3.134 \(\int (a \cosh ^4(x))^{5/2} \, dx\)

Optimal. Leaf size=132 \[ \frac{1}{10} a^2 \sinh (x) \cosh ^7(x) \sqrt{a \cosh ^4(x)}+\frac{9}{80} a^2 \sinh (x) \cosh ^5(x) \sqrt{a \cosh ^4(x)}+\frac{21}{160} a^2 \sinh (x) \cosh ^3(x) \sqrt{a \cosh ^4(x)}+\frac{21}{128} a^2 \sinh (x) \cosh (x) \sqrt{a \cosh ^4(x)}+\frac{63}{256} a^2 \tanh (x) \sqrt{a \cosh ^4(x)}+\frac{63}{256} a^2 x \text{sech}^2(x) \sqrt{a \cosh ^4(x)} \]

[Out]

(63*a^2*x*Sqrt[a*Cosh[x]^4]*Sech[x]^2)/256 + (21*a^2*Cosh[x]*Sqrt[a*Cosh[x]^4]*Sinh[x])/128 + (21*a^2*Cosh[x]^
3*Sqrt[a*Cosh[x]^4]*Sinh[x])/160 + (9*a^2*Cosh[x]^5*Sqrt[a*Cosh[x]^4]*Sinh[x])/80 + (a^2*Cosh[x]^7*Sqrt[a*Cosh
[x]^4]*Sinh[x])/10 + (63*a^2*Sqrt[a*Cosh[x]^4]*Tanh[x])/256

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Rubi [A]  time = 0.0529856, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3207, 2635, 8} \[ \frac{1}{10} a^2 \sinh (x) \cosh ^7(x) \sqrt{a \cosh ^4(x)}+\frac{9}{80} a^2 \sinh (x) \cosh ^5(x) \sqrt{a \cosh ^4(x)}+\frac{21}{160} a^2 \sinh (x) \cosh ^3(x) \sqrt{a \cosh ^4(x)}+\frac{21}{128} a^2 \sinh (x) \cosh (x) \sqrt{a \cosh ^4(x)}+\frac{63}{256} a^2 \tanh (x) \sqrt{a \cosh ^4(x)}+\frac{63}{256} a^2 x \text{sech}^2(x) \sqrt{a \cosh ^4(x)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cosh[x]^4)^(5/2),x]

[Out]

(63*a^2*x*Sqrt[a*Cosh[x]^4]*Sech[x]^2)/256 + (21*a^2*Cosh[x]*Sqrt[a*Cosh[x]^4]*Sinh[x])/128 + (21*a^2*Cosh[x]^
3*Sqrt[a*Cosh[x]^4]*Sinh[x])/160 + (9*a^2*Cosh[x]^5*Sqrt[a*Cosh[x]^4]*Sinh[x])/80 + (a^2*Cosh[x]^7*Sqrt[a*Cosh
[x]^4]*Sinh[x])/10 + (63*a^2*Sqrt[a*Cosh[x]^4]*Tanh[x])/256

Rule 3207

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Sin[e + f*x]^n)^FracPart[p])/(Sin[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \left (a \cosh ^4(x)\right )^{5/2} \, dx &=\left (a^2 \sqrt{a \cosh ^4(x)} \text{sech}^2(x)\right ) \int \cosh ^{10}(x) \, dx\\ &=\frac{1}{10} a^2 \cosh ^7(x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{1}{10} \left (9 a^2 \sqrt{a \cosh ^4(x)} \text{sech}^2(x)\right ) \int \cosh ^8(x) \, dx\\ &=\frac{9}{80} a^2 \cosh ^5(x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{1}{10} a^2 \cosh ^7(x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{1}{80} \left (63 a^2 \sqrt{a \cosh ^4(x)} \text{sech}^2(x)\right ) \int \cosh ^6(x) \, dx\\ &=\frac{21}{160} a^2 \cosh ^3(x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{9}{80} a^2 \cosh ^5(x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{1}{10} a^2 \cosh ^7(x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{1}{32} \left (21 a^2 \sqrt{a \cosh ^4(x)} \text{sech}^2(x)\right ) \int \cosh ^4(x) \, dx\\ &=\frac{21}{128} a^2 \cosh (x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{21}{160} a^2 \cosh ^3(x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{9}{80} a^2 \cosh ^5(x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{1}{10} a^2 \cosh ^7(x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{1}{128} \left (63 a^2 \sqrt{a \cosh ^4(x)} \text{sech}^2(x)\right ) \int \cosh ^2(x) \, dx\\ &=\frac{21}{128} a^2 \cosh (x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{21}{160} a^2 \cosh ^3(x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{9}{80} a^2 \cosh ^5(x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{1}{10} a^2 \cosh ^7(x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{63}{256} a^2 \sqrt{a \cosh ^4(x)} \tanh (x)+\frac{1}{256} \left (63 a^2 \sqrt{a \cosh ^4(x)} \text{sech}^2(x)\right ) \int 1 \, dx\\ &=\frac{63}{256} a^2 x \sqrt{a \cosh ^4(x)} \text{sech}^2(x)+\frac{21}{128} a^2 \cosh (x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{21}{160} a^2 \cosh ^3(x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{9}{80} a^2 \cosh ^5(x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{1}{10} a^2 \cosh ^7(x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{63}{256} a^2 \sqrt{a \cosh ^4(x)} \tanh (x)\\ \end{align*}

Mathematica [A]  time = 0.116271, size = 53, normalized size = 0.4 \[ \frac{a (2520 x+2100 \sinh (2 x)+600 \sinh (4 x)+150 \sinh (6 x)+25 \sinh (8 x)+2 \sinh (10 x)) \text{sech}^6(x) \left (a \cosh ^4(x)\right )^{3/2}}{10240} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cosh[x]^4)^(5/2),x]

[Out]

(a*(a*Cosh[x]^4)^(3/2)*Sech[x]^6*(2520*x + 2100*Sinh[2*x] + 600*Sinh[4*x] + 150*Sinh[6*x] + 25*Sinh[8*x] + 2*S
inh[10*x]))/10240

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Maple [A]  time = 0.138, size = 177, normalized size = 1.3 \begin{align*}{\frac{\sqrt{8} \left ( \cosh \left ( 2\,x \right ) +1 \right ) \sqrt{2}}{10240\,\sinh \left ( 2\,x \right ) }\sqrt{a \left ( -1+\cosh \left ( 2\,x \right ) \right ) \left ( \cosh \left ( 2\,x \right ) +1 \right ) }{a}^{{\frac{3}{2}}} \left ( 8\,\sqrt{a \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}}\sqrt{a} \left ( \sinh \left ( 2\,x \right ) \right ) ^{4}+50\,\sqrt{a \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}}\sqrt{a}\cosh \left ( 2\,x \right ) \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}+160\,\sqrt{a \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}}\sqrt{a} \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}+325\,\cosh \left ( 2\,x \right ) \sqrt{a \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}}\sqrt{a}+640\,\sqrt{a \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}}\sqrt{a}+315\,\ln \left ( \sqrt{a}\cosh \left ( 2\,x \right ) +\sqrt{a \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}} \right ) a \right ){\frac{1}{\sqrt{a \left ( \cosh \left ( 2\,x \right ) +1 \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*cosh(x)^4)^(5/2),x)

[Out]

1/10240*8^(1/2)*(cosh(2*x)+1)*(a*(-1+cosh(2*x))*(cosh(2*x)+1))^(1/2)*2^(1/2)*a^(3/2)*(8*(a*sinh(2*x)^2)^(1/2)*
a^(1/2)*sinh(2*x)^4+50*(a*sinh(2*x)^2)^(1/2)*a^(1/2)*cosh(2*x)*sinh(2*x)^2+160*(a*sinh(2*x)^2)^(1/2)*a^(1/2)*s
inh(2*x)^2+325*cosh(2*x)*(a*sinh(2*x)^2)^(1/2)*a^(1/2)+640*(a*sinh(2*x)^2)^(1/2)*a^(1/2)+315*ln(a^(1/2)*cosh(2
*x)+(a*sinh(2*x)^2)^(1/2))*a)/sinh(2*x)/(a*(cosh(2*x)+1)^2)^(1/2)

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Maxima [A]  time = 1.72198, size = 135, normalized size = 1.02 \begin{align*} \frac{63}{256} \, a^{\frac{5}{2}} x + \frac{1}{20480} \,{\left (25 \, a^{\frac{5}{2}} e^{\left (-2 \, x\right )} + 150 \, a^{\frac{5}{2}} e^{\left (-4 \, x\right )} + 600 \, a^{\frac{5}{2}} e^{\left (-6 \, x\right )} + 2100 \, a^{\frac{5}{2}} e^{\left (-8 \, x\right )} - 2100 \, a^{\frac{5}{2}} e^{\left (-12 \, x\right )} - 600 \, a^{\frac{5}{2}} e^{\left (-14 \, x\right )} - 150 \, a^{\frac{5}{2}} e^{\left (-16 \, x\right )} - 25 \, a^{\frac{5}{2}} e^{\left (-18 \, x\right )} - 2 \, a^{\frac{5}{2}} e^{\left (-20 \, x\right )} + 2 \, a^{\frac{5}{2}}\right )} e^{\left (10 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cosh(x)^4)^(5/2),x, algorithm="maxima")

[Out]

63/256*a^(5/2)*x + 1/20480*(25*a^(5/2)*e^(-2*x) + 150*a^(5/2)*e^(-4*x) + 600*a^(5/2)*e^(-6*x) + 2100*a^(5/2)*e
^(-8*x) - 2100*a^(5/2)*e^(-12*x) - 600*a^(5/2)*e^(-14*x) - 150*a^(5/2)*e^(-16*x) - 25*a^(5/2)*e^(-18*x) - 2*a^
(5/2)*e^(-20*x) + 2*a^(5/2))*e^(10*x)

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Fricas [B]  time = 2.13916, size = 4963, normalized size = 37.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cosh(x)^4)^(5/2),x, algorithm="fricas")

[Out]

1/20480*(40*a^2*cosh(x)*e^(2*x)*sinh(x)^19 + 2*a^2*e^(2*x)*sinh(x)^20 + 5*(76*a^2*cosh(x)^2 + 5*a^2)*e^(2*x)*s
inh(x)^18 + 30*(76*a^2*cosh(x)^3 + 15*a^2*cosh(x))*e^(2*x)*sinh(x)^17 + 15*(646*a^2*cosh(x)^4 + 255*a^2*cosh(x
)^2 + 10*a^2)*e^(2*x)*sinh(x)^16 + 48*(646*a^2*cosh(x)^5 + 425*a^2*cosh(x)^3 + 50*a^2*cosh(x))*e^(2*x)*sinh(x)
^15 + 60*(1292*a^2*cosh(x)^6 + 1275*a^2*cosh(x)^4 + 300*a^2*cosh(x)^2 + 10*a^2)*e^(2*x)*sinh(x)^14 + 120*(1292
*a^2*cosh(x)^7 + 1785*a^2*cosh(x)^5 + 700*a^2*cosh(x)^3 + 70*a^2*cosh(x))*e^(2*x)*sinh(x)^13 + 60*(4199*a^2*co
sh(x)^8 + 7735*a^2*cosh(x)^6 + 4550*a^2*cosh(x)^4 + 910*a^2*cosh(x)^2 + 35*a^2)*e^(2*x)*sinh(x)^12 + 80*(4199*
a^2*cosh(x)^9 + 9945*a^2*cosh(x)^7 + 8190*a^2*cosh(x)^5 + 2730*a^2*cosh(x)^3 + 315*a^2*cosh(x))*e^(2*x)*sinh(x
)^11 + 2*(184756*a^2*cosh(x)^10 + 546975*a^2*cosh(x)^8 + 600600*a^2*cosh(x)^6 + 300300*a^2*cosh(x)^4 + 69300*a
^2*cosh(x)^2 + 2520*a^2*x)*e^(2*x)*sinh(x)^10 + 20*(16796*a^2*cosh(x)^11 + 60775*a^2*cosh(x)^9 + 85800*a^2*cos
h(x)^7 + 60060*a^2*cosh(x)^5 + 23100*a^2*cosh(x)^3 + 2520*a^2*x*cosh(x))*e^(2*x)*sinh(x)^9 + 30*(8398*a^2*cosh
(x)^12 + 36465*a^2*cosh(x)^10 + 64350*a^2*cosh(x)^8 + 60060*a^2*cosh(x)^6 + 34650*a^2*cosh(x)^4 + 7560*a^2*x*c
osh(x)^2 - 70*a^2)*e^(2*x)*sinh(x)^8 + 240*(646*a^2*cosh(x)^13 + 3315*a^2*cosh(x)^11 + 7150*a^2*cosh(x)^9 + 85
80*a^2*cosh(x)^7 + 6930*a^2*cosh(x)^5 + 2520*a^2*x*cosh(x)^3 - 70*a^2*cosh(x))*e^(2*x)*sinh(x)^7 + 60*(1292*a^
2*cosh(x)^14 + 7735*a^2*cosh(x)^12 + 20020*a^2*cosh(x)^10 + 30030*a^2*cosh(x)^8 + 32340*a^2*cosh(x)^6 + 17640*
a^2*x*cosh(x)^4 - 980*a^2*cosh(x)^2 - 10*a^2)*e^(2*x)*sinh(x)^6 + 24*(1292*a^2*cosh(x)^15 + 8925*a^2*cosh(x)^1
3 + 27300*a^2*cosh(x)^11 + 50050*a^2*cosh(x)^9 + 69300*a^2*cosh(x)^7 + 52920*a^2*x*cosh(x)^5 - 4900*a^2*cosh(x
)^3 - 150*a^2*cosh(x))*e^(2*x)*sinh(x)^5 + 30*(323*a^2*cosh(x)^16 + 2550*a^2*cosh(x)^14 + 9100*a^2*cosh(x)^12
+ 20020*a^2*cosh(x)^10 + 34650*a^2*cosh(x)^8 + 35280*a^2*x*cosh(x)^6 - 4900*a^2*cosh(x)^4 - 300*a^2*cosh(x)^2
- 5*a^2)*e^(2*x)*sinh(x)^4 + 120*(19*a^2*cosh(x)^17 + 170*a^2*cosh(x)^15 + 700*a^2*cosh(x)^13 + 1820*a^2*cosh(
x)^11 + 3850*a^2*cosh(x)^9 + 5040*a^2*x*cosh(x)^7 - 980*a^2*cosh(x)^5 - 100*a^2*cosh(x)^3 - 5*a^2*cosh(x))*e^(
2*x)*sinh(x)^3 + 5*(76*a^2*cosh(x)^18 + 765*a^2*cosh(x)^16 + 3600*a^2*cosh(x)^14 + 10920*a^2*cosh(x)^12 + 2772
0*a^2*cosh(x)^10 + 45360*a^2*x*cosh(x)^8 - 11760*a^2*cosh(x)^6 - 1800*a^2*cosh(x)^4 - 180*a^2*cosh(x)^2 - 5*a^
2)*e^(2*x)*sinh(x)^2 + 10*(4*a^2*cosh(x)^19 + 45*a^2*cosh(x)^17 + 240*a^2*cosh(x)^15 + 840*a^2*cosh(x)^13 + 25
20*a^2*cosh(x)^11 + 5040*a^2*x*cosh(x)^9 - 1680*a^2*cosh(x)^7 - 360*a^2*cosh(x)^5 - 60*a^2*cosh(x)^3 - 5*a^2*c
osh(x))*e^(2*x)*sinh(x) + (2*a^2*cosh(x)^20 + 25*a^2*cosh(x)^18 + 150*a^2*cosh(x)^16 + 600*a^2*cosh(x)^14 + 21
00*a^2*cosh(x)^12 + 5040*a^2*x*cosh(x)^10 - 2100*a^2*cosh(x)^8 - 600*a^2*cosh(x)^6 - 150*a^2*cosh(x)^4 - 25*a^
2*cosh(x)^2 - 2*a^2)*e^(2*x))*sqrt(a*e^(8*x) + 4*a*e^(6*x) + 6*a*e^(4*x) + 4*a*e^(2*x) + a)*e^(-2*x)/(cosh(x)^
10*e^(4*x) + 2*cosh(x)^10*e^(2*x) + (e^(4*x) + 2*e^(2*x) + 1)*sinh(x)^10 + cosh(x)^10 + 10*(cosh(x)*e^(4*x) +
2*cosh(x)*e^(2*x) + cosh(x))*sinh(x)^9 + 45*(cosh(x)^2*e^(4*x) + 2*cosh(x)^2*e^(2*x) + cosh(x)^2)*sinh(x)^8 +
120*(cosh(x)^3*e^(4*x) + 2*cosh(x)^3*e^(2*x) + cosh(x)^3)*sinh(x)^7 + 210*(cosh(x)^4*e^(4*x) + 2*cosh(x)^4*e^(
2*x) + cosh(x)^4)*sinh(x)^6 + 252*(cosh(x)^5*e^(4*x) + 2*cosh(x)^5*e^(2*x) + cosh(x)^5)*sinh(x)^5 + 210*(cosh(
x)^6*e^(4*x) + 2*cosh(x)^6*e^(2*x) + cosh(x)^6)*sinh(x)^4 + 120*(cosh(x)^7*e^(4*x) + 2*cosh(x)^7*e^(2*x) + cos
h(x)^7)*sinh(x)^3 + 45*(cosh(x)^8*e^(4*x) + 2*cosh(x)^8*e^(2*x) + cosh(x)^8)*sinh(x)^2 + 10*(cosh(x)^9*e^(4*x)
+ 2*cosh(x)^9*e^(2*x) + cosh(x)^9)*sinh(x))

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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((a*cosh(x)**4)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.16287, size = 154, normalized size = 1.17 \begin{align*} \frac{1}{20480} \,{\left (5040 \, a^{2} x + 2 \, a^{2} e^{\left (10 \, x\right )} + 25 \, a^{2} e^{\left (8 \, x\right )} + 150 \, a^{2} e^{\left (6 \, x\right )} + 600 \, a^{2} e^{\left (4 \, x\right )} + 2100 \, a^{2} e^{\left (2 \, x\right )} -{\left (5754 \, a^{2} e^{\left (10 \, x\right )} + 2100 \, a^{2} e^{\left (8 \, x\right )} + 600 \, a^{2} e^{\left (6 \, x\right )} + 150 \, a^{2} e^{\left (4 \, x\right )} + 25 \, a^{2} e^{\left (2 \, x\right )} + 2 \, a^{2}\right )} e^{\left (-10 \, x\right )}\right )} \sqrt{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cosh(x)^4)^(5/2),x, algorithm="giac")

[Out]

1/20480*(5040*a^2*x + 2*a^2*e^(10*x) + 25*a^2*e^(8*x) + 150*a^2*e^(6*x) + 600*a^2*e^(4*x) + 2100*a^2*e^(2*x) -
(5754*a^2*e^(10*x) + 2100*a^2*e^(8*x) + 600*a^2*e^(6*x) + 150*a^2*e^(4*x) + 25*a^2*e^(2*x) + 2*a^2)*e^(-10*x)
)*sqrt(a)