3.342 \(\int \frac{\cosh ^3(c+d x)}{(a+b \sinh ^2(c+d x))^3} \, dx\)

Optimal. Leaf size=117 \[ \frac{(a+3 b) \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} b^{3/2} d}+\frac{(a+3 b) \sinh (c+d x)}{8 a^2 b d \left (a+b \sinh ^2(c+d x)\right )}-\frac{(a-b) \sinh (c+d x)}{4 a b d \left (a+b \sinh ^2(c+d x)\right )^2} \]

[Out]

((a + 3*b)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*b^(3/2)*d) - ((a - b)*Sinh[c + d*x])/(4*a*b*d*(
a + b*Sinh[c + d*x]^2)^2) + ((a + 3*b)*Sinh[c + d*x])/(8*a^2*b*d*(a + b*Sinh[c + d*x]^2))

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Rubi [A]  time = 0.0925692, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3190, 385, 199, 205} \[ \frac{(a+3 b) \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} b^{3/2} d}+\frac{(a+3 b) \sinh (c+d x)}{8 a^2 b d \left (a+b \sinh ^2(c+d x)\right )}-\frac{(a-b) \sinh (c+d x)}{4 a b d \left (a+b \sinh ^2(c+d x)\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]^3/(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

((a + 3*b)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*b^(3/2)*d) - ((a - b)*Sinh[c + d*x])/(4*a*b*d*(
a + b*Sinh[c + d*x]^2)^2) + ((a + 3*b)*Sinh[c + d*x])/(8*a^2*b*d*(a + b*Sinh[c + d*x]^2))

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 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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 ^3(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1+x^2}{\left (a+b x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=-\frac{(a-b) \sinh (c+d x)}{4 a b d \left (a+b \sinh ^2(c+d x)\right )^2}+\frac{(a+3 b) \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{4 a b d}\\ &=-\frac{(a-b) \sinh (c+d x)}{4 a b d \left (a+b \sinh ^2(c+d x)\right )^2}+\frac{(a+3 b) \sinh (c+d x)}{8 a^2 b d \left (a+b \sinh ^2(c+d x)\right )}+\frac{(a+3 b) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sinh (c+d x)\right )}{8 a^2 b d}\\ &=\frac{(a+3 b) \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} b^{3/2} d}-\frac{(a-b) \sinh (c+d x)}{4 a b d \left (a+b \sinh ^2(c+d x)\right )^2}+\frac{(a+3 b) \sinh (c+d x)}{8 a^2 b d \left (a+b \sinh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.665558, size = 114, normalized size = 0.97 \[ \frac{(a+3 b) \left (\frac{\sinh (c+d x) \left (5 a+3 b \sinh ^2(c+d x)\right )}{8 a^2 \left (a+b \sinh ^2(c+d x)\right )^2}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b}}\right )-\frac{\sinh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2}}{3 b d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]^3/(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

(-(Sinh[c + d*x]/(a + b*Sinh[c + d*x]^2)^2) + (a + 3*b)*((3*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(8*a^(5/2
)*Sqrt[b]) + (Sinh[c + d*x]*(5*a + 3*b*Sinh[c + d*x]^2))/(8*a^2*(a + b*Sinh[c + d*x]^2)^2)))/(3*b*d)

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Maple [B]  time = 0.068, size = 1348, normalized size = 11.5 \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)^3/(a+b*sinh(d*x+c)^2)^3,x)

[Out]

1/4/d/b/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2*tanh(1/2*d*x+1/2*c)^
7-5/4/d/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/a*tanh(1/2*d*x+1/2*c
)^7-3/4/d/b/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2*tanh(1/2*d*x+1/2
*c)^5+11/4/d/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/a*tanh(1/2*d*x+
1/2*c)^5-3/d*b/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/a^2*tanh(1/2*
d*x+1/2*c)^5+3/4/d/b/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2*tanh(1/
2*d*x+1/2*c)^3-11/4/d/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/a*tanh
(1/2*d*x+1/2*c)^3+3/d*b/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/a^2*
tanh(1/2*d*x+1/2*c)^3-1/4/d/b/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^
2*tanh(1/2*d*x+1/2*c)+5/4/d/(tanh(1/2*d*x+1/2*c)^4*a-2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/
a*tanh(1/2*d*x+1/2*c)-1/8/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))-1/8/d/b/a/((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))-1/4/d/a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*a
rctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))-1/8/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))+1/8/d/b/a/((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))-1/4/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))-3
/8/d/a^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))+3/8/d*b/a^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))+3/8/d/a^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))+3/8/d*b/a^2/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tan
h(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 (a b e^{\left (7 \, c\right )} + 3 \, b^{2} e^{\left (7 \, c\right )}\right )} e^{\left (7 \, d x\right )} -{\left (4 \, a^{2} e^{\left (5 \, c\right )} - 17 \, a b e^{\left (5 \, c\right )} + 9 \, b^{2} e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} +{\left (4 \, a^{2} e^{\left (3 \, c\right )} - 17 \, a b e^{\left (3 \, c\right )} + 9 \, b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} -{\left (a b e^{c} + 3 \, b^{2} e^{c}\right )} e^{\left (d x\right )}}{4 \,{\left (a^{2} b^{3} d e^{\left (8 \, d x + 8 \, c\right )} + a^{2} b^{3} d + 4 \,{\left (2 \, a^{3} b^{2} d e^{\left (6 \, c\right )} - a^{2} b^{3} d e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 2 \,{\left (8 \, a^{4} b d e^{\left (4 \, c\right )} - 8 \, a^{3} b^{2} d e^{\left (4 \, c\right )} + 3 \, a^{2} b^{3} d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 4 \,{\left (2 \, a^{3} b^{2} d e^{\left (2 \, c\right )} - a^{2} b^{3} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}} + \frac{1}{8} \, \int \frac{2 \,{\left ({\left (a e^{\left (3 \, c\right )} + 3 \, b e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} +{\left (a e^{c} + 3 \, b e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + a^{2} b^{2} + 2 \,{\left (2 \, a^{3} b e^{\left (2 \, c\right )} - a^{2} b^{2} 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)^3/(a+b*sinh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

1/4*((a*b*e^(7*c) + 3*b^2*e^(7*c))*e^(7*d*x) - (4*a^2*e^(5*c) - 17*a*b*e^(5*c) + 9*b^2*e^(5*c))*e^(5*d*x) + (4
*a^2*e^(3*c) - 17*a*b*e^(3*c) + 9*b^2*e^(3*c))*e^(3*d*x) - (a*b*e^c + 3*b^2*e^c)*e^(d*x))/(a^2*b^3*d*e^(8*d*x
+ 8*c) + a^2*b^3*d + 4*(2*a^3*b^2*d*e^(6*c) - a^2*b^3*d*e^(6*c))*e^(6*d*x) + 2*(8*a^4*b*d*e^(4*c) - 8*a^3*b^2*
d*e^(4*c) + 3*a^2*b^3*d*e^(4*c))*e^(4*d*x) + 4*(2*a^3*b^2*d*e^(2*c) - a^2*b^3*d*e^(2*c))*e^(2*d*x)) + 1/8*inte
grate(2*((a*e^(3*c) + 3*b*e^(3*c))*e^(3*d*x) + (a*e^c + 3*b*e^c)*e^(d*x))/(a^2*b^2*e^(4*d*x + 4*c) + a^2*b^2 +
2*(2*a^3*b*e^(2*c) - a^2*b^2*e^(2*c))*e^(2*d*x)), x)

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Fricas [B]  time = 2.33105, size = 11478, normalized size = 98.1 \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)^3/(a+b*sinh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[1/16*(4*(a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^7 + 28*(a^2*b^2 + 3*a*b^3)*cosh(d*x + c)*sinh(d*x + c)^6 + 4*(a^2*b
^2 + 3*a*b^3)*sinh(d*x + c)^7 - 4*(4*a^3*b - 17*a^2*b^2 + 9*a*b^3)*cosh(d*x + c)^5 - 4*(4*a^3*b - 17*a^2*b^2 +
9*a*b^3 - 21*(a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 20*(7*(a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^3
- (4*a^3*b - 17*a^2*b^2 + 9*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(4*a^3*b - 17*a^2*b^2 + 9*a*b^3)*cosh(d
*x + c)^3 + 4*(35*(a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^4 + 4*a^3*b - 17*a^2*b^2 + 9*a*b^3 - 10*(4*a^3*b - 17*a^2*
b^2 + 9*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 4*(21*(a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^5 - 10*(4*a^3*b - 17
*a^2*b^2 + 9*a*b^3)*cosh(d*x + c)^3 + 3*(4*a^3*b - 17*a^2*b^2 + 9*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 - ((a*
b^2 + 3*b^3)*cosh(d*x + c)^8 + 8*(a*b^2 + 3*b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (a*b^2 + 3*b^3)*sinh(d*x + c)
^8 + 4*(2*a^2*b + 5*a*b^2 - 3*b^3)*cosh(d*x + c)^6 + 4*(2*a^2*b + 5*a*b^2 - 3*b^3 + 7*(a*b^2 + 3*b^3)*cosh(d*x
+ c)^2)*sinh(d*x + c)^6 + 8*(7*(a*b^2 + 3*b^3)*cosh(d*x + c)^3 + 3*(2*a^2*b + 5*a*b^2 - 3*b^3)*cosh(d*x + c))
*sinh(d*x + c)^5 + 2*(8*a^3 + 16*a^2*b - 21*a*b^2 + 9*b^3)*cosh(d*x + c)^4 + 2*(35*(a*b^2 + 3*b^3)*cosh(d*x +
c)^4 + 8*a^3 + 16*a^2*b - 21*a*b^2 + 9*b^3 + 30*(2*a^2*b + 5*a*b^2 - 3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 +
8*(7*(a*b^2 + 3*b^3)*cosh(d*x + c)^5 + 10*(2*a^2*b + 5*a*b^2 - 3*b^3)*cosh(d*x + c)^3 + (8*a^3 + 16*a^2*b - 2
1*a*b^2 + 9*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + a*b^2 + 3*b^3 + 4*(2*a^2*b + 5*a*b^2 - 3*b^3)*cosh(d*x + c)^
2 + 4*(7*(a*b^2 + 3*b^3)*cosh(d*x + c)^6 + 15*(2*a^2*b + 5*a*b^2 - 3*b^3)*cosh(d*x + c)^4 + 2*a^2*b + 5*a*b^2
- 3*b^3 + 3*(8*a^3 + 16*a^2*b - 21*a*b^2 + 9*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a*b^2 + 3*b^3)*cosh(d
*x + c)^7 + 3*(2*a^2*b + 5*a*b^2 - 3*b^3)*cosh(d*x + c)^5 + (8*a^3 + 16*a^2*b - 21*a*b^2 + 9*b^3)*cosh(d*x + c
)^3 + (2*a^2*b + 5*a*b^2 - 3*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a*b)*log((b*cosh(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) - 4*(cosh(d*x + c)^3 + 3*cosh(
d*x + c)*sinh(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*cosh(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)) - 4*(a^2*b^2 + 3*a*b^3)*cosh(d*x + c) + 4*(7*(a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^6 - 5*(4*a^3*b - 1
7*a^2*b^2 + 9*a*b^3)*cosh(d*x + c)^4 - a^2*b^2 - 3*a*b^3 + 3*(4*a^3*b - 17*a^2*b^2 + 9*a*b^3)*cosh(d*x + c)^2)
*sinh(d*x + c))/(a^3*b^4*d*cosh(d*x + c)^8 + 8*a^3*b^4*d*cosh(d*x + c)*sinh(d*x + c)^7 + a^3*b^4*d*sinh(d*x +
c)^8 + a^3*b^4*d + 4*(2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^6 + 4*(7*a^3*b^4*d*cosh(d*x + c)^2 + (2*a^4*b^3 - a
^3*b^4)*d)*sinh(d*x + c)^6 + 2*(8*a^5*b^2 - 8*a^4*b^3 + 3*a^3*b^4)*d*cosh(d*x + c)^4 + 8*(7*a^3*b^4*d*cosh(d*x
+ c)^3 + 3*(2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*a^3*b^4*d*cosh(d*x + c)^4 + 30*(2*a
^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^2 + (8*a^5*b^2 - 8*a^4*b^3 + 3*a^3*b^4)*d)*sinh(d*x + c)^4 + 4*(2*a^4*b^3 -
a^3*b^4)*d*cosh(d*x + c)^2 + 8*(7*a^3*b^4*d*cosh(d*x + c)^5 + 10*(2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^3 + (8*
a^5*b^2 - 8*a^4*b^3 + 3*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*a^3*b^4*d*cosh(d*x + c)^6 + 15*(2*a^4
*b^3 - a^3*b^4)*d*cosh(d*x + c)^4 + 3*(8*a^5*b^2 - 8*a^4*b^3 + 3*a^3*b^4)*d*cosh(d*x + c)^2 + (2*a^4*b^3 - a^3
*b^4)*d)*sinh(d*x + c)^2 + 8*(a^3*b^4*d*cosh(d*x + c)^7 + 3*(2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^5 + (8*a^5*b
^2 - 8*a^4*b^3 + 3*a^3*b^4)*d*cosh(d*x + c)^3 + (2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)), 1/8*(2*
(a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^7 + 14*(a^2*b^2 + 3*a*b^3)*cosh(d*x + c)*sinh(d*x + c)^6 + 2*(a^2*b^2 + 3*a*
b^3)*sinh(d*x + c)^7 - 2*(4*a^3*b - 17*a^2*b^2 + 9*a*b^3)*cosh(d*x + c)^5 - 2*(4*a^3*b - 17*a^2*b^2 + 9*a*b^3
- 21*(a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 10*(7*(a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^3 - (4*a^3
*b - 17*a^2*b^2 + 9*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 2*(4*a^3*b - 17*a^2*b^2 + 9*a*b^3)*cosh(d*x + c)^3
+ 2*(35*(a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^4 + 4*a^3*b - 17*a^2*b^2 + 9*a*b^3 - 10*(4*a^3*b - 17*a^2*b^2 + 9*a
*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(21*(a^2*b^2 + 3*a*b^3)*cosh(d*x + c)^5 - 10*(4*a^3*b - 17*a^2*b^2
+ 9*a*b^3)*cosh(d*x + c)^3 + 3*(4*a^3*b - 17*a^2*b^2 + 9*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + ((a*b^2 + 3*b
^3)*cosh(d*x + c)^8 + 8*(a*b^2 + 3*b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (a*b^2 + 3*b^3)*sinh(d*x + c)^8 + 4*(2
*a^2*b + 5*a*b^2 - 3*b^3)*cosh(d*x + c)^6 + 4*(2*a^2*b + 5*a*b^2 - 3*b^3 + 7*(a*b^2 + 3*b^3)*cosh(d*x + c)^2)*
sinh(d*x + c)^6 + 8*(7*(a*b^2 + 3*b^3)*cosh(d*x + c)^3 + 3*(2*a^2*b + 5*a*b^2 - 3*b^3)*cosh(d*x + c))*sinh(d*x
+ c)^5 + 2*(8*a^3 + 16*a^2*b - 21*a*b^2 + 9*b^3)*cosh(d*x + c)^4 + 2*(35*(a*b^2 + 3*b^3)*cosh(d*x + c)^4 + 8*
a^3 + 16*a^2*b - 21*a*b^2 + 9*b^3 + 30*(2*a^2*b + 5*a*b^2 - 3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(a*
b^2 + 3*b^3)*cosh(d*x + c)^5 + 10*(2*a^2*b + 5*a*b^2 - 3*b^3)*cosh(d*x + c)^3 + (8*a^3 + 16*a^2*b - 21*a*b^2 +
9*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + a*b^2 + 3*b^3 + 4*(2*a^2*b + 5*a*b^2 - 3*b^3)*cosh(d*x + c)^2 + 4*(7*
(a*b^2 + 3*b^3)*cosh(d*x + c)^6 + 15*(2*a^2*b + 5*a*b^2 - 3*b^3)*cosh(d*x + c)^4 + 2*a^2*b + 5*a*b^2 - 3*b^3 +
3*(8*a^3 + 16*a^2*b - 21*a*b^2 + 9*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a*b^2 + 3*b^3)*cosh(d*x + c)^7
+ 3*(2*a^2*b + 5*a*b^2 - 3*b^3)*cosh(d*x + c)^5 + (8*a^3 + 16*a^2*b - 21*a*b^2 + 9*b^3)*cosh(d*x + c)^3 + (2*
a^2*b + 5*a*b^2 - 3*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a*b)*arctan(1/2*sqrt(a*b)*(cosh(d*x + c) + sinh(d*
x + c))/a) + ((a*b^2 + 3*b^3)*cosh(d*x + c)^8 + 8*(a*b^2 + 3*b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (a*b^2 + 3*b
^3)*sinh(d*x + c)^8 + 4*(2*a^2*b + 5*a*b^2 - 3*b^3)*cosh(d*x + c)^6 + 4*(2*a^2*b + 5*a*b^2 - 3*b^3 + 7*(a*b^2
+ 3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a*b^2 + 3*b^3)*cosh(d*x + c)^3 + 3*(2*a^2*b + 5*a*b^2 - 3*b^
3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(8*a^3 + 16*a^2*b - 21*a*b^2 + 9*b^3)*cosh(d*x + c)^4 + 2*(35*(a*b^2 + 3
*b^3)*cosh(d*x + c)^4 + 8*a^3 + 16*a^2*b - 21*a*b^2 + 9*b^3 + 30*(2*a^2*b + 5*a*b^2 - 3*b^3)*cosh(d*x + c)^2)*
sinh(d*x + c)^4 + 8*(7*(a*b^2 + 3*b^3)*cosh(d*x + c)^5 + 10*(2*a^2*b + 5*a*b^2 - 3*b^3)*cosh(d*x + c)^3 + (8*a
^3 + 16*a^2*b - 21*a*b^2 + 9*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + a*b^2 + 3*b^3 + 4*(2*a^2*b + 5*a*b^2 - 3*b^
3)*cosh(d*x + c)^2 + 4*(7*(a*b^2 + 3*b^3)*cosh(d*x + c)^6 + 15*(2*a^2*b + 5*a*b^2 - 3*b^3)*cosh(d*x + c)^4 + 2
*a^2*b + 5*a*b^2 - 3*b^3 + 3*(8*a^3 + 16*a^2*b - 21*a*b^2 + 9*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a*b^
2 + 3*b^3)*cosh(d*x + c)^7 + 3*(2*a^2*b + 5*a*b^2 - 3*b^3)*cosh(d*x + c)^5 + (8*a^3 + 16*a^2*b - 21*a*b^2 + 9*
b^3)*cosh(d*x + c)^3 + (2*a^2*b + 5*a*b^2 - 3*b^3)*cosh(d*x + c))*sinh(d*x + c))*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)) - 2*(a^2*b^2 + 3*a*b^3)*cosh(d*x + c) + 2*(7*(a^2*b^2 + 3*a*b
^3)*cosh(d*x + c)^6 - 5*(4*a^3*b - 17*a^2*b^2 + 9*a*b^3)*cosh(d*x + c)^4 - a^2*b^2 - 3*a*b^3 + 3*(4*a^3*b - 17
*a^2*b^2 + 9*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/(a^3*b^4*d*cosh(d*x + c)^8 + 8*a^3*b^4*d*cosh(d*x + c)*sin
h(d*x + c)^7 + a^3*b^4*d*sinh(d*x + c)^8 + a^3*b^4*d + 4*(2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^6 + 4*(7*a^3*b^
4*d*cosh(d*x + c)^2 + (2*a^4*b^3 - a^3*b^4)*d)*sinh(d*x + c)^6 + 2*(8*a^5*b^2 - 8*a^4*b^3 + 3*a^3*b^4)*d*cosh(
d*x + c)^4 + 8*(7*a^3*b^4*d*cosh(d*x + c)^3 + 3*(2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35
*a^3*b^4*d*cosh(d*x + c)^4 + 30*(2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^2 + (8*a^5*b^2 - 8*a^4*b^3 + 3*a^3*b^4)*
d)*sinh(d*x + c)^4 + 4*(2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^2 + 8*(7*a^3*b^4*d*cosh(d*x + c)^5 + 10*(2*a^4*b^
3 - a^3*b^4)*d*cosh(d*x + c)^3 + (8*a^5*b^2 - 8*a^4*b^3 + 3*a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*a
^3*b^4*d*cosh(d*x + c)^6 + 15*(2*a^4*b^3 - a^3*b^4)*d*cosh(d*x + c)^4 + 3*(8*a^5*b^2 - 8*a^4*b^3 + 3*a^3*b^4)*
d*cosh(d*x + c)^2 + (2*a^4*b^3 - a^3*b^4)*d)*sinh(d*x + c)^2 + 8*(a^3*b^4*d*cosh(d*x + c)^7 + 3*(2*a^4*b^3 - a
^3*b^4)*d*cosh(d*x + c)^5 + (8*a^5*b^2 - 8*a^4*b^3 + 3*a^3*b^4)*d*cosh(d*x + c)^3 + (2*a^4*b^3 - a^3*b^4)*d*co
sh(d*x + c))*sinh(d*x + c))]

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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)**3/(a+b*sinh(d*x+c)**2)**3,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)^3/(a+b*sinh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError