3.9.10 \(\int \frac {1}{(\cosh ^2(x)+\sinh ^2(x))^3} \, dx\) [810]

Optimal. Leaf size=26 \[ \frac {1}{2} \text {ArcTan}(\tanh (x))+\frac {\text {sech}^2(x) \tanh (x)}{2 \left (1+\tanh ^2(x)\right )^2} \]

[Out]

1/2*arctan(tanh(x))+1/2*sech(x)^2*tanh(x)/(1+tanh(x)^2)^2

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Rubi [A]
time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, 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 = {424, 21, 209} \begin {gather*} \frac {1}{2} \text {ArcTan}(\tanh (x))+\frac {\tanh (x) \text {sech}^2(x)}{2 \left (\tanh ^2(x)+1\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Cosh[x]^2 + Sinh[x]^2)^(-3),x]

[Out]

ArcTan[Tanh[x]]/2 + (Sech[x]^2*Tanh[x])/(2*(1 + Tanh[x]^2)^2)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 209

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

Rule 424

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a*d - c*b)*x*(a + b*x^n)^(
p + 1)*((c + d*x^n)^(q - 1)/(a*b*n*(p + 1))), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)
^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /;
FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q
, x]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 22, normalized size = 0.85 \begin {gather*} \frac {1}{4} \text {ArcTan}(\sinh (2 x))+\frac {1}{4} \text {sech}(2 x) \tanh (2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Cosh[x]^2 + Sinh[x]^2)^(-3),x]

[Out]

ArcTan[Sinh[2*x]]/4 + (Sech[2*x]*Tanh[2*x])/4

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(122\) vs. \(2(22)=44\).
time = 1.36, size = 123, normalized size = 4.73

method	result	size
risch	\(\frac {{\mathrm e}^{2 x} \left ({\mathrm e}^{4 x}-1\right )}{2 \left (1+{\mathrm e}^{4 x}\right )^{2}}+\frac {i \ln \left ({\mathrm e}^{2 x}+i\right )}{4}-\frac {i \ln \left ({\mathrm e}^{2 x}-i\right )}{4}\)	\(44\)
default	\(-\frac {2 \left (-\frac {\left (\tanh ^{7}\left (\frac {x}{2}\right )\right )}{2}+\frac {\left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{2}+\frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{2}-\frac {\tanh \left (\frac {x}{2}\right )}{2}\right )}{\left (\tanh ^{4}\left (\frac {x}{2}\right )+6 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+1\right )^{2}}-\frac {\left (2+\sqrt {2}\right ) \sqrt {2}\, \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right )}{2+2 \sqrt {2}}\right )}{2 \left (2+2 \sqrt {2}\right )}-\frac {\left (-2+\sqrt {2}\right ) \sqrt {2}\, \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right )}{-2+2 \sqrt {2}}\right )}{2 \left (-2+2 \sqrt {2}\right )}\)	\(123\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)^2+cosh(x)^2)^3,x,method=_RETURNVERBOSE)

[Out]

-2*(-1/2*tanh(1/2*x)^7+1/2*tanh(1/2*x)^5+1/2*tanh(1/2*x)^3-1/2*tanh(1/2*x))/(tanh(1/2*x)^4+6*tanh(1/2*x)^2+1)^
2-1/2*(2+2^(1/2))*2^(1/2)/(2+2*2^(1/2))*arctan(2*tanh(1/2*x)/(2+2*2^(1/2)))-1/2*(-2+2^(1/2))*2^(1/2)/(-2+2*2^(
1/2))*arctan(2*tanh(1/2*x)/(-2+2*2^(1/2)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (22) = 44\).
time = 0.47, size = 64, normalized size = 2.46 \begin {gather*} \frac {e^{\left (-2 \, x\right )} - e^{\left (-6 \, x\right )}}{2 \, {\left (2 \, e^{\left (-4 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} + \frac {1}{2} \, \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{\left (-x\right )}\right )}\right ) - \frac {1}{2} \, \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{\left (-x\right )}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cosh(x)^2+sinh(x)^2)^3,x, algorithm="maxima")

[Out]

1/2*(e^(-2*x) - e^(-6*x))/(2*e^(-4*x) + e^(-8*x) + 1) + 1/2*arctan(1/2*sqrt(2)*(sqrt(2) + 2*e^(-x))) - 1/2*arc
tan(-1/2*sqrt(2)*(sqrt(2) - 2*e^(-x)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (22) = 44\).
time = 0.41, size = 304, normalized size = 11.69 \begin {gather*} \frac {\cosh \left (x\right )^{6} + 20 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{3} + 15 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + {\left (15 \, \cosh \left (x\right )^{4} - 1\right )} \sinh \left (x\right )^{2} - {\left (\cosh \left (x\right )^{8} + 56 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{5} + 28 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{6} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} + 1\right )} \sinh \left (x\right )^{4} + 2 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} + \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} + 3 \, \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} + \cosh \left (x\right )^{3}\right )} \sinh \left (x\right ) + 1\right )} \arctan \left (-\frac {\cosh \left (x\right ) + \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - \cosh \left (x\right )^{2} + 2 \, {\left (3 \, \cosh \left (x\right )^{5} - \cosh \left (x\right )\right )} \sinh \left (x\right )}{2 \, {\left (\cosh \left (x\right )^{8} + 56 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{5} + 28 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{6} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} + 1\right )} \sinh \left (x\right )^{4} + 2 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} + \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} + 3 \, \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} + \cosh \left (x\right )^{3}\right )} \sinh \left (x\right ) + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cosh(x)^2+sinh(x)^2)^3,x, algorithm="fricas")

[Out]

1/2*(cosh(x)^6 + 20*cosh(x)^3*sinh(x)^3 + 15*cosh(x)^2*sinh(x)^4 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6 + (15*cosh(
x)^4 - 1)*sinh(x)^2 - (cosh(x)^8 + 56*cosh(x)^3*sinh(x)^5 + 28*cosh(x)^2*sinh(x)^6 + 8*cosh(x)*sinh(x)^7 + sin
h(x)^8 + 2*(35*cosh(x)^4 + 1)*sinh(x)^4 + 2*cosh(x)^4 + 8*(7*cosh(x)^5 + cosh(x))*sinh(x)^3 + 4*(7*cosh(x)^6 +
3*cosh(x)^2)*sinh(x)^2 + 8*(cosh(x)^7 + cosh(x)^3)*sinh(x) + 1)*arctan(-(cosh(x) + sinh(x))/(cosh(x) - sinh(x
))) - cosh(x)^2 + 2*(3*cosh(x)^5 - cosh(x))*sinh(x))/(cosh(x)^8 + 56*cosh(x)^3*sinh(x)^5 + 28*cosh(x)^2*sinh(x
)^6 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + 2*(35*cosh(x)^4 + 1)*sinh(x)^4 + 2*cosh(x)^4 + 8*(7*cosh(x)^5 + cosh(x
))*sinh(x)^3 + 4*(7*cosh(x)^6 + 3*cosh(x)^2)*sinh(x)^2 + 8*(cosh(x)^7 + cosh(x)^3)*sinh(x) + 1)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 3602 vs. \(2 (24) = 48\).
time = 143.40, size = 3602, normalized size = 138.54 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cosh(x)**2+sinh(x)**2)**3,x)

[Out]

1939450125521*sqrt(3 - 2*sqrt(2))*tanh(x/2)**8*atan(tanh(x/2)/sqrt(3 - 2*sqrt(2)))/(1136103579984*sqrt(2)*tanh
(x/2)**8 + 1606693091074*tanh(x/2)**8 + 13633242959808*sqrt(2)*tanh(x/2)**6 + 19280317092888*tanh(x/2)**6 + 43
171936039392*sqrt(2)*tanh(x/2)**4 + 61054337460812*tanh(x/2)**4 + 13633242959808*sqrt(2)*tanh(x/2)**2 + 192803
17092888*tanh(x/2)**2 + 1136103579984*sqrt(2) + 1606693091074) + 1371398335529*sqrt(2)*sqrt(3 - 2*sqrt(2))*tan
h(x/2)**8*atan(tanh(x/2)/sqrt(3 - 2*sqrt(2)))/(1136103579984*sqrt(2)*tanh(x/2)**8 + 1606693091074*tanh(x/2)**8
+ 13633242959808*sqrt(2)*tanh(x/2)**6 + 19280317092888*tanh(x/2)**6 + 43171936039392*sqrt(2)*tanh(x/2)**4 + 6
1054337460812*tanh(x/2)**4 + 13633242959808*sqrt(2)*tanh(x/2)**2 + 19280317092888*tanh(x/2)**2 + 1136103579984
*sqrt(2) + 1606693091074) - 235294755545*sqrt(2)*sqrt(2*sqrt(2) + 3)*tanh(x/2)**8*atan(tanh(x/2)/sqrt(2*sqrt(2
) + 3))/(1136103579984*sqrt(2)*tanh(x/2)**8 + 1606693091074*tanh(x/2)**8 + 13633242959808*sqrt(2)*tanh(x/2)**6
+ 19280317092888*tanh(x/2)**6 + 43171936039392*sqrt(2)*tanh(x/2)**4 + 61054337460812*tanh(x/2)**4 + 136332429
59808*sqrt(2)*tanh(x/2)**2 + 19280317092888*tanh(x/2)**2 + 1136103579984*sqrt(2) + 1606693091074) - 3327570344
47*sqrt(2*sqrt(2) + 3)*tanh(x/2)**8*atan(tanh(x/2)/sqrt(2*sqrt(2) + 3))/(1136103579984*sqrt(2)*tanh(x/2)**8 +
1606693091074*tanh(x/2)**8 + 13633242959808*sqrt(2)*tanh(x/2)**6 + 19280317092888*tanh(x/2)**6 + 4317193603939
2*sqrt(2)*tanh(x/2)**4 + 61054337460812*tanh(x/2)**4 + 13633242959808*sqrt(2)*tanh(x/2)**2 + 19280317092888*ta
nh(x/2)**2 + 1136103579984*sqrt(2) + 1606693091074) + 1136103579984*sqrt(2)*tanh(x/2)**7/(1136103579984*sqrt(2
)*tanh(x/2)**8 + 1606693091074*tanh(x/2)**8 + 13633242959808*sqrt(2)*tanh(x/2)**6 + 19280317092888*tanh(x/2)**
6 + 43171936039392*sqrt(2)*tanh(x/2)**4 + 61054337460812*tanh(x/2)**4 + 13633242959808*sqrt(2)*tanh(x/2)**2 +
19280317092888*tanh(x/2)**2 + 1136103579984*sqrt(2) + 1606693091074) + 1606693091074*tanh(x/2)**7/(11361035799
84*sqrt(2)*tanh(x/2)**8 + 1606693091074*tanh(x/2)**8 + 13633242959808*sqrt(2)*tanh(x/2)**6 + 19280317092888*ta
nh(x/2)**6 + 43171936039392*sqrt(2)*tanh(x/2)**4 + 61054337460812*tanh(x/2)**4 + 13633242959808*sqrt(2)*tanh(x
/2)**2 + 19280317092888*tanh(x/2)**2 + 1136103579984*sqrt(2) + 1606693091074) + 23273401506252*sqrt(3 - 2*sqrt
(2))*tanh(x/2)**6*atan(tanh(x/2)/sqrt(3 - 2*sqrt(2)))/(1136103579984*sqrt(2)*tanh(x/2)**8 + 1606693091074*tanh
(x/2)**8 + 13633242959808*sqrt(2)*tanh(x/2)**6 + 19280317092888*tanh(x/2)**6 + 43171936039392*sqrt(2)*tanh(x/2
)**4 + 61054337460812*tanh(x/2)**4 + 13633242959808*sqrt(2)*tanh(x/2)**2 + 19280317092888*tanh(x/2)**2 + 11361
03579984*sqrt(2) + 1606693091074) + 16456780026348*sqrt(2)*sqrt(3 - 2*sqrt(2))*tanh(x/2)**6*atan(tanh(x/2)/sqr
t(3 - 2*sqrt(2)))/(1136103579984*sqrt(2)*tanh(x/2)**8 + 1606693091074*tanh(x/2)**8 + 13633242959808*sqrt(2)*ta
nh(x/2)**6 + 19280317092888*tanh(x/2)**6 + 43171936039392*sqrt(2)*tanh(x/2)**4 + 61054337460812*tanh(x/2)**4 +
13633242959808*sqrt(2)*tanh(x/2)**2 + 19280317092888*tanh(x/2)**2 + 1136103579984*sqrt(2) + 1606693091074) -
2823537066540*sqrt(2)*sqrt(2*sqrt(2) + 3)*tanh(x/2)**6*atan(tanh(x/2)/sqrt(2*sqrt(2) + 3))/(1136103579984*sqrt
(2)*tanh(x/2)**8 + 1606693091074*tanh(x/2)**8 + 13633242959808*sqrt(2)*tanh(x/2)**6 + 19280317092888*tanh(x/2)
**6 + 43171936039392*sqrt(2)*tanh(x/2)**4 + 61054337460812*tanh(x/2)**4 + 13633242959808*sqrt(2)*tanh(x/2)**2
+ 19280317092888*tanh(x/2)**2 + 1136103579984*sqrt(2) + 1606693091074) - 3993084413364*sqrt(2*sqrt(2) + 3)*tan
h(x/2)**6*atan(tanh(x/2)/sqrt(2*sqrt(2) + 3))/(1136103579984*sqrt(2)*tanh(x/2)**8 + 1606693091074*tanh(x/2)**8
+ 13633242959808*sqrt(2)*tanh(x/2)**6 + 19280317092888*tanh(x/2)**6 + 43171936039392*sqrt(2)*tanh(x/2)**4 + 6
1054337460812*tanh(x/2)**4 + 13633242959808*sqrt(2)*tanh(x/2)**2 + 19280317092888*tanh(x/2)**2 + 1136103579984
*sqrt(2) + 1606693091074) - 1606693091074*tanh(x/2)**5/(1136103579984*sqrt(2)*tanh(x/2)**8 + 1606693091074*tan
h(x/2)**8 + 13633242959808*sqrt(2)*tanh(x/2)**6 + 19280317092888*tanh(x/2)**6 + 43171936039392*sqrt(2)*tanh(x/
2)**4 + 61054337460812*tanh(x/2)**4 + 13633242959808*sqrt(2)*tanh(x/2)**2 + 19280317092888*tanh(x/2)**2 + 1136
103579984*sqrt(2) + 1606693091074) - 1136103579984*sqrt(2)*tanh(x/2)**5/(1136103579984*sqrt(2)*tanh(x/2)**8 +
1606693091074*tanh(x/2)**8 + 13633242959808*sqrt(2)*tanh(x/2)**6 + 19280317092888*tanh(x/2)**6 + 4317193603939
2*sqrt(2)*tanh(x/2)**4 + 61054337460812*tanh(x/2)**4 + 13633242959808*sqrt(2)*tanh(x/2)**2 + 19280317092888*ta
nh(x/2)**2 + 1136103579984*sqrt(2) + 1606693091074) + 73699104769798*sqrt(3 - 2*sqrt(2))*tanh(x/2)**4*atan(tan
h(x/2)/sqrt(3 - 2*sqrt(2)))/(1136103579984*sqrt(2)*tanh(x/2)**8 + 1606693091074*tanh(x/2)**8 + 13633242959808*
sqrt(2)*tanh(x/2)**6 + 19280317092888*tanh(x/2)**6 + 43171936039392*sqrt(2)*tanh(x/2)**4 + 61054337460812*tanh
(x/2)**4 + 13633242959808*sqrt(2)*tanh(x/2)**2 + 19280317092888*tanh(x/2)**2 + 1136103579984*sqrt(2) + 1606693
091074) + 52113136750102*sqrt(2)*sqrt(3 - 2*sqr...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (22) = 44\).
time = 0.41, size = 46, normalized size = 1.77 \begin {gather*} \frac {e^{\left (2 \, x\right )} - e^{\left (-2 \, x\right )}}{2 \, {\left ({\left (e^{\left (2 \, x\right )} - e^{\left (-2 \, x\right )}\right )}^{2} + 4\right )}} + \frac {1}{4} \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (4 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cosh(x)^2+sinh(x)^2)^3,x, algorithm="giac")

[Out]

1/2*(e^(2*x) - e^(-2*x))/((e^(2*x) - e^(-2*x))^2 + 4) + 1/4*arctan(1/2*(e^(4*x) - 1)*e^(-2*x))

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Mupad [B]
time = 1.56, size = 28, normalized size = 1.08 \begin {gather*} \frac {\mathrm {atan}\left ({\mathrm {e}}^{2\,x}\right )}{2}-\frac {{\mathrm {e}}^{-2\,x}}{4\,{\mathrm {cosh}\left (2\,x\right )}^2}+\frac {1}{4\,\mathrm {cosh}\left (2\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(x)^2 + sinh(x)^2)^3,x)

[Out]

atan(exp(2*x))/2 - exp(-2*x)/(4*cosh(2*x)^2) + 1/(4*cosh(2*x))

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