3.5.11 \(\int \frac {\cosh ^5(c+d x)}{a+b \sqrt {\sinh (c+d x)}} \, dx\) [411]

Optimal. Leaf size=259 \[ -\frac {2 a \left (a^4+b^4\right )^2 \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{b^{10} d}+\frac {2 \left (a^4+b^4\right )^2 \sqrt {\sinh (c+d x)}}{b^9 d}-\frac {a^3 \left (a^4+2 b^4\right ) \sinh (c+d x)}{b^8 d}+\frac {2 a^2 \left (a^4+2 b^4\right ) \sinh ^{\frac {3}{2}}(c+d x)}{3 b^7 d}-\frac {a \left (a^4+2 b^4\right ) \sinh ^2(c+d x)}{2 b^6 d}+\frac {2 \left (a^4+2 b^4\right ) \sinh ^{\frac {5}{2}}(c+d x)}{5 b^5 d}-\frac {a^3 \sinh ^3(c+d x)}{3 b^4 d}+\frac {2 a^2 \sinh ^{\frac {7}{2}}(c+d x)}{7 b^3 d}-\frac {a \sinh ^4(c+d x)}{4 b^2 d}+\frac {2 \sinh ^{\frac {9}{2}}(c+d x)}{9 b d} \]

[Out]

-2*a*(a^4+b^4)^2*ln(a+b*sinh(d*x+c)^(1/2))/b^10/d-a^3*(a^4+2*b^4)*sinh(d*x+c)/b^8/d+2/3*a^2*(a^4+2*b^4)*sinh(d
*x+c)^(3/2)/b^7/d-1/2*a*(a^4+2*b^4)*sinh(d*x+c)^2/b^6/d+2/5*(a^4+2*b^4)*sinh(d*x+c)^(5/2)/b^5/d-1/3*a^3*sinh(d
*x+c)^3/b^4/d+2/7*a^2*sinh(d*x+c)^(7/2)/b^3/d-1/4*a*sinh(d*x+c)^4/b^2/d+2/9*sinh(d*x+c)^(9/2)/b/d+2*(a^4+b^4)^
2*sinh(d*x+c)^(1/2)/b^9/d

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Rubi [A]
time = 0.20, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3302, 1904, 1634} \begin {gather*} -\frac {2 a \left (a^4+b^4\right )^2 \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{b^{10} d}+\frac {2 \left (a^4+b^4\right )^2 \sqrt {\sinh (c+d x)}}{b^9 d}-\frac {a \left (a^4+2 b^4\right ) \sinh ^2(c+d x)}{2 b^6 d}+\frac {2 \left (a^4+2 b^4\right ) \sinh ^{\frac {5}{2}}(c+d x)}{5 b^5 d}-\frac {a^3 \sinh ^3(c+d x)}{3 b^4 d}+\frac {2 a^2 \sinh ^{\frac {7}{2}}(c+d x)}{7 b^3 d}-\frac {a^3 \left (a^4+2 b^4\right ) \sinh (c+d x)}{b^8 d}+\frac {2 a^2 \left (a^4+2 b^4\right ) \sinh ^{\frac {3}{2}}(c+d x)}{3 b^7 d}-\frac {a \sinh ^4(c+d x)}{4 b^2 d}+\frac {2 \sinh ^{\frac {9}{2}}(c+d x)}{9 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]^5/(a + b*Sqrt[Sinh[c + d*x]]),x]

[Out]

(-2*a*(a^4 + b^4)^2*Log[a + b*Sqrt[Sinh[c + d*x]]])/(b^10*d) + (2*(a^4 + b^4)^2*Sqrt[Sinh[c + d*x]])/(b^9*d) -
(a^3*(a^4 + 2*b^4)*Sinh[c + d*x])/(b^8*d) + (2*a^2*(a^4 + 2*b^4)*Sinh[c + d*x]^(3/2))/(3*b^7*d) - (a*(a^4 + 2
*b^4)*Sinh[c + d*x]^2)/(2*b^6*d) + (2*(a^4 + 2*b^4)*Sinh[c + d*x]^(5/2))/(5*b^5*d) - (a^3*Sinh[c + d*x]^3)/(3*
b^4*d) + (2*a^2*Sinh[c + d*x]^(7/2))/(7*b^3*d) - (a*Sinh[c + d*x]^4)/(4*b^2*d) + (2*Sinh[c + d*x]^(9/2))/(9*b*
d)

Rule 1634

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rule 1904

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{g = Denominator[n]}, Dist[g, Subst[Int[x^(g - 1)*(
Pq /. x -> x^g)*(a + b*x^(g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && FractionQ[n]

Rule 3302

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With
[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x]
, x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (EqQ[n, 4] || GtQ[m, 0
] || IGtQ[p, 0] || IntegersQ[m, p])

Rubi steps

\begin {align*} \int \frac {\cosh ^5(c+d x)}{a+b \sqrt {\sinh (c+d x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{a+b \sqrt {x}} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {2 \text {Subst}\left (\int \frac {x \left (1+x^4\right )^2}{a+b x} \, dx,x,\sqrt {\sinh (c+d x)}\right )}{d}\\ &=\frac {2 \text {Subst}\left (\int \left (\frac {\left (a^4+b^4\right )^2}{b^9}-\frac {a^3 \left (a^4+2 b^4\right ) x}{b^8}+\frac {a^2 \left (a^4+2 b^4\right ) x^2}{b^7}-\frac {a \left (a^4+2 b^4\right ) x^3}{b^6}+\frac {\left (a^4+2 b^4\right ) x^4}{b^5}-\frac {a^3 x^5}{b^4}+\frac {a^2 x^6}{b^3}-\frac {a x^7}{b^2}+\frac {x^8}{b}-\frac {a \left (a^4+b^4\right )^2}{b^9 (a+b x)}\right ) \, dx,x,\sqrt {\sinh (c+d x)}\right )}{d}\\ &=-\frac {2 a \left (a^4+b^4\right )^2 \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{b^{10} d}+\frac {2 \left (a^4+b^4\right )^2 \sqrt {\sinh (c+d x)}}{b^9 d}-\frac {a^3 \left (a^4+2 b^4\right ) \sinh (c+d x)}{b^8 d}+\frac {2 a^2 \left (a^4+2 b^4\right ) \sinh ^{\frac {3}{2}}(c+d x)}{3 b^7 d}-\frac {a \left (a^4+2 b^4\right ) \sinh ^2(c+d x)}{2 b^6 d}+\frac {2 \left (a^4+2 b^4\right ) \sinh ^{\frac {5}{2}}(c+d x)}{5 b^5 d}-\frac {a^3 \sinh ^3(c+d x)}{3 b^4 d}+\frac {2 a^2 \sinh ^{\frac {7}{2}}(c+d x)}{7 b^3 d}-\frac {a \sinh ^4(c+d x)}{4 b^2 d}+\frac {2 \sinh ^{\frac {9}{2}}(c+d x)}{9 b d}\\ \end {align*}

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Mathematica [A]
time = 0.41, size = 220, normalized size = 0.85 \begin {gather*} \frac {-2520 a \left (a^4+b^4\right )^2 \log \left (a+b \sqrt {\sinh (c+d x)}\right )+2520 b \left (a^4+b^4\right )^2 \sqrt {\sinh (c+d x)}-1260 a^3 b^2 \left (a^4+2 b^4\right ) \sinh (c+d x)+840 a^2 b^3 \left (a^4+2 b^4\right ) \sinh ^{\frac {3}{2}}(c+d x)-630 a b^4 \left (a^4+2 b^4\right ) \sinh ^2(c+d x)+504 b^5 \left (a^4+2 b^4\right ) \sinh ^{\frac {5}{2}}(c+d x)-420 a^3 b^6 \sinh ^3(c+d x)+360 a^2 b^7 \sinh ^{\frac {7}{2}}(c+d x)-315 a b^8 \sinh ^4(c+d x)+280 b^9 \sinh ^{\frac {9}{2}}(c+d x)}{1260 b^{10} d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2520*a*(a^4 + b^4)^2*Log[a + b*Sqrt[Sinh[c + d*x]]] + 2520*b*(a^4 + b^4)^2*Sqrt[Sinh[c + d*x]] - 1260*a^3*b^
2*(a^4 + 2*b^4)*Sinh[c + d*x] + 840*a^2*b^3*(a^4 + 2*b^4)*Sinh[c + d*x]^(3/2) - 630*a*b^4*(a^4 + 2*b^4)*Sinh[c
+ d*x]^2 + 504*b^5*(a^4 + 2*b^4)*Sinh[c + d*x]^(5/2) - 420*a^3*b^6*Sinh[c + d*x]^3 + 360*a^2*b^7*Sinh[c + d*x
]^(7/2) - 315*a*b^8*Sinh[c + d*x]^4 + 280*b^9*Sinh[c + d*x]^(9/2))/(1260*b^10*d)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.51, size = 435, normalized size = 1.68

method	result	size
default	\(\frac {a \left (\frac {2 \left (-\frac {1}{2} a^{8}-b^{4} a^{4}-\frac {1}{2} b^{8}\right ) \ln \left (a^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a^{2}\right )}{b^{10}}-\frac {1}{4 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {-2 a^{2}+3 b^{2}}{6 b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {4 a^{4}-4 a^{2} b^{2}+9 b^{4}}{8 b^{6} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (a^{8}+2 b^{4} a^{4}+b^{8}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{10}}-\frac {-8 a^{6}+4 a^{4} b^{2}-16 a^{2} b^{4}+7 b^{6}}{8 b^{8} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{4 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {-2 a^{2}-3 b^{2}}{6 b^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {4 a^{4}+4 a^{2} b^{2}+9 b^{4}}{8 b^{6} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (a^{8}+2 b^{4} a^{4}+b^{8}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{10}}-\frac {-8 a^{6}-4 a^{4} b^{2}-16 a^{2} b^{4}-7 b^{6}}{8 b^{8} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\right )}{d}+\frac {\mathit {`\,int/indef0`\,}\left (-\frac {\left (\cosh ^{4}\left (d x +c \right )\right ) b \left (\sqrt {\sinh }\left (d x +c \right )\right )}{-b^{2} \sinh \left (d x +c \right )+a^{2}}, \sinh \left (d x +c \right )\right )}{d}\)	\(435\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)^5/(a+b*sinh(d*x+c)^(1/2)),x,method=_RETURNVERBOSE)

[Out]

a/d*(2/b^10*(-1/2*a^8-b^4*a^4-1/2*b^8)*ln(a^2*tanh(1/2*d*x+1/2*c)^2+2*b^2*tanh(1/2*d*x+1/2*c)-a^2)-1/4/b^2/(ta
nh(1/2*d*x+1/2*c)-1)^4-1/6*(-2*a^2+3*b^2)/b^4/(tanh(1/2*d*x+1/2*c)-1)^3-1/8*(4*a^4-4*a^2*b^2+9*b^4)/b^6/(tanh(
1/2*d*x+1/2*c)-1)^2+(a^8+2*a^4*b^4+b^8)/b^10*ln(tanh(1/2*d*x+1/2*c)-1)-1/8*(-8*a^6+4*a^4*b^2-16*a^2*b^4+7*b^6)
/b^8/(tanh(1/2*d*x+1/2*c)-1)-1/4/b^2/(tanh(1/2*d*x+1/2*c)+1)^4-1/6*(-2*a^2-3*b^2)/b^4/(tanh(1/2*d*x+1/2*c)+1)^
3-1/8*(4*a^4+4*a^2*b^2+9*b^4)/b^6/(tanh(1/2*d*x+1/2*c)+1)^2+(a^8+2*a^4*b^4+b^8)/b^10*ln(tanh(1/2*d*x+1/2*c)+1)
-1/8*(-8*a^6-4*a^4*b^2-16*a^2*b^4-7*b^6)/b^8/(tanh(1/2*d*x+1/2*c)+1))+`int/indef0`(-cosh(d*x+c)^4*b*sinh(d*x+c
)^(1/2)/(-b^2*sinh(d*x+c)+a^2),sinh(d*x+c))/d

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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(cosh(d*x+c)^5/(a+b*sinh(d*x+c)^(1/2)),x, algorithm="maxima")

[Out]

integrate(cosh(d*x + c)^5/(b*sqrt(sinh(d*x + c)) + a), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2595 vs. \(2 (233) = 466\).
time = 1.32, size = 2595, normalized size = 10.02 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^5/(a+b*sinh(d*x+c)^(1/2)),x, algorithm="fricas")

[Out]

-1/20160*(315*a*b^8*cosh(d*x + c)^8 + 315*a*b^8*sinh(d*x + c)^8 + 840*a^3*b^6*cosh(d*x + c)^7 - 840*a^3*b^6*co
sh(d*x + c) + 315*a*b^8 + 840*(3*a*b^8*cosh(d*x + c) + a^3*b^6)*sinh(d*x + c)^7 + 1260*(2*a^5*b^4 + 3*a*b^8)*c
osh(d*x + c)^6 + 420*(21*a*b^8*cosh(d*x + c)^2 + 14*a^3*b^6*cosh(d*x + c) + 6*a^5*b^4 + 9*a*b^8)*sinh(d*x + c)
^6 + 2520*(4*a^7*b^2 + 7*a^3*b^6)*cosh(d*x + c)^5 + 2520*(7*a*b^8*cosh(d*x + c)^3 + 7*a^3*b^6*cosh(d*x + c)^2
+ 4*a^7*b^2 + 7*a^3*b^6 + 3*(2*a^5*b^4 + 3*a*b^8)*cosh(d*x + c))*sinh(d*x + c)^5 - 20160*((a^9 + 2*a^5*b^4 + a
*b^8)*d*x + (a^9 + 2*a^5*b^4 + a*b^8)*c)*cosh(d*x + c)^4 + 210*(105*a*b^8*cosh(d*x + c)^4 + 140*a^3*b^6*cosh(d
*x + c)^3 - 96*(a^9 + 2*a^5*b^4 + a*b^8)*d*x + 90*(2*a^5*b^4 + 3*a*b^8)*cosh(d*x + c)^2 - 96*(a^9 + 2*a^5*b^4
+ a*b^8)*c + 60*(4*a^7*b^2 + 7*a^3*b^6)*cosh(d*x + c))*sinh(d*x + c)^4 - 2520*(4*a^7*b^2 + 7*a^3*b^6)*cosh(d*x
+ c)^3 + 840*(21*a*b^8*cosh(d*x + c)^5 + 35*a^3*b^6*cosh(d*x + c)^4 - 12*a^7*b^2 - 21*a^3*b^6 + 30*(2*a^5*b^4
+ 3*a*b^8)*cosh(d*x + c)^3 + 30*(4*a^7*b^2 + 7*a^3*b^6)*cosh(d*x + c)^2 - 96*((a^9 + 2*a^5*b^4 + a*b^8)*d*x +
(a^9 + 2*a^5*b^4 + a*b^8)*c)*cosh(d*x + c))*sinh(d*x + c)^3 + 1260*(2*a^5*b^4 + 3*a*b^8)*cosh(d*x + c)^2 + 12
60*(7*a*b^8*cosh(d*x + c)^6 + 14*a^3*b^6*cosh(d*x + c)^5 + 2*a^5*b^4 + 3*a*b^8 + 15*(2*a^5*b^4 + 3*a*b^8)*cosh
(d*x + c)^4 + 20*(4*a^7*b^2 + 7*a^3*b^6)*cosh(d*x + c)^3 - 96*((a^9 + 2*a^5*b^4 + a*b^8)*d*x + (a^9 + 2*a^5*b^
4 + a*b^8)*c)*cosh(d*x + c)^2 - 6*(4*a^7*b^2 + 7*a^3*b^6)*cosh(d*x + c))*sinh(d*x + c)^2 - 20160*((a^9 + 2*a^5
*b^4 + a*b^8)*cosh(d*x + c)^4 + 4*(a^9 + 2*a^5*b^4 + a*b^8)*cosh(d*x + c)^3*sinh(d*x + c) + 6*(a^9 + 2*a^5*b^4
+ a*b^8)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^9 + 2*a^5*b^4 + a*b^8)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^9 +
2*a^5*b^4 + a*b^8)*sinh(d*x + c)^4)*log((b^2*cosh(d*x + c)^2 + b^2*sinh(d*x + c)^2 + 2*a^2*cosh(d*x + c) - b^
2 + 2*(b^2*cosh(d*x + c) + a^2)*sinh(d*x + c) - 4*(a*b*cosh(d*x + c) + a*b*sinh(d*x + c))*sqrt(sinh(d*x + c)))
/(b^2*cosh(d*x + c)^2 + b^2*sinh(d*x + c)^2 - 2*a^2*cosh(d*x + c) - b^2 + 2*(b^2*cosh(d*x + c) - a^2)*sinh(d*x
+ c))) + 20160*((a^9 + 2*a^5*b^4 + a*b^8)*cosh(d*x + c)^4 + 4*(a^9 + 2*a^5*b^4 + a*b^8)*cosh(d*x + c)^3*sinh(
d*x + c) + 6*(a^9 + 2*a^5*b^4 + a*b^8)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^9 + 2*a^5*b^4 + a*b^8)*cosh(d*x
+ c)*sinh(d*x + c)^3 + (a^9 + 2*a^5*b^4 + a*b^8)*sinh(d*x + c)^4)*log(2*(b^2*sinh(d*x + c) - a^2)/(cosh(d*x +
c) - sinh(d*x + c))) + 840*(3*a*b^8*cosh(d*x + c)^7 + 7*a^3*b^6*cosh(d*x + c)^6 - a^3*b^6 + 9*(2*a^5*b^4 + 3*a
*b^8)*cosh(d*x + c)^5 + 15*(4*a^7*b^2 + 7*a^3*b^6)*cosh(d*x + c)^4 - 96*((a^9 + 2*a^5*b^4 + a*b^8)*d*x + (a^9
+ 2*a^5*b^4 + a*b^8)*c)*cosh(d*x + c)^3 - 9*(4*a^7*b^2 + 7*a^3*b^6)*cosh(d*x + c)^2 + 3*(2*a^5*b^4 + 3*a*b^8)*
cosh(d*x + c))*sinh(d*x + c) - 8*(35*b^9*cosh(d*x + c)^8 + 35*b^9*sinh(d*x + c)^8 + 90*a^2*b^7*cosh(d*x + c)^7
- 90*a^2*b^7*cosh(d*x + c) + 35*b^9 + 10*(28*b^9*cosh(d*x + c) + 9*a^2*b^7)*sinh(d*x + c)^7 + 28*(9*a^4*b^5 +
13*b^9)*cosh(d*x + c)^6 + 14*(70*b^9*cosh(d*x + c)^2 + 45*a^2*b^7*cosh(d*x + c) + 18*a^4*b^5 + 26*b^9)*sinh(d
*x + c)^6 + 30*(28*a^6*b^3 + 47*a^2*b^7)*cosh(d*x + c)^5 + 2*(980*b^9*cosh(d*x + c)^3 + 945*a^2*b^7*cosh(d*x +
c)^2 + 420*a^6*b^3 + 705*a^2*b^7 + 84*(9*a^4*b^5 + 13*b^9)*cosh(d*x + c))*sinh(d*x + c)^5 + 42*(120*a^8*b + 2
28*a^4*b^5 + 101*b^9)*cosh(d*x + c)^4 + 2*(1225*b^9*cosh(d*x + c)^4 + 1575*a^2*b^7*cosh(d*x + c)^3 + 2520*a^8*
b + 4788*a^4*b^5 + 2121*b^9 + 210*(9*a^4*b^5 + 13*b^9)*cosh(d*x + c)^2 + 75*(28*a^6*b^3 + 47*a^2*b^7)*cosh(d*x
+ c))*sinh(d*x + c)^4 - 30*(28*a^6*b^3 + 47*a^2*b^7)*cosh(d*x + c)^3 + 2*(980*b^9*cosh(d*x + c)^5 + 1575*a^2*
b^7*cosh(d*x + c)^4 - 420*a^6*b^3 - 705*a^2*b^7 + 280*(9*a^4*b^5 + 13*b^9)*cosh(d*x + c)^3 + 150*(28*a^6*b^3 +
47*a^2*b^7)*cosh(d*x + c)^2 + 84*(120*a^8*b + 228*a^4*b^5 + 101*b^9)*cosh(d*x + c))*sinh(d*x + c)^3 + 28*(9*a
^4*b^5 + 13*b^9)*cosh(d*x + c)^2 + 2*(490*b^9*cosh(d*x + c)^6 + 945*a^2*b^7*cosh(d*x + c)^5 + 126*a^4*b^5 + 18
2*b^9 + 210*(9*a^4*b^5 + 13*b^9)*cosh(d*x + c)^4 + 150*(28*a^6*b^3 + 47*a^2*b^7)*cosh(d*x + c)^3 + 126*(120*a^
8*b + 228*a^4*b^5 + 101*b^9)*cosh(d*x + c)^2 - 45*(28*a^6*b^3 + 47*a^2*b^7)*cosh(d*x + c))*sinh(d*x + c)^2 + 2
*(140*b^9*cosh(d*x + c)^7 + 315*a^2*b^7*cosh(d*x + c)^6 - 45*a^2*b^7 + 84*(9*a^4*b^5 + 13*b^9)*cosh(d*x + c)^5
+ 75*(28*a^6*b^3 + 47*a^2*b^7)*cosh(d*x + c)^4 + 84*(120*a^8*b + 228*a^4*b^5 + 101*b^9)*cosh(d*x + c)^3 - 45*
(28*a^6*b^3 + 47*a^2*b^7)*cosh(d*x + c)^2 + 28*(9*a^4*b^5 + 13*b^9)*cosh(d*x + c))*sinh(d*x + c))*sqrt(sinh(d*
x + c)))/(b^10*d*cosh(d*x + c)^4 + 4*b^10*d*cosh(d*x + c)^3*sinh(d*x + c) + 6*b^10*d*cosh(d*x + c)^2*sinh(d*x
+ c)^2 + 4*b^10*d*cosh(d*x + c)*sinh(d*x + c)^3 + b^10*d*sinh(d*x + c)^4)

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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(cosh(d*x+c)**5/(a+b*sinh(d*x+c)**(1/2)),x)

[Out]

Timed out

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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(cosh(d*x+c)^5/(a+b*sinh(d*x+c)^(1/2)),x, algorithm="giac")

[Out]

integrate(cosh(d*x + c)^5/(b*sqrt(sinh(d*x + c)) + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^5}{a+b\,\sqrt {\mathrm {sinh}\left (c+d\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c + d*x)^5/(a + b*sinh(c + d*x)^(1/2)),x)

[Out]

int(cosh(c + d*x)^5/(a + b*sinh(c + d*x)^(1/2)), x)

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