3.593 \(\int \frac{\sinh (x)}{\left (-9+4 \cosh ^2(x)\right )^{5/2}} \, dx\)

Optimal. Leaf size=37 \[ \frac{2 \cosh (x)}{243 \sqrt{4 \cosh ^2(x)-9}}-\frac{\cosh (x)}{27 \left (4 \cosh ^2(x)-9\right )^{3/2}} \]

[Out]

-Cosh[x]/(27*(-9 + 4*Cosh[x]^2)^(3/2)) + (2*Cosh[x])/(243*Sqrt[-9 + 4*Cosh[x]^2]
)

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Rubi [A]  time = 0.0691723, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 \cosh (x)}{243 \sqrt{4 \cosh ^2(x)-9}}-\frac{\cosh (x)}{27 \left (4 \cosh ^2(x)-9\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sinh[x]/(-9 + 4*Cosh[x]^2)^(5/2),x]

[Out]

-Cosh[x]/(27*(-9 + 4*Cosh[x]^2)^(3/2)) + (2*Cosh[x])/(243*Sqrt[-9 + 4*Cosh[x]^2]
)

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Rubi in Sympy [A]  time = 3.11215, size = 34, normalized size = 0.92 \[ \frac{2 \cosh{\left (x \right )}}{243 \sqrt{4 \cosh ^{2}{\left (x \right )} - 9}} - \frac{\cosh{\left (x \right )}}{27 \left (4 \cosh ^{2}{\left (x \right )} - 9\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sinh(x)/(-9+4*cosh(x)**2)**(5/2),x)

[Out]

2*cosh(x)/(243*sqrt(4*cosh(x)**2 - 9)) - cosh(x)/(27*(4*cosh(x)**2 - 9)**(3/2))

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Mathematica [A]  time = 0.0611392, size = 26, normalized size = 0.7 \[ \frac{\cosh (x) (4 \cosh (2 x)-23)}{243 (2 \cosh (2 x)-7)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sinh[x]/(-9 + 4*Cosh[x]^2)^(5/2),x]

[Out]

(Cosh[x]*(-23 + 4*Cosh[2*x]))/(243*(-7 + 2*Cosh[2*x])^(3/2))

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Maple [A]  time = 0.019, size = 30, normalized size = 0.8 \[ -{\frac{\cosh \left ( x \right ) }{27} \left ( -9+4\, \left ( \cosh \left ( x \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,\cosh \left ( x \right ) }{243}{\frac{1}{\sqrt{-9+4\, \left ( \cosh \left ( x \right ) \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sinh(x)/(-9+4*cosh(x)^2)^(5/2),x)

[Out]

-1/27*cosh(x)/(-9+4*cosh(x)^2)^(3/2)+2/243*cosh(x)/(-9+4*cosh(x)^2)^(1/2)

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Maxima [A]  time = 1.43846, size = 169, normalized size = 4.57 \[ -\frac{1855 \, e^{\left (-2 \, x\right )} - 8485 \, e^{\left (-4 \, x\right )} + 5285 \, e^{\left (-6 \, x\right )} - 980 \, e^{\left (-8 \, x\right )} + 56 \, e^{\left (-10 \, x\right )} - 106}{12150 \,{\left (3 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right )}^{\frac{5}{2}}{\left (-3 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right )}^{\frac{5}{2}}} + \frac{980 \, e^{\left (-2 \, x\right )} - 5285 \, e^{\left (-4 \, x\right )} + 8485 \, e^{\left (-6 \, x\right )} - 1855 \, e^{\left (-8 \, x\right )} + 106 \, e^{\left (-10 \, x\right )} - 56}{12150 \,{\left (3 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right )}^{\frac{5}{2}}{\left (-3 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sinh(x)/(4*cosh(x)^2 - 9)^(5/2),x, algorithm="maxima")

[Out]

-1/12150*(1855*e^(-2*x) - 8485*e^(-4*x) + 5285*e^(-6*x) - 980*e^(-8*x) + 56*e^(-
10*x) - 106)/((3*e^(-x) + e^(-2*x) + 1)^(5/2)*(-3*e^(-x) + e^(-2*x) + 1)^(5/2))
+ 1/12150*(980*e^(-2*x) - 5285*e^(-4*x) + 8485*e^(-6*x) - 1855*e^(-8*x) + 106*e^
(-10*x) - 56)/((3*e^(-x) + e^(-2*x) + 1)^(5/2)*(-3*e^(-x) + e^(-2*x) + 1)^(5/2))

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Fricas [A]  time = 0.274774, size = 3929, normalized size = 106.19 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sinh(x)/(4*cosh(x)^2 - 9)^(5/2),x, algorithm="fricas")

[Out]

1/2*(256*cosh(x)^18 + 3584*cosh(x)*sinh(x)^17 + 256*sinh(x)^18 + 256*(89*cosh(x)
^2 - 24)*sinh(x)^16 - 6144*cosh(x)^16 + 12288*(7*cosh(x)^3 - 6*cosh(x))*sinh(x)^
15 + 16*(13120*cosh(x)^4 - 24576*cosh(x)^2 + 3617)*sinh(x)^14 + 57872*cosh(x)^14
+ 32*(10304*cosh(x)^5 - 37632*cosh(x)^3 + 18117*cosh(x))*sinh(x)^13 + 16*(17472
*cosh(x)^6 - 139776*cosh(x)^4 + 156299*cosh(x)^2 - 17013)*sinh(x)^12 - 272208*co
sh(x)^12 - 64*(832*cosh(x)^7 + 34944*cosh(x)^5 - 91481*cosh(x)^3 + 34410*cosh(x)
)*sinh(x)^11 - 2*(256256*cosh(x)^8 - 3613896*cosh(x)^4 + 3661584*cosh(x)^2 - 342
187)*sinh(x)^10 + 684374*cosh(x)^10 - 4*(183040*cosh(x)^9 - 878592*cosh(x)^7 - 4
45016*cosh(x)^5 + 2998560*cosh(x)^3 - 1083665*cosh(x))*sinh(x)^9 - 2*(256256*cos
h(x)^10 - 2635776*cosh(x)^8 + 4368936*cosh(x)^6 + 3516120*cosh(x)^4 - 5362095*co
sh(x)^2 + 484868)*sinh(x)^8 - 969736*cosh(x)^8 - 16*(3328*cosh(x)^11 - 219648*co
sh(x)^9 + 895752*cosh(x)^7 - 494064*cosh(x)^5 - 741915*cosh(x)^3 + 306774*cosh(x
))*sinh(x)^7 + 4*(69888*cosh(x)^12 - 2184468*cosh(x)^8 + 4168080*cosh(x)^6 + 802
515*cosh(x)^4 - 2513896*cosh(x)^2 + 228401)*sinh(x)^6 + 913604*cosh(x)^6 + 8*(41
216*cosh(x)^13 - 279552*cosh(x)^11 + 222508*cosh(x)^9 + 988128*cosh(x)^7 - 39666
9*cosh(x)^5 - 1445332*cosh(x)^3 + 515771*cosh(x))*sinh(x)^5 + 2*(104960*cosh(x)^
14 - 1118208*cosh(x)^12 + 3613896*cosh(x)^10 - 3516120*cosh(x)^8 + 1605030*cosh(
x)^6 - 5445720*cosh(x)^4 + 4141118*cosh(x)^2 - 325331)*sinh(x)^4 - 650662*cosh(x
)^4 + 16*(5376*cosh(x)^15 - 75264*cosh(x)^13 + 365924*cosh(x)^11 - 749640*cosh(x
)^9 + 741915*cosh(x)^7 - 722666*cosh(x)^5 + 633709*cosh(x)^3 - 145279*cosh(x))*s
inh(x)^3 + 2*(11392*cosh(x)^16 - 196608*cosh(x)^14 + 1250392*cosh(x)^12 - 366158
4*cosh(x)^10 + 5362095*cosh(x)^8 - 5027792*cosh(x)^6 + 4141118*cosh(x)^4 - 16738
02*cosh(x)^2 + 112164)*sinh(x)^2 + 224328*cosh(x)^2 + 4*(896*cosh(x)^17 - 18432*
cosh(x)^15 + 144936*cosh(x)^13 - 550560*cosh(x)^11 + 1083665*cosh(x)^9 - 1227096
*cosh(x)^7 + 1031542*cosh(x)^5 - 581116*cosh(x)^3 + 107790*cosh(x))*sinh(x) - (5
12*cosh(x)^16 + 6144*cosh(x)*sinh(x)^15 + 512*sinh(x)^16 + 256*(128*cosh(x)^2 -
41)*sinh(x)^14 - 10496*cosh(x)^14 + 512*(196*cosh(x)^3 - 205*cosh(x))*sinh(x)^13
+ 32*(5824*cosh(x)^4 - 14104*cosh(x)^2 + 2527)*sinh(x)^12 + 80864*cosh(x)^12 +
256*(728*cosh(x)^5 - 4100*cosh(x)^3 + 2527*cosh(x))*sinh(x)^11 - 16*(79376*cosh(
x)^4 - 131404*cosh(x)^2 + 18085)*sinh(x)^10 - 289360*cosh(x)^10 - 32*(9152*cosh(
x)^7 + 7216*cosh(x)^5 - 101080*cosh(x)^3 + 54255*cosh(x))*sinh(x)^9 - 2*(219648*
cosh(x)^8 - 865920*cosh(x)^6 - 606480*cosh(x)^4 + 1880840*cosh(x)^2 - 240425)*si
nh(x)^8 + 480850*cosh(x)^8 - 8*(36608*cosh(x)^9 - 346368*cosh(x)^7 + 485184*cosh
(x)^5 + 289360*cosh(x)^3 - 240425*cosh(x))*sinh(x)^7 + (1731840*cosh(x)^8 - 6792
576*cosh(x)^6 + 4051040*cosh(x)^4 + 1923400*cosh(x)^2 - 326917)*sinh(x)^6 - 3269
17*cosh(x)^6 + 2*(93184*cosh(x)^11 - 115456*cosh(x)^9 - 1940736*cosh(x)^7 + 4051
040*cosh(x)^5 - 961700*cosh(x)^3 - 326917*cosh(x))*sinh(x)^5 + (186368*cosh(x)^1
2 - 1270016*cosh(x)^10 + 1212960*cosh(x)^8 + 4051040*cosh(x)^6 - 4808500*cosh(x)
^4 + 326917*cosh(x)^2 + 72171)*sinh(x)^4 + 72171*cosh(x)^4 + 4*(25088*cosh(x)^13
- 262400*cosh(x)^11 + 808640*cosh(x)^9 - 578720*cosh(x)^7 - 480850*cosh(x)^5 +
326917*cosh(x)^3)*sinh(x)^3 + (32768*cosh(x)^14 - 451328*cosh(x)^12 + 2102464*co
sh(x)^10 - 3761680*cosh(x)^8 + 1923400*cosh(x)^6 + 326917*cosh(x)^4 - 144342*cos
h(x)^2 - 4374)*sinh(x)^2 - 4374*cosh(x)^2 + 2*(3072*cosh(x)^15 - 52480*cosh(x)^1
3 + 323456*cosh(x)^11 - 868080*cosh(x)^9 + 961700*cosh(x)^7 - 326917*cosh(x)^5 +
4374*cosh(x))*sinh(x))*sqrt((2*cosh(x)^2 + 2*sinh(x)^2 - 7)/(cosh(x)^2 - 2*cosh
(x)*sinh(x) + sinh(x)^2)) - 17934)/(1024*cosh(x)^22 + 18432*cosh(x)*sinh(x)^21 +
1024*sinh(x)^22 + 1024*(151*cosh(x)^2 - 35)*sinh(x)^20 - 35840*cosh(x)^20 + 204
80*(39*cosh(x)^3 - 28*cosh(x))*sinh(x)^19 + 128*(22040*cosh(x)^4 - 33040*cosh(x)
^2 + 4111)*sinh(x)^18 + 526208*cosh(x)^18 + 256*(27816*cosh(x)^5 - 73920*cosh(x)
^3 + 28793*cosh(x))*sinh(x)^17 + 128*(100776*cosh(x)^6 - 442680*cosh(x)^4 + 3663
91*cosh(x)^2 - 32844)*sinh(x)^16 - 4204032*cosh(x)^16 + 2048*(7752*cosh(x)^7 - 5
7120*cosh(x)^5 + 86571*cosh(x)^3 - 24703*cosh(x))*sinh(x)^15 + 4*(2480640*cosh(x
)^8 - 40212480*cosh(x)^6 + 108446080*cosh(x)^4 - 67766272*cosh(x)^2 + 4969137)*s
inh(x)^14 + 19876548*cosh(x)^14 - 8*(826880*cosh(x)^9 + 14622720*cosh(x)^7 - 856
51328*cosh(x)^5 + 104629504*cosh(x)^3 - 25107253*cosh(x))*sinh(x)^13 - 4*(644966
4*cosh(x)^10 - 11880960*cosh(x)^8 - 148127616*cosh(x)^6 + 395612672*cosh(x)^4 -
219950523*cosh(x)^2 + 14276325)*sinh(x)^12 - 57105300*cosh(x)^12 - 16*(2149888*c
osh(x)^11 - 15841280*cosh(x)^9 + 4790656*cosh(x)^7 + 104610688*cosh(x)^5 - 13286
0169*cosh(x)^3 + 29580362*cosh(x))*sinh(x)^11 - 4*(6449664*cosh(x)^12 - 87127040
*cosh(x)^10 + 251652544*cosh(x)^8 + 71751680*cosh(x)^6 - 716355497*cosh(x)^4 + 4
12292930*cosh(x)^2 - 25234392)*sinh(x)^10 + 100937568*cosh(x)^10 - 8*(826880*cos
h(x)^13 - 31682560*cosh(x)^11 + 181529920*cosh(x)^9 - 246774528*cosh(x)^7 - 1841
36667*cosh(x)^5 + 378020580*cosh(x)^3 - 85057450*cosh(x))*sinh(x)^9 + 4*(2480640
*cosh(x)^14 + 11880960*cosh(x)^12 - 251652544*cosh(x)^10 + 778761984*cosh(x)^8 -
405583893*cosh(x)^6 - 707446635*cosh(x)^4 + 477715480*cosh(x)^2 - 28833252)*sin
h(x)^8 - 115333008*cosh(x)^8 + 16*(992256*cosh(x)^15 - 7311360*cosh(x)^13 - 4790
656*cosh(x)^11 + 123387264*cosh(x)^9 - 207118626*cosh(x)^7 - 44503620*cosh(x)^5
+ 181428620*cosh(x)^3 - 41154953*cosh(x))*sinh(x)^7 + 2*(6449664*cosh(x)^16 - 80
424960*cosh(x)^14 + 296255232*cosh(x)^12 - 143503360*cosh(x)^10 - 811167786*cosh
(x)^8 + 326555208*cosh(x)^6 + 1388794400*cosh(x)^4 - 822107664*cosh(x)^2 + 47077
041)*sinh(x)^6 + 94154082*cosh(x)^6 + 8*(890112*cosh(x)^17 - 14622720*cosh(x)^15
+ 85651328*cosh(x)^13 - 209221376*cosh(x)^11 + 184136667*cosh(x)^9 - 89007240*c
osh(x)^7 + 301517692*cosh(x)^5 - 311984526*cosh(x)^3 + 58966813*cosh(x))*sinh(x)
^5 + (2821120*cosh(x)^18 - 56663040*cosh(x)^16 + 433784320*cosh(x)^14 - 15824506
88*cosh(x)^12 + 2865421988*cosh(x)^10 - 2829786540*cosh(x)^8 + 2777588800*cosh(x
)^6 - 2789614240*cosh(x)^4 + 1039551278*cosh(x)^2 - 53093873)*sinh(x)^4 - 530938
73*cosh(x)^4 + 8*(99840*cosh(x)^19 - 2365440*cosh(x)^17 + 22162176*cosh(x)^15 -
104629504*cosh(x)^13 + 265720338*cosh(x)^11 - 378020580*cosh(x)^9 + 362857240*co
sh(x)^7 - 311984526*cosh(x)^5 + 165492714*cosh(x)^3 - 24496633*cosh(x))*sinh(x)^
3 + (154624*cosh(x)^20 - 4229120*cosh(x)^18 + 46898048*cosh(x)^16 - 271065088*co
sh(x)^14 + 879802092*cosh(x)^12 - 1649171720*cosh(x)^10 + 1910861920*cosh(x)^8 -
1644215328*cosh(x)^6 + 1039551278*cosh(x)^4 - 285758382*cosh(x)^2 + 14806051)*s
inh(x)^2 + 14806051*cosh(x)^2 + 2*(9216*cosh(x)^21 - 286720*cosh(x)^19 + 3685504
*cosh(x)^17 - 25295872*cosh(x)^15 + 100429012*cosh(x)^13 - 236642896*cosh(x)^11
+ 340229800*cosh(x)^9 - 329239624*cosh(x)^7 + 235867252*cosh(x)^5 - 97986532*cos
h(x)^3 + 14258985*cosh(x))*sinh(x) - 4*(512*cosh(x)^20 + 8192*cosh(x)*sinh(x)^19
+ 512*sinh(x)^20 + 256*(236*cosh(x)^2 - 63)*sinh(x)^18 - 16128*cosh(x)^18 + 153
6*(176*cosh(x)^3 - 147*cosh(x))*sinh(x)^17 + 384*(2108*cosh(x)^4 - 3738*cosh(x)^
2 + 543)*sinh(x)^16 + 208512*cosh(x)^16 + 1536*(1088*cosh(x)^5 - 3528*cosh(x)^3
+ 1629*cosh(x))*sinh(x)^15 + 192*(11968*cosh(x)^6 - 68880*cosh(x)^4 + 69504*cosh
(x)^2 - 7399)*sinh(x)^14 - 1420608*cosh(x)^14 + 384*(4352*cosh(x)^7 - 54096*cosh
(x)^5 + 106428*cosh(x)^3 - 36995*cosh(x))*sinh(x)^13 - 6*(113152*cosh(x)^8 + 293
5296*cosh(x)^6 - 12649728*cosh(x)^4 + 10181024*cosh(x)^2 - 907017)*sinh(x)^12 +
5442102*cosh(x)^12 - 16*(226304*cosh(x)^9 - 209664*cosh(x)^7 - 4743648*cosh(x)^5
+ 8878800*cosh(x)^3 - 2721051*cosh(x))*sinh(x)^11 - (4978688*cosh(x)^10 - 32288
256*cosh(x)^8 + 171893568*cosh(x)^4 - 141494652*cosh(x)^2 + 11566737)*sinh(x)^10
- 11566737*cosh(x)^10 - 2*(1810432*cosh(x)^11 - 23063040*cosh(x)^9 + 59634432*c
osh(x)^7 + 15626688*cosh(x)^5 - 108842040*cosh(x)^3 + 34700211*cosh(x))*sinh(x)^
9 - 3*(226304*cosh(x)^12 - 10762752*cosh(x)^10 + 59634432*cosh(x)^8 - 78133440*c
osh(x)^6 - 27210510*cosh(x)^4 + 50122527*cosh(x)^2 - 4200776)*sinh(x)^8 + 126023
28*cosh(x)^8 + 24*(69632*cosh(x)^13 + 139776*cosh(x)^11 - 4969536*cosh(x)^9 + 15
626688*cosh(x)^7 - 10884204*cosh(x)^5 - 3855579*cosh(x)^3 + 2100388*cosh(x))*sin
h(x)^7 + 3*(765952*cosh(x)^14 - 5870592*cosh(x)^12 + 78133440*cosh(x)^8 - 152378
856*cosh(x)^6 + 53978106*cosh(x)^4 + 16803104*cosh(x)^2 - 2010729)*sinh(x)^6 - 6
032187*cosh(x)^6 + 6*(278528*cosh(x)^15 - 3462144*cosh(x)^13 + 12649728*cosh(x)^
11 - 5208896*cosh(x)^9 - 43536816*cosh(x)^7 + 53978106*cosh(x)^5 - 8401552*cosh(
x)^3 - 2010729*cosh(x))*sinh(x)^5 + 3*(269824*cosh(x)^16 - 4408320*cosh(x)^14 +
25299456*cosh(x)^12 - 57297856*cosh(x)^10 + 27210510*cosh(x)^8 + 53978106*cosh(x
)^6 - 42007760*cosh(x)^4 + 2010729*cosh(x)^2 + 362436)*sinh(x)^4 + 1087308*cosh(
x)^4 + 12*(22528*cosh(x)^17 - 451584*cosh(x)^15 + 3405696*cosh(x)^13 - 11838400*
cosh(x)^11 + 18140340*cosh(x)^9 - 7711158*cosh(x)^7 - 4200776*cosh(x)^5 + 201072
9*cosh(x)^3)*sinh(x)^3 + (60416*cosh(x)^18 - 1435392*cosh(x)^16 + 13344768*cosh(
x)^14 - 61086144*cosh(x)^12 + 141494652*cosh(x)^10 - 150367581*cosh(x)^8 + 50409
312*cosh(x)^6 + 6032187*cosh(x)^4 - 2174616*cosh(x)^2 - 64477)*sinh(x)^2 - 64477
*cosh(x)^2 + 2*(4096*cosh(x)^19 - 112896*cosh(x)^17 + 1251072*cosh(x)^15 - 71030
40*cosh(x)^13 + 21768408*cosh(x)^11 - 34700211*cosh(x)^9 + 25204656*cosh(x)^7 -
6032187*cosh(x)^5 + 64477*cosh(x))*sinh(x))*sqrt((2*cosh(x)^2 + 2*sinh(x)^2 - 7)
/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) - 1276303)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sinh(x)/(-9+4*cosh(x)**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.224391, size = 54, normalized size = 1.46 \[ \frac{{\left ({\left (2 \, e^{\left (2 \, x\right )} - 21\right )} e^{\left (2 \, x\right )} - 21\right )} e^{\left (2 \, x\right )} + 2}{486 \,{\left (e^{\left (4 \, x\right )} - 7 \, e^{\left (2 \, x\right )} + 1\right )}^{\frac{3}{2}}} - \frac{1}{243} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sinh(x)/(4*cosh(x)^2 - 9)^(5/2),x, algorithm="giac")

[Out]

1/486*(((2*e^(2*x) - 21)*e^(2*x) - 21)*e^(2*x) + 2)/(e^(4*x) - 7*e^(2*x) + 1)^(3
/2) - 1/243