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

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

[Out]

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

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Rubi [A]  time = 0.11, 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 = {3190, 390, 205} \[ \frac {\left (a^2+3 a b+3 b^2\right ) \cosh (x)}{b^3}-\frac {(a+3 b) \cosh ^3(x)}{3 b^2}-\frac {(a+b)^3 \tan ^{-1}\left (\frac {\sqrt {b} \cosh (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{7/2}}+\frac {\cosh ^5(x)}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]^7/(a + b*Cosh[x]^2),x]

[Out]

-(((a + b)^3*ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a]])/(Sqrt[a]*b^(7/2))) + ((a^2 + 3*a*b + 3*b^2)*Cosh[x])/b^3 - ((a
+ 3*b)*Cosh[x]^3)/(3*b^2) + Cosh[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 3190

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

Rubi steps

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

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Mathematica [C]  time = 0.25, size = 148, normalized size = 1.90 \[ \frac {\left (8 a^2+22 a b+19 b^2\right ) \cosh (x)}{8 b^3}-\frac {(a+b)^3 \tan ^{-1}\left (\frac {\sqrt {b}-i \sqrt {a+b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a}}\right )}{\sqrt {a} b^{7/2}}-\frac {(a+b)^3 \tan ^{-1}\left (\frac {\sqrt {b}+i \sqrt {a+b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a}}\right )}{\sqrt {a} b^{7/2}}-\frac {(4 a+9 b) \cosh (3 x)}{48 b^2}+\frac {\cosh (5 x)}{80 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((a + b)^3*ArcTan[(Sqrt[b] - I*Sqrt[a + b]*Tanh[x/2])/Sqrt[a]])/(Sqrt[a]*b^(7/2))) - ((a + b)^3*ArcTan[(Sqrt
[b] + I*Sqrt[a + b]*Tanh[x/2])/Sqrt[a]])/(Sqrt[a]*b^(7/2)) + ((8*a^2 + 22*a*b + 19*b^2)*Cosh[x])/(8*b^3) - ((4
*a + 9*b)*Cosh[3*x])/(48*b^2) + Cosh[5*x]/(80*b)

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fricas [B]  time = 0.45, size = 2346, normalized size = 30.08 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/480*(3*a*b^3*cosh(x)^10 + 30*a*b^3*cosh(x)*sinh(x)^9 + 3*a*b^3*sinh(x)^10 - 5*(4*a^2*b^2 + 9*a*b^3)*cosh(x)
^8 + 5*(27*a*b^3*cosh(x)^2 - 4*a^2*b^2 - 9*a*b^3)*sinh(x)^8 + 40*(9*a*b^3*cosh(x)^3 - (4*a^2*b^2 + 9*a*b^3)*co
sh(x))*sinh(x)^7 + 30*(8*a^3*b + 22*a^2*b^2 + 19*a*b^3)*cosh(x)^6 + 10*(63*a*b^3*cosh(x)^4 + 24*a^3*b + 66*a^2
*b^2 + 57*a*b^3 - 14*(4*a^2*b^2 + 9*a*b^3)*cosh(x)^2)*sinh(x)^6 + 4*(189*a*b^3*cosh(x)^5 - 70*(4*a^2*b^2 + 9*a
*b^3)*cosh(x)^3 + 45*(8*a^3*b + 22*a^2*b^2 + 19*a*b^3)*cosh(x))*sinh(x)^5 + 30*(8*a^3*b + 22*a^2*b^2 + 19*a*b^
3)*cosh(x)^4 + 10*(63*a*b^3*cosh(x)^6 - 35*(4*a^2*b^2 + 9*a*b^3)*cosh(x)^4 + 24*a^3*b + 66*a^2*b^2 + 57*a*b^3
+ 45*(8*a^3*b + 22*a^2*b^2 + 19*a*b^3)*cosh(x)^2)*sinh(x)^4 + 3*a*b^3 + 40*(9*a*b^3*cosh(x)^7 - 7*(4*a^2*b^2 +
9*a*b^3)*cosh(x)^5 + 15*(8*a^3*b + 22*a^2*b^2 + 19*a*b^3)*cosh(x)^3 + 3*(8*a^3*b + 22*a^2*b^2 + 19*a*b^3)*cos
h(x))*sinh(x)^3 - 5*(4*a^2*b^2 + 9*a*b^3)*cosh(x)^2 + 5*(27*a*b^3*cosh(x)^8 - 28*(4*a^2*b^2 + 9*a*b^3)*cosh(x)
^6 + 90*(8*a^3*b + 22*a^2*b^2 + 19*a*b^3)*cosh(x)^4 - 4*a^2*b^2 - 9*a*b^3 + 36*(8*a^3*b + 22*a^2*b^2 + 19*a*b^
3)*cosh(x)^2)*sinh(x)^2 - 240*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^5 + 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*c
osh(x)^4*sinh(x) + 10*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^3*sinh(x)^2 + 10*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)
*cosh(x)^2*sinh(x)^3 + 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)*sinh(x)^4 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*s
inh(x)^5)*sqrt(-a*b)*log((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*c
osh(x)^2 - 2*a + b)*sinh(x)^2 + 4*(b*cosh(x)^3 - (2*a - 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)/(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)) + 10*(3*a*b^3*cosh(x)^9 - 4*(4*a^2*b^2 + 9*a*b^3)*cosh(x)^7 + 18*(8*a^3*b + 22*a^2*b^2 + 19*a
*b^3)*cosh(x)^5 + 12*(8*a^3*b + 22*a^2*b^2 + 19*a*b^3)*cosh(x)^3 - (4*a^2*b^2 + 9*a*b^3)*cosh(x))*sinh(x))/(a*
b^4*cosh(x)^5 + 5*a*b^4*cosh(x)^4*sinh(x) + 10*a*b^4*cosh(x)^3*sinh(x)^2 + 10*a*b^4*cosh(x)^2*sinh(x)^3 + 5*a*
b^4*cosh(x)*sinh(x)^4 + a*b^4*sinh(x)^5), 1/480*(3*a*b^3*cosh(x)^10 + 30*a*b^3*cosh(x)*sinh(x)^9 + 3*a*b^3*sin
h(x)^10 - 5*(4*a^2*b^2 + 9*a*b^3)*cosh(x)^8 + 5*(27*a*b^3*cosh(x)^2 - 4*a^2*b^2 - 9*a*b^3)*sinh(x)^8 + 40*(9*a
*b^3*cosh(x)^3 - (4*a^2*b^2 + 9*a*b^3)*cosh(x))*sinh(x)^7 + 30*(8*a^3*b + 22*a^2*b^2 + 19*a*b^3)*cosh(x)^6 + 1
0*(63*a*b^3*cosh(x)^4 + 24*a^3*b + 66*a^2*b^2 + 57*a*b^3 - 14*(4*a^2*b^2 + 9*a*b^3)*cosh(x)^2)*sinh(x)^6 + 4*(
189*a*b^3*cosh(x)^5 - 70*(4*a^2*b^2 + 9*a*b^3)*cosh(x)^3 + 45*(8*a^3*b + 22*a^2*b^2 + 19*a*b^3)*cosh(x))*sinh(
x)^5 + 30*(8*a^3*b + 22*a^2*b^2 + 19*a*b^3)*cosh(x)^4 + 10*(63*a*b^3*cosh(x)^6 - 35*(4*a^2*b^2 + 9*a*b^3)*cosh
(x)^4 + 24*a^3*b + 66*a^2*b^2 + 57*a*b^3 + 45*(8*a^3*b + 22*a^2*b^2 + 19*a*b^3)*cosh(x)^2)*sinh(x)^4 + 3*a*b^3
+ 40*(9*a*b^3*cosh(x)^7 - 7*(4*a^2*b^2 + 9*a*b^3)*cosh(x)^5 + 15*(8*a^3*b + 22*a^2*b^2 + 19*a*b^3)*cosh(x)^3
+ 3*(8*a^3*b + 22*a^2*b^2 + 19*a*b^3)*cosh(x))*sinh(x)^3 - 5*(4*a^2*b^2 + 9*a*b^3)*cosh(x)^2 + 5*(27*a*b^3*cos
h(x)^8 - 28*(4*a^2*b^2 + 9*a*b^3)*cosh(x)^6 + 90*(8*a^3*b + 22*a^2*b^2 + 19*a*b^3)*cosh(x)^4 - 4*a^2*b^2 - 9*a
*b^3 + 36*(8*a^3*b + 22*a^2*b^2 + 19*a*b^3)*cosh(x)^2)*sinh(x)^2 - 480*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x
)^5 + 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^4*sinh(x) + 10*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^3*sinh(
x)^2 + 10*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^2*sinh(x)^3 + 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)*sinh
(x)^4 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sinh(x)^5)*sqrt(a*b)*arctan(1/2*sqrt(a*b)*(cosh(x) + sinh(x))/a) + 480
*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^5 + 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^4*sinh(x) + 10*(a^3 +
3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^3*sinh(x)^2 + 10*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)^2*sinh(x)^3 + 5*(a^3
+ 3*a^2*b + 3*a*b^2 + b^3)*cosh(x)*sinh(x)^4 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sinh(x)^5)*sqrt(a*b)*arctan(1/
2*(b*cosh(x)^3 + 3*b*cosh(x)*sinh(x)^2 + b*sinh(x)^3 + (4*a + b)*cosh(x) + (3*b*cosh(x)^2 + 4*a + b)*sinh(x))*
sqrt(a*b)/(a*b)) + 10*(3*a*b^3*cosh(x)^9 - 4*(4*a^2*b^2 + 9*a*b^3)*cosh(x)^7 + 18*(8*a^3*b + 22*a^2*b^2 + 19*a
*b^3)*cosh(x)^5 + 12*(8*a^3*b + 22*a^2*b^2 + 19*a*b^3)*cosh(x)^3 - (4*a^2*b^2 + 9*a*b^3)*cosh(x))*sinh(x))/(a*
b^4*cosh(x)^5 + 5*a*b^4*cosh(x)^4*sinh(x) + 10*a*b^4*cosh(x)^3*sinh(x)^2 + 10*a*b^4*cosh(x)^2*sinh(x)^3 + 5*a*
b^4*cosh(x)*sinh(x)^4 + a*b^4*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(sinh(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 = 0.09, size = 395, normalized size = 5.06 \[ -\frac {1}{5 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{5}}-\frac {1}{2 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {a}{2 b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {7}{8 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {a}{3 b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {1}{4 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {a^{2}}{b^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {5 a}{2 b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {15}{8 b \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {1}{5 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {1}{2 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {a}{2 b^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {7}{8 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {a}{3 b^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {1}{4 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {a^{2}}{b^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {5 a}{2 b^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {15}{8 b \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {\arctan \left (\frac {2 \left (a +b \right ) \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 a +2 b}{4 \sqrt {a b}}\right ) a^{3}}{b^{3} \sqrt {a b}}-\frac {3 \arctan \left (\frac {2 \left (a +b \right ) \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 a +2 b}{4 \sqrt {a b}}\right ) a^{2}}{b^{2} \sqrt {a b}}-\frac {3 \arctan \left (\frac {2 \left (a +b \right ) \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 a +2 b}{4 \sqrt {a b}}\right ) a}{b \sqrt {a b}}-\frac {\arctan \left (\frac {2 \left (a +b \right ) \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 a +2 b}{4 \sqrt {a b}}\right )}{\sqrt {a b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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+1/3/b^2/(
tanh(1/2*x)-1)^3*a+1/4/b/(tanh(1/2*x)-1)^3-1/b^3/(tanh(1/2*x)-1)*a^2-5/2/b^2/(tanh(1/2*x)-1)*a-15/8/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)^2-1/
3/b^2/(tanh(1/2*x)+1)^3*a-1/4/b/(tanh(1/2*x)+1)^3+1/b^3/(tanh(1/2*x)+1)*a^2+5/2/b^2/(tanh(1/2*x)+1)*a+15/8/b/(
tanh(1/2*x)+1)-1/b^3/(a*b)^(1/2)*arctan(1/4*(2*(a+b)*tanh(1/2*x)^2-2*a+2*b)/(a*b)^(1/2))*a^3-3/b^2/(a*b)^(1/2)
*arctan(1/4*(2*(a+b)*tanh(1/2*x)^2-2*a+2*b)/(a*b)^(1/2))*a^2-3/b/(a*b)^(1/2)*arctan(1/4*(2*(a+b)*tanh(1/2*x)^2
-2*a+2*b)/(a*b)^(1/2))*a-1/(a*b)^(1/2)*arctan(1/4*(2*(a+b)*tanh(1/2*x)^2-2*a+2*b)/(a*b)^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (3 \, b^{2} e^{\left (10 \, x\right )} + 3 \, b^{2} - 5 \, {\left (4 \, a b + 9 \, b^{2}\right )} e^{\left (8 \, x\right )} + 30 \, {\left (8 \, a^{2} + 22 \, a b + 19 \, b^{2}\right )} e^{\left (6 \, x\right )} + 30 \, {\left (8 \, a^{2} + 22 \, a b + 19 \, b^{2}\right )} e^{\left (4 \, x\right )} - 5 \, {\left (4 \, a b + 9 \, b^{2}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-5 \, x\right )}}{480 \, b^{3}} - \frac {1}{128} \, \int \frac {256 \, {\left ({\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} e^{\left (3 \, x\right )} - {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} e^{x}\right )}}{b^{4} e^{\left (4 \, x\right )} + b^{4} + 2 \, {\left (2 \, a b^{3} + b^{4}\right )} e^{\left (2 \, x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(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 + 9*b^2)*e^(8*x) + 30*(8*a^2 + 22*a*b + 19*b^2)*e^(6*x) + 30*(8*a^2 +
22*a*b + 19*b^2)*e^(4*x) - 5*(4*a*b + 9*b^2)*e^(2*x))*e^(-5*x)/b^3 - 1/128*integrate(256*((a^3 + 3*a^2*b + 3*
a*b^2 + b^3)*e^(3*x) - (a^3 + 3*a^2*b + 3*a*b^2 + b^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 = 1.55, size = 805, normalized size = 10.32 \[ \frac {{\mathrm {e}}^{-5\,x}}{160\,b}+\frac {{\mathrm {e}}^{5\,x}}{160\,b}+\frac {{\mathrm {e}}^{-x}\,\left (8\,a^2+22\,a\,b+19\,b^2\right )}{16\,b^3}-\frac {\left (2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,{\left (a+b\right )}^3\,\sqrt {a\,b^7}}{2\,a\,b^3\,\sqrt {{\left (a+b\right )}^6}}\right )-2\,\mathrm {atan}\left (\frac {2\,{\mathrm {e}}^{3\,x}\,\left (a^7\,\sqrt {a\,b^7}+b^7\,\sqrt {a\,b^7}+7\,a\,b^6\,\sqrt {a\,b^7}+7\,a^6\,b\,\sqrt {a\,b^7}+21\,a^2\,b^5\,\sqrt {a\,b^7}+35\,a^3\,b^4\,\sqrt {a\,b^7}+35\,a^4\,b^3\,\sqrt {a\,b^7}+21\,a^5\,b^2\,\sqrt {a\,b^7}\right )}{a\,b^3\,\sqrt {{\left (a+b\right )}^6}\,\left (4\,a^4+16\,a^3\,b+24\,a^2\,b^2+16\,a\,b^3+4\,b^4\right )}+\frac {a\,b^8\,{\mathrm {e}}^x\,\sqrt {a\,b^7}\,\left (\frac {4\,\left (2\,a\,b^7\,\sqrt {a^6+6\,a^5\,b+15\,a^4\,b^2+20\,a^3\,b^3+15\,a^2\,b^4+6\,a\,b^5+b^6}+8\,a^2\,b^6\,\sqrt {a^6+6\,a^5\,b+15\,a^4\,b^2+20\,a^3\,b^3+15\,a^2\,b^4+6\,a\,b^5+b^6}+12\,a^3\,b^5\,\sqrt {a^6+6\,a^5\,b+15\,a^4\,b^2+20\,a^3\,b^3+15\,a^2\,b^4+6\,a\,b^5+b^6}+8\,a^4\,b^4\,\sqrt {a^6+6\,a^5\,b+15\,a^4\,b^2+20\,a^3\,b^3+15\,a^2\,b^4+6\,a\,b^5+b^6}+2\,a^5\,b^3\,\sqrt {a^6+6\,a^5\,b+15\,a^4\,b^2+20\,a^3\,b^3+15\,a^2\,b^4+6\,a\,b^5+b^6}\right )}{a^2\,b^{15}\,{\left (a+b\right )}^3}+\frac {2\,\left (a^7\,\sqrt {a\,b^7}+b^7\,\sqrt {a\,b^7}+7\,a\,b^6\,\sqrt {a\,b^7}+7\,a^6\,b\,\sqrt {a\,b^7}+21\,a^2\,b^5\,\sqrt {a\,b^7}+35\,a^3\,b^4\,\sqrt {a\,b^7}+35\,a^4\,b^3\,\sqrt {a\,b^7}+21\,a^5\,b^2\,\sqrt {a\,b^7}\right )}{a^2\,b^{11}\,\sqrt {a\,b^7}\,\sqrt {{\left (a+b\right )}^6}}\right )}{4\,a^4+16\,a^3\,b+24\,a^2\,b^2+16\,a\,b^3+4\,b^4}\right )\right )\,\sqrt {a^6+6\,a^5\,b+15\,a^4\,b^2+20\,a^3\,b^3+15\,a^2\,b^4+6\,a\,b^5+b^6}}{2\,\sqrt {a\,b^7}}-\frac {{\mathrm {e}}^{-3\,x}\,\left (4\,a+9\,b\right )}{96\,b^2}-\frac {{\mathrm {e}}^{3\,x}\,\left (4\,a+9\,b\right )}{96\,b^2}+\frac {{\mathrm {e}}^x\,\left (8\,a^2+22\,a\,b+19\,b^2\right )}{16\,b^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)^7/(a + b*cosh(x)^2),x)

[Out]

exp(-5*x)/(160*b) + exp(5*x)/(160*b) + (exp(-x)*(22*a*b + 8*a^2 + 19*b^2))/(16*b^3) - ((2*atan((exp(x)*(a + b)
^3*(a*b^7)^(1/2))/(2*a*b^3*((a + b)^6)^(1/2))) - 2*atan((2*exp(3*x)*(a^7*(a*b^7)^(1/2) + b^7*(a*b^7)^(1/2) + 7
*a*b^6*(a*b^7)^(1/2) + 7*a^6*b*(a*b^7)^(1/2) + 21*a^2*b^5*(a*b^7)^(1/2) + 35*a^3*b^4*(a*b^7)^(1/2) + 35*a^4*b^
3*(a*b^7)^(1/2) + 21*a^5*b^2*(a*b^7)^(1/2)))/(a*b^3*((a + b)^6)^(1/2)*(16*a*b^3 + 16*a^3*b + 4*a^4 + 4*b^4 + 2
4*a^2*b^2)) + (a*b^8*exp(x)*(a*b^7)^(1/2)*((4*(2*a*b^7*(6*a*b^5 + 6*a^5*b + a^6 + b^6 + 15*a^2*b^4 + 20*a^3*b^
3 + 15*a^4*b^2)^(1/2) + 8*a^2*b^6*(6*a*b^5 + 6*a^5*b + a^6 + b^6 + 15*a^2*b^4 + 20*a^3*b^3 + 15*a^4*b^2)^(1/2)
+ 12*a^3*b^5*(6*a*b^5 + 6*a^5*b + a^6 + b^6 + 15*a^2*b^4 + 20*a^3*b^3 + 15*a^4*b^2)^(1/2) + 8*a^4*b^4*(6*a*b^
5 + 6*a^5*b + a^6 + b^6 + 15*a^2*b^4 + 20*a^3*b^3 + 15*a^4*b^2)^(1/2) + 2*a^5*b^3*(6*a*b^5 + 6*a^5*b + a^6 + b
^6 + 15*a^2*b^4 + 20*a^3*b^3 + 15*a^4*b^2)^(1/2)))/(a^2*b^15*(a + b)^3) + (2*(a^7*(a*b^7)^(1/2) + b^7*(a*b^7)^
(1/2) + 7*a*b^6*(a*b^7)^(1/2) + 7*a^6*b*(a*b^7)^(1/2) + 21*a^2*b^5*(a*b^7)^(1/2) + 35*a^3*b^4*(a*b^7)^(1/2) +
35*a^4*b^3*(a*b^7)^(1/2) + 21*a^5*b^2*(a*b^7)^(1/2)))/(a^2*b^11*(a*b^7)^(1/2)*((a + b)^6)^(1/2))))/(16*a*b^3 +
16*a^3*b + 4*a^4 + 4*b^4 + 24*a^2*b^2)))*(6*a*b^5 + 6*a^5*b + a^6 + b^6 + 15*a^2*b^4 + 20*a^3*b^3 + 15*a^4*b^
2)^(1/2))/(2*(a*b^7)^(1/2)) - (exp(-3*x)*(4*a + 9*b))/(96*b^2) - (exp(3*x)*(4*a + 9*b))/(96*b^2) + (exp(x)*(22
*a*b + 8*a^2 + 19*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(sinh(x)**7/(a+b*cosh(x)**2),x)

[Out]

Timed out

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