3.415 \(\int \frac{\cosh ^5(c+d x)}{(a+b \sqrt{\sinh (c+d x)})^2} \, dx\)
Optimal. Leaf size=270 \[ \frac{a^2 \sinh ^3(c+d x)}{b^4 d}-\frac{8 a^3 \sinh ^{\frac{5}{2}}(c+d x)}{5 b^5 d}+\frac{\left (5 a^4+2 b^4\right ) \sinh ^2(c+d x)}{2 b^6 d}-\frac{4 a \left (3 a^4+2 b^4\right ) \sinh ^{\frac{3}{2}}(c+d x)}{3 b^7 d}+\frac{a^2 \left (7 a^4+6 b^4\right ) \sinh (c+d x)}{b^8 d}-\frac{16 a^3 \left (a^4+b^4\right ) \sqrt{\sinh (c+d x)}}{b^9 d}+\frac{2 a \left (a^4+b^4\right )^2}{b^{10} d \left (a+b \sqrt{\sinh (c+d x)}\right )}+\frac{2 \left (a^4+b^4\right ) \left (9 a^4+b^4\right ) \log \left (a+b \sqrt{\sinh (c+d x)}\right )}{b^{10} d}-\frac{4 a \sinh ^{\frac{7}{2}}(c+d x)}{7 b^3 d}+\frac{\sinh ^4(c+d x)}{4 b^2 d} \]
[Out]
(2*(a^4 + b^4)*(9*a^4 + b^4)*Log[a + b*Sqrt[Sinh[c + d*x]]])/(b^10*d) + (2*a*(a^4 + b^4)^2)/(b^10*d*(a + b*Sqr
t[Sinh[c + d*x]])) - (16*a^3*(a^4 + b^4)*Sqrt[Sinh[c + d*x]])/(b^9*d) + (a^2*(7*a^4 + 6*b^4)*Sinh[c + d*x])/(b
^8*d) - (4*a*(3*a^4 + 2*b^4)*Sinh[c + d*x]^(3/2))/(3*b^7*d) + ((5*a^4 + 2*b^4)*Sinh[c + d*x]^2)/(2*b^6*d) - (8
*a^3*Sinh[c + d*x]^(5/2))/(5*b^5*d) + (a^2*Sinh[c + d*x]^3)/(b^4*d) - (4*a*Sinh[c + d*x]^(7/2))/(7*b^3*d) + Si
nh[c + d*x]^4/(4*b^2*d)
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Rubi [A] time = 0.323083, antiderivative size = 270, 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} \[ \frac{a^2 \sinh ^3(c+d x)}{b^4 d}-\frac{8 a^3 \sinh ^{\frac{5}{2}}(c+d x)}{5 b^5 d}+\frac{\left (5 a^4+2 b^4\right ) \sinh ^2(c+d x)}{2 b^6 d}-\frac{4 a \left (3 a^4+2 b^4\right ) \sinh ^{\frac{3}{2}}(c+d x)}{3 b^7 d}+\frac{a^2 \left (7 a^4+6 b^4\right ) \sinh (c+d x)}{b^8 d}-\frac{16 a^3 \left (a^4+b^4\right ) \sqrt{\sinh (c+d x)}}{b^9 d}+\frac{2 a \left (a^4+b^4\right )^2}{b^{10} d \left (a+b \sqrt{\sinh (c+d x)}\right )}+\frac{2 \left (a^4+b^4\right ) \left (9 a^4+b^4\right ) \log \left (a+b \sqrt{\sinh (c+d x)}\right )}{b^{10} d}-\frac{4 a \sinh ^{\frac{7}{2}}(c+d x)}{7 b^3 d}+\frac{\sinh ^4(c+d x)}{4 b^2 d} \]
Antiderivative was successfully verified.
[In]
Int[Cosh[c + d*x]^5/(a + b*Sqrt[Sinh[c + d*x]])^2,x]
[Out]
(2*(a^4 + b^4)*(9*a^4 + b^4)*Log[a + b*Sqrt[Sinh[c + d*x]]])/(b^10*d) + (2*a*(a^4 + b^4)^2)/(b^10*d*(a + b*Sqr
t[Sinh[c + d*x]])) - (16*a^3*(a^4 + b^4)*Sqrt[Sinh[c + d*x]])/(b^9*d) + (a^2*(7*a^4 + 6*b^4)*Sinh[c + d*x])/(b
^8*d) - (4*a*(3*a^4 + 2*b^4)*Sinh[c + d*x]^(3/2))/(3*b^7*d) + ((5*a^4 + 2*b^4)*Sinh[c + d*x]^2)/(2*b^6*d) - (8
*a^3*Sinh[c + d*x]^(5/2))/(5*b^5*d) + (a^2*Sinh[c + d*x]^3)/(b^4*d) - (4*a*Sinh[c + d*x]^(7/2))/(7*b^3*d) + Si
nh[c + d*x]^4/(4*b^2*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)}{\left (a+b \sqrt{\sinh (c+d x)}\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{\left (a+b \sqrt{x}\right )^2} \, 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)^2} \, dx,x,\sqrt{\sinh (c+d x)}\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (-\frac{8 a^3 \left (a^4+b^4\right )}{b^9}+\frac{a^2 \left (7 a^4+6 b^4\right ) x}{b^8}-\frac{2 a \left (3 a^4+2 b^4\right ) x^2}{b^7}+\frac{\left (5 a^4+2 b^4\right ) x^3}{b^6}-\frac{4 a^3 x^4}{b^5}+\frac{3 a^2 x^5}{b^4}-\frac{2 a x^6}{b^3}+\frac{x^7}{b^2}-\frac{a \left (a^4+b^4\right )^2}{b^9 (a+b x)^2}+\frac{9 a^8+10 a^4 b^4+b^8}{b^9 (a+b x)}\right ) \, dx,x,\sqrt{\sinh (c+d x)}\right )}{d}\\ &=\frac{2 \left (a^4+b^4\right ) \left (9 a^4+b^4\right ) \log \left (a+b \sqrt{\sinh (c+d x)}\right )}{b^{10} d}+\frac{2 a \left (a^4+b^4\right )^2}{b^{10} d \left (a+b \sqrt{\sinh (c+d x)}\right )}-\frac{16 a^3 \left (a^4+b^4\right ) \sqrt{\sinh (c+d x)}}{b^9 d}+\frac{a^2 \left (7 a^4+6 b^4\right ) \sinh (c+d x)}{b^8 d}-\frac{4 a \left (3 a^4+2 b^4\right ) \sinh ^{\frac{3}{2}}(c+d x)}{3 b^7 d}+\frac{\left (5 a^4+2 b^4\right ) \sinh ^2(c+d x)}{2 b^6 d}-\frac{8 a^3 \sinh ^{\frac{5}{2}}(c+d x)}{5 b^5 d}+\frac{a^2 \sinh ^3(c+d x)}{b^4 d}-\frac{4 a \sinh ^{\frac{7}{2}}(c+d x)}{7 b^3 d}+\frac{\sinh ^4(c+d x)}{4 b^2 d}\\ \end{align*}
Mathematica [A] time = 0.618676, size = 288, normalized size = 1.07 \[ \frac{180 a^2 b^7 \sinh ^{\frac{7}{2}}(c+d x)-252 a^3 b^6 \sinh ^3(c+d x)+42 b^5 \left (9 a^4+10 b^4\right ) \sinh ^{\frac{5}{2}}(c+d x)-70 a b^4 \left (9 a^4+10 b^4\right ) \sinh ^2(c+d x)+140 a^2 b^3 \left (9 a^4+10 b^4\right ) \sinh ^{\frac{3}{2}}(c+d x)-420 a^3 b^2 \left (9 a^4+10 b^4\right ) \sinh (c+d x)+840 b \left (a^4+b^4\right ) \sqrt{\sinh (c+d x)} \left (\left (9 a^4+b^4\right ) \log \left (a+b \sqrt{\sinh (c+d x)}\right )-8 a^4\right )+840 a \left (a^4+b^4\right ) \left (\left (9 a^4+b^4\right ) \log \left (a+b \sqrt{\sinh (c+d x)}\right )+a^4+b^4\right )-135 a b^8 \sinh ^4(c+d x)+105 b^9 \sinh ^{\frac{9}{2}}(c+d x)}{420 b^{10} d \left (a+b \sqrt{\sinh (c+d x)}\right )} \]
Antiderivative was successfully verified.
[In]
Integrate[Cosh[c + d*x]^5/(a + b*Sqrt[Sinh[c + d*x]])^2,x]
[Out]
(840*a*(a^4 + b^4)*(a^4 + b^4 + (9*a^4 + b^4)*Log[a + b*Sqrt[Sinh[c + d*x]]]) + 840*b*(a^4 + b^4)*(-8*a^4 + (9
*a^4 + b^4)*Log[a + b*Sqrt[Sinh[c + d*x]]])*Sqrt[Sinh[c + d*x]] - 420*a^3*b^2*(9*a^4 + 10*b^4)*Sinh[c + d*x] +
140*a^2*b^3*(9*a^4 + 10*b^4)*Sinh[c + d*x]^(3/2) - 70*a*b^4*(9*a^4 + 10*b^4)*Sinh[c + d*x]^2 + 42*b^5*(9*a^4
+ 10*b^4)*Sinh[c + d*x]^(5/2) - 252*a^3*b^6*Sinh[c + d*x]^3 + 180*a^2*b^7*Sinh[c + d*x]^(7/2) - 135*a*b^8*Sinh
[c + d*x]^4 + 105*b^9*Sinh[c + d*x]^(9/2))/(420*b^10*d*(a + b*Sqrt[Sinh[c + d*x]]))
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Maple [C] time = 0.218, size = 955, normalized size = 3.5 \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))^2,x)
[Out]
9/8/d/b^2/(tanh(1/2*d*x+1/2*c)-1)^2+9/8/d/b^2/(tanh(1/2*d*x+1/2*c)+1)^2-7/d/b^8/(tanh(1/2*d*x+1/2*c)-1)*a^6+5/
2/d/b^6/(tanh(1/2*d*x+1/2*c)-1)*a^4-6/d/b^4/(tanh(1/2*d*x+1/2*c)-1)*a^2-1/d/b^4/(tanh(1/2*d*x+1/2*c)-1)^3*a^2+
5/2/d/b^6/(tanh(1/2*d*x+1/2*c)-1)^2*a^4-3/2/d/b^4/(tanh(1/2*d*x+1/2*c)-1)^2*a^2-9/d/b^10*ln(tanh(1/2*d*x+1/2*c
)-1)*a^8-10/d/b^6*ln(tanh(1/2*d*x+1/2*c)-1)*a^4+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)*a^8+10/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)*a^4-7/d/b^8/(tanh(1/2*d*x+1/2*
c)+1)*a^6-5/2/d/b^6/(tanh(1/2*d*x+1/2*c)+1)*a^4-6/d/b^4/(tanh(1/2*d*x+1/2*c)+1)*a^2-1/d/b^4/(tanh(1/2*d*x+1/2*
c)+1)^3*a^2+5/2/d/b^6/(tanh(1/2*d*x+1/2*c)+1)^2*a^4+3/2/d/b^4/(tanh(1/2*d*x+1/2*c)+1)^2*a^2-9/d/b^10*ln(tanh(1
/2*d*x+1/2*c)+1)*a^8-10/d/b^6*ln(tanh(1/2*d*x+1/2*c)+1)*a^4-1/d/b^2*ln(tanh(1/2*d*x+1/2*c)+1)-1/d/b^2*ln(tanh(
1/2*d*x+1/2*c)-1)-4/d*tanh(1/2*d*x+1/2*c)/(a^2*tanh(1/2*d*x+1/2*c)^2+2*b^2*tanh(1/2*d*x+1/2*c)-a^2)-1/2/d/b^2/
(tanh(1/2*d*x+1/2*c)+1)^3+1/4/d/b^2/(tanh(1/2*d*x+1/2*c)-1)^4+1/2/d/b^2/(tanh(1/2*d*x+1/2*c)-1)^3+1/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/d/b^2/(tanh(1/2*d*x+1/2*c)+1)^4-7/8/d/b^2/(tanh(1/
2*d*x+1/2*c)+1)+7/8/d/b^2/(tanh(1/2*d*x+1/2*c)-1)+`int/indef0`(-2*cosh(d*x+c)^4*a*b*sinh(d*x+c)^(1/2)/(b^4*sin
h(d*x+c)^2-2*a^2*b^2*sinh(d*x+c)+a^4),sinh(d*x+c))/d-8/d/b^4*tanh(1/2*d*x+1/2*c)/(a^2*tanh(1/2*d*x+1/2*c)^2+2*
b^2*tanh(1/2*d*x+1/2*c)-a^2)*a^4-4/d/b^8*tanh(1/2*d*x+1/2*c)/(a^2*tanh(1/2*d*x+1/2*c)^2+2*b^2*tanh(1/2*d*x+1/2
*c)-a^2)*a^8
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Maxima [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))^2,x, algorithm="maxima")
[Out]
Timed out
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Fricas [B] time = 8.8936, size = 12573, normalized size = 46.57 \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))^2,x, algorithm="fricas")
[Out]
1/6720*(105*b^10*cosh(d*x + c)^10 + 105*b^10*sinh(d*x + c)^10 + 630*a^2*b^8*cosh(d*x + c)^9 + 630*a^2*b^8*cosh
(d*x + c) - 105*b^10 + 210*(5*b^10*cosh(d*x + c) + 3*a^2*b^8)*sinh(d*x + c)^9 + 105*(24*a^4*b^6 + 11*b^10)*cos
h(d*x + c)^8 + 105*(45*b^10*cosh(d*x + c)^2 + 54*a^2*b^8*cosh(d*x + c) + 24*a^4*b^6 + 11*b^10)*sinh(d*x + c)^8
+ 840*(18*a^6*b^4 + 17*a^2*b^8)*cosh(d*x + c)^7 + 840*(15*b^10*cosh(d*x + c)^3 + 27*a^2*b^8*cosh(d*x + c)^2 +
18*a^6*b^4 + 17*a^2*b^8 + (24*a^4*b^6 + 11*b^10)*cosh(d*x + c))*sinh(d*x + c)^7 - 420*(112*a^8*b^2 + 94*a^4*b
^6 + 3*b^10 + 16*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*d*x + 16*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*c)*cosh(d*x + c)^6 +
210*(105*b^10*cosh(d*x + c)^4 + 252*a^2*b^8*cosh(d*x + c)^3 - 224*a^8*b^2 - 188*a^4*b^6 - 6*b^10 - 32*(9*a^8*
b^2 + 10*a^4*b^6 + b^10)*d*x + 14*(24*a^4*b^6 + 11*b^10)*cosh(d*x + c)^2 - 32*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*
c + 28*(18*a^6*b^4 + 17*a^2*b^8)*cosh(d*x + c))*sinh(d*x + c)^6 - 1680*(16*a^10 + 60*a^6*b^4 + 37*a^2*b^8 - 8*
(9*a^10 + 10*a^6*b^4 + a^2*b^8)*d*x - 8*(9*a^10 + 10*a^6*b^4 + a^2*b^8)*c)*cosh(d*x + c)^5 + 420*(63*b^10*cosh
(d*x + c)^5 + 189*a^2*b^8*cosh(d*x + c)^4 - 64*a^10 - 240*a^6*b^4 - 148*a^2*b^8 + 14*(24*a^4*b^6 + 11*b^10)*co
sh(d*x + c)^3 + 32*(9*a^10 + 10*a^6*b^4 + a^2*b^8)*d*x + 42*(18*a^6*b^4 + 17*a^2*b^8)*cosh(d*x + c)^2 + 32*(9*
a^10 + 10*a^6*b^4 + a^2*b^8)*c - 6*(112*a^8*b^2 + 94*a^4*b^6 + 3*b^10 + 16*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*d*x
+ 16*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*c)*cosh(d*x + c))*sinh(d*x + c)^5 + 420*(112*a^8*b^2 + 94*a^4*b^6 + 3*b^
10 + 16*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*d*x + 16*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*c)*cosh(d*x + c)^4 + 210*(105
*b^10*cosh(d*x + c)^6 + 378*a^2*b^8*cosh(d*x + c)^5 + 224*a^8*b^2 + 188*a^4*b^6 + 6*b^10 + 35*(24*a^4*b^6 + 11
*b^10)*cosh(d*x + c)^4 + 140*(18*a^6*b^4 + 17*a^2*b^8)*cosh(d*x + c)^3 + 32*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*d*
x - 30*(112*a^8*b^2 + 94*a^4*b^6 + 3*b^10 + 16*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*d*x + 16*(9*a^8*b^2 + 10*a^4*b^
6 + b^10)*c)*cosh(d*x + c)^2 + 32*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*c - 40*(16*a^10 + 60*a^6*b^4 + 37*a^2*b^8 -
8*(9*a^10 + 10*a^6*b^4 + a^2*b^8)*d*x - 8*(9*a^10 + 10*a^6*b^4 + a^2*b^8)*c)*cosh(d*x + c))*sinh(d*x + c)^4 +
840*(18*a^6*b^4 + 17*a^2*b^8)*cosh(d*x + c)^3 + 840*(15*b^10*cosh(d*x + c)^7 + 63*a^2*b^8*cosh(d*x + c)^6 + 18
*a^6*b^4 + 17*a^2*b^8 + 7*(24*a^4*b^6 + 11*b^10)*cosh(d*x + c)^5 + 35*(18*a^6*b^4 + 17*a^2*b^8)*cosh(d*x + c)^
4 - 10*(112*a^8*b^2 + 94*a^4*b^6 + 3*b^10 + 16*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*d*x + 16*(9*a^8*b^2 + 10*a^4*b^
6 + b^10)*c)*cosh(d*x + c)^3 - 20*(16*a^10 + 60*a^6*b^4 + 37*a^2*b^8 - 8*(9*a^10 + 10*a^6*b^4 + a^2*b^8)*d*x -
8*(9*a^10 + 10*a^6*b^4 + a^2*b^8)*c)*cosh(d*x + c)^2 + 2*(112*a^8*b^2 + 94*a^4*b^6 + 3*b^10 + 16*(9*a^8*b^2 +
10*a^4*b^6 + b^10)*d*x + 16*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*c)*cosh(d*x + c))*sinh(d*x + c)^3 - 105*(24*a^4*b
^6 + 11*b^10)*cosh(d*x + c)^2 + 105*(45*b^10*cosh(d*x + c)^8 + 216*a^2*b^8*cosh(d*x + c)^7 - 24*a^4*b^6 - 11*b
^10 + 28*(24*a^4*b^6 + 11*b^10)*cosh(d*x + c)^6 + 168*(18*a^6*b^4 + 17*a^2*b^8)*cosh(d*x + c)^5 - 60*(112*a^8*
b^2 + 94*a^4*b^6 + 3*b^10 + 16*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*d*x + 16*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*c)*cos
h(d*x + c)^4 - 160*(16*a^10 + 60*a^6*b^4 + 37*a^2*b^8 - 8*(9*a^10 + 10*a^6*b^4 + a^2*b^8)*d*x - 8*(9*a^10 + 10
*a^6*b^4 + a^2*b^8)*c)*cosh(d*x + c)^3 + 24*(112*a^8*b^2 + 94*a^4*b^6 + 3*b^10 + 16*(9*a^8*b^2 + 10*a^4*b^6 +
b^10)*d*x + 16*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*c)*cosh(d*x + c)^2 + 24*(18*a^6*b^4 + 17*a^2*b^8)*cosh(d*x + c)
)*sinh(d*x + c)^2 + 6720*((9*a^8*b^2 + 10*a^4*b^6 + b^10)*cosh(d*x + c)^6 + (9*a^8*b^2 + 10*a^4*b^6 + b^10)*si
nh(d*x + c)^6 - 2*(9*a^10 + 10*a^6*b^4 + a^2*b^8)*cosh(d*x + c)^5 - 2*(9*a^10 + 10*a^6*b^4 + a^2*b^8 - 3*(9*a^
8*b^2 + 10*a^4*b^6 + b^10)*cosh(d*x + c))*sinh(d*x + c)^5 - (9*a^8*b^2 + 10*a^4*b^6 + b^10)*cosh(d*x + c)^4 -
(9*a^8*b^2 + 10*a^4*b^6 + b^10 - 15*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*cosh(d*x + c)^2 + 10*(9*a^10 + 10*a^6*b^4
+ a^2*b^8)*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(5*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*cosh(d*x + c)^3 - 5*(9*a^10 +
10*a^6*b^4 + a^2*b^8)*cosh(d*x + c)^2 - (9*a^8*b^2 + 10*a^4*b^6 + b^10)*cosh(d*x + c))*sinh(d*x + c)^3 + (15*
(9*a^8*b^2 + 10*a^4*b^6 + b^10)*cosh(d*x + c)^4 - 20*(9*a^10 + 10*a^6*b^4 + a^2*b^8)*cosh(d*x + c)^3 - 6*(9*a^
8*b^2 + 10*a^4*b^6 + b^10)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(3*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*cosh(d*x +
c)^5 - 5*(9*a^10 + 10*a^6*b^4 + a^2*b^8)*cosh(d*x + c)^4 - 2*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*cosh(d*x + c)^3)*
sinh(d*x + c))*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))) + 6720*((9*a^
8*b^2 + 10*a^4*b^6 + b^10)*cosh(d*x + c)^6 + (9*a^8*b^2 + 10*a^4*b^6 + b^10)*sinh(d*x + c)^6 - 2*(9*a^10 + 10*
a^6*b^4 + a^2*b^8)*cosh(d*x + c)^5 - 2*(9*a^10 + 10*a^6*b^4 + a^2*b^8 - 3*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*cosh
(d*x + c))*sinh(d*x + c)^5 - (9*a^8*b^2 + 10*a^4*b^6 + b^10)*cosh(d*x + c)^4 - (9*a^8*b^2 + 10*a^4*b^6 + b^10
- 15*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*cosh(d*x + c)^2 + 10*(9*a^10 + 10*a^6*b^4 + a^2*b^8)*cosh(d*x + c))*sinh(
d*x + c)^4 + 4*(5*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*cosh(d*x + c)^3 - 5*(9*a^10 + 10*a^6*b^4 + a^2*b^8)*cosh(d*x
+ c)^2 - (9*a^8*b^2 + 10*a^4*b^6 + b^10)*cosh(d*x + c))*sinh(d*x + c)^3 + (15*(9*a^8*b^2 + 10*a^4*b^6 + b^10)
*cosh(d*x + c)^4 - 20*(9*a^10 + 10*a^6*b^4 + a^2*b^8)*cosh(d*x + c)^3 - 6*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*cosh
(d*x + c)^2)*sinh(d*x + c)^2 + 2*(3*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*cosh(d*x + c)^5 - 5*(9*a^10 + 10*a^6*b^4 +
a^2*b^8)*cosh(d*x + c)^4 - 2*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*cosh(d*x + c)^3)*sinh(d*x + c))*log(2*(b^2*sinh(
d*x + c) - a^2)/(cosh(d*x + c) - sinh(d*x + c))) + 210*(5*b^10*cosh(d*x + c)^9 + 27*a^2*b^8*cosh(d*x + c)^8 +
3*a^2*b^8 + 4*(24*a^4*b^6 + 11*b^10)*cosh(d*x + c)^7 + 28*(18*a^6*b^4 + 17*a^2*b^8)*cosh(d*x + c)^6 - 12*(112*
a^8*b^2 + 94*a^4*b^6 + 3*b^10 + 16*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*d*x + 16*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*c)
*cosh(d*x + c)^5 - 40*(16*a^10 + 60*a^6*b^4 + 37*a^2*b^8 - 8*(9*a^10 + 10*a^6*b^4 + a^2*b^8)*d*x - 8*(9*a^10 +
10*a^6*b^4 + a^2*b^8)*c)*cosh(d*x + c)^4 + 8*(112*a^8*b^2 + 94*a^4*b^6 + 3*b^10 + 16*(9*a^8*b^2 + 10*a^4*b^6
+ b^10)*d*x + 16*(9*a^8*b^2 + 10*a^4*b^6 + b^10)*c)*cosh(d*x + c)^3 + 12*(18*a^6*b^4 + 17*a^2*b^8)*cosh(d*x +
c)^2 - (24*a^4*b^6 + 11*b^10)*cosh(d*x + c))*sinh(d*x + c) - 32*(15*a*b^9*cosh(d*x + c)^9 + 15*a*b^9*sinh(d*x
+ c)^9 + 54*a^3*b^7*cosh(d*x + c)^8 - 54*a^3*b^7*cosh(d*x + c)^2 + 15*a*b^9*cosh(d*x + c) + 27*(5*a*b^9*cosh(d
*x + c) + 2*a^3*b^7)*sinh(d*x + c)^8 + 4*(63*a^5*b^5 + 55*a*b^9)*cosh(d*x + c)^7 + 4*(135*a*b^9*cosh(d*x + c)^
2 + 108*a^3*b^7*cosh(d*x + c) + 63*a^5*b^5 + 55*a*b^9)*sinh(d*x + c)^7 + 2*(1260*a^7*b^3 + 1319*a^3*b^7)*cosh(
d*x + c)^6 + 2*(630*a*b^9*cosh(d*x + c)^3 + 756*a^3*b^7*cosh(d*x + c)^2 + 1260*a^7*b^3 + 1319*a^3*b^7 + 14*(63
*a^5*b^5 + 55*a*b^9)*cosh(d*x + c))*sinh(d*x + c)^6 - 2*(3780*a^9*b + 4452*a^5*b^5 + 655*a*b^9)*cosh(d*x + c)^
5 + 2*(945*a*b^9*cosh(d*x + c)^4 + 1512*a^3*b^7*cosh(d*x + c)^3 - 3780*a^9*b - 4452*a^5*b^5 - 655*a*b^9 + 42*(
63*a^5*b^5 + 55*a*b^9)*cosh(d*x + c)^2 + 6*(1260*a^7*b^3 + 1319*a^3*b^7)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(1
260*a^7*b^3 + 1319*a^3*b^7)*cosh(d*x + c)^4 + 2*(945*a*b^9*cosh(d*x + c)^5 + 1890*a^3*b^7*cosh(d*x + c)^4 - 12
60*a^7*b^3 - 1319*a^3*b^7 + 70*(63*a^5*b^5 + 55*a*b^9)*cosh(d*x + c)^3 + 15*(1260*a^7*b^3 + 1319*a^3*b^7)*cosh
(d*x + c)^2 - 5*(3780*a^9*b + 4452*a^5*b^5 + 655*a*b^9)*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(63*a^5*b^5 + 55*a*
b^9)*cosh(d*x + c)^3 + 4*(315*a*b^9*cosh(d*x + c)^6 + 756*a^3*b^7*cosh(d*x + c)^5 + 63*a^5*b^5 + 55*a*b^9 + 35
*(63*a^5*b^5 + 55*a*b^9)*cosh(d*x + c)^4 + 10*(1260*a^7*b^3 + 1319*a^3*b^7)*cosh(d*x + c)^3 - 5*(3780*a^9*b +
4452*a^5*b^5 + 655*a*b^9)*cosh(d*x + c)^2 - 2*(1260*a^7*b^3 + 1319*a^3*b^7)*cosh(d*x + c))*sinh(d*x + c)^3 + 2
*(270*a*b^9*cosh(d*x + c)^7 + 756*a^3*b^7*cosh(d*x + c)^6 - 27*a^3*b^7 + 42*(63*a^5*b^5 + 55*a*b^9)*cosh(d*x +
c)^5 + 15*(1260*a^7*b^3 + 1319*a^3*b^7)*cosh(d*x + c)^4 - 10*(3780*a^9*b + 4452*a^5*b^5 + 655*a*b^9)*cosh(d*x
+ c)^3 - 6*(1260*a^7*b^3 + 1319*a^3*b^7)*cosh(d*x + c)^2 + 6*(63*a^5*b^5 + 55*a*b^9)*cosh(d*x + c))*sinh(d*x
+ c)^2 + (135*a*b^9*cosh(d*x + c)^8 + 432*a^3*b^7*cosh(d*x + c)^7 - 108*a^3*b^7*cosh(d*x + c) + 15*a*b^9 + 28*
(63*a^5*b^5 + 55*a*b^9)*cosh(d*x + c)^6 + 12*(1260*a^7*b^3 + 1319*a^3*b^7)*cosh(d*x + c)^5 - 10*(3780*a^9*b +
4452*a^5*b^5 + 655*a*b^9)*cosh(d*x + c)^4 - 8*(1260*a^7*b^3 + 1319*a^3*b^7)*cosh(d*x + c)^3 + 12*(63*a^5*b^5 +
55*a*b^9)*cosh(d*x + c)^2)*sinh(d*x + c))*sqrt(sinh(d*x + c)))/(b^12*d*cosh(d*x + c)^6 + b^12*d*sinh(d*x + c)
^6 - 2*a^2*b^10*d*cosh(d*x + c)^5 - b^12*d*cosh(d*x + c)^4 + 2*(3*b^12*d*cosh(d*x + c) - a^2*b^10*d)*sinh(d*x
+ c)^5 + (15*b^12*d*cosh(d*x + c)^2 - 10*a^2*b^10*d*cosh(d*x + c) - b^12*d)*sinh(d*x + c)^4 + 4*(5*b^12*d*cosh
(d*x + c)^3 - 5*a^2*b^10*d*cosh(d*x + c)^2 - b^12*d*cosh(d*x + c))*sinh(d*x + c)^3 + (15*b^12*d*cosh(d*x + c)^
4 - 20*a^2*b^10*d*cosh(d*x + c)^3 - 6*b^12*d*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(3*b^12*d*cosh(d*x + c)^5 -
5*a^2*b^10*d*cosh(d*x + c)^4 - 2*b^12*d*cosh(d*x + c)^3)*sinh(d*x + c))
________________________________________________________________________________________
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))**2,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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))^2,x, algorithm="giac")
[Out]
Exception raised: TypeError