3.20 \(\int \frac {\cosh ^7(x)}{a+b \cosh ^2(x)} \, dx\)

Optimal. Leaf size=78 \[ -\frac {a^3 \tan ^{-1}\left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{b^{7/2} \sqrt {a+b}}+\frac {\left (a^2-a b+b^2\right ) \sinh (x)}{b^3}-\frac {(a-2 b) \sinh ^3(x)}{3 b^2}+\frac {\sinh ^5(x)}{5 b} \]

[Out]

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

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Rubi [A]  time = 0.09, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3186, 390, 205} \[ \frac {\left (a^2-a b+b^2\right ) \sinh (x)}{b^3}-\frac {a^3 \tan ^{-1}\left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{b^{7/2} \sqrt {a+b}}-\frac {(a-2 b) \sinh ^3(x)}{3 b^2}+\frac {\sinh ^5(x)}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]^7/(a + b*Cosh[x]^2),x]

[Out]

-((a^3*ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a + b]])/(b^(7/2)*Sqrt[a + b])) + ((a^2 - a*b + b^2)*Sinh[x])/b^3 - ((a -
2*b)*Sinh[x]^3)/(3*b^2) + Sinh[x]^5/(5*b)

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 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 3186

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 {\cosh ^7(x)}{a+b \cosh ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^3}{a+b+b x^2} \, dx,x,\sinh (x)\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {a^2-a b+b^2}{b^3}-\frac {(a-2 b) x^2}{b^2}+\frac {x^4}{b}-\frac {a^3}{b^3 \left (a+b+b x^2\right )}\right ) \, dx,x,\sinh (x)\right )\\ &=\frac {\left (a^2-a b+b^2\right ) \sinh (x)}{b^3}-\frac {(a-2 b) \sinh ^3(x)}{3 b^2}+\frac {\sinh ^5(x)}{5 b}-\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{a+b+b x^2} \, dx,x,\sinh (x)\right )}{b^3}\\ &=-\frac {a^3 \tan ^{-1}\left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{b^{7/2} \sqrt {a+b}}+\frac {\left (a^2-a b+b^2\right ) \sinh (x)}{b^3}-\frac {(a-2 b) \sinh ^3(x)}{3 b^2}+\frac {\sinh ^5(x)}{5 b}\\ \end {align*}

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Mathematica [A]  time = 0.29, size = 86, normalized size = 1.10 \[ \frac {a^3 \tan ^{-1}\left (\frac {\sqrt {a+b} \text {csch}(x)}{\sqrt {b}}\right )}{b^{7/2} \sqrt {a+b}}+\frac {\left (8 a^2-6 a b+5 b^2\right ) \sinh (x)}{8 b^3}-\frac {(4 a-5 b) \sinh (3 x)}{48 b^2}+\frac {\sinh (5 x)}{80 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]^7/(a + b*Cosh[x]^2),x]

[Out]

(a^3*ArcTan[(Sqrt[a + b]*Csch[x])/Sqrt[b]])/(b^(7/2)*Sqrt[a + b]) + ((8*a^2 - 6*a*b + 5*b^2)*Sinh[x])/(8*b^3)
- ((4*a - 5*b)*Sinh[3*x])/(48*b^2) + Sinh[5*x]/(80*b)

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fricas [B]  time = 0.52, size = 2508, normalized size = 32.15 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^7/(a+b*cosh(x)^2),x, algorithm="fricas")

[Out]

[1/480*(3*(a*b^3 + b^4)*cosh(x)^10 + 30*(a*b^3 + b^4)*cosh(x)*sinh(x)^9 + 3*(a*b^3 + b^4)*sinh(x)^10 - 5*(4*a^
2*b^2 - a*b^3 - 5*b^4)*cosh(x)^8 - 5*(4*a^2*b^2 - a*b^3 - 5*b^4 - 27*(a*b^3 + b^4)*cosh(x)^2)*sinh(x)^8 + 40*(
9*(a*b^3 + b^4)*cosh(x)^3 - (4*a^2*b^2 - a*b^3 - 5*b^4)*cosh(x))*sinh(x)^7 + 30*(8*a^3*b + 2*a^2*b^2 - a*b^3 +
5*b^4)*cosh(x)^6 + 10*(63*(a*b^3 + b^4)*cosh(x)^4 + 24*a^3*b + 6*a^2*b^2 - 3*a*b^3 + 15*b^4 - 14*(4*a^2*b^2 -
a*b^3 - 5*b^4)*cosh(x)^2)*sinh(x)^6 + 4*(189*(a*b^3 + b^4)*cosh(x)^5 - 70*(4*a^2*b^2 - a*b^3 - 5*b^4)*cosh(x)
^3 + 45*(8*a^3*b + 2*a^2*b^2 - a*b^3 + 5*b^4)*cosh(x))*sinh(x)^5 - 30*(8*a^3*b + 2*a^2*b^2 - a*b^3 + 5*b^4)*co
sh(x)^4 + 10*(63*(a*b^3 + b^4)*cosh(x)^6 - 35*(4*a^2*b^2 - a*b^3 - 5*b^4)*cosh(x)^4 - 24*a^3*b - 6*a^2*b^2 + 3
*a*b^3 - 15*b^4 + 45*(8*a^3*b + 2*a^2*b^2 - a*b^3 + 5*b^4)*cosh(x)^2)*sinh(x)^4 - 3*a*b^3 - 3*b^4 + 40*(9*(a*b
^3 + b^4)*cosh(x)^7 - 7*(4*a^2*b^2 - a*b^3 - 5*b^4)*cosh(x)^5 + 15*(8*a^3*b + 2*a^2*b^2 - a*b^3 + 5*b^4)*cosh(
x)^3 - 3*(8*a^3*b + 2*a^2*b^2 - a*b^3 + 5*b^4)*cosh(x))*sinh(x)^3 + 5*(4*a^2*b^2 - a*b^3 - 5*b^4)*cosh(x)^2 +
5*(27*(a*b^3 + b^4)*cosh(x)^8 - 28*(4*a^2*b^2 - a*b^3 - 5*b^4)*cosh(x)^6 + 90*(8*a^3*b + 2*a^2*b^2 - a*b^3 + 5
*b^4)*cosh(x)^4 + 4*a^2*b^2 - a*b^3 - 5*b^4 - 36*(8*a^3*b + 2*a^2*b^2 - a*b^3 + 5*b^4)*cosh(x)^2)*sinh(x)^2 -
240*(a^3*cosh(x)^5 + 5*a^3*cosh(x)^4*sinh(x) + 10*a^3*cosh(x)^3*sinh(x)^2 + 10*a^3*cosh(x)^2*sinh(x)^3 + 5*a^3
*cosh(x)*sinh(x)^4 + a^3*sinh(x)^5)*sqrt(-a*b - b^2)*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*(cosh(x)^3 + 3*cosh(x)*sinh(x)^2 + sinh(x)^3 + (3*cosh(x)^2 - 1)*sinh(x) - cosh(x))*sqrt(-a*b - b^2) +
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)*s
inh(x)^2 + 4*(b*cosh(x)^3 + (2*a + b)*cosh(x))*sinh(x) + b)) + 10*(3*(a*b^3 + b^4)*cosh(x)^9 - 4*(4*a^2*b^2 -
a*b^3 - 5*b^4)*cosh(x)^7 + 18*(8*a^3*b + 2*a^2*b^2 - a*b^3 + 5*b^4)*cosh(x)^5 - 12*(8*a^3*b + 2*a^2*b^2 - a*b^
3 + 5*b^4)*cosh(x)^3 + (4*a^2*b^2 - a*b^3 - 5*b^4)*cosh(x))*sinh(x))/((a*b^4 + b^5)*cosh(x)^5 + 5*(a*b^4 + b^5
)*cosh(x)^4*sinh(x) + 10*(a*b^4 + b^5)*cosh(x)^3*sinh(x)^2 + 10*(a*b^4 + b^5)*cosh(x)^2*sinh(x)^3 + 5*(a*b^4 +
b^5)*cosh(x)*sinh(x)^4 + (a*b^4 + b^5)*sinh(x)^5), 1/480*(3*(a*b^3 + b^4)*cosh(x)^10 + 30*(a*b^3 + b^4)*cosh(
x)*sinh(x)^9 + 3*(a*b^3 + b^4)*sinh(x)^10 - 5*(4*a^2*b^2 - a*b^3 - 5*b^4)*cosh(x)^8 - 5*(4*a^2*b^2 - a*b^3 - 5
*b^4 - 27*(a*b^3 + b^4)*cosh(x)^2)*sinh(x)^8 + 40*(9*(a*b^3 + b^4)*cosh(x)^3 - (4*a^2*b^2 - a*b^3 - 5*b^4)*cos
h(x))*sinh(x)^7 + 30*(8*a^3*b + 2*a^2*b^2 - a*b^3 + 5*b^4)*cosh(x)^6 + 10*(63*(a*b^3 + b^4)*cosh(x)^4 + 24*a^3
*b + 6*a^2*b^2 - 3*a*b^3 + 15*b^4 - 14*(4*a^2*b^2 - a*b^3 - 5*b^4)*cosh(x)^2)*sinh(x)^6 + 4*(189*(a*b^3 + b^4)
*cosh(x)^5 - 70*(4*a^2*b^2 - a*b^3 - 5*b^4)*cosh(x)^3 + 45*(8*a^3*b + 2*a^2*b^2 - a*b^3 + 5*b^4)*cosh(x))*sinh
(x)^5 - 30*(8*a^3*b + 2*a^2*b^2 - a*b^3 + 5*b^4)*cosh(x)^4 + 10*(63*(a*b^3 + b^4)*cosh(x)^6 - 35*(4*a^2*b^2 -
a*b^3 - 5*b^4)*cosh(x)^4 - 24*a^3*b - 6*a^2*b^2 + 3*a*b^3 - 15*b^4 + 45*(8*a^3*b + 2*a^2*b^2 - a*b^3 + 5*b^4)*
cosh(x)^2)*sinh(x)^4 - 3*a*b^3 - 3*b^4 + 40*(9*(a*b^3 + b^4)*cosh(x)^7 - 7*(4*a^2*b^2 - a*b^3 - 5*b^4)*cosh(x)
^5 + 15*(8*a^3*b + 2*a^2*b^2 - a*b^3 + 5*b^4)*cosh(x)^3 - 3*(8*a^3*b + 2*a^2*b^2 - a*b^3 + 5*b^4)*cosh(x))*sin
h(x)^3 + 5*(4*a^2*b^2 - a*b^3 - 5*b^4)*cosh(x)^2 + 5*(27*(a*b^3 + b^4)*cosh(x)^8 - 28*(4*a^2*b^2 - a*b^3 - 5*b
^4)*cosh(x)^6 + 90*(8*a^3*b + 2*a^2*b^2 - a*b^3 + 5*b^4)*cosh(x)^4 + 4*a^2*b^2 - a*b^3 - 5*b^4 - 36*(8*a^3*b +
2*a^2*b^2 - a*b^3 + 5*b^4)*cosh(x)^2)*sinh(x)^2 - 480*(a^3*cosh(x)^5 + 5*a^3*cosh(x)^4*sinh(x) + 10*a^3*cosh(
x)^3*sinh(x)^2 + 10*a^3*cosh(x)^2*sinh(x)^3 + 5*a^3*cosh(x)*sinh(x)^4 + a^3*sinh(x)^5)*sqrt(a*b + b^2)*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)*sin
h(x))/sqrt(a*b + b^2)) - 480*(a^3*cosh(x)^5 + 5*a^3*cosh(x)^4*sinh(x) + 10*a^3*cosh(x)^3*sinh(x)^2 + 10*a^3*co
sh(x)^2*sinh(x)^3 + 5*a^3*cosh(x)*sinh(x)^4 + a^3*sinh(x)^5)*sqrt(a*b + b^2)*arctan(1/2*sqrt(a*b + b^2)*(cosh(
x) + sinh(x))/(a + b)) + 10*(3*(a*b^3 + b^4)*cosh(x)^9 - 4*(4*a^2*b^2 - a*b^3 - 5*b^4)*cosh(x)^7 + 18*(8*a^3*b
+ 2*a^2*b^2 - a*b^3 + 5*b^4)*cosh(x)^5 - 12*(8*a^3*b + 2*a^2*b^2 - a*b^3 + 5*b^4)*cosh(x)^3 + (4*a^2*b^2 - a*
b^3 - 5*b^4)*cosh(x))*sinh(x))/((a*b^4 + b^5)*cosh(x)^5 + 5*(a*b^4 + b^5)*cosh(x)^4*sinh(x) + 10*(a*b^4 + b^5)
*cosh(x)^3*sinh(x)^2 + 10*(a*b^4 + b^5)*cosh(x)^2*sinh(x)^3 + 5*(a*b^4 + b^5)*cosh(x)*sinh(x)^4 + (a*b^4 + b^5
)*sinh(x)^5)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^7/(a+b*cosh(x)^2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [a,b]=[-54,60]Warning, need to choose a branch for the root of a polynomial with parameters. Thi
s might be wrong.The choice was done assuming [a,b]=[-64,24]Undef/Unsigned Inf encountered in limitLimit: Max
order reached or unable to make series expansion Error: Bad Argument Value

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maple [B]  time = 0.12, size = 317, normalized size = 4.06 \[ -\frac {1}{2 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}-\frac {1}{5 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{5}}+\frac {a}{2 b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {7}{8 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {11}{12 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {a}{3 b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {a^{2}}{b^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {a}{b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {1}{b \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {1}{5 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {1}{2 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {a}{2 b^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {7}{8 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {11}{12 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {a}{3 b^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {a^{2}}{b^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {a}{b^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {1}{b \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {a^{3} \arctan \left (\frac {-2 \sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )+2 \sqrt {a}}{2 \sqrt {b}}\right )}{b^{\frac {7}{2}} \sqrt {a +b}}-\frac {a^{3} \arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )+2 \sqrt {a}}{2 \sqrt {b}}\right )}{b^{\frac {7}{2}} \sqrt {a +b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^7/(a+b*cosh(x)^2),x)

[Out]

-1/2/b/(tanh(1/2*x)-1)^4-1/5/b/(tanh(1/2*x)-1)^5+1/2/b^2/(tanh(1/2*x)-1)^2*a-7/8/b/(tanh(1/2*x)-1)^2-11/12/b/(
tanh(1/2*x)-1)^3+1/3/b^2/(tanh(1/2*x)-1)^3*a-1/b^3/(tanh(1/2*x)-1)*a^2+1/b^2/(tanh(1/2*x)-1)*a-1/b/(tanh(1/2*x
)-1)-1/5/b/(tanh(1/2*x)+1)^5+1/2/b/(tanh(1/2*x)+1)^4-1/2/b^2/(tanh(1/2*x)+1)^2*a+7/8/b/(tanh(1/2*x)+1)^2-11/12
/b/(tanh(1/2*x)+1)^3+1/3/b^2/(tanh(1/2*x)+1)^3*a-1/b^3/(tanh(1/2*x)+1)*a^2+1/b^2/(tanh(1/2*x)+1)*a-1/b/(tanh(1
/2*x)+1)+a^3/b^(7/2)/(a+b)^(1/2)*arctan(1/2*(-2*(a+b)^(1/2)*tanh(1/2*x)+2*a^(1/2))/b^(1/2))-a^3/b^(7/2)/(a+b)^
(1/2)*arctan(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 \[ \frac {{\left (3 \, b^{2} e^{\left (10 \, x\right )} - 3 \, b^{2} - 5 \, {\left (4 \, a b - 5 \, b^{2}\right )} e^{\left (8 \, x\right )} + 30 \, {\left (8 \, a^{2} - 6 \, a b + 5 \, b^{2}\right )} e^{\left (6 \, x\right )} - 30 \, {\left (8 \, a^{2} - 6 \, a b + 5 \, b^{2}\right )} e^{\left (4 \, x\right )} + 5 \, {\left (4 \, a b - 5 \, b^{2}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-5 \, x\right )}}{480 \, b^{3}} - \frac {1}{128} \, \int \frac {256 \, {\left (a^{3} e^{\left (3 \, x\right )} + a^{3} e^{x}\right )}}{b^{4} e^{\left (4 \, x\right )} + b^{4} + 2 \, {\left (2 \, a b^{3} + b^{4}\right )} e^{\left (2 \, x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^7/(a+b*cosh(x)^2),x, algorithm="maxima")

[Out]

1/480*(3*b^2*e^(10*x) - 3*b^2 - 5*(4*a*b - 5*b^2)*e^(8*x) + 30*(8*a^2 - 6*a*b + 5*b^2)*e^(6*x) - 30*(8*a^2 - 6
*a*b + 5*b^2)*e^(4*x) + 5*(4*a*b - 5*b^2)*e^(2*x))*e^(-5*x)/b^3 - 1/128*integrate(256*(a^3*e^(3*x) + a^3*e^x)/
(b^4*e^(4*x) + b^4 + 2*(2*a*b^3 + b^4)*e^(2*x)), x)

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mupad [B]  time = 1.39, size = 293, normalized size = 3.76 \[ \frac {{\mathrm {e}}^{5\,x}}{160\,b}-\frac {{\mathrm {e}}^{-5\,x}}{160\,b}-\frac {{\mathrm {e}}^{-x}\,\left (8\,a^2-6\,a\,b+5\,b^2\right )}{16\,b^3}+\frac {\left (2\,\mathrm {atan}\left (\frac {\left (b^9\,\sqrt {b^8+a\,b^7}+a\,b^8\,\sqrt {b^8+a\,b^7}\right )\,\left ({\mathrm {e}}^x\,\left (\frac {2\,a^7}{b^{11}\,{\left (a+b\right )}^2\,\sqrt {a^6}}-\frac {4\,\left (2\,a^4\,b^4\,\sqrt {a^6}+2\,a^5\,b^3\,\sqrt {a^6}\right )}{a^3\,b^8\,\left (a+b\right )\,\sqrt {b^7\,\left (a+b\right )}\,\sqrt {b^8+a\,b^7}}\right )-\frac {2\,a^7\,{\mathrm {e}}^{3\,x}}{b^{11}\,{\left (a+b\right )}^2\,\sqrt {a^6}}\right )}{4\,a^4}\right )-2\,\mathrm {atan}\left (\frac {a^3\,{\mathrm {e}}^x\,\sqrt {b^7\,\left (a+b\right )}}{2\,b^3\,\left (a+b\right )\,\sqrt {a^6}}\right )\right )\,\sqrt {a^6}}{2\,\sqrt {b^8+a\,b^7}}+\frac {{\mathrm {e}}^{-3\,x}\,\left (4\,a-5\,b\right )}{96\,b^2}-\frac {{\mathrm {e}}^{3\,x}\,\left (4\,a-5\,b\right )}{96\,b^2}+\frac {{\mathrm {e}}^x\,\left (8\,a^2-6\,a\,b+5\,b^2\right )}{16\,b^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^7/(a + b*cosh(x)^2),x)

[Out]

exp(5*x)/(160*b) - exp(-5*x)/(160*b) - (exp(-x)*(8*a^2 - 6*a*b + 5*b^2))/(16*b^3) + ((2*atan(((b^9*(a*b^7 + b^
8)^(1/2) + a*b^8*(a*b^7 + b^8)^(1/2))*(exp(x)*((2*a^7)/(b^11*(a + b)^2*(a^6)^(1/2)) - (4*(2*a^4*b^4*(a^6)^(1/2
) + 2*a^5*b^3*(a^6)^(1/2)))/(a^3*b^8*(a + b)*(b^7*(a + b))^(1/2)*(a*b^7 + b^8)^(1/2))) - (2*a^7*exp(3*x))/(b^1
1*(a + b)^2*(a^6)^(1/2))))/(4*a^4)) - 2*atan((a^3*exp(x)*(b^7*(a + b))^(1/2))/(2*b^3*(a + b)*(a^6)^(1/2))))*(a
^6)^(1/2))/(2*(a*b^7 + b^8)^(1/2)) + (exp(-3*x)*(4*a - 5*b))/(96*b^2) - (exp(3*x)*(4*a - 5*b))/(96*b^2) + (exp
(x)*(8*a^2 - 6*a*b + 5*b^2))/(16*b^3)

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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(cosh(x)**7/(a+b*cosh(x)**2),x)

[Out]

Timed out

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