\(\int \frac {\sinh ^7(x)}{a+a \cosh (x)} \, dx\) [153]

Optimal result

Integrand size = 13, antiderivative size = 46 \[ \int \frac {\sinh ^7(x)}{a+a \cosh (x)} \, dx=\frac {(a-a \cosh (x))^4}{a^5}-\frac {4 (a-a \cosh (x))^5}{5 a^6}+\frac {(a-a \cosh (x))^6}{6 a^7} \]

[Out]

(a-a*cosh(x))^4/a^5-4/5*(a-a*cosh(x))^5/a^6+1/6*(a-a*cosh(x))^6/a^7

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2746, 45} \[ \int \frac {\sinh ^7(x)}{a+a \cosh (x)} \, dx=\frac {(a-a \cosh (x))^6}{6 a^7}-\frac {4 (a-a \cosh (x))^5}{5 a^6}+\frac {(a-a \cosh (x))^4}{a^5} \]

[In]

Int[Sinh[x]^7/(a + a*Cosh[x]),x]

[Out]

(a - a*Cosh[x])^4/a^5 - (4*(a - a*Cosh[x])^5)/(5*a^6) + (a - a*Cosh[x])^6/(6*a^7)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2746

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 + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int (a-x)^3 (a+x)^2 \, dx,x,a \cosh (x)\right )}{a^7} \\ & = -\frac {\text {Subst}\left (\int \left (4 a^2 (a-x)^3-4 a (a-x)^4+(a-x)^5\right ) \, dx,x,a \cosh (x)\right )}{a^7} \\ & = \frac {(a-a \cosh (x))^4}{a^5}-\frac {4 (a-a \cosh (x))^5}{5 a^6}+\frac {(a-a \cosh (x))^6}{6 a^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.59 \[ \int \frac {\sinh ^7(x)}{a+a \cosh (x)} \, dx=\frac {4 (27+28 \cosh (x)+5 \cosh (2 x)) \sinh ^8\left (\frac {x}{2}\right )}{15 a} \]

[In]

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

[Out]

(4*(27 + 28*Cosh[x] + 5*Cosh[2*x])*Sinh[x/2]^8)/(15*a)

Maple [A] (verified)

Time = 120.53 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(\frac {\frac {\cosh \left (x \right )^{6}}{6}-\frac {\cosh \left (x \right )^{5}}{5}-\frac {\cosh \left (x \right )^{4}}{2}+\frac {2 \cosh \left (x \right )^{3}}{3}+\frac {\cosh \left (x \right )^{2}}{2}-\cosh \left (x \right )}{a}\)	\(40\)
default	\(\frac {\frac {\cosh \left (x \right )^{6}}{6}-\frac {\cosh \left (x \right )^{5}}{5}-\frac {\cosh \left (x \right )^{4}}{2}+\frac {2 \cosh \left (x \right )^{3}}{3}+\frac {\cosh \left (x \right )^{2}}{2}-\cosh \left (x \right )}{a}\)	\(40\)
risch	\(\frac {{\mathrm e}^{6 x}}{384 a}-\frac {{\mathrm e}^{5 x}}{160 a}-\frac {{\mathrm e}^{4 x}}{64 a}+\frac {5 \,{\mathrm e}^{3 x}}{96 a}+\frac {5 \,{\mathrm e}^{2 x}}{128 a}-\frac {5 \,{\mathrm e}^{x}}{16 a}-\frac {5 \,{\mathrm e}^{-x}}{16 a}+\frac {5 \,{\mathrm e}^{-2 x}}{128 a}+\frac {5 \,{\mathrm e}^{-3 x}}{96 a}-\frac {{\mathrm e}^{-4 x}}{64 a}-\frac {{\mathrm e}^{-5 x}}{160 a}+\frac {{\mathrm e}^{-6 x}}{384 a}\)	\(108\)

[In]

int(sinh(x)^7/(a+a*cosh(x)),x,method=_RETURNVERBOSE)

[Out]

1/a*(1/6*cosh(x)^6-1/5*cosh(x)^5-1/2*cosh(x)^4+2/3*cosh(x)^3+1/2*cosh(x)^2-cosh(x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (45) = 90\).

Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.04 \[ \int \frac {\sinh ^7(x)}{a+a \cosh (x)} \, dx=\frac {5 \, \cosh \left (x\right )^{6} + 5 \, \sinh \left (x\right )^{6} - 12 \, \cosh \left (x\right )^{5} + 15 \, {\left (5 \, \cosh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) - 2\right )} \sinh \left (x\right )^{4} - 30 \, \cosh \left (x\right )^{4} + 100 \, \cosh \left (x\right )^{3} + 15 \, {\left (5 \, \cosh \left (x\right )^{4} - 8 \, \cosh \left (x\right )^{3} - 12 \, \cosh \left (x\right )^{2} + 20 \, \cosh \left (x\right ) + 5\right )} \sinh \left (x\right )^{2} + 75 \, \cosh \left (x\right )^{2} - 600 \, \cosh \left (x\right )}{960 \, a} \]

[In]

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

[Out]

1/960*(5*cosh(x)^6 + 5*sinh(x)^6 - 12*cosh(x)^5 + 15*(5*cosh(x)^2 - 4*cosh(x) - 2)*sinh(x)^4 - 30*cosh(x)^4 +
100*cosh(x)^3 + 15*(5*cosh(x)^4 - 8*cosh(x)^3 - 12*cosh(x)^2 + 20*cosh(x) + 5)*sinh(x)^2 + 75*cosh(x)^2 - 600*
cosh(x))/a

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (39) = 78\).

Time = 1.83 (sec) , antiderivative size = 284, normalized size of antiderivative = 6.17 \[ \int \frac {\sinh ^7(x)}{a+a \cosh (x)} \, dx=\frac {320 \tanh ^{6}{\left (\frac {x}{2} \right )}}{15 a \tanh ^{12}{\left (\frac {x}{2} \right )} - 90 a \tanh ^{10}{\left (\frac {x}{2} \right )} + 225 a \tanh ^{8}{\left (\frac {x}{2} \right )} - 300 a \tanh ^{6}{\left (\frac {x}{2} \right )} + 225 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 90 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 15 a} - \frac {240 \tanh ^{4}{\left (\frac {x}{2} \right )}}{15 a \tanh ^{12}{\left (\frac {x}{2} \right )} - 90 a \tanh ^{10}{\left (\frac {x}{2} \right )} + 225 a \tanh ^{8}{\left (\frac {x}{2} \right )} - 300 a \tanh ^{6}{\left (\frac {x}{2} \right )} + 225 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 90 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 15 a} + \frac {96 \tanh ^{2}{\left (\frac {x}{2} \right )}}{15 a \tanh ^{12}{\left (\frac {x}{2} \right )} - 90 a \tanh ^{10}{\left (\frac {x}{2} \right )} + 225 a \tanh ^{8}{\left (\frac {x}{2} \right )} - 300 a \tanh ^{6}{\left (\frac {x}{2} \right )} + 225 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 90 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 15 a} - \frac {16}{15 a \tanh ^{12}{\left (\frac {x}{2} \right )} - 90 a \tanh ^{10}{\left (\frac {x}{2} \right )} + 225 a \tanh ^{8}{\left (\frac {x}{2} \right )} - 300 a \tanh ^{6}{\left (\frac {x}{2} \right )} + 225 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 90 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 15 a} \]

[In]

integrate(sinh(x)**7/(a+a*cosh(x)),x)

[Out]

320*tanh(x/2)**6/(15*a*tanh(x/2)**12 - 90*a*tanh(x/2)**10 + 225*a*tanh(x/2)**8 - 300*a*tanh(x/2)**6 + 225*a*ta
nh(x/2)**4 - 90*a*tanh(x/2)**2 + 15*a) - 240*tanh(x/2)**4/(15*a*tanh(x/2)**12 - 90*a*tanh(x/2)**10 + 225*a*tan
h(x/2)**8 - 300*a*tanh(x/2)**6 + 225*a*tanh(x/2)**4 - 90*a*tanh(x/2)**2 + 15*a) + 96*tanh(x/2)**2/(15*a*tanh(x
/2)**12 - 90*a*tanh(x/2)**10 + 225*a*tanh(x/2)**8 - 300*a*tanh(x/2)**6 + 225*a*tanh(x/2)**4 - 90*a*tanh(x/2)**
2 + 15*a) - 16/(15*a*tanh(x/2)**12 - 90*a*tanh(x/2)**10 + 225*a*tanh(x/2)**8 - 300*a*tanh(x/2)**6 + 225*a*tanh
(x/2)**4 - 90*a*tanh(x/2)**2 + 15*a)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.83 \[ \int \frac {\sinh ^7(x)}{a+a \cosh (x)} \, dx=-\frac {{\left (12 \, e^{\left (-x\right )} + 30 \, e^{\left (-2 \, x\right )} - 100 \, e^{\left (-3 \, x\right )} - 75 \, e^{\left (-4 \, x\right )} + 600 \, e^{\left (-5 \, x\right )} - 5\right )} e^{\left (6 \, x\right )}}{1920 \, a} - \frac {600 \, e^{\left (-x\right )} - 75 \, e^{\left (-2 \, x\right )} - 100 \, e^{\left (-3 \, x\right )} + 30 \, e^{\left (-4 \, x\right )} + 12 \, e^{\left (-5 \, x\right )} - 5 \, e^{\left (-6 \, x\right )}}{1920 \, a} \]

[In]

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

[Out]

-1/1920*(12*e^(-x) + 30*e^(-2*x) - 100*e^(-3*x) - 75*e^(-4*x) + 600*e^(-5*x) - 5)*e^(6*x)/a - 1/1920*(600*e^(-
x) - 75*e^(-2*x) - 100*e^(-3*x) + 30*e^(-4*x) + 12*e^(-5*x) - 5*e^(-6*x))/a

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.63 \[ \int \frac {\sinh ^7(x)}{a+a \cosh (x)} \, dx=-\frac {{\left (600 \, e^{\left (5 \, x\right )} - 75 \, e^{\left (4 \, x\right )} - 100 \, e^{\left (3 \, x\right )} + 30 \, e^{\left (2 \, x\right )} + 12 \, e^{x} - 5\right )} e^{\left (-6 \, x\right )} - 5 \, e^{\left (6 \, x\right )} + 12 \, e^{\left (5 \, x\right )} + 30 \, e^{\left (4 \, x\right )} - 100 \, e^{\left (3 \, x\right )} - 75 \, e^{\left (2 \, x\right )} + 600 \, e^{x}}{1920 \, a} \]

[In]

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

[Out]

-1/1920*((600*e^(5*x) - 75*e^(4*x) - 100*e^(3*x) + 30*e^(2*x) + 12*e^x - 5)*e^(-6*x) - 5*e^(6*x) + 12*e^(5*x)
+ 30*e^(4*x) - 100*e^(3*x) - 75*e^(2*x) + 600*e^x)/a

Mupad [B] (verification not implemented)

Time = 1.83 (sec) , antiderivative size = 107, normalized size of antiderivative = 2.33 \[ \int \frac {\sinh ^7(x)}{a+a \cosh (x)} \, dx=\frac {5\,{\mathrm {e}}^{-2\,x}}{128\,a}-\frac {5\,{\mathrm {e}}^{-x}}{16\,a}+\frac {5\,{\mathrm {e}}^{2\,x}}{128\,a}+\frac {5\,{\mathrm {e}}^{-3\,x}}{96\,a}+\frac {5\,{\mathrm {e}}^{3\,x}}{96\,a}-\frac {{\mathrm {e}}^{-4\,x}}{64\,a}-\frac {{\mathrm {e}}^{4\,x}}{64\,a}-\frac {{\mathrm {e}}^{-5\,x}}{160\,a}-\frac {{\mathrm {e}}^{5\,x}}{160\,a}+\frac {{\mathrm {e}}^{-6\,x}}{384\,a}+\frac {{\mathrm {e}}^{6\,x}}{384\,a}-\frac {5\,{\mathrm {e}}^x}{16\,a} \]

[In]

int(sinh(x)^7/(a + a*cosh(x)),x)

[Out]

(5*exp(-2*x))/(128*a) - (5*exp(-x))/(16*a) + (5*exp(2*x))/(128*a) + (5*exp(-3*x))/(96*a) + (5*exp(3*x))/(96*a)
- exp(-4*x)/(64*a) - exp(4*x)/(64*a) - exp(-5*x)/(160*a) - exp(5*x)/(160*a) + exp(-6*x)/(384*a) + exp(6*x)/(3
84*a) - (5*exp(x))/(16*a)