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

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

[Out]

(-2*a*(a^4 + b^4)^2*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)

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Rubi [A]  time = 0.296316, antiderivative size = 259, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {3223, 1890, 1620} \[ \frac{2 a^2 \sinh ^{\frac{7}{2}}(c+d x)}{7 b^3 d}-\frac{a^3 \sinh ^3(c+d x)}{3 b^4 d}+\frac{2 \left (a^4+2 b^4\right ) \sinh ^{\frac{5}{2}}(c+d x)}{5 b^5 d}-\frac{a \left (a^4+2 b^4\right ) \sinh ^2(c+d x)}{2 b^6 d}+\frac{2 a^2 \left (a^4+2 b^4\right ) \sinh ^{\frac{3}{2}}(c+d x)}{3 b^7 d}-\frac{a^3 \left (a^4+2 b^4\right ) \sinh (c+d x)}{b^8 d}+\frac{2 \left (a^4+b^4\right )^2 \sqrt{\sinh (c+d x)}}{b^9 d}-\frac{2 a \left (a^4+b^4\right )^2 \log \left (a+b \sqrt{\sinh (c+d x)}\right )}{b^{10} d}-\frac{a \sinh ^4(c+d x)}{4 b^2 d}+\frac{2 \sinh ^{\frac{9}{2}}(c+d x)}{9 b d} \]

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 3223

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])

Rule 1890

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 1620

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]

Rubi steps

\begin{align*} \int \frac{\cosh ^5(c+d x)}{a+b \sqrt{\sinh (c+d x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{a+b \sqrt{x}} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{2 \operatorname{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 \operatorname{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*}

Mathematica [A]  time = 0.443508, size = 220, normalized size = 0.85 \[ \frac{360 a^2 b^7 \sinh ^{\frac{7}{2}}(c+d x)-420 a^3 b^6 \sinh ^3(c+d x)+504 b^5 \left (a^4+2 b^4\right ) \sinh ^{\frac{5}{2}}(c+d x)-630 a b^4 \left (a^4+2 b^4\right ) \sinh ^2(c+d x)+840 a^2 b^3 \left (a^4+2 b^4\right ) \sinh ^{\frac{3}{2}}(c+d x)-1260 a^3 b^2 \left (a^4+2 b^4\right ) \sinh (c+d x)+2520 b \left (a^4+b^4\right )^2 \sqrt{\sinh (c+d x)}-2520 a \left (a^4+b^4\right )^2 \log \left (a+b \sqrt{\sinh (c+d x)}\right )-315 a b^8 \sinh ^4(c+d x)+280 b^9 \sinh ^{\frac{9}{2}}(c+d x)}{1260 b^{10} d} \]

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]  time = 0.105, size = 780, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

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)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (d x + c\right )^{5}}{b \sqrt{\sinh \left (d x + c\right )} + a}\,{d x} \end{align*}

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]  time = 8.19125, size = 6338, normalized size = 24.47 \begin{align*} \text{result too large to display} \end{align*}

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)]  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(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., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (d x + c\right )^{5}}{b \sqrt{\sinh \left (d x + c\right )} + a}\,{d x} \end{align*}

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)