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

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

[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)

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Rubi [A]  time = 0.105302, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 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]

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 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]

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*}

Mathematica [C]  time = 0.235471, size = 148, normalized size = 1.9 \[ \frac{\left (8 a^2+22 a b+19 b^2\right ) \cosh (x)}{8 b^3}-\frac{(4 a+9 b) \cosh (3 x)}{48 b^2}-\frac{(a+b)^3 \tan ^{-1}\left (\frac{\sqrt{b}-i \sqrt{a+b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a}}\right )}{\sqrt{a} b^{7/2}}-\frac{(a+b)^3 \tan ^{-1}\left (\frac{\sqrt{b}+i \sqrt{a+b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a}}\right )}{\sqrt{a} b^{7/2}}+\frac{\cosh (5 x)}{80 b} \]

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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Maple [B]  time = 0.033, size = 395, normalized size = 5.1 \begin{align*} \text{result too large to display} \end{align*}

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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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Fricas [B]  time = 2.24972, size = 6149, normalized size = 78.83 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

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