])/Sqrt[6 - (2*I)*Sqrt[3]]]) - Sqrt
[6 - (2*I)*Sqrt[3]]*(-3*I + Sqrt[3])*ArcTan[((3 - I*Sqrt[3])*Tanh[x/2])/Sqrt[6 + (2*I)*Sqrt[3]]] + 6*Tanh[x/2]
)/18

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fricas [B]  time = 0.54, size = 602, normalized size = 6.62 \[ \frac {4 \, {\left (3^{\frac {3}{4}} e^{x} + 3^{\frac {3}{4}}\right )} \sqrt {-4 \, \sqrt {3} + 8} \arctan \left (\frac {1}{12} \, {\left (\sqrt {3} {\left (\sqrt {3} + 3\right )} - 3 \, \sqrt {3} + 9\right )} e^{x} - \frac {1}{48} \, {\left (2 \, \sqrt {3} {\left (\sqrt {3} + 3\right )} - {\left (3^{\frac {3}{4}} {\left (3 \, \sqrt {3} + 5\right )} + 3 \cdot 3^{\frac {1}{4}} {\left (\sqrt {3} + 1\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} - 6 \, \sqrt {3} + 18\right )} \sqrt {2 \, {\left (3^{\frac {1}{4}} {\left (\sqrt {3} + 2\right )} - 3^{\frac {1}{4}} e^{x}\right )} \sqrt {-4 \, \sqrt {3} + 8} + 4 \, \sqrt {3} + 4 \, e^{\left (2 \, x\right )} - 4 \, e^{x} + 4} + \frac {1}{12} \, \sqrt {3} {\left (\sqrt {3} - 3\right )} - \frac {1}{24} \, {\left ({\left (3^{\frac {3}{4}} {\left (3 \, \sqrt {3} + 5\right )} + 3 \cdot 3^{\frac {1}{4}} {\left (\sqrt {3} + 1\right )}\right )} e^{x} - 3^{\frac {3}{4}} {\left (\sqrt {3} + 1\right )} - 3 \cdot 3^{\frac {1}{4}} {\left (\sqrt {3} - 1\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} + \frac {1}{4} \, \sqrt {3} - \frac {1}{4}\right ) + 4 \, {\left (3^{\frac {3}{4}} e^{x} + 3^{\frac {3}{4}}\right )} \sqrt {-4 \, \sqrt {3} + 8} \arctan \left (-\frac {1}{12} \, {\left (\sqrt {3} {\left (\sqrt {3} + 3\right )} - 3 \, \sqrt {3} + 9\right )} e^{x} + \frac {1}{48} \, {\left (2 \, \sqrt {3} {\left (\sqrt {3} + 3\right )} + {\left (3^{\frac {3}{4}} {\left (3 \, \sqrt {3} + 5\right )} + 3 \cdot 3^{\frac {1}{4}} {\left (\sqrt {3} + 1\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} - 6 \, \sqrt {3} + 18\right )} \sqrt {-2 \, {\left (3^{\frac {1}{4}} {\left (\sqrt {3} + 2\right )} - 3^{\frac {1}{4}} e^{x}\right )} \sqrt {-4 \, \sqrt {3} + 8} + 4 \, \sqrt {3} + 4 \, e^{\left (2 \, x\right )} - 4 \, e^{x} + 4} - \frac {1}{12} \, \sqrt {3} {\left (\sqrt {3} - 3\right )} - \frac {1}{24} \, {\left ({\left (3^{\frac {3}{4}} {\left (3 \, \sqrt {3} + 5\right )} + 3 \cdot 3^{\frac {1}{4}} {\left (\sqrt {3} + 1\right )}\right )} e^{x} - 3^{\frac {3}{4}} {\left (\sqrt {3} + 1\right )} - 3 \cdot 3^{\frac {1}{4}} {\left (\sqrt {3} - 1\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} - \frac {1}{4} \, \sqrt {3} + \frac {1}{4}\right ) - {\left (3^{\frac {1}{4}} {\left (2 \, \sqrt {3} + 3\right )} e^{x} + 3^{\frac {1}{4}} {\left (2 \, \sqrt {3} + 3\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (2 \, {\left (3^{\frac {1}{4}} {\left (\sqrt {3} + 2\right )} - 3^{\frac {1}{4}} e^{x}\right )} \sqrt {-4 \, \sqrt {3} + 8} + 4 \, \sqrt {3} + 4 \, e^{\left (2 \, x\right )} - 4 \, e^{x} + 4\right ) + {\left (3^{\frac {1}{4}} {\left (2 \, \sqrt {3} + 3\right )} e^{x} + 3^{\frac {1}{4}} {\left (2 \, \sqrt {3} + 3\right )}\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (-2 \, {\left (3^{\frac {1}{4}} {\left (\sqrt {3} + 2\right )} - 3^{\frac {1}{4}} e^{x}\right )} \sqrt {-4 \, \sqrt {3} + 8} + 4 \, \sqrt {3} + 4 \, e^{\left (2 \, x\right )} - 4 \, e^{x} + 4\right ) - 24}{36 \, {\left (e^{x} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cosh(x)^3),x, algorithm="fricas")

[Out]

1/36*(4*(3^(3/4)*e^x + 3^(3/4))*sqrt(-4*sqrt(3) + 8)*arctan(1/12*(sqrt(3)*(sqrt(3) + 3) - 3*sqrt(3) + 9)*e^x -
1/48*(2*sqrt(3)*(sqrt(3) + 3) - (3^(3/4)*(3*sqrt(3) + 5) + 3*3^(1/4)*(sqrt(3) + 1))*sqrt(-4*sqrt(3) + 8) - 6*
sqrt(3) + 18)*sqrt(2*(3^(1/4)*(sqrt(3) + 2) - 3^(1/4)*e^x)*sqrt(-4*sqrt(3) + 8) + 4*sqrt(3) + 4*e^(2*x) - 4*e^
x + 4) + 1/12*sqrt(3)*(sqrt(3) - 3) - 1/24*((3^(3/4)*(3*sqrt(3) + 5) + 3*3^(1/4)*(sqrt(3) + 1))*e^x - 3^(3/4)*
(sqrt(3) + 1) - 3*3^(1/4)*(sqrt(3) - 1))*sqrt(-4*sqrt(3) + 8) + 1/4*sqrt(3) - 1/4) + 4*(3^(3/4)*e^x + 3^(3/4))
*sqrt(-4*sqrt(3) + 8)*arctan(-1/12*(sqrt(3)*(sqrt(3) + 3) - 3*sqrt(3) + 9)*e^x + 1/48*(2*sqrt(3)*(sqrt(3) + 3)
+ (3^(3/4)*(3*sqrt(3) + 5) + 3*3^(1/4)*(sqrt(3) + 1))*sqrt(-4*sqrt(3) + 8) - 6*sqrt(3) + 18)*sqrt(-2*(3^(1/4)
*(sqrt(3) + 2) - 3^(1/4)*e^x)*sqrt(-4*sqrt(3) + 8) + 4*sqrt(3) + 4*e^(2*x) - 4*e^x + 4) - 1/12*sqrt(3)*(sqrt(3
) - 3) - 1/24*((3^(3/4)*(3*sqrt(3) + 5) + 3*3^(1/4)*(sqrt(3) + 1))*e^x - 3^(3/4)*(sqrt(3) + 1) - 3*3^(1/4)*(sq
rt(3) - 1))*sqrt(-4*sqrt(3) + 8) - 1/4*sqrt(3) + 1/4) - (3^(1/4)*(2*sqrt(3) + 3)*e^x + 3^(1/4)*(2*sqrt(3) + 3)
)*sqrt(-4*sqrt(3) + 8)*log(2*(3^(1/4)*(sqrt(3) + 2) - 3^(1/4)*e^x)*sqrt(-4*sqrt(3) + 8) + 4*sqrt(3) + 4*e^(2*x
) - 4*e^x + 4) + (3^(1/4)*(2*sqrt(3) + 3)*e^x + 3^(1/4)*(2*sqrt(3) + 3))*sqrt(-4*sqrt(3) + 8)*log(-2*(3^(1/4)*
(sqrt(3) + 2) - 3^(1/4)*e^x)*sqrt(-4*sqrt(3) + 8) + 4*sqrt(3) + 4*e^(2*x) - 4*e^x + 4) - 24)/(e^x + 1)

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giac [B]  time = 0.14, size = 275, normalized size = 3.02 \[ \frac {1}{18} \, \sqrt {6 \, \sqrt {3} + 9} \log \left (4 \, {\left (2 \, \sqrt {3} \sqrt {6 \, \sqrt {3} + 9} - 3 \, \sqrt {6 \, \sqrt {3} + 9} + 6 \, e^{x} - 3\right )}^{2} + 4 \, {\left (\sqrt {3} \sqrt {6 \, \sqrt {3} + 9} - 3 \, \sqrt {3}\right )}^{2}\right ) - \frac {1}{18} \, \sqrt {6 \, \sqrt {3} + 9} \log \left (4 \, {\left (2 \, \sqrt {3} \sqrt {6 \, \sqrt {3} + 9} - 3 \, \sqrt {6 \, \sqrt {3} + 9} - 6 \, e^{x} + 3\right )}^{2} + 4 \, {\left (\sqrt {3} \sqrt {6 \, \sqrt {3} + 9} + 3 \, \sqrt {3}\right )}^{2}\right ) - \frac {\sqrt {3} \sqrt {6 \, \sqrt {3} + 9} \arctan \left (\frac {3 \, {\left (\sqrt {2 \, \sqrt {3} - 3} + 2 \, e^{x} - 1\right )}}{\sqrt {3} \sqrt {6 \, \sqrt {3} + 9} - 3 \, \sqrt {3}}\right )}{9 \, {\left (2 \, \sqrt {3} + 3\right )}} - \frac {\sqrt {3} \sqrt {6 \, \sqrt {3} + 9} \arctan \left (-\frac {3 \, {\left (\sqrt {2 \, \sqrt {3} - 3} - 2 \, e^{x} + 1\right )}}{\sqrt {3} \sqrt {6 \, \sqrt {3} + 9} + 3 \, \sqrt {3}}\right )}{9 \, {\left (2 \, \sqrt {3} + 3\right )}} - \frac {2}{3 \, {\left (e^{x} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cosh(x)^3),x, algorithm="giac")

[Out]

1/18*sqrt(6*sqrt(3) + 9)*log(4*(2*sqrt(3)*sqrt(6*sqrt(3) + 9) - 3*sqrt(6*sqrt(3) + 9) + 6*e^x - 3)^2 + 4*(sqrt
(3)*sqrt(6*sqrt(3) + 9) - 3*sqrt(3))^2) - 1/18*sqrt(6*sqrt(3) + 9)*log(4*(2*sqrt(3)*sqrt(6*sqrt(3) + 9) - 3*sq
rt(6*sqrt(3) + 9) - 6*e^x + 3)^2 + 4*(sqrt(3)*sqrt(6*sqrt(3) + 9) + 3*sqrt(3))^2) - 1/9*sqrt(3)*sqrt(6*sqrt(3)
+ 9)*arctan(3*(sqrt(2*sqrt(3) - 3) + 2*e^x - 1)/(sqrt(3)*sqrt(6*sqrt(3) + 9) - 3*sqrt(3)))/(2*sqrt(3) + 3) -
1/9*sqrt(3)*sqrt(6*sqrt(3) + 9)*arctan(-3*(sqrt(2*sqrt(3) - 3) - 2*e^x + 1)/(sqrt(3)*sqrt(6*sqrt(3) + 9) + 3*s
qrt(3)))/(2*sqrt(3) + 3) - 2/3/(e^x + 1)

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maple [B]  time = 0.05, size = 216, normalized size = 2.37 \[ \frac {\tanh \left (\frac {x}{2}\right )}{3}+\frac {3^{\frac {3}{4}} \sqrt {2}\, \arctan \left (\sqrt {2}\, 3^{\frac {1}{4}} \tanh \left (\frac {x}{2}\right )-1\right )}{18}+\frac {3^{\frac {3}{4}} \sqrt {2}\, \ln \left (\frac {\tanh ^{2}\left (\frac {x}{2}\right )+\frac {3^{\frac {3}{4}} \tanh \left (\frac {x}{2}\right ) \sqrt {2}}{3}+\frac {\sqrt {3}}{3}}{\tanh ^{2}\left (\frac {x}{2}\right )-\frac {3^{\frac {3}{4}} \tanh \left (\frac {x}{2}\right ) \sqrt {2}}{3}+\frac {\sqrt {3}}{3}}\right )}{36}+\frac {3^{\frac {3}{4}} \sqrt {2}\, \arctan \left (\sqrt {2}\, 3^{\frac {1}{4}} \tanh \left (\frac {x}{2}\right )+1\right )}{18}-\frac {\sqrt {2}\, 3^{\frac {1}{4}} \ln \left (\frac {\tanh ^{2}\left (\frac {x}{2}\right )-\frac {3^{\frac {3}{4}} \tanh \left (\frac {x}{2}\right ) \sqrt {2}}{3}+\frac {\sqrt {3}}{3}}{\tanh ^{2}\left (\frac {x}{2}\right )+\frac {3^{\frac {3}{4}} \tanh \left (\frac {x}{2}\right ) \sqrt {2}}{3}+\frac {\sqrt {3}}{3}}\right )}{12}-\frac {\sqrt {2}\, 3^{\frac {1}{4}} \arctan \left (\sqrt {2}\, 3^{\frac {1}{4}} \tanh \left (\frac {x}{2}\right )-1\right )}{6}-\frac {\sqrt {2}\, 3^{\frac {1}{4}} \arctan \left (\sqrt {2}\, 3^{\frac {1}{4}} \tanh \left (\frac {x}{2}\right )+1\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+cosh(x)^3),x)

[Out]

1/3*tanh(1/2*x)+1/18*3^(3/4)*2^(1/2)*arctan(2^(1/2)*3^(1/4)*tanh(1/2*x)-1)+1/36*3^(3/4)*2^(1/2)*ln((tanh(1/2*x
)^2+1/3*3^(3/4)*tanh(1/2*x)*2^(1/2)+1/3*3^(1/2))/(tanh(1/2*x)^2-1/3*3^(3/4)*tanh(1/2*x)*2^(1/2)+1/3*3^(1/2)))+
1/18*3^(3/4)*2^(1/2)*arctan(2^(1/2)*3^(1/4)*tanh(1/2*x)+1)-1/12*2^(1/2)*3^(1/4)*ln((tanh(1/2*x)^2-1/3*3^(3/4)*
tanh(1/2*x)*2^(1/2)+1/3*3^(1/2))/(tanh(1/2*x)^2+1/3*3^(3/4)*tanh(1/2*x)*2^(1/2)+1/3*3^(1/2)))-1/6*2^(1/2)*3^(1
/4)*arctan(2^(1/2)*3^(1/4)*tanh(1/2*x)-1)-1/6*2^(1/2)*3^(1/4)*arctan(2^(1/2)*3^(1/4)*tanh(1/2*x)+1)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2}{3 \, {\left (e^{x} + 1\right )}} - \int \frac {2 \, {\left (e^{\left (3 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + e^{x}\right )}}{3 \, {\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (3 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 2 \, e^{x} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cosh(x)^3),x, algorithm="maxima")

[Out]

-2/3/(e^x + 1) - integrate(2/3*(e^(3*x) - 4*e^(2*x) + e^x)/(e^(4*x) - 2*e^(3*x) + 6*e^(2*x) - 2*e^x + 1), x)

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mupad [B]  time = 3.43, size = 291, normalized size = 3.20 \[ \ln \left (\frac {128}{9}+\sqrt {\frac {1}{18}-\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}\,\left (\frac {160}{3}+\sqrt {\frac {1}{18}-\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}\,\left (384\,{\mathrm {e}}^x+\sqrt {\frac {1}{18}-\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}\,\left (1152\,{\mathrm {e}}^x-864\right )-192\right )-\frac {32\,{\mathrm {e}}^x}{3}\right )-\frac {32\,{\mathrm {e}}^x}{3}\right )\,\sqrt {\frac {1}{18}-\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}+\ln \left (\frac {128}{9}+\sqrt {\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}\,\left (\frac {160}{3}+\sqrt {\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}\,\left (384\,{\mathrm {e}}^x+\sqrt {\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}\,\left (1152\,{\mathrm {e}}^x-864\right )-192\right )-\frac {32\,{\mathrm {e}}^x}{3}\right )-\frac {32\,{\mathrm {e}}^x}{3}\right )\,\sqrt {\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}-\ln \left (\frac {128}{9}-\sqrt {\frac {1}{18}-\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}\,\left (\frac {160}{3}+\sqrt {\frac {1}{18}-\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}\,\left (192+\sqrt {\frac {1}{18}-\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}\,\left (1152\,{\mathrm {e}}^x-864\right )-384\,{\mathrm {e}}^x\right )-\frac {32\,{\mathrm {e}}^x}{3}\right )-\frac {32\,{\mathrm {e}}^x}{3}\right )\,\sqrt {\frac {1}{18}-\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}-\ln \left (\frac {128}{9}-\sqrt {\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}\,\left (\frac {160}{3}+\sqrt {\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}\,\left (192+\sqrt {\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}\,\left (1152\,{\mathrm {e}}^x-864\right )-384\,{\mathrm {e}}^x\right )-\frac {32\,{\mathrm {e}}^x}{3}\right )-\frac {32\,{\mathrm {e}}^x}{3}\right )\,\sqrt {\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}}-\frac {2}{3\,\left ({\mathrm {e}}^x+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(x)^3 + 1),x)

[Out]

log((1/18 - (3^(1/2)*1i)/54)^(1/2)*((1/18 - (3^(1/2)*1i)/54)^(1/2)*(384*exp(x) + (1/18 - (3^(1/2)*1i)/54)^(1/2
)*(1152*exp(x) - 864) - 192) - (32*exp(x))/3 + 160/3) - (32*exp(x))/3 + 128/9)*(1/18 - (3^(1/2)*1i)/54)^(1/2)
+ log(((3^(1/2)*1i)/54 + 1/18)^(1/2)*(((3^(1/2)*1i)/54 + 1/18)^(1/2)*(384*exp(x) + ((3^(1/2)*1i)/54 + 1/18)^(1
/2)*(1152*exp(x) - 864) - 192) - (32*exp(x))/3 + 160/3) - (32*exp(x))/3 + 128/9)*((3^(1/2)*1i)/54 + 1/18)^(1/2
) - log(128/9 - (1/18 - (3^(1/2)*1i)/54)^(1/2)*((1/18 - (3^(1/2)*1i)/54)^(1/2)*((1/18 - (3^(1/2)*1i)/54)^(1/2)
*(1152*exp(x) - 864) - 384*exp(x) + 192) - (32*exp(x))/3 + 160/3) - (32*exp(x))/3)*(1/18 - (3^(1/2)*1i)/54)^(1
/2) - log(128/9 - ((3^(1/2)*1i)/54 + 1/18)^(1/2)*(((3^(1/2)*1i)/54 + 1/18)^(1/2)*(((3^(1/2)*1i)/54 + 1/18)^(1/
2)*(1152*exp(x) - 864) - 384*exp(x) + 192) - (32*exp(x))/3 + 160/3) - (32*exp(x))/3)*((3^(1/2)*1i)/54 + 1/18)^
(1/2) - 2/(3*(exp(x) + 1))

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sympy [B]  time = 3.26, size = 330, normalized size = 3.63 \[ - \frac {2 \sqrt {2} \cdot 3^{\frac {3}{4}} \log {\left (36 \tanh ^{2}{\left (\frac {x}{2} \right )} - 12 \sqrt {2} \cdot 3^{\frac {3}{4}} \tanh {\left (\frac {x}{2} \right )} + 12 \sqrt {3} \right )}}{18 + 18 \sqrt {3}} - \frac {3 \sqrt {2} \sqrt [4]{3} \log {\left (36 \tanh ^{2}{\left (\frac {x}{2} \right )} - 12 \sqrt {2} \cdot 3^{\frac {3}{4}} \tanh {\left (\frac {x}{2} \right )} + 12 \sqrt {3} \right )}}{18 + 18 \sqrt {3}} + \frac {3 \sqrt {2} \sqrt [4]{3} \log {\left (36 \tanh ^{2}{\left (\frac {x}{2} \right )} + 12 \sqrt {2} \cdot 3^{\frac {3}{4}} \tanh {\left (\frac {x}{2} \right )} + 12 \sqrt {3} \right )}}{18 + 18 \sqrt {3}} + \frac {2 \sqrt {2} \cdot 3^{\frac {3}{4}} \log {\left (36 \tanh ^{2}{\left (\frac {x}{2} \right )} + 12 \sqrt {2} \cdot 3^{\frac {3}{4}} \tanh {\left (\frac {x}{2} \right )} + 12 \sqrt {3} \right )}}{18 + 18 \sqrt {3}} + \frac {6 \tanh {\left (\frac {x}{2} \right )}}{18 + 18 \sqrt {3}} + \frac {6 \sqrt {3} \tanh {\left (\frac {x}{2} \right )}}{18 + 18 \sqrt {3}} - \frac {2 \sqrt {2} \cdot 3^{\frac {3}{4}} \operatorname {atan}{\left (\sqrt {2} \sqrt [4]{3} \tanh {\left (\frac {x}{2} \right )} - 1 \right )}}{18 + 18 \sqrt {3}} - \frac {2 \sqrt {2} \cdot 3^{\frac {3}{4}} \operatorname {atan}{\left (\sqrt {2} \sqrt [4]{3} \tanh {\left (\frac {x}{2} \right )} + 1 \right )}}{18 + 18 \sqrt {3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cosh(x)**3),x)

[Out]

-2*sqrt(2)*3**(3/4)*log(36*tanh(x/2)**2 - 12*sqrt(2)*3**(3/4)*tanh(x/2) + 12*sqrt(3))/(18 + 18*sqrt(3)) - 3*sq
rt(2)*3**(1/4)*log(36*tanh(x/2)**2 - 12*sqrt(2)*3**(3/4)*tanh(x/2) + 12*sqrt(3))/(18 + 18*sqrt(3)) + 3*sqrt(2)
*3**(1/4)*log(36*tanh(x/2)**2 + 12*sqrt(2)*3**(3/4)*tanh(x/2) + 12*sqrt(3))/(18 + 18*sqrt(3)) + 2*sqrt(2)*3**(
3/4)*log(36*tanh(x/2)**2 + 12*sqrt(2)*3**(3/4)*tanh(x/2) + 12*sqrt(3))/(18 + 18*sqrt(3)) + 6*tanh(x/2)/(18 + 1
8*sqrt(3)) + 6*sqrt(3)*tanh(x/2)/(18 + 18*sqrt(3)) - 2*sqrt(2)*3**(3/4)*atan(sqrt(2)*3**(1/4)*tanh(x/2) - 1)/(
18 + 18*sqrt(3)) - 2*sqrt(2)*3**(3/4)*atan(sqrt(2)*3**(1/4)*tanh(x/2) + 1)/(18 + 18*sqrt(3))

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