3.316 \(\int \frac{\cosh ^7(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\)

Optimal. Leaf size=108 \[ \frac{\left (a^2-3 a b+3 b^2\right ) \sinh (c+d x)}{b^3 d}-\frac{(a-3 b) \sinh ^3(c+d x)}{3 b^2 d}-\frac{(a-b)^3 \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} b^{7/2} d}+\frac{\sinh ^5(c+d x)}{5 b d} \]

[Out]

-(((a - b)^3*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(7/2)*d)) + ((a^2 - 3*a*b + 3*b^2)*Sinh[c + d
*x])/(b^3*d) - ((a - 3*b)*Sinh[c + d*x]^3)/(3*b^2*d) + Sinh[c + d*x]^5/(5*b*d)

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Rubi [A]  time = 0.112928, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3190, 390, 205} \[ \frac{\left (a^2-3 a b+3 b^2\right ) \sinh (c+d x)}{b^3 d}-\frac{(a-3 b) \sinh ^3(c+d x)}{3 b^2 d}-\frac{(a-b)^3 \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} b^{7/2} d}+\frac{\sinh ^5(c+d x)}{5 b d} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]^7/(a + b*Sinh[c + d*x]^2),x]

[Out]

-(((a - b)^3*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(7/2)*d)) + ((a^2 - 3*a*b + 3*b^2)*Sinh[c + d
*x])/(b^3*d) - ((a - 3*b)*Sinh[c + d*x]^3)/(3*b^2*d) + Sinh[c + d*x]^5/(5*b*d)

Rule 3190

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e +
f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\cosh ^7(c+d x)}{a+b \sinh ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^3}{a+b x^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^2-3 a b+3 b^2}{b^3}-\frac{(a-3 b) x^2}{b^2}+\frac{x^4}{b}+\frac{-a^3+3 a^2 b-3 a b^2+b^3}{b^3 \left (a+b x^2\right )}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{\left (a^2-3 a b+3 b^2\right ) \sinh (c+d x)}{b^3 d}-\frac{(a-3 b) \sinh ^3(c+d x)}{3 b^2 d}+\frac{\sinh ^5(c+d x)}{5 b d}-\frac{(a-b)^3 \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sinh (c+d x)\right )}{b^3 d}\\ &=-\frac{(a-b)^3 \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} b^{7/2} d}+\frac{\left (a^2-3 a b+3 b^2\right ) \sinh (c+d x)}{b^3 d}-\frac{(a-3 b) \sinh ^3(c+d x)}{3 b^2 d}+\frac{\sinh ^5(c+d x)}{5 b d}\\ \end{align*}

Mathematica [A]  time = 0.476694, size = 117, normalized size = 1.08 \[ \frac{30 \sqrt{b} \left (8 a^2-22 a b+19 b^2\right ) \sinh (c+d x)+5 b^{3/2} (9 b-4 a) \sinh (3 (c+d x))+\frac{3 \left (\sqrt{a} b^{5/2} \sinh (5 (c+d x))+80 (a-b)^3 \tan ^{-1}\left (\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right )\right )}{\sqrt{a}}}{240 b^{7/2} d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]^7/(a + b*Sinh[c + d*x]^2),x]

[Out]

(30*Sqrt[b]*(8*a^2 - 22*a*b + 19*b^2)*Sinh[c + d*x] + 5*b^(3/2)*(-4*a + 9*b)*Sinh[3*(c + d*x)] + (3*(80*(a - b
)^3*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]] + Sqrt[a]*b^(5/2)*Sinh[5*(c + d*x)]))/Sqrt[a])/(240*b^(7/2)*d)

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Maple [B]  time = 0.081, size = 1656, normalized size = 15.3 \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)^7/(a+b*sinh(d*x+c)^2),x)

[Out]

11/8/d/b/(tanh(1/2*d*x+1/2*c)+1)^2-11/8/d/b/(tanh(1/2*d*x+1/2*c)-1)^2-4/d*a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1
/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))+3/d*a/b/((2*(-b*(a-b))
^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))-1/5/d/b/(tanh(1/2*d
*x+1/2*c)-1)^5-1/d/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2
*b)*a)^(1/2))+1/d/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b
)*a)^(1/2))+1/d/b^3*a^4/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2
*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))+1/d/b^3*a^4/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a
*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))-1/2/d/b^2/(tanh(1/2*d*x+1/2*c)+1)^2*a+1/2/d/b^2/(ta
nh(1/2*d*x+1/2*c)-1)^2*a+3/d/b^2/(tanh(1/2*d*x+1/2*c)+1)*a+3/d/b^2/(tanh(1/2*d*x+1/2*c)-1)*a-4/d*a^3/b^2/(-b*(
a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(
1/2))-4/d*a^3/b^2/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*
(a-b))^(1/2)+a-2*b)*a)^(1/2))+6/d*a^2/b/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1
/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))+6/d*a^2/b/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^
(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))+1/d*b/(-b*(a-b))^(1/2)/((2*(-b*(a-b))
^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))+1/d*b/(-b*(a-b))^(1
/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))+1/
3/d/b^2/(tanh(1/2*d*x+1/2*c)+1)^3*a-1/d/b^3/(tanh(1/2*d*x+1/2*c)+1)*a^2+1/3/d/b^2/(tanh(1/2*d*x+1/2*c)-1)^3*a-
1/d/b^3/(tanh(1/2*d*x+1/2*c)-1)*a^2-1/2/d/b/(tanh(1/2*d*x+1/2*c)-1)^4+1/2/d/b/(tanh(1/2*d*x+1/2*c)+1)^4+3/d*a^
2/b^2/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))+
1/d*a^3/b^3/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^
(1/2))-3/d*a^2/b^2/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2
*b)*a)^(1/2))-5/4/d/b/(tanh(1/2*d*x+1/2*c)+1)^3-5/4/d/b/(tanh(1/2*d*x+1/2*c)-1)^3-3/d/b/(tanh(1/2*d*x+1/2*c)+1
)-3/d/b/(tanh(1/2*d*x+1/2*c)-1)-3/d*a/b/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*
(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))-4/d*a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/
2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))-1/5/d/b/(tanh(1/2*d*x+1/2*c)+1)^5-1/d*a^3/b^3/((2*(-b*(a-b)
)^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (3 \, b^{2} e^{\left (10 \, d x + 10 \, c\right )} - 3 \, b^{2} - 5 \,{\left (4 \, a b e^{\left (8 \, c\right )} - 9 \, b^{2} e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} + 30 \,{\left (8 \, a^{2} e^{\left (6 \, c\right )} - 22 \, a b e^{\left (6 \, c\right )} + 19 \, b^{2} e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} - 30 \,{\left (8 \, a^{2} e^{\left (4 \, c\right )} - 22 \, a b e^{\left (4 \, c\right )} + 19 \, b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 5 \,{\left (4 \, a b e^{\left (2 \, c\right )} - 9 \, b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{480 \, b^{3} d} - \frac{1}{128} \, \int \frac{256 \,{\left ({\left (a^{3} e^{\left (3 \, c\right )} - 3 \, a^{2} b e^{\left (3 \, c\right )} + 3 \, a b^{2} e^{\left (3 \, c\right )} - b^{3} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} +{\left (a^{3} e^{c} - 3 \, a^{2} b e^{c} + 3 \, a b^{2} e^{c} - b^{3} e^{c}\right )} e^{\left (d x\right )}\right )}}{b^{4} e^{\left (4 \, d x + 4 \, c\right )} + b^{4} + 2 \,{\left (2 \, a b^{3} e^{\left (2 \, c\right )} - b^{4} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^7/(a+b*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

1/480*(3*b^2*e^(10*d*x + 10*c) - 3*b^2 - 5*(4*a*b*e^(8*c) - 9*b^2*e^(8*c))*e^(8*d*x) + 30*(8*a^2*e^(6*c) - 22*
a*b*e^(6*c) + 19*b^2*e^(6*c))*e^(6*d*x) - 30*(8*a^2*e^(4*c) - 22*a*b*e^(4*c) + 19*b^2*e^(4*c))*e^(4*d*x) + 5*(
4*a*b*e^(2*c) - 9*b^2*e^(2*c))*e^(2*d*x))*e^(-5*d*x - 5*c)/(b^3*d) - 1/128*integrate(256*((a^3*e^(3*c) - 3*a^2
*b*e^(3*c) + 3*a*b^2*e^(3*c) - b^3*e^(3*c))*e^(3*d*x) + (a^3*e^c - 3*a^2*b*e^c + 3*a*b^2*e^c - b^3*e^c)*e^(d*x
))/(b^4*e^(4*d*x + 4*c) + b^4 + 2*(2*a*b^3*e^(2*c) - b^4*e^(2*c))*e^(2*d*x)), x)

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Fricas [B]  time = 1.85931, size = 7526, normalized size = 69.69 \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)^7/(a+b*sinh(d*x+c)^2),x, algorithm="fricas")

[Out]

[1/480*(3*a*b^3*cosh(d*x + c)^10 + 30*a*b^3*cosh(d*x + c)*sinh(d*x + c)^9 + 3*a*b^3*sinh(d*x + c)^10 - 5*(4*a^
2*b^2 - 9*a*b^3)*cosh(d*x + c)^8 + 5*(27*a*b^3*cosh(d*x + c)^2 - 4*a^2*b^2 + 9*a*b^3)*sinh(d*x + c)^8 + 40*(9*
a*b^3*cosh(d*x + c)^3 - (4*a^2*b^2 - 9*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 + 30*(8*a^3*b - 22*a^2*b^2 + 19*a
*b^3)*cosh(d*x + c)^6 + 10*(63*a*b^3*cosh(d*x + c)^4 + 24*a^3*b - 66*a^2*b^2 + 57*a*b^3 - 14*(4*a^2*b^2 - 9*a*
b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(189*a*b^3*cosh(d*x + c)^5 - 70*(4*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^
3 + 45*(8*a^3*b - 22*a^2*b^2 + 19*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 - 30*(8*a^3*b - 22*a^2*b^2 + 19*a*b^3)
*cosh(d*x + c)^4 + 10*(63*a*b^3*cosh(d*x + c)^6 - 35*(4*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^4 - 24*a^3*b + 66*a^2
*b^2 - 57*a*b^3 + 45*(8*a^3*b - 22*a^2*b^2 + 19*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 - 3*a*b^3 + 40*(9*a*b^
3*cosh(d*x + c)^7 - 7*(4*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^5 + 15*(8*a^3*b - 22*a^2*b^2 + 19*a*b^3)*cosh(d*x +
c)^3 - 3*(8*a^3*b - 22*a^2*b^2 + 19*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 5*(4*a^2*b^2 - 9*a*b^3)*cosh(d*x +
c)^2 + 5*(27*a*b^3*cosh(d*x + c)^8 - 28*(4*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^6 + 90*(8*a^3*b - 22*a^2*b^2 + 19
*a*b^3)*cosh(d*x + c)^4 + 4*a^2*b^2 - 9*a*b^3 - 36*(8*a^3*b - 22*a^2*b^2 + 19*a*b^3)*cosh(d*x + c)^2)*sinh(d*x
+ c)^2 + 240*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cosh(d*x + c)^5 + 5*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cosh(d*x +
c)^4*sinh(d*x + c) + 10*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cosh(d*x + c)^3*sinh(d*x + c)^2 + 10*(a^3 - 3*a^2*b +
3*a*b^2 - b^3)*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cosh(d*x + c)*sinh(d*x + c)
^4 + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*sinh(d*x + c)^5)*sqrt(-a*b)*log((b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*si
nh(d*x + c)^3 + b*sinh(d*x + c)^4 - 2*(2*a + b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 - 2*a - b)*sinh(d*x +
c)^2 + 4*(b*cosh(d*x + c)^3 - (2*a + b)*cosh(d*x + c))*sinh(d*x + c) - 4*(cosh(d*x + c)^3 + 3*cosh(d*x + c)*s
inh(d*x + c)^2 + sinh(d*x + c)^3 + (3*cosh(d*x + c)^2 - 1)*sinh(d*x + c) - cosh(d*x + c))*sqrt(-a*b) + b)/(b*c
osh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 + 2*(2*a - b)*cosh(d*x + c)^2 + 2*(3*b*
cosh(d*x + c)^2 + 2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + (2*a - b)*cosh(d*x + c))*sinh(d*x + c) + b
)) + 10*(3*a*b^3*cosh(d*x + c)^9 - 4*(4*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^7 + 18*(8*a^3*b - 22*a^2*b^2 + 19*a*b
^3)*cosh(d*x + c)^5 - 12*(8*a^3*b - 22*a^2*b^2 + 19*a*b^3)*cosh(d*x + c)^3 + (4*a^2*b^2 - 9*a*b^3)*cosh(d*x +
c))*sinh(d*x + c))/(a*b^4*d*cosh(d*x + c)^5 + 5*a*b^4*d*cosh(d*x + c)^4*sinh(d*x + c) + 10*a*b^4*d*cosh(d*x +
c)^3*sinh(d*x + c)^2 + 10*a*b^4*d*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*a*b^4*d*cosh(d*x + c)*sinh(d*x + c)^4 +
a*b^4*d*sinh(d*x + c)^5), 1/480*(3*a*b^3*cosh(d*x + c)^10 + 30*a*b^3*cosh(d*x + c)*sinh(d*x + c)^9 + 3*a*b^3*s
inh(d*x + c)^10 - 5*(4*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^8 + 5*(27*a*b^3*cosh(d*x + c)^2 - 4*a^2*b^2 + 9*a*b^3)
*sinh(d*x + c)^8 + 40*(9*a*b^3*cosh(d*x + c)^3 - (4*a^2*b^2 - 9*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 + 30*(8*
a^3*b - 22*a^2*b^2 + 19*a*b^3)*cosh(d*x + c)^6 + 10*(63*a*b^3*cosh(d*x + c)^4 + 24*a^3*b - 66*a^2*b^2 + 57*a*b
^3 - 14*(4*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(189*a*b^3*cosh(d*x + c)^5 - 70*(4*a^2*b^2
- 9*a*b^3)*cosh(d*x + c)^3 + 45*(8*a^3*b - 22*a^2*b^2 + 19*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 - 30*(8*a^3*b
- 22*a^2*b^2 + 19*a*b^3)*cosh(d*x + c)^4 + 10*(63*a*b^3*cosh(d*x + c)^6 - 35*(4*a^2*b^2 - 9*a*b^3)*cosh(d*x +
c)^4 - 24*a^3*b + 66*a^2*b^2 - 57*a*b^3 + 45*(8*a^3*b - 22*a^2*b^2 + 19*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)
^4 - 3*a*b^3 + 40*(9*a*b^3*cosh(d*x + c)^7 - 7*(4*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^5 + 15*(8*a^3*b - 22*a^2*b^
2 + 19*a*b^3)*cosh(d*x + c)^3 - 3*(8*a^3*b - 22*a^2*b^2 + 19*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 5*(4*a^2*
b^2 - 9*a*b^3)*cosh(d*x + c)^2 + 5*(27*a*b^3*cosh(d*x + c)^8 - 28*(4*a^2*b^2 - 9*a*b^3)*cosh(d*x + c)^6 + 90*(
8*a^3*b - 22*a^2*b^2 + 19*a*b^3)*cosh(d*x + c)^4 + 4*a^2*b^2 - 9*a*b^3 - 36*(8*a^3*b - 22*a^2*b^2 + 19*a*b^3)*
cosh(d*x + c)^2)*sinh(d*x + c)^2 - 480*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cosh(d*x + c)^5 + 5*(a^3 - 3*a^2*b + 3
*a*b^2 - b^3)*cosh(d*x + c)^4*sinh(d*x + c) + 10*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cosh(d*x + c)^3*sinh(d*x + c)
^2 + 10*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*co
sh(d*x + c)*sinh(d*x + c)^4 + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*sinh(d*x + c)^5)*sqrt(a*b)*arctan(1/2*sqrt(a*b)*
(cosh(d*x + c) + sinh(d*x + c))/a) - 480*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cosh(d*x + c)^5 + 5*(a^3 - 3*a^2*b +
3*a*b^2 - b^3)*cosh(d*x + c)^4*sinh(d*x + c) + 10*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cosh(d*x + c)^3*sinh(d*x +
c)^2 + 10*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*
cosh(d*x + c)*sinh(d*x + c)^4 + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*sinh(d*x + c)^5)*sqrt(a*b)*arctan(1/2*(b*cosh(
d*x + c)^3 + 3*b*cosh(d*x + c)*sinh(d*x + c)^2 + b*sinh(d*x + c)^3 + (4*a - b)*cosh(d*x + c) + (3*b*cosh(d*x +
c)^2 + 4*a - b)*sinh(d*x + c))*sqrt(a*b)/(a*b)) + 10*(3*a*b^3*cosh(d*x + c)^9 - 4*(4*a^2*b^2 - 9*a*b^3)*cosh(
d*x + c)^7 + 18*(8*a^3*b - 22*a^2*b^2 + 19*a*b^3)*cosh(d*x + c)^5 - 12*(8*a^3*b - 22*a^2*b^2 + 19*a*b^3)*cosh(
d*x + c)^3 + (4*a^2*b^2 - 9*a*b^3)*cosh(d*x + c))*sinh(d*x + c))/(a*b^4*d*cosh(d*x + c)^5 + 5*a*b^4*d*cosh(d*x
+ c)^4*sinh(d*x + c) + 10*a*b^4*d*cosh(d*x + c)^3*sinh(d*x + c)^2 + 10*a*b^4*d*cosh(d*x + c)^2*sinh(d*x + c)^
3 + 5*a*b^4*d*cosh(d*x + c)*sinh(d*x + c)^4 + a*b^4*d*sinh(d*x + c)^5)]

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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)**7/(a+b*sinh(d*x+c)**2),x)

[Out]

Timed out

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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)^7/(a+b*sinh(d*x+c)^2),x, algorithm="giac")

[Out]

Exception raised: TypeError