∫ 1_____ (a cosh3(x))5∕2 dx [133]

3.2.33 \(\int \frac {1}{(a \cosh ^3(x))^{5/2}} \, dx\) [133]

Optimal. Leaf size=121 \[ \frac {154 i \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right )}{195 a^2 \sqrt {a \cosh ^3(x)}}+\frac {154 \cosh (x) \sinh (x)}{195 a^2 \sqrt {a \cosh ^3(x)}}+\frac {154 \tanh (x)}{585 a^2 \sqrt {a \cosh ^3(x)}}+\frac {22 \text {sech}^2(x) \tanh (x)}{117 a^2 \sqrt {a \cosh ^3(x)}}+\frac {2 \text {sech}^4(x) \tanh (x)}{13 a^2 \sqrt {a \cosh ^3(x)}} \]

[Out]

154/195*I*cosh(x)^(3/2)*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticE(I*sinh(1/2*x),2^(1/2))/a^2/(a*cosh(x)^3)^(

1/2)+154/195*cosh(x)*sinh(x)/a^2/(a*cosh(x)^3)^(1/2)+154/585*tanh(x)/a^2/(a*cosh(x)^3)^(1/2)+22/117*sech(x)^2*

tanh(x)/a^2/(a*cosh(x)^3)^(1/2)+2/13*sech(x)^4*tanh(x)/a^2/(a*cosh(x)^3)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3286, 2716, 2719} \begin {gather*} \frac {154 \sinh (x) \cosh (x)}{195 a^2 \sqrt {a \cosh ^3(x)}}+\frac {154 \tanh (x)}{585 a^2 \sqrt {a \cosh ^3(x)}}+\frac {154 i \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right )}{195 a^2 \sqrt {a \cosh ^3(x)}}+\frac {2 \tanh (x) \text {sech}^4(x)}{13 a^2 \sqrt {a \cosh ^3(x)}}+\frac {22 \tanh (x) \text {sech}^2(x)}{117 a^2 \sqrt {a \cosh ^3(x)}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[(a*Cosh[x]^3)^(-5/2),x]

[Out]

(((154*I)/195)*Cosh[x]^(3/2)*EllipticE[(I/2)*x, 2])/(a^2*Sqrt[a*Cosh[x]^3]) + (154*Cosh[x]*Sinh[x])/(195*a^2*S

qrt[a*Cosh[x]^3]) + (154*Tanh[x])/(585*a^2*Sqrt[a*Cosh[x]^3]) + (22*Sech[x]^2*Tanh[x])/(117*a^2*Sqrt[a*Cosh[x]

^3]) + (2*Sech[x]^4*Tanh[x])/(13*a^2*Sqrt[a*Cosh[x]^3])

Rule 2716

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1

))), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1

] && IntegerQ[2*n]

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{

c, d}, x]

Rule 3286

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

]])

Rubi steps

\begin {align*} \int \frac {1}{\left (a \cosh ^3(x)\right )^{5/2}} \, dx &=\frac {\cosh ^{\frac {3}{2}}(x) \int \frac {1}{\cosh ^{\frac {15}{2}}(x)} \, dx}{a^2 \sqrt {a \cosh ^3(x)}}\\ &=\frac {2 \text {sech}^4(x) \tanh (x)}{13 a^2 \sqrt {a \cosh ^3(x)}}+\frac {\left (11 \cosh ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\cosh ^{\frac {11}{2}}(x)} \, dx}{13 a^2 \sqrt {a \cosh ^3(x)}}\\ &=\frac {22 \text {sech}^2(x) \tanh (x)}{117 a^2 \sqrt {a \cosh ^3(x)}}+\frac {2 \text {sech}^4(x) \tanh (x)}{13 a^2 \sqrt {a \cosh ^3(x)}}+\frac {\left (77 \cosh ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\cosh ^{\frac {7}{2}}(x)} \, dx}{117 a^2 \sqrt {a \cosh ^3(x)}}\\ &=\frac {154 \tanh (x)}{585 a^2 \sqrt {a \cosh ^3(x)}}+\frac {22 \text {sech}^2(x) \tanh (x)}{117 a^2 \sqrt {a \cosh ^3(x)}}+\frac {2 \text {sech}^4(x) \tanh (x)}{13 a^2 \sqrt {a \cosh ^3(x)}}+\frac {\left (77 \cosh ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\cosh ^{\frac {3}{2}}(x)} \, dx}{195 a^2 \sqrt {a \cosh ^3(x)}}\\ &=\frac {154 \cosh (x) \sinh (x)}{195 a^2 \sqrt {a \cosh ^3(x)}}+\frac {154 \tanh (x)}{585 a^2 \sqrt {a \cosh ^3(x)}}+\frac {22 \text {sech}^2(x) \tanh (x)}{117 a^2 \sqrt {a \cosh ^3(x)}}+\frac {2 \text {sech}^4(x) \tanh (x)}{13 a^2 \sqrt {a \cosh ^3(x)}}-\frac {\left (77 \cosh ^{\frac {3}{2}}(x)\right ) \int \sqrt {\cosh (x)} \, dx}{195 a^2 \sqrt {a \cosh ^3(x)}}\\ &=\frac {154 i \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right )}{195 a^2 \sqrt {a \cosh ^3(x)}}+\frac {154 \cosh (x) \sinh (x)}{195 a^2 \sqrt {a \cosh ^3(x)}}+\frac {154 \tanh (x)}{585 a^2 \sqrt {a \cosh ^3(x)}}+\frac {22 \text {sech}^2(x) \tanh (x)}{117 a^2 \sqrt {a \cosh ^3(x)}}+\frac {2 \text {sech}^4(x) \tanh (x)}{13 a^2 \sqrt {a \cosh ^3(x)}}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 61, normalized size = 0.50 \begin {gather*} \frac {462 i \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right )+462 \cosh (x) \sinh (x)+2 \left (77+55 \text {sech}^2(x)+45 \text {sech}^4(x)\right ) \tanh (x)}{585 a^2 \sqrt {a \cosh ^3(x)}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a*Cosh[x]^3)^(-5/2),x]

[Out]

((462*I)*Cosh[x]^(3/2)*EllipticE[(I/2)*x, 2] + 462*Cosh[x]*Sinh[x] + 2*(77 + 55*Sech[x]^2 + 45*Sech[x]^4)*Tanh

[x])/(585*a^2*Sqrt[a*Cosh[x]^3])

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Maple [F]
time = 0.92, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \left (\cosh ^{3}\left (x \right )\right )\right )^{\frac {5}{2}}}\, dx\]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*cosh(x)^3)^(5/2),x)

[Out]

int(1/(a*cosh(x)^3)^(5/2),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x)^3)^(5/2),x, algorithm="maxima")

[Out]

integrate((a*cosh(x)^3)^(-5/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.14, size = 1668, normalized size = 13.79 \begin {gather*} \text {Too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x)^3)^(5/2),x, algorithm="fricas")

[Out]

2/585*(231*(sqrt(2)*cosh(x)^14 + 14*sqrt(2)*cosh(x)*sinh(x)^13 + sqrt(2)*sinh(x)^14 + 7*(13*sqrt(2)*cosh(x)^2

+ sqrt(2))*sinh(x)^12 + 7*sqrt(2)*cosh(x)^12 + 28*(13*sqrt(2)*cosh(x)^3 + 3*sqrt(2)*cosh(x))*sinh(x)^11 + 7*(1

43*sqrt(2)*cosh(x)^4 + 66*sqrt(2)*cosh(x)^2 + 3*sqrt(2))*sinh(x)^10 + 21*sqrt(2)*cosh(x)^10 + 14*(143*sqrt(2)*

cosh(x)^5 + 110*sqrt(2)*cosh(x)^3 + 15*sqrt(2)*cosh(x))*sinh(x)^9 + 7*(429*sqrt(2)*cosh(x)^6 + 495*sqrt(2)*cos

h(x)^4 + 135*sqrt(2)*cosh(x)^2 + 5*sqrt(2))*sinh(x)^8 + 35*sqrt(2)*cosh(x)^8 + 8*(429*sqrt(2)*cosh(x)^7 + 693*

sqrt(2)*cosh(x)^5 + 315*sqrt(2)*cosh(x)^3 + 35*sqrt(2)*cosh(x))*sinh(x)^7 + 7*(429*sqrt(2)*cosh(x)^8 + 924*sqr

t(2)*cosh(x)^6 + 630*sqrt(2)*cosh(x)^4 + 140*sqrt(2)*cosh(x)^2 + 5*sqrt(2))*sinh(x)^6 + 35*sqrt(2)*cosh(x)^6 +

14*(143*sqrt(2)*cosh(x)^9 + 396*sqrt(2)*cosh(x)^7 + 378*sqrt(2)*cosh(x)^5 + 140*sqrt(2)*cosh(x)^3 + 15*sqrt(2

)*cosh(x))*sinh(x)^5 + 7*(143*sqrt(2)*cosh(x)^10 + 495*sqrt(2)*cosh(x)^8 + 630*sqrt(2)*cosh(x)^6 + 350*sqrt(2)

*cosh(x)^4 + 75*sqrt(2)*cosh(x)^2 + 3*sqrt(2))*sinh(x)^4 + 21*sqrt(2)*cosh(x)^4 + 28*(13*sqrt(2)*cosh(x)^11 +

55*sqrt(2)*cosh(x)^9 + 90*sqrt(2)*cosh(x)^7 + 70*sqrt(2)*cosh(x)^5 + 25*sqrt(2)*cosh(x)^3 + 3*sqrt(2)*cosh(x))

*sinh(x)^3 + 7*(13*sqrt(2)*cosh(x)^12 + 66*sqrt(2)*cosh(x)^10 + 135*sqrt(2)*cosh(x)^8 + 140*sqrt(2)*cosh(x)^6

+ 75*sqrt(2)*cosh(x)^4 + 18*sqrt(2)*cosh(x)^2 + sqrt(2))*sinh(x)^2 + 7*sqrt(2)*cosh(x)^2 + 14*(sqrt(2)*cosh(x)

^13 + 6*sqrt(2)*cosh(x)^11 + 15*sqrt(2)*cosh(x)^9 + 20*sqrt(2)*cosh(x)^7 + 15*sqrt(2)*cosh(x)^5 + 6*sqrt(2)*co

sh(x)^3 + sqrt(2)*cosh(x))*sinh(x) + sqrt(2))*sqrt(a)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cosh(x

) + sinh(x))) + 2*(231*cosh(x)^14 + 3234*cosh(x)*sinh(x)^13 + 231*sinh(x)^14 + 77*(273*cosh(x)^2 + 20)*sinh(x)

^12 + 1540*cosh(x)^12 + 924*(91*cosh(x)^3 + 20*cosh(x))*sinh(x)^11 + 11*(21021*cosh(x)^4 + 9240*cosh(x)^2 + 39

7)*sinh(x)^10 + 4367*cosh(x)^10 + 22*(21021*cosh(x)^5 + 15400*cosh(x)^3 + 1985*cosh(x))*sinh(x)^9 + (693693*co

sh(x)^6 + 762300*cosh(x)^4 + 196515*cosh(x)^2 + 6808)*sinh(x)^8 + 6808*cosh(x)^8 + 8*(99099*cosh(x)^7 + 152460

*cosh(x)^5 + 65505*cosh(x)^3 + 6808*cosh(x))*sinh(x)^7 + (693693*cosh(x)^8 + 1422960*cosh(x)^6 + 917070*cosh(x

)^4 + 190624*cosh(x)^2 + 1277)*sinh(x)^6 + 1277*cosh(x)^6 + 2*(231231*cosh(x)^9 + 609840*cosh(x)^7 + 550242*co

sh(x)^5 + 190624*cosh(x)^3 + 3831*cosh(x))*sinh(x)^5 + (231231*cosh(x)^10 + 762300*cosh(x)^8 + 917070*cosh(x)^

6 + 476560*cosh(x)^4 + 19155*cosh(x)^2 + 484)*sinh(x)^4 + 484*cosh(x)^4 + 4*(21021*cosh(x)^11 + 84700*cosh(x)^

9 + 131010*cosh(x)^7 + 95312*cosh(x)^5 + 6385*cosh(x)^3 + 484*cosh(x))*sinh(x)^3 + (21021*cosh(x)^12 + 101640*

cosh(x)^10 + 196515*cosh(x)^8 + 190624*cosh(x)^6 + 19155*cosh(x)^4 + 2904*cosh(x)^2 + 77)*sinh(x)^2 + 77*cosh(

x)^2 + 2*(1617*cosh(x)^13 + 9240*cosh(x)^11 + 21835*cosh(x)^9 + 27232*cosh(x)^7 + 3831*cosh(x)^5 + 968*cosh(x)

^3 + 77*cosh(x))*sinh(x))*sqrt(a*cosh(x)))/(a^3*cosh(x)^14 + 14*a^3*cosh(x)*sinh(x)^13 + a^3*sinh(x)^14 + 7*a^

3*cosh(x)^12 + 21*a^3*cosh(x)^10 + 7*(13*a^3*cosh(x)^2 + a^3)*sinh(x)^12 + 28*(13*a^3*cosh(x)^3 + 3*a^3*cosh(x

))*sinh(x)^11 + 35*a^3*cosh(x)^8 + 7*(143*a^3*cosh(x)^4 + 66*a^3*cosh(x)^2 + 3*a^3)*sinh(x)^10 + 14*(143*a^3*c

osh(x)^5 + 110*a^3*cosh(x)^3 + 15*a^3*cosh(x))*sinh(x)^9 + 35*a^3*cosh(x)^6 + 7*(429*a^3*cosh(x)^6 + 495*a^3*c

osh(x)^4 + 135*a^3*cosh(x)^2 + 5*a^3)*sinh(x)^8 + 8*(429*a^3*cosh(x)^7 + 693*a^3*cosh(x)^5 + 315*a^3*cosh(x)^3

+ 35*a^3*cosh(x))*sinh(x)^7 + 21*a^3*cosh(x)^4 + 7*(429*a^3*cosh(x)^8 + 924*a^3*cosh(x)^6 + 630*a^3*cosh(x)^4

+ 140*a^3*cosh(x)^2 + 5*a^3)*sinh(x)^6 + 14*(143*a^3*cosh(x)^9 + 396*a^3*cosh(x)^7 + 378*a^3*cosh(x)^5 + 140*

a^3*cosh(x)^3 + 15*a^3*cosh(x))*sinh(x)^5 + 7*a^3*cosh(x)^2 + 7*(143*a^3*cosh(x)^10 + 495*a^3*cosh(x)^8 + 630*

a^3*cosh(x)^6 + 350*a^3*cosh(x)^4 + 75*a^3*cosh(x)^2 + 3*a^3)*sinh(x)^4 + 28*(13*a^3*cosh(x)^11 + 55*a^3*cosh(

x)^9 + 90*a^3*cosh(x)^7 + 70*a^3*cosh(x)^5 + 25*a^3*cosh(x)^3 + 3*a^3*cosh(x))*sinh(x)^3 + a^3 + 7*(13*a^3*cos

h(x)^12 + 66*a^3*cosh(x)^10 + 135*a^3*cosh(x)^8 + 140*a^3*cosh(x)^6 + 75*a^3*cosh(x)^4 + 18*a^3*cosh(x)^2 + a^

3)*sinh(x)^2 + 14*(a^3*cosh(x)^13 + 6*a^3*cosh(x)^11 + 15*a^3*cosh(x)^9 + 20*a^3*cosh(x)^7 + 15*a^3*cosh(x)^5

+ 6*a^3*cosh(x)^3 + a^3*cosh(x))*sinh(x))

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x)**3)**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3880 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x)^3)^(5/2),x, algorithm="giac")

[Out]

integrate((a*cosh(x)^3)^(-5/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (a\,{\mathrm {cosh}\left (x\right )}^3\right )}^{5/2}} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*cosh(x)^3)^(5/2),x)

[Out]

int(1/(a*cosh(x)^3)^(5/2), x)

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