3.1.30 \(\int \frac {\text {sech}^3(x)}{a+b \cosh ^2(x)} \, dx\) [30]

Optimal. Leaf size=59 \[ \frac {(a-2 b) \text {ArcTan}(\sinh (x))}{2 a^2}+\frac {b^{3/2} \text {ArcTan}\left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{a^2 \sqrt {a+b}}+\frac {\text {sech}(x) \tanh (x)}{2 a} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3265, 425, 536, 209, 211} \begin {gather*} \frac {b^{3/2} \text {ArcTan}\left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{a^2 \sqrt {a+b}}+\frac {(a-2 b) \text {ArcTan}(\sinh (x))}{2 a^2}+\frac {\tanh (x) \text {sech}(x)}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sech[x]^3/(a + b*Cosh[x]^2),x]

[Out]

((a - 2*b)*ArcTan[Sinh[x]])/(2*a^2) + (b^(3/2)*ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a + b]])/(a^2*Sqrt[a + b]) + (Sec
h[x]*Tanh[x])/(2*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 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}^3(x)}{a+b \cosh ^2(x)} \, dx &=\text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^2 \left (a+b+b x^2\right )} \, dx,x,\sinh (x)\right )\\ &=\frac {\text {sech}(x) \tanh (x)}{2 a}-\frac {\text {Subst}\left (\int \frac {-a+b-b x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\sinh (x)\right )}{2 a}\\ &=\frac {\text {sech}(x) \tanh (x)}{2 a}+\frac {(a-2 b) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (x)\right )}{2 a^2}+\frac {b^2 \text {Subst}\left (\int \frac {1}{a+b+b x^2} \, dx,x,\sinh (x)\right )}{a^2}\\ &=\frac {(a-2 b) \tan ^{-1}(\sinh (x))}{2 a^2}+\frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{a^2 \sqrt {a+b}}+\frac {\text {sech}(x) \tanh (x)}{2 a}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 58, normalized size = 0.98 \begin {gather*} \frac {-\frac {2 b^{3/2} \text {ArcTan}\left (\frac {\sqrt {a+b} \text {csch}(x)}{\sqrt {b}}\right )}{\sqrt {a+b}}+2 (a-2 b) \text {ArcTan}\left (\tanh \left (\frac {x}{2}\right )\right )+a \text {sech}(x) \tanh (x)}{2 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]^3/(a + b*Cosh[x]^2),x]

[Out]

((-2*b^(3/2)*ArcTan[(Sqrt[a + b]*Csch[x])/Sqrt[b]])/Sqrt[a + b] + 2*(a - 2*b)*ArcTan[Tanh[x/2]] + a*Sech[x]*Ta
nh[x])/(2*a^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(122\) vs. \(2(47)=94\).
time = 0.82, size = 123, normalized size = 2.08

method	result	size
default	\(\frac {\frac {2 \left (-\frac {a \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{2}+\frac {a \tanh \left (\frac {x}{2}\right )}{2}\right )}{\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}+\left (-2 b +a \right ) \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{a^{2}}+\frac {2 b^{2} \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )+2 \sqrt {a}}{2 \sqrt {b}}\right )}{2 \sqrt {a +b}\, \sqrt {b}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )-2 \sqrt {a}}{2 \sqrt {b}}\right )}{2 \sqrt {a +b}\, \sqrt {b}}\right )}{a^{2}}\)	\(123\)
risch	\(\frac {{\mathrm e}^{x} \left ({\mathrm e}^{2 x}-1\right )}{\left (1+{\mathrm e}^{2 x}\right )^{2} a}-\frac {i b \ln \left ({\mathrm e}^{x}+i\right )}{a^{2}}+\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{2 a}+\frac {i b \ln \left ({\mathrm e}^{x}-i\right )}{a^{2}}-\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{2 a}+\frac {\sqrt {-b \left (a +b \right )}\, b \ln \left ({\mathrm e}^{2 x}+\frac {2 \sqrt {-b \left (a +b \right )}\, {\mathrm e}^{x}}{b}-1\right )}{2 \left (a +b \right ) a^{2}}-\frac {\sqrt {-b \left (a +b \right )}\, b \ln \left ({\mathrm e}^{2 x}-\frac {2 \sqrt {-b \left (a +b \right )}\, {\mathrm e}^{x}}{b}-1\right )}{2 \left (a +b \right ) a^{2}}\)	\(154\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^3/(a+b*cosh(x)^2),x,method=_RETURNVERBOSE)

[Out]

2/a^2*((-1/2*a*tanh(1/2*x)^3+1/2*a*tanh(1/2*x))/(tanh(1/2*x)^2+1)^2+1/2*(-2*b+a)*arctan(tanh(1/2*x)))+2*b^2/a^
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))+1/2/(a+b)^(1/2)/b^(1/2)*a
rctan(1/2*(2*(a+b)^(1/2)*tanh(1/2*x)-2*a^(1/2))/b^(1/2)))

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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)^3/(a+b*cosh(x)^2),x, algorithm="maxima")

[Out]

(e^(3*x) - e^x)/(a*e^(4*x) + 2*a*e^(2*x) + a) + (a - 2*b)*arctan(e^x)/a^2 + 8*integrate(1/4*(b^2*e^(3*x) + b^2
*e^x)/(a^2*b*e^(4*x) + a^2*b + 2*(2*a^3 + a^2*b)*e^(2*x)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (47) = 94\).
time = 0.42, size = 963, normalized size = 16.32 \begin {gather*} \left [\frac {2 \, a \cosh \left (x\right )^{3} + 6 \, a \cosh \left (x\right ) \sinh \left (x\right )^{2} + 2 \, a \sinh \left (x\right )^{3} + {\left (b \cosh \left (x\right )^{4} + 4 \, b \cosh \left (x\right ) \sinh \left (x\right )^{3} + b \sinh \left (x\right )^{4} + 2 \, b \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b \cosh \left (x\right )^{2} + b\right )} \sinh \left (x\right )^{2} + 4 \, {\left (b \cosh \left (x\right )^{3} + b \cosh \left (x\right )\right )} \sinh \left (x\right ) + b\right )} \sqrt {-\frac {b}{a + b}} \log \left (\frac {b \cosh \left (x\right )^{4} + 4 \, b \cosh \left (x\right ) \sinh \left (x\right )^{3} + b \sinh \left (x\right )^{4} - 2 \, {\left (2 \, a + 3 \, b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b \cosh \left (x\right )^{2} - 2 \, a - 3 \, b\right )} \sinh \left (x\right )^{2} + 4 \, {\left (b \cosh \left (x\right )^{3} - {\left (2 \, a + 3 \, b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 4 \, {\left ({\left (a + b\right )} \cosh \left (x\right )^{3} + 3 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + {\left (a + b\right )} \sinh \left (x\right )^{3} - {\left (a + b\right )} \cosh \left (x\right ) + {\left (3 \, {\left (a + b\right )} \cosh \left (x\right )^{2} - a - b\right )} \sinh \left (x\right )\right )} \sqrt {-\frac {b}{a + b}} + b}{b \cosh \left (x\right )^{4} + 4 \, b \cosh \left (x\right ) \sinh \left (x\right )^{3} + b \sinh \left (x\right )^{4} + 2 \, {\left (2 \, a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b \cosh \left (x\right )^{2} + 2 \, a + b\right )} \sinh \left (x\right )^{2} + 4 \, {\left (b \cosh \left (x\right )^{3} + {\left (2 \, a + b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + b}\right ) + 2 \, {\left ({\left (a - 2 \, b\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a - 2 \, b\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a - 2 \, b\right )} \sinh \left (x\right )^{4} + 2 \, {\left (a - 2 \, b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, {\left (a - 2 \, b\right )} \cosh \left (x\right )^{2} + a - 2 \, b\right )} \sinh \left (x\right )^{2} + 4 \, {\left ({\left (a - 2 \, b\right )} \cosh \left (x\right )^{3} + {\left (a - 2 \, b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + a - 2 \, b\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - 2 \, a \cosh \left (x\right ) + 2 \, {\left (3 \, a \cosh \left (x\right )^{2} - a\right )} \sinh \left (x\right )}{2 \, {\left (a^{2} \cosh \left (x\right )^{4} + 4 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + a^{2} \sinh \left (x\right )^{4} + 2 \, a^{2} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, a^{2} \cosh \left (x\right )^{2} + a^{2}\right )} \sinh \left (x\right )^{2} + a^{2} + 4 \, {\left (a^{2} \cosh \left (x\right )^{3} + a^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}}, \frac {a \cosh \left (x\right )^{3} + 3 \, a \cosh \left (x\right ) \sinh \left (x\right )^{2} + a \sinh \left (x\right )^{3} + {\left (b \cosh \left (x\right )^{4} + 4 \, b \cosh \left (x\right ) \sinh \left (x\right )^{3} + b \sinh \left (x\right )^{4} + 2 \, b \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b \cosh \left (x\right )^{2} + b\right )} \sinh \left (x\right )^{2} + 4 \, {\left (b \cosh \left (x\right )^{3} + b \cosh \left (x\right )\right )} \sinh \left (x\right ) + b\right )} \sqrt {\frac {b}{a + b}} \arctan \left (\frac {1}{2} \, \sqrt {\frac {b}{a + b}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}\right ) + {\left (b \cosh \left (x\right )^{4} + 4 \, b \cosh \left (x\right ) \sinh \left (x\right )^{3} + b \sinh \left (x\right )^{4} + 2 \, b \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b \cosh \left (x\right )^{2} + b\right )} \sinh \left (x\right )^{2} + 4 \, {\left (b \cosh \left (x\right )^{3} + b \cosh \left (x\right )\right )} \sinh \left (x\right ) + b\right )} \sqrt {\frac {b}{a + b}} \arctan \left (\frac {{\left (b \cosh \left (x\right )^{3} + 3 \, b \cosh \left (x\right ) \sinh \left (x\right )^{2} + b \sinh \left (x\right )^{3} + {\left (4 \, a + 3 \, b\right )} \cosh \left (x\right ) + {\left (3 \, b \cosh \left (x\right )^{2} + 4 \, a + 3 \, b\right )} \sinh \left (x\right )\right )} \sqrt {\frac {b}{a + b}}}{2 \, b}\right ) + {\left ({\left (a - 2 \, b\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a - 2 \, b\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a - 2 \, b\right )} \sinh \left (x\right )^{4} + 2 \, {\left (a - 2 \, b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, {\left (a - 2 \, b\right )} \cosh \left (x\right )^{2} + a - 2 \, b\right )} \sinh \left (x\right )^{2} + 4 \, {\left ({\left (a - 2 \, b\right )} \cosh \left (x\right )^{3} + {\left (a - 2 \, b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + a - 2 \, b\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - a \cosh \left (x\right ) + {\left (3 \, a \cosh \left (x\right )^{2} - a\right )} \sinh \left (x\right )}{a^{2} \cosh \left (x\right )^{4} + 4 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + a^{2} \sinh \left (x\right )^{4} + 2 \, a^{2} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, a^{2} \cosh \left (x\right )^{2} + a^{2}\right )} \sinh \left (x\right )^{2} + a^{2} + 4 \, {\left (a^{2} \cosh \left (x\right )^{3} + a^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^3/(a+b*cosh(x)^2),x, algorithm="fricas")

[Out]

[1/2*(2*a*cosh(x)^3 + 6*a*cosh(x)*sinh(x)^2 + 2*a*sinh(x)^3 + (b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)
^4 + 2*b*cosh(x)^2 + 2*(3*b*cosh(x)^2 + b)*sinh(x)^2 + 4*(b*cosh(x)^3 + b*cosh(x))*sinh(x) + b)*sqrt(-b/(a + b
))*log((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)*sin
h(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)) + 2*((a - 2*b)*cosh(x)^4 + 4*(a - 2*b)*cosh(x)*sinh(x)^
3 + (a - 2*b)*sinh(x)^4 + 2*(a - 2*b)*cosh(x)^2 + 2*(3*(a - 2*b)*cosh(x)^2 + a - 2*b)*sinh(x)^2 + 4*((a - 2*b)
*cosh(x)^3 + (a - 2*b)*cosh(x))*sinh(x) + a - 2*b)*arctan(cosh(x) + sinh(x)) - 2*a*cosh(x) + 2*(3*a*cosh(x)^2
- a)*sinh(x))/(a^2*cosh(x)^4 + 4*a^2*cosh(x)*sinh(x)^3 + a^2*sinh(x)^4 + 2*a^2*cosh(x)^2 + 2*(3*a^2*cosh(x)^2
+ a^2)*sinh(x)^2 + a^2 + 4*(a^2*cosh(x)^3 + a^2*cosh(x))*sinh(x)), (a*cosh(x)^3 + 3*a*cosh(x)*sinh(x)^2 + a*si
nh(x)^3 + (b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 + 2*b*cosh(x)^2 + 2*(3*b*cosh(x)^2 + b)*sinh(x)^2
+ 4*(b*cosh(x)^3 + b*cosh(x))*sinh(x) + b)*sqrt(b/(a + b))*arctan(1/2*sqrt(b/(a + b))*(cosh(x) + sinh(x))) +
(b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 + 2*b*cosh(x)^2 + 2*(3*b*cosh(x)^2 + b)*sinh(x)^2 + 4*(b*co
sh(x)^3 + b*cosh(x))*sinh(x) + b)*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)*cosh(x) + (3*b*cosh(x)^2 + 4*a + 3*b)*sinh(x))*sqrt(b/(a + b))/b) + ((a - 2*b)*cosh(x)^4 + 4*(
a - 2*b)*cosh(x)*sinh(x)^3 + (a - 2*b)*sinh(x)^4 + 2*(a - 2*b)*cosh(x)^2 + 2*(3*(a - 2*b)*cosh(x)^2 + a - 2*b)
*sinh(x)^2 + 4*((a - 2*b)*cosh(x)^3 + (a - 2*b)*cosh(x))*sinh(x) + a - 2*b)*arctan(cosh(x) + sinh(x)) - a*cosh
(x) + (3*a*cosh(x)^2 - a)*sinh(x))/(a^2*cosh(x)^4 + 4*a^2*cosh(x)*sinh(x)^3 + a^2*sinh(x)^4 + 2*a^2*cosh(x)^2
+ 2*(3*a^2*cosh(x)^2 + a^2)*sinh(x)^2 + a^2 + 4*(a^2*cosh(x)^3 + a^2*cosh(x))*sinh(x))]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {sech}^{3}{\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)**3/(a+b*cosh(x)**2),x)

[Out]

Integral(sech(x)**3/(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)^3/(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 = 1.56, size = 447, normalized size = 7.58 \begin {gather*} \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\left (a^7\,{\left (a^4\right )}^{3/2}-12\,b^3\,{\left (a^4\right )}^{5/2}-18\,b^7\,{\left (a^4\right )}^{3/2}+36\,a^2\,b^5\,{\left (a^4\right )}^{3/2}-30\,a^3\,b^4\,{\left (a^4\right )}^{3/2}+21\,a^5\,b^2\,{\left (a^4\right )}^{3/2}+9\,a\,b^6\,{\left (a^4\right )}^{3/2}-8\,a^6\,b\,{\left (a^4\right )}^{3/2}\right )}{a^{12}\,\sqrt {a^2-4\,a\,b+4\,b^2}-6\,a^{11}\,b\,\sqrt {a^2-4\,a\,b+4\,b^2}+9\,a^6\,b^6\,\sqrt {a^2-4\,a\,b+4\,b^2}-18\,a^8\,b^4\,\sqrt {a^2-4\,a\,b+4\,b^2}+6\,a^9\,b^3\,\sqrt {a^2-4\,a\,b+4\,b^2}+9\,a^{10}\,b^2\,\sqrt {a^2-4\,a\,b+4\,b^2}}\right )\,\sqrt {a^2-4\,a\,b+4\,b^2}}{\sqrt {a^4}}-\frac {2\,{\mathrm {e}}^x}{a\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )}+\frac {{\mathrm {e}}^x}{a\,\left ({\mathrm {e}}^{2\,x}+1\right )}-\frac {{\left (-b\right )}^{3/2}\,\ln \left (\frac {64\,\left ({\mathrm {e}}^{2\,x}-1\right )\,\left (a^3-3\,a^2\,b+3\,b^3\right )}{a^5\,{\left (a+b\right )}^2}-\frac {128\,{\mathrm {e}}^x\,\left (a^3-3\,a^2\,b+3\,b^3\right )}{a^5\,\sqrt {-b}\,{\left (a+b\right )}^{3/2}}\right )}{2\,a^2\,\sqrt {a+b}}+\frac {{\left (-b\right )}^{3/2}\,\ln \left (\frac {64\,\left ({\mathrm {e}}^{2\,x}-1\right )\,\left (a^3-3\,a^2\,b+3\,b^3\right )}{a^5\,{\left (a+b\right )}^2}+\frac {128\,{\mathrm {e}}^x\,\left (a^3-3\,a^2\,b+3\,b^3\right )}{a^5\,\sqrt {-b}\,{\left (a+b\right )}^{3/2}}\right )}{2\,a^2\,\sqrt {a+b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(x)^3*(a + b*cosh(x)^2)),x)

[Out]

(atan((exp(x)*(a^7*(a^4)^(3/2) - 12*b^3*(a^4)^(5/2) - 18*b^7*(a^4)^(3/2) + 36*a^2*b^5*(a^4)^(3/2) - 30*a^3*b^4
*(a^4)^(3/2) + 21*a^5*b^2*(a^4)^(3/2) + 9*a*b^6*(a^4)^(3/2) - 8*a^6*b*(a^4)^(3/2)))/(a^12*(a^2 - 4*a*b + 4*b^2
)^(1/2) - 6*a^11*b*(a^2 - 4*a*b + 4*b^2)^(1/2) + 9*a^6*b^6*(a^2 - 4*a*b + 4*b^2)^(1/2) - 18*a^8*b^4*(a^2 - 4*a
*b + 4*b^2)^(1/2) + 6*a^9*b^3*(a^2 - 4*a*b + 4*b^2)^(1/2) + 9*a^10*b^2*(a^2 - 4*a*b + 4*b^2)^(1/2)))*(a^2 - 4*
a*b + 4*b^2)^(1/2))/(a^4)^(1/2) - (2*exp(x))/(a*(2*exp(2*x) + exp(4*x) + 1)) + exp(x)/(a*(exp(2*x) + 1)) - ((-
b)^(3/2)*log((64*(exp(2*x) - 1)*(a^3 - 3*a^2*b + 3*b^3))/(a^5*(a + b)^2) - (128*exp(x)*(a^3 - 3*a^2*b + 3*b^3)
)/(a^5*(-b)^(1/2)*(a + b)^(3/2))))/(2*a^2*(a + b)^(1/2)) + ((-b)^(3/2)*log((64*(exp(2*x) - 1)*(a^3 - 3*a^2*b +
3*b^3))/(a^5*(a + b)^2) + (128*exp(x)*(a^3 - 3*a^2*b + 3*b^3))/(a^5*(-b)^(1/2)*(a + b)^(3/2))))/(2*a^2*(a + b
)^(1/2))

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