3.2.67 \(\int \frac {\sinh ^5(x)}{a+b \cosh (x)} \, dx\) [167]

Optimal. Leaf size=83 \[ -\frac {a \left (a^2-2 b^2\right ) \cosh (x)}{b^4}+\frac {\left (a^2-2 b^2\right ) \cosh ^2(x)}{2 b^3}-\frac {a \cosh ^3(x)}{3 b^2}+\frac {\cosh ^4(x)}{4 b}+\frac {\left (a^2-b^2\right )^2 \log (a+b \cosh (x))}{b^5} \]

[Out]

-a*(a^2-2*b^2)*cosh(x)/b^4+1/2*(a^2-2*b^2)*cosh(x)^2/b^3-1/3*a*cosh(x)^3/b^2+1/4*cosh(x)^4/b+(a^2-b^2)^2*ln(a+
b*cosh(x))/b^5

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2747, 711} \begin {gather*} \frac {\left (a^2-b^2\right )^2 \log (a+b \cosh (x))}{b^5}-\frac {a \left (a^2-2 b^2\right ) \cosh (x)}{b^4}+\frac {\left (a^2-2 b^2\right ) \cosh ^2(x)}{2 b^3}-\frac {a \cosh ^3(x)}{3 b^2}+\frac {\cosh ^4(x)}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[x]^5/(a + b*Cosh[x]),x]

[Out]

-((a*(a^2 - 2*b^2)*Cosh[x])/b^4) + ((a^2 - 2*b^2)*Cosh[x]^2)/(2*b^3) - (a*Cosh[x]^3)/(3*b^2) + Cosh[x]^4/(4*b)
+ ((a^2 - b^2)^2*Log[a + b*Cosh[x]])/b^5

Rule 711

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rule 2747

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.09, size = 84, normalized size = 1.01 \begin {gather*} \frac {-24 a b \left (4 a^2-7 b^2\right ) \cosh (x)-12 b^2 \left (-2 a^2+3 b^2\right ) \cosh (2 x)-8 a b^3 \cosh (3 x)+3 b^4 \cosh (4 x)+96 \left (a^2-b^2\right )^2 \log (a+b \cosh (x))}{96 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-24*a*b*(4*a^2 - 7*b^2)*Cosh[x] - 12*b^2*(-2*a^2 + 3*b^2)*Cosh[2*x] - 8*a*b^3*Cosh[3*x] + 3*b^4*Cosh[4*x] + 9
6*(a^2 - b^2)^2*Log[a + b*Cosh[x]])/(96*b^5)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(327\) vs. \(2(77)=154\).
time = 0.58, size = 328, normalized size = 3.95

method	result	size
risch	\(-\frac {x \,a^{4}}{b^{5}}+\frac {2 x \,a^{2}}{b^{3}}-\frac {x}{b}+\frac {{\mathrm e}^{4 x}}{64 b}-\frac {a \,{\mathrm e}^{3 x}}{24 b^{2}}+\frac {{\mathrm e}^{2 x} a^{2}}{8 b^{3}}-\frac {3 \,{\mathrm e}^{2 x}}{16 b}-\frac {a^{3} {\mathrm e}^{x}}{2 b^{4}}+\frac {7 a \,{\mathrm e}^{x}}{8 b^{2}}-\frac {a^{3} {\mathrm e}^{-x}}{2 b^{4}}+\frac {7 a \,{\mathrm e}^{-x}}{8 b^{2}}+\frac {{\mathrm e}^{-2 x} a^{2}}{8 b^{3}}-\frac {3 \,{\mathrm e}^{-2 x}}{16 b}-\frac {a \,{\mathrm e}^{-3 x}}{24 b^{2}}+\frac {{\mathrm e}^{-4 x}}{64 b}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}+1\right ) a^{4}}{b^{5}}-\frac {2 \ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}+1\right ) a^{2}}{b^{3}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}+1\right )}{b}\)	\(210\)
default	\(\frac {\left (a^{5}-a^{4} b -2 a^{3} b^{2}+2 a^{2} b^{3}+a \,b^{4}-b^{5}\right ) \ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-a -b \right )}{b^{5} \left (a -b \right )}+\frac {1}{4 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}-\frac {-2 a -3 b}{6 b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {-4 a^{2}-4 a b +3 b^{2}}{8 b^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {\left (-a^{4}+2 a^{2} b^{2}-b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b^{5}}-\frac {-8 a^{3}-4 a^{2} b +12 a \,b^{2}+5 b^{3}}{8 b^{4} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {1}{4 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {2 a +3 b}{6 b^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {-4 a^{2}-4 a b +3 b^{2}}{8 b^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {\left (-a^{4}+2 a^{2} b^{2}-b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b^{5}}-\frac {8 a^{3}+4 a^{2} b -12 a \,b^{2}-5 b^{3}}{8 b^{4} \left (\tanh \left (\frac {x}{2}\right )+1\right )}\)	\(328\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^5-a^4*b-2*a^3*b^2+2*a^2*b^3+a*b^4-b^5)/b^5/(a-b)*ln(a*tanh(1/2*x)^2-b*tanh(1/2*x)^2-a-b)+1/4/b/(tanh(1/2*x)
-1)^4-1/6*(-2*a-3*b)/b^2/(tanh(1/2*x)-1)^3-1/8*(-4*a^2-4*a*b+3*b^2)/b^3/(tanh(1/2*x)-1)^2+(-a^4+2*a^2*b^2-b^4)
/b^5*ln(tanh(1/2*x)-1)-1/8*(-8*a^3-4*a^2*b+12*a*b^2+5*b^3)/b^4/(tanh(1/2*x)-1)+1/4/b/(tanh(1/2*x)+1)^4-1/6*(2*
a+3*b)/b^2/(tanh(1/2*x)+1)^3-1/8*(-4*a^2-4*a*b+3*b^2)/b^3/(tanh(1/2*x)+1)^2+(-a^4+2*a^2*b^2-b^4)/b^5*ln(tanh(1
/2*x)+1)-1/8*(8*a^3+4*a^2*b-12*a*b^2-5*b^3)/b^4/(tanh(1/2*x)+1)

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (77) = 154\).
time = 0.26, size = 178, normalized size = 2.14 \begin {gather*} -\frac {{\left (8 \, a b^{2} e^{\left (-x\right )} - 3 \, b^{3} - 12 \, {\left (2 \, a^{2} b - 3 \, b^{3}\right )} e^{\left (-2 \, x\right )} + 24 \, {\left (4 \, a^{3} - 7 \, a b^{2}\right )} e^{\left (-3 \, x\right )}\right )} e^{\left (4 \, x\right )}}{192 \, b^{4}} - \frac {8 \, a b^{2} e^{\left (-3 \, x\right )} - 3 \, b^{3} e^{\left (-4 \, x\right )} + 24 \, {\left (4 \, a^{3} - 7 \, a b^{2}\right )} e^{\left (-x\right )} - 12 \, {\left (2 \, a^{2} b - 3 \, b^{3}\right )} e^{\left (-2 \, x\right )}}{192 \, b^{4}} + \frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} x}{b^{5}} + \frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} + b\right )}{b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*(8*a*b^2*e^(-x) - 3*b^3 - 12*(2*a^2*b - 3*b^3)*e^(-2*x) + 24*(4*a^3 - 7*a*b^2)*e^(-3*x))*e^(4*x)/b^4 -
1/192*(8*a*b^2*e^(-3*x) - 3*b^3*e^(-4*x) + 24*(4*a^3 - 7*a*b^2)*e^(-x) - 12*(2*a^2*b - 3*b^3)*e^(-2*x))/b^4 +
(a^4 - 2*a^2*b^2 + b^4)*x/b^5 + (a^4 - 2*a^2*b^2 + b^4)*log(2*a*e^(-x) + b*e^(-2*x) + b)/b^5

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 866 vs. \(2 (77) = 154\).
time = 0.38, size = 866, normalized size = 10.43 \begin {gather*} \frac {3 \, b^{4} \cosh \left (x\right )^{8} + 3 \, b^{4} \sinh \left (x\right )^{8} - 8 \, a b^{3} \cosh \left (x\right )^{7} + 8 \, {\left (3 \, b^{4} \cosh \left (x\right ) - a b^{3}\right )} \sinh \left (x\right )^{7} + 12 \, {\left (2 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cosh \left (x\right )^{6} + 4 \, {\left (21 \, b^{4} \cosh \left (x\right )^{2} - 14 \, a b^{3} \cosh \left (x\right ) + 6 \, a^{2} b^{2} - 9 \, b^{4}\right )} \sinh \left (x\right )^{6} - 192 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} x \cosh \left (x\right )^{4} - 24 \, {\left (4 \, a^{3} b - 7 \, a b^{3}\right )} \cosh \left (x\right )^{5} + 24 \, {\left (7 \, b^{4} \cosh \left (x\right )^{3} - 7 \, a b^{3} \cosh \left (x\right )^{2} - 4 \, a^{3} b + 7 \, a b^{3} + 3 \, {\left (2 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} - 8 \, a b^{3} \cosh \left (x\right ) + 2 \, {\left (105 \, b^{4} \cosh \left (x\right )^{4} - 140 \, a b^{3} \cosh \left (x\right )^{3} + 90 \, {\left (2 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cosh \left (x\right )^{2} - 96 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} x - 60 \, {\left (4 \, a^{3} b - 7 \, a b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{4} + 3 \, b^{4} - 24 \, {\left (4 \, a^{3} b - 7 \, a b^{3}\right )} \cosh \left (x\right )^{3} + 8 \, {\left (21 \, b^{4} \cosh \left (x\right )^{5} - 35 \, a b^{3} \cosh \left (x\right )^{4} - 12 \, a^{3} b + 21 \, a b^{3} + 30 \, {\left (2 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cosh \left (x\right )^{3} - 96 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} x \cosh \left (x\right ) - 30 \, {\left (4 \, a^{3} b - 7 \, a b^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{3} + 12 \, {\left (2 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cosh \left (x\right )^{2} + 12 \, {\left (7 \, b^{4} \cosh \left (x\right )^{6} - 14 \, a b^{3} \cosh \left (x\right )^{5} + 15 \, {\left (2 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cosh \left (x\right )^{4} + 2 \, a^{2} b^{2} - 3 \, b^{4} - 96 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} x \cosh \left (x\right )^{2} - 20 \, {\left (4 \, a^{3} b - 7 \, a b^{3}\right )} \cosh \left (x\right )^{3} - 6 \, {\left (4 \, a^{3} b - 7 \, a b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 192 \, {\left ({\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )^{4}\right )} \log \left (\frac {2 \, {\left (b \cosh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 8 \, {\left (3 \, b^{4} \cosh \left (x\right )^{7} - 7 \, a b^{3} \cosh \left (x\right )^{6} + 9 \, {\left (2 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cosh \left (x\right )^{5} - 96 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} x \cosh \left (x\right )^{3} - 15 \, {\left (4 \, a^{3} b - 7 \, a b^{3}\right )} \cosh \left (x\right )^{4} - a b^{3} - 9 \, {\left (4 \, a^{3} b - 7 \, a b^{3}\right )} \cosh \left (x\right )^{2} + 3 \, {\left (2 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{192 \, {\left (b^{5} \cosh \left (x\right )^{4} + 4 \, b^{5} \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, b^{5} \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, b^{5} \cosh \left (x\right ) \sinh \left (x\right )^{3} + b^{5} \sinh \left (x\right )^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(3*b^4*cosh(x)^8 + 3*b^4*sinh(x)^8 - 8*a*b^3*cosh(x)^7 + 8*(3*b^4*cosh(x) - a*b^3)*sinh(x)^7 + 12*(2*a^2
*b^2 - 3*b^4)*cosh(x)^6 + 4*(21*b^4*cosh(x)^2 - 14*a*b^3*cosh(x) + 6*a^2*b^2 - 9*b^4)*sinh(x)^6 - 192*(a^4 - 2
*a^2*b^2 + b^4)*x*cosh(x)^4 - 24*(4*a^3*b - 7*a*b^3)*cosh(x)^5 + 24*(7*b^4*cosh(x)^3 - 7*a*b^3*cosh(x)^2 - 4*a
^3*b + 7*a*b^3 + 3*(2*a^2*b^2 - 3*b^4)*cosh(x))*sinh(x)^5 - 8*a*b^3*cosh(x) + 2*(105*b^4*cosh(x)^4 - 140*a*b^3
*cosh(x)^3 + 90*(2*a^2*b^2 - 3*b^4)*cosh(x)^2 - 96*(a^4 - 2*a^2*b^2 + b^4)*x - 60*(4*a^3*b - 7*a*b^3)*cosh(x))
*sinh(x)^4 + 3*b^4 - 24*(4*a^3*b - 7*a*b^3)*cosh(x)^3 + 8*(21*b^4*cosh(x)^5 - 35*a*b^3*cosh(x)^4 - 12*a^3*b +
21*a*b^3 + 30*(2*a^2*b^2 - 3*b^4)*cosh(x)^3 - 96*(a^4 - 2*a^2*b^2 + b^4)*x*cosh(x) - 30*(4*a^3*b - 7*a*b^3)*co
sh(x)^2)*sinh(x)^3 + 12*(2*a^2*b^2 - 3*b^4)*cosh(x)^2 + 12*(7*b^4*cosh(x)^6 - 14*a*b^3*cosh(x)^5 + 15*(2*a^2*b
^2 - 3*b^4)*cosh(x)^4 + 2*a^2*b^2 - 3*b^4 - 96*(a^4 - 2*a^2*b^2 + b^4)*x*cosh(x)^2 - 20*(4*a^3*b - 7*a*b^3)*co
sh(x)^3 - 6*(4*a^3*b - 7*a*b^3)*cosh(x))*sinh(x)^2 + 192*((a^4 - 2*a^2*b^2 + b^4)*cosh(x)^4 + 4*(a^4 - 2*a^2*b
^2 + b^4)*cosh(x)^3*sinh(x) + 6*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2*sinh(x)^2 + 4*(a^4 - 2*a^2*b^2 + b^4)*cosh(x
)*sinh(x)^3 + (a^4 - 2*a^2*b^2 + b^4)*sinh(x)^4)*log(2*(b*cosh(x) + a)/(cosh(x) - sinh(x))) + 8*(3*b^4*cosh(x)
^7 - 7*a*b^3*cosh(x)^6 + 9*(2*a^2*b^2 - 3*b^4)*cosh(x)^5 - 96*(a^4 - 2*a^2*b^2 + b^4)*x*cosh(x)^3 - 15*(4*a^3*
b - 7*a*b^3)*cosh(x)^4 - a*b^3 - 9*(4*a^3*b - 7*a*b^3)*cosh(x)^2 + 3*(2*a^2*b^2 - 3*b^4)*cosh(x))*sinh(x))/(b^
5*cosh(x)^4 + 4*b^5*cosh(x)^3*sinh(x) + 6*b^5*cosh(x)^2*sinh(x)^2 + 4*b^5*cosh(x)*sinh(x)^3 + b^5*sinh(x)^4)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 0.41, size = 124, normalized size = 1.49 \begin {gather*} \frac {3 \, b^{3} {\left (e^{\left (-x\right )} + e^{x}\right )}^{4} - 8 \, a b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} + 24 \, a^{2} b {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 48 \, b^{3} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 96 \, a^{3} {\left (e^{\left (-x\right )} + e^{x}\right )} + 192 \, a b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}}{192 \, b^{4}} + \frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | b {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, a \right |}\right )}{b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(3*b^3*(e^(-x) + e^x)^4 - 8*a*b^2*(e^(-x) + e^x)^3 + 24*a^2*b*(e^(-x) + e^x)^2 - 48*b^3*(e^(-x) + e^x)^2
- 96*a^3*(e^(-x) + e^x) + 192*a*b^2*(e^(-x) + e^x))/b^4 + (a^4 - 2*a^2*b^2 + b^4)*log(abs(b*(e^(-x) + e^x) +
2*a))/b^5

________________________________________________________________________________________

Mupad [B]
time = 1.31, size = 169, normalized size = 2.04 \begin {gather*} \frac {{\mathrm {e}}^{-4\,x}}{64\,b}+\frac {{\mathrm {e}}^{4\,x}}{64\,b}-\frac {x\,{\left (a^2-b^2\right )}^2}{b^5}+\frac {{\mathrm {e}}^{-x}\,\left (7\,a\,b^2-4\,a^3\right )}{8\,b^4}+\frac {\ln \left (b+2\,a\,{\mathrm {e}}^x+b\,{\mathrm {e}}^{2\,x}\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}{b^5}-\frac {a\,{\mathrm {e}}^{-3\,x}}{24\,b^2}-\frac {a\,{\mathrm {e}}^{3\,x}}{24\,b^2}+\frac {{\mathrm {e}}^{-2\,x}\,\left (2\,a^2-3\,b^2\right )}{16\,b^3}+\frac {{\mathrm {e}}^{2\,x}\,\left (2\,a^2-3\,b^2\right )}{16\,b^3}+\frac {{\mathrm {e}}^x\,\left (7\,a\,b^2-4\,a^3\right )}{8\,b^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)^5/(a + b*cosh(x)),x)

[Out]

exp(-4*x)/(64*b) + exp(4*x)/(64*b) - (x*(a^2 - b^2)^2)/b^5 + (exp(-x)*(7*a*b^2 - 4*a^3))/(8*b^4) + (log(b + 2*
a*exp(x) + b*exp(2*x))*(a^4 + b^4 - 2*a^2*b^2))/b^5 - (a*exp(-3*x))/(24*b^2) - (a*exp(3*x))/(24*b^2) + (exp(-2
*x)*(2*a^2 - 3*b^2))/(16*b^3) + (exp(2*x)*(2*a^2 - 3*b^2))/(16*b^3) + (exp(x)*(7*a*b^2 - 4*a^3))/(8*b^4)

________________________________________________________________________________________