3.1.32 \(\int \frac {\text {sech}^5(x)}{a+b \cosh ^2(x)} \, dx\) [32]

Optimal. Leaf size=90 \[ \frac {\left (3 a^2-4 a b+8 b^2\right ) \text {ArcTan}(\sinh (x))}{8 a^3}-\frac {b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{a^3 \sqrt {a+b}}+\frac {(3 a-4 b) \text {sech}(x) \tanh (x)}{8 a^2}+\frac {\text {sech}^3(x) \tanh (x)}{4 a} \]

[Out]

1/8*(3*a^2-4*a*b+8*b^2)*arctan(sinh(x))/a^3-b^(5/2)*arctan(sinh(x)*b^(1/2)/(a+b)^(1/2))/a^3/(a+b)^(1/2)+1/8*(3
*a-4*b)*sech(x)*tanh(x)/a^2+1/4*sech(x)^3*tanh(x)/a

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Rubi [A]
time = 0.10, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3265, 425, 541, 536, 209, 211} \begin {gather*} -\frac {b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{a^3 \sqrt {a+b}}+\frac {(3 a-4 b) \tanh (x) \text {sech}(x)}{8 a^2}+\frac {\left (3 a^2-4 a b+8 b^2\right ) \text {ArcTan}(\sinh (x))}{8 a^3}+\frac {\tanh (x) \text {sech}^3(x)}{4 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sech[x]^5/(a + b*Cosh[x]^2),x]

[Out]

((3*a^2 - 4*a*b + 8*b^2)*ArcTan[Sinh[x]])/(8*a^3) - (b^(5/2)*ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a + b]])/(a^3*Sqrt[
a + b]) + ((3*a - 4*b)*Sech[x]*Tanh[x])/(8*a^2) + (Sech[x]^3*Tanh[x])/(4*a)

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 211

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

Rule 425

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

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 541

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

Rule 3265

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos
[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {\text {sech}^5(x)}{a+b \cosh ^2(x)} \, dx &=\text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^3 \left (a+b+b x^2\right )} \, dx,x,\sinh (x)\right )\\ &=\frac {\text {sech}^3(x) \tanh (x)}{4 a}-\frac {\text {Subst}\left (\int \frac {-3 a+b-3 b x^2}{\left (1+x^2\right )^2 \left (a+b+b x^2\right )} \, dx,x,\sinh (x)\right )}{4 a}\\ &=\frac {(3 a-4 b) \text {sech}(x) \tanh (x)}{8 a^2}+\frac {\text {sech}^3(x) \tanh (x)}{4 a}+\frac {\text {Subst}\left (\int \frac {3 a^2-a b+4 b^2+(3 a-4 b) b x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\sinh (x)\right )}{8 a^2}\\ &=\frac {(3 a-4 b) \text {sech}(x) \tanh (x)}{8 a^2}+\frac {\text {sech}^3(x) \tanh (x)}{4 a}-\frac {b^3 \text {Subst}\left (\int \frac {1}{a+b+b x^2} \, dx,x,\sinh (x)\right )}{a^3}+\frac {\left (3 a^2-4 a b+8 b^2\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (x)\right )}{8 a^3}\\ &=\frac {\left (3 a^2-4 a b+8 b^2\right ) \tan ^{-1}(\sinh (x))}{8 a^3}-\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{a^3 \sqrt {a+b}}+\frac {(3 a-4 b) \text {sech}(x) \tanh (x)}{8 a^2}+\frac {\text {sech}^3(x) \tanh (x)}{4 a}\\ \end {align*}

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Mathematica [A]
time = 0.21, size = 86, normalized size = 0.96 \begin {gather*} \frac {\frac {8 b^{5/2} \text {ArcTan}\left (\frac {\sqrt {a+b} \text {csch}(x)}{\sqrt {b}}\right )}{\sqrt {a+b}}+2 \left (3 a^2-4 a b+8 b^2\right ) \text {ArcTan}\left (\tanh \left (\frac {x}{2}\right )\right )+a (3 a-4 b) \text {sech}(x) \tanh (x)+2 a^2 \text {sech}^3(x) \tanh (x)}{8 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]^5/(a + b*Cosh[x]^2),x]

[Out]

((8*b^(5/2)*ArcTan[(Sqrt[a + b]*Csch[x])/Sqrt[b]])/Sqrt[a + b] + 2*(3*a^2 - 4*a*b + 8*b^2)*ArcTan[Tanh[x/2]] +
a*(3*a - 4*b)*Sech[x]*Tanh[x] + 2*a^2*Sech[x]^3*Tanh[x])/(8*a^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(182\) vs. \(2(76)=152\).
time = 0.81, size = 183, normalized size = 2.03

method	result	size
default	\(-\frac {2 b^{3} \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )+2 \sqrt {a}}{2 \sqrt {b}}\right )}{2 \sqrt {a +b}\, \sqrt {b}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )-2 \sqrt {a}}{2 \sqrt {b}}\right )}{2 \sqrt {a +b}\, \sqrt {b}}\right )}{a^{3}}+\frac {\frac {2 \left (\left (-\frac {5}{8} a^{2}+\frac {1}{2} a b \right ) \left (\tanh ^{7}\left (\frac {x}{2}\right )\right )+\left (\frac {3}{8} a^{2}+\frac {1}{2} a b \right ) \left (\tanh ^{5}\left (\frac {x}{2}\right )\right )+\left (-\frac {3}{8} a^{2}-\frac {1}{2} a b \right ) \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )+\left (\frac {5}{8} a^{2}-\frac {1}{2} a b \right ) \tanh \left (\frac {x}{2}\right )\right )}{\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}+\frac {\left (3 a^{2}-4 a b +8 b^{2}\right ) \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{4}}{a^{3}}\)	\(183\)
risch	\(\frac {{\mathrm e}^{x} \left (3 a \,{\mathrm e}^{6 x}-4 \,{\mathrm e}^{6 x} b +11 a \,{\mathrm e}^{4 x}-4 \,{\mathrm e}^{4 x} b -11 \,{\mathrm e}^{2 x} a +4 b \,{\mathrm e}^{2 x}-3 a +4 b \right )}{4 \left (1+{\mathrm e}^{2 x}\right )^{4} a^{2}}+\frac {3 i \ln \left ({\mathrm e}^{x}+i\right )}{8 a}-\frac {i b \ln \left ({\mathrm e}^{x}+i\right )}{2 a^{2}}+\frac {i \ln \left ({\mathrm e}^{x}+i\right ) b^{2}}{a^{3}}-\frac {3 i \ln \left ({\mathrm e}^{x}-i\right )}{8 a}+\frac {i b \ln \left ({\mathrm e}^{x}-i\right )}{2 a^{2}}-\frac {i \ln \left ({\mathrm e}^{x}-i\right ) b^{2}}{a^{3}}+\frac {\sqrt {-b \left (a +b \right )}\, b^{2} \ln \left ({\mathrm e}^{2 x}-\frac {2 \sqrt {-b \left (a +b \right )}\, {\mathrm e}^{x}}{b}-1\right )}{2 \left (a +b \right ) a^{3}}-\frac {\sqrt {-b \left (a +b \right )}\, b^{2} \ln \left ({\mathrm e}^{2 x}+\frac {2 \sqrt {-b \left (a +b \right )}\, {\mathrm e}^{x}}{b}-1\right )}{2 \left (a +b \right ) a^{3}}\)	\(232\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^5/(a+b*cosh(x)^2),x,method=_RETURNVERBOSE)

[Out]

-2*b^3/a^3*(1/2/(a+b)^(1/2)/b^(1/2)*arctan(1/2*(2*(a+b)^(1/2)*tanh(1/2*x)+2*a^(1/2))/b^(1/2))+1/2/(a+b)^(1/2)/
b^(1/2)*arctan(1/2*(2*(a+b)^(1/2)*tanh(1/2*x)-2*a^(1/2))/b^(1/2)))+2/a^3*(((-5/8*a^2+1/2*a*b)*tanh(1/2*x)^7+(3
/8*a^2+1/2*a*b)*tanh(1/2*x)^5+(-3/8*a^2-1/2*a*b)*tanh(1/2*x)^3+(5/8*a^2-1/2*a*b)*tanh(1/2*x))/(tanh(1/2*x)^2+1
)^4+1/8*(3*a^2-4*a*b+8*b^2)*arctan(tanh(1/2*x)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^5/(a+b*cosh(x)^2),x, algorithm="maxima")

[Out]

1/4*((3*a - 4*b)*e^(7*x) + (11*a - 4*b)*e^(5*x) - (11*a - 4*b)*e^(3*x) - (3*a - 4*b)*e^x)/(a^2*e^(8*x) + 4*a^2
*e^(6*x) + 6*a^2*e^(4*x) + 4*a^2*e^(2*x) + a^2) + 1/4*(3*a^2 - 4*a*b + 8*b^2)*arctan(e^x)/a^3 - 32*integrate(1
/16*(b^3*e^(3*x) + b^3*e^x)/(a^3*b*e^(4*x) + a^3*b + 2*(2*a^4 + a^3*b)*e^(2*x)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1673 vs. \(2 (76) = 152\).
time = 0.46, size = 3239, normalized size = 35.99 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^5/(a+b*cosh(x)^2),x, algorithm="fricas")

[Out]

[1/4*((3*a^2 - 4*a*b)*cosh(x)^7 + 7*(3*a^2 - 4*a*b)*cosh(x)*sinh(x)^6 + (3*a^2 - 4*a*b)*sinh(x)^7 + (11*a^2 -
4*a*b)*cosh(x)^5 + (21*(3*a^2 - 4*a*b)*cosh(x)^2 + 11*a^2 - 4*a*b)*sinh(x)^5 + 5*(7*(3*a^2 - 4*a*b)*cosh(x)^3
+ (11*a^2 - 4*a*b)*cosh(x))*sinh(x)^4 - (11*a^2 - 4*a*b)*cosh(x)^3 + (35*(3*a^2 - 4*a*b)*cosh(x)^4 + 10*(11*a^
2 - 4*a*b)*cosh(x)^2 - 11*a^2 + 4*a*b)*sinh(x)^3 + (21*(3*a^2 - 4*a*b)*cosh(x)^5 + 10*(11*a^2 - 4*a*b)*cosh(x)
^3 - 3*(11*a^2 - 4*a*b)*cosh(x))*sinh(x)^2 + 2*(b^2*cosh(x)^8 + 8*b^2*cosh(x)*sinh(x)^7 + b^2*sinh(x)^8 + 4*b^
2*cosh(x)^6 + 4*(7*b^2*cosh(x)^2 + b^2)*sinh(x)^6 + 6*b^2*cosh(x)^4 + 8*(7*b^2*cosh(x)^3 + 3*b^2*cosh(x))*sinh
(x)^5 + 2*(35*b^2*cosh(x)^4 + 30*b^2*cosh(x)^2 + 3*b^2)*sinh(x)^4 + 4*b^2*cosh(x)^2 + 8*(7*b^2*cosh(x)^5 + 10*
b^2*cosh(x)^3 + 3*b^2*cosh(x))*sinh(x)^3 + 4*(7*b^2*cosh(x)^6 + 15*b^2*cosh(x)^4 + 9*b^2*cosh(x)^2 + b^2)*sinh
(x)^2 + b^2 + 8*(b^2*cosh(x)^7 + 3*b^2*cosh(x)^5 + 3*b^2*cosh(x)^3 + b^2*cosh(x))*sinh(x))*sqrt(-b/(a + b))*lo
g((b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 - 2*(2*a + 3*b)*cosh(x)^2 + 2*(3*b*cosh(x)^2 - 2*a - 3*b)
*sinh(x)^2 + 4*(b*cosh(x)^3 - (2*a + 3*b)*cosh(x))*sinh(x) - 4*((a + b)*cosh(x)^3 + 3*(a + b)*cosh(x)*sinh(x)^
2 + (a + b)*sinh(x)^3 - (a + b)*cosh(x) + (3*(a + b)*cosh(x)^2 - a - b)*sinh(x))*sqrt(-b/(a + b)) + b)/(b*cosh
(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 + 2*(2*a + b)*cosh(x)^2 + 2*(3*b*cosh(x)^2 + 2*a + b)*sinh(x)^2 +
4*(b*cosh(x)^3 + (2*a + b)*cosh(x))*sinh(x) + b)) + ((3*a^2 - 4*a*b + 8*b^2)*cosh(x)^8 + 8*(3*a^2 - 4*a*b + 8*
b^2)*cosh(x)*sinh(x)^7 + (3*a^2 - 4*a*b + 8*b^2)*sinh(x)^8 + 4*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^6 + 4*(7*(3*a^2
- 4*a*b + 8*b^2)*cosh(x)^2 + 3*a^2 - 4*a*b + 8*b^2)*sinh(x)^6 + 8*(7*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^3 + 3*(3
*a^2 - 4*a*b + 8*b^2)*cosh(x))*sinh(x)^5 + 6*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^4 + 2*(35*(3*a^2 - 4*a*b + 8*b^2)
*cosh(x)^4 + 30*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^2 + 9*a^2 - 12*a*b + 24*b^2)*sinh(x)^4 + 8*(7*(3*a^2 - 4*a*b +
8*b^2)*cosh(x)^5 + 10*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^3 + 3*(3*a^2 - 4*a*b + 8*b^2)*cosh(x))*sinh(x)^3 + 4*(3
*a^2 - 4*a*b + 8*b^2)*cosh(x)^2 + 4*(7*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^6 + 15*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^
4 + 9*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^2 + 3*a^2 - 4*a*b + 8*b^2)*sinh(x)^2 + 3*a^2 - 4*a*b + 8*b^2 + 8*((3*a^2
- 4*a*b + 8*b^2)*cosh(x)^7 + 3*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^5 + 3*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^3 + (3*a
^2 - 4*a*b + 8*b^2)*cosh(x))*sinh(x))*arctan(cosh(x) + sinh(x)) - (3*a^2 - 4*a*b)*cosh(x) + (7*(3*a^2 - 4*a*b)
*cosh(x)^6 + 5*(11*a^2 - 4*a*b)*cosh(x)^4 - 3*(11*a^2 - 4*a*b)*cosh(x)^2 - 3*a^2 + 4*a*b)*sinh(x))/(a^3*cosh(x
)^8 + 8*a^3*cosh(x)*sinh(x)^7 + a^3*sinh(x)^8 + 4*a^3*cosh(x)^6 + 6*a^3*cosh(x)^4 + 4*(7*a^3*cosh(x)^2 + a^3)*
sinh(x)^6 + 8*(7*a^3*cosh(x)^3 + 3*a^3*cosh(x))*sinh(x)^5 + 4*a^3*cosh(x)^2 + 2*(35*a^3*cosh(x)^4 + 30*a^3*cos
h(x)^2 + 3*a^3)*sinh(x)^4 + 8*(7*a^3*cosh(x)^5 + 10*a^3*cosh(x)^3 + 3*a^3*cosh(x))*sinh(x)^3 + a^3 + 4*(7*a^3*
cosh(x)^6 + 15*a^3*cosh(x)^4 + 9*a^3*cosh(x)^2 + a^3)*sinh(x)^2 + 8*(a^3*cosh(x)^7 + 3*a^3*cosh(x)^5 + 3*a^3*c
osh(x)^3 + a^3*cosh(x))*sinh(x)), 1/4*((3*a^2 - 4*a*b)*cosh(x)^7 + 7*(3*a^2 - 4*a*b)*cosh(x)*sinh(x)^6 + (3*a^
2 - 4*a*b)*sinh(x)^7 + (11*a^2 - 4*a*b)*cosh(x)^5 + (21*(3*a^2 - 4*a*b)*cosh(x)^2 + 11*a^2 - 4*a*b)*sinh(x)^5
+ 5*(7*(3*a^2 - 4*a*b)*cosh(x)^3 + (11*a^2 - 4*a*b)*cosh(x))*sinh(x)^4 - (11*a^2 - 4*a*b)*cosh(x)^3 + (35*(3*a
^2 - 4*a*b)*cosh(x)^4 + 10*(11*a^2 - 4*a*b)*cosh(x)^2 - 11*a^2 + 4*a*b)*sinh(x)^3 + (21*(3*a^2 - 4*a*b)*cosh(x
)^5 + 10*(11*a^2 - 4*a*b)*cosh(x)^3 - 3*(11*a^2 - 4*a*b)*cosh(x))*sinh(x)^2 - 4*(b^2*cosh(x)^8 + 8*b^2*cosh(x)
*sinh(x)^7 + b^2*sinh(x)^8 + 4*b^2*cosh(x)^6 + 4*(7*b^2*cosh(x)^2 + b^2)*sinh(x)^6 + 6*b^2*cosh(x)^4 + 8*(7*b^
2*cosh(x)^3 + 3*b^2*cosh(x))*sinh(x)^5 + 2*(35*b^2*cosh(x)^4 + 30*b^2*cosh(x)^2 + 3*b^2)*sinh(x)^4 + 4*b^2*cos
h(x)^2 + 8*(7*b^2*cosh(x)^5 + 10*b^2*cosh(x)^3 + 3*b^2*cosh(x))*sinh(x)^3 + 4*(7*b^2*cosh(x)^6 + 15*b^2*cosh(x
)^4 + 9*b^2*cosh(x)^2 + b^2)*sinh(x)^2 + b^2 + 8*(b^2*cosh(x)^7 + 3*b^2*cosh(x)^5 + 3*b^2*cosh(x)^3 + b^2*cosh
(x))*sinh(x))*sqrt(b/(a + b))*arctan(1/2*sqrt(b/(a + b))*(cosh(x) + sinh(x))) - 4*(b^2*cosh(x)^8 + 8*b^2*cosh(
x)*sinh(x)^7 + b^2*sinh(x)^8 + 4*b^2*cosh(x)^6 + 4*(7*b^2*cosh(x)^2 + b^2)*sinh(x)^6 + 6*b^2*cosh(x)^4 + 8*(7*
b^2*cosh(x)^3 + 3*b^2*cosh(x))*sinh(x)^5 + 2*(35*b^2*cosh(x)^4 + 30*b^2*cosh(x)^2 + 3*b^2)*sinh(x)^4 + 4*b^2*c
osh(x)^2 + 8*(7*b^2*cosh(x)^5 + 10*b^2*cosh(x)^3 + 3*b^2*cosh(x))*sinh(x)^3 + 4*(7*b^2*cosh(x)^6 + 15*b^2*cosh
(x)^4 + 9*b^2*cosh(x)^2 + b^2)*sinh(x)^2 + b^2 + 8*(b^2*cosh(x)^7 + 3*b^2*cosh(x)^5 + 3*b^2*cosh(x)^3 + b^2*co
sh(x))*sinh(x))*sqrt(b/(a + b))*arctan(1/2*(b*cosh(x)^3 + 3*b*cosh(x)*sinh(x)^2 + b*sinh(x)^3 + (4*a + 3*b)*co
sh(x) + (3*b*cosh(x)^2 + 4*a + 3*b)*sinh(x))*sqrt(b/(a + b))/b) + ((3*a^2 - 4*a*b + 8*b^2)*cosh(x)^8 + 8*(3*a^
2 - 4*a*b + 8*b^2)*cosh(x)*sinh(x)^7 + (3*a^2 - 4*a*b + 8*b^2)*sinh(x)^8 + 4*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^6
+ 4*(7*(3*a^2 - 4*a*b + 8*b^2)*cosh(x)^2 + 3*a...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {sech}^{5}{\left (x \right )}}{a + b \cosh ^{2}{\left (x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)**5/(a+b*cosh(x)**2),x)

[Out]

Integral(sech(x)**5/(a + b*cosh(x)**2), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^5/(a+b*cosh(x)^2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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Mupad [B]
time = 36.10, size = 1305, normalized size = 14.50 \begin {gather*} \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\left (243\,a^{12}\,\sqrt {a^6}+5024\,b^6\,{\left (a^6\right )}^{3/2}+18432\,b^{12}\,\sqrt {a^6}+6912\,a^2\,b^{10}\,\sqrt {a^6}+30720\,a^3\,b^9\,\sqrt {a^6}-26880\,a^4\,b^8\,\sqrt {a^6}+24192\,a^5\,b^7\,\sqrt {a^6}-13408\,a^7\,b^5\,\sqrt {a^6}+17160\,a^8\,b^4\,\sqrt {a^6}-9540\,a^9\,b^3\,\sqrt {a^6}+4563\,a^{10}\,b^2\,\sqrt {a^6}-9216\,a\,b^{11}\,\sqrt {a^6}-1134\,a^{11}\,b\,\sqrt {a^6}\right )}{81\,a^{13}\,\sqrt {9\,a^4-24\,a^3\,b+64\,a^2\,b^2-64\,a\,b^3+64\,b^4}-270\,a^{12}\,b\,\sqrt {9\,a^4-24\,a^3\,b+64\,a^2\,b^2-64\,a\,b^3+64\,b^4}+2304\,a^3\,b^{10}\,\sqrt {9\,a^4-24\,a^3\,b+64\,a^2\,b^2-64\,a\,b^3+64\,b^4}+3840\,a^6\,b^7\,\sqrt {9\,a^4-24\,a^3\,b+64\,a^2\,b^2-64\,a\,b^3+64\,b^4}-1440\,a^7\,b^6\,\sqrt {9\,a^4-24\,a^3\,b+64\,a^2\,b^2-64\,a\,b^3+64\,b^4}+864\,a^8\,b^5\,\sqrt {9\,a^4-24\,a^3\,b+64\,a^2\,b^2-64\,a\,b^3+64\,b^4}+1600\,a^9\,b^4\,\sqrt {9\,a^4-24\,a^3\,b+64\,a^2\,b^2-64\,a\,b^3+64\,b^4}-1200\,a^{10}\,b^3\,\sqrt {9\,a^4-24\,a^3\,b+64\,a^2\,b^2-64\,a\,b^3+64\,b^4}+945\,a^{11}\,b^2\,\sqrt {9\,a^4-24\,a^3\,b+64\,a^2\,b^2-64\,a\,b^3+64\,b^4}}\right )\,\sqrt {9\,a^4-24\,a^3\,b+64\,a^2\,b^2-64\,a\,b^3+64\,b^4}}{4\,\sqrt {a^6}}-\frac {6\,{\mathrm {e}}^x}{a\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )}+\frac {\sqrt {b^5}\,\left (2\,\mathrm {atan}\left (\left ({\mathrm {e}}^x\,\left (\frac {2\,\left (48\,b^8\,\sqrt {a^7+b\,a^6}+40\,a^3\,b^5\,\sqrt {a^7+b\,a^6}-15\,a^4\,b^4\,\sqrt {a^7+b\,a^6}+9\,a^5\,b^3\,\sqrt {a^7+b\,a^6}\right )}{a^{11}\,b\,\left (a+b\right )\,\left (a^2+b\,a\right )\,\sqrt {a^7+b\,a^6}\,\sqrt {b^5}\,\left (9\,a^6-6\,a^5\,b+25\,a^4\,b^2+40\,a^3\,b^3+48\,a\,b^5+48\,b^6\right )}-\frac {4\,\left (96\,a^4\,{\left (b^5\right )}^{3/2}+18\,a^9\,\sqrt {b^5}+80\,a^6\,b^3\,\sqrt {b^5}+50\,a^7\,b^2\,\sqrt {b^5}+96\,a^3\,b\,{\left (b^5\right )}^{3/2}-12\,a^8\,b\,\sqrt {b^5}\right )}{a^8\,b^4\,\left (a+b\right )\,\left (a^2+b\,a\right )\,\sqrt {a^6\,\left (a+b\right )}\,\sqrt {a^7+b\,a^6}\,\left (9\,a^5-15\,a^4\,b+40\,a^3\,b^2+48\,b^5\right )}\right )-\frac {2\,{\mathrm {e}}^{3\,x}\,\left (48\,b^8\,\sqrt {a^7+b\,a^6}+40\,a^3\,b^5\,\sqrt {a^7+b\,a^6}-15\,a^4\,b^4\,\sqrt {a^7+b\,a^6}+9\,a^5\,b^3\,\sqrt {a^7+b\,a^6}\right )}{a^{11}\,b\,\left (a+b\right )\,\left (a^2+b\,a\right )\,\sqrt {a^7+b\,a^6}\,\sqrt {b^5}\,\left (9\,a^6-6\,a^5\,b+25\,a^4\,b^2+40\,a^3\,b^3+48\,a\,b^5+48\,b^6\right )}\right )\,\left (\frac {a^{11}\,b\,\sqrt {a^7+b\,a^6}}{4}+\frac {a^9\,b^3\,\sqrt {a^7+b\,a^6}}{4}+\frac {a^{10}\,b^2\,\sqrt {a^7+b\,a^6}}{2}\right )\right )-2\,\mathrm {atan}\left (\frac {b^3\,{\mathrm {e}}^x\,\sqrt {a^6\,\left (a+b\right )}\,\left (9\,a^5-15\,a^4\,b+40\,a^3\,b^2+48\,b^5\right )}{2\,a^3\,\sqrt {b^5}\,\left (9\,a^6-6\,a^5\,b+25\,a^4\,b^2+40\,a^3\,b^3+48\,a\,b^5+48\,b^6\right )}\right )\right )}{2\,\sqrt {a^7+b\,a^6}}+\frac {4\,{\mathrm {e}}^x}{a\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}+\frac {{\mathrm {e}}^x\,\left (a+4\,b\right )}{2\,a^2\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )}-\frac {{\mathrm {e}}^x\,\left (4\,a\,b-3\,a^2\right )}{4\,a^3\,\left ({\mathrm {e}}^{2\,x}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(x)^5*(a + b*cosh(x)^2)),x)

[Out]

(atan((exp(x)*(243*a^12*(a^6)^(1/2) + 5024*b^6*(a^6)^(3/2) + 18432*b^12*(a^6)^(1/2) + 6912*a^2*b^10*(a^6)^(1/2
) + 30720*a^3*b^9*(a^6)^(1/2) - 26880*a^4*b^8*(a^6)^(1/2) + 24192*a^5*b^7*(a^6)^(1/2) - 13408*a^7*b^5*(a^6)^(1
/2) + 17160*a^8*b^4*(a^6)^(1/2) - 9540*a^9*b^3*(a^6)^(1/2) + 4563*a^10*b^2*(a^6)^(1/2) - 9216*a*b^11*(a^6)^(1/
2) - 1134*a^11*b*(a^6)^(1/2)))/(81*a^13*(9*a^4 - 24*a^3*b - 64*a*b^3 + 64*b^4 + 64*a^2*b^2)^(1/2) - 270*a^12*b
*(9*a^4 - 24*a^3*b - 64*a*b^3 + 64*b^4 + 64*a^2*b^2)^(1/2) + 2304*a^3*b^10*(9*a^4 - 24*a^3*b - 64*a*b^3 + 64*b
^4 + 64*a^2*b^2)^(1/2) + 3840*a^6*b^7*(9*a^4 - 24*a^3*b - 64*a*b^3 + 64*b^4 + 64*a^2*b^2)^(1/2) - 1440*a^7*b^6
*(9*a^4 - 24*a^3*b - 64*a*b^3 + 64*b^4 + 64*a^2*b^2)^(1/2) + 864*a^8*b^5*(9*a^4 - 24*a^3*b - 64*a*b^3 + 64*b^4
+ 64*a^2*b^2)^(1/2) + 1600*a^9*b^4*(9*a^4 - 24*a^3*b - 64*a*b^3 + 64*b^4 + 64*a^2*b^2)^(1/2) - 1200*a^10*b^3*
(9*a^4 - 24*a^3*b - 64*a*b^3 + 64*b^4 + 64*a^2*b^2)^(1/2) + 945*a^11*b^2*(9*a^4 - 24*a^3*b - 64*a*b^3 + 64*b^4
+ 64*a^2*b^2)^(1/2)))*(9*a^4 - 24*a^3*b - 64*a*b^3 + 64*b^4 + 64*a^2*b^2)^(1/2))/(4*(a^6)^(1/2)) - (6*exp(x))
/(a*(3*exp(2*x) + 3*exp(4*x) + exp(6*x) + 1)) + ((b^5)^(1/2)*(2*atan((exp(x)*((2*(48*b^8*(a^6*b + a^7)^(1/2) +
40*a^3*b^5*(a^6*b + a^7)^(1/2) - 15*a^4*b^4*(a^6*b + a^7)^(1/2) + 9*a^5*b^3*(a^6*b + a^7)^(1/2)))/(a^11*b*(a
+ b)*(a*b + a^2)*(a^6*b + a^7)^(1/2)*(b^5)^(1/2)*(48*a*b^5 - 6*a^5*b + 9*a^6 + 48*b^6 + 40*a^3*b^3 + 25*a^4*b^
2)) - (4*(96*a^4*(b^5)^(3/2) + 18*a^9*(b^5)^(1/2) + 80*a^6*b^3*(b^5)^(1/2) + 50*a^7*b^2*(b^5)^(1/2) + 96*a^3*b
*(b^5)^(3/2) - 12*a^8*b*(b^5)^(1/2)))/(a^8*b^4*(a + b)*(a*b + a^2)*(a^6*(a + b))^(1/2)*(a^6*b + a^7)^(1/2)*(9*
a^5 - 15*a^4*b + 48*b^5 + 40*a^3*b^2))) - (2*exp(3*x)*(48*b^8*(a^6*b + a^7)^(1/2) + 40*a^3*b^5*(a^6*b + a^7)^(
1/2) - 15*a^4*b^4*(a^6*b + a^7)^(1/2) + 9*a^5*b^3*(a^6*b + a^7)^(1/2)))/(a^11*b*(a + b)*(a*b + a^2)*(a^6*b + a
^7)^(1/2)*(b^5)^(1/2)*(48*a*b^5 - 6*a^5*b + 9*a^6 + 48*b^6 + 40*a^3*b^3 + 25*a^4*b^2)))*((a^11*b*(a^6*b + a^7)
^(1/2))/4 + (a^9*b^3*(a^6*b + a^7)^(1/2))/4 + (a^10*b^2*(a^6*b + a^7)^(1/2))/2)) - 2*atan((b^3*exp(x)*(a^6*(a
+ b))^(1/2)*(9*a^5 - 15*a^4*b + 48*b^5 + 40*a^3*b^2))/(2*a^3*(b^5)^(1/2)*(48*a*b^5 - 6*a^5*b + 9*a^6 + 48*b^6
+ 40*a^3*b^3 + 25*a^4*b^2)))))/(2*(a^6*b + a^7)^(1/2)) + (4*exp(x))/(a*(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) +
exp(8*x) + 1)) + (exp(x)*(a + 4*b))/(2*a^2*(2*exp(2*x) + exp(4*x) + 1)) - (exp(x)*(4*a*b - 3*a^2))/(4*a^3*(ex
p(2*x) + 1))

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