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

Optimal. Leaf size=118 \[ \frac{3 \cosh (c+d x)}{8 d (a-b)^2 \left (a+b \cosh ^2(c+d x)-b\right )}+\frac{\cosh (c+d x)}{4 d (a-b) \left (a+b \cosh ^2(c+d x)-b\right )^2}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right )}{8 \sqrt{b} d (a-b)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.0831678, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3186, 199, 205} \[ \frac{3 \cosh (c+d x)}{8 d (a-b)^2 \left (a+b \cosh ^2(c+d x)-b\right )}+\frac{\cosh (c+d x)}{4 d (a-b) \left (a+b \cosh ^2(c+d x)-b\right )^2}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right )}{8 \sqrt{b} d (a-b)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3186

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

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

Mathematica [C]  time = 0.746678, size = 149, normalized size = 1.26 \[ \frac{\frac{2 \cosh (c+d x) (10 a+3 b \cosh (2 (c+d x))-7 b)}{(a-b)^2 (2 a+b \cosh (2 (c+d x))-b)^2}+\frac{3 \left (\tan ^{-1}\left (\frac{\sqrt{b}-i \sqrt{a} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a-b}}\right )+\tan ^{-1}\left (\frac{\sqrt{b}+i \sqrt{a} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a-b}}\right )\right )}{\sqrt{b} (a-b)^{5/2}}}{8 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*(ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]] + ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/
2])/Sqrt[a - b]]))/((a - b)^(5/2)*Sqrt[b]) + (2*Cosh[c + d*x]*(10*a - 7*b + 3*b*Cosh[2*(c + d*x)]))/((a - b)^2
*(2*a - b + b*Cosh[2*(c + d*x)])^2))/(8*d)

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Maple [B]  time = 0.034, size = 964, normalized size = 8.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)/(a+b*sinh(d*x+c)^2)^3,x)

[Out]

-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/(a^2-2*a*b+b^2)*tan
h(1/2*d*x+1/2*c)^6+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^2-
2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^6*b-2/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/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^6*b^2+15/4/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/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4-23/2/d/(tanh(1/2*d*x+1/2*c)^4*a-2*ta
nh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4+14/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*b^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c
)^4-4/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^2/(a^2-2*a*b+b^2)*
tanh(1/2*d*x+1/2*c)^4*b^3-15/4/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^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^2*a+8/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*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^2-2/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/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^2*b^2+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/(a^2-2*a*b+b^2)-1/2/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^2-2*a*b+b^2)*b+3/8/d/(a^2-2*a*b+b^2)/(a*b-b^2)^(1/2)
*arctan(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a-2*a+4*b)/(a*b-b^2)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (20 \, a e^{\left (5 \, c\right )} - 11 \, b e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} +{\left (20 \, a e^{\left (3 \, c\right )} - 11 \, b e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + 3 \, b e^{\left (7 \, d x + 7 \, c\right )} + 3 \, b e^{\left (d x + c\right )}}{4 \,{\left (a^{2} b^{2} d - 2 \, a b^{3} d + b^{4} d +{\left (a^{2} b^{2} d e^{\left (8 \, c\right )} - 2 \, a b^{3} d e^{\left (8 \, c\right )} + b^{4} d e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} + 4 \,{\left (2 \, a^{3} b d e^{\left (6 \, c\right )} - 5 \, a^{2} b^{2} d e^{\left (6 \, c\right )} + 4 \, a b^{3} d e^{\left (6 \, c\right )} - b^{4} d e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 2 \,{\left (8 \, a^{4} d e^{\left (4 \, c\right )} - 24 \, a^{3} b d e^{\left (4 \, c\right )} + 27 \, a^{2} b^{2} d e^{\left (4 \, c\right )} - 14 \, a b^{3} d e^{\left (4 \, c\right )} + 3 \, b^{4} d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 4 \,{\left (2 \, a^{3} b d e^{\left (2 \, c\right )} - 5 \, a^{2} b^{2} d e^{\left (2 \, c\right )} + 4 \, a b^{3} d e^{\left (2 \, c\right )} - b^{4} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}} + \frac{1}{2} \, \int \frac{3 \,{\left (e^{\left (3 \, d x + 3 \, c\right )} - e^{\left (d x + c\right )}\right )}}{2 \,{\left (a^{2} b - 2 \, a b^{2} + b^{3} +{\left (a^{2} b e^{\left (4 \, c\right )} - 2 \, a b^{2} e^{\left (4 \, c\right )} + b^{3} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (2 \, a^{3} e^{\left (2 \, c\right )} - 5 \, a^{2} b e^{\left (2 \, c\right )} + 4 \, a b^{2} e^{\left (2 \, c\right )} - b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((20*a*e^(5*c) - 11*b*e^(5*c))*e^(5*d*x) + (20*a*e^(3*c) - 11*b*e^(3*c))*e^(3*d*x) + 3*b*e^(7*d*x + 7*c) +
3*b*e^(d*x + c))/(a^2*b^2*d - 2*a*b^3*d + b^4*d + (a^2*b^2*d*e^(8*c) - 2*a*b^3*d*e^(8*c) + b^4*d*e^(8*c))*e^(
8*d*x) + 4*(2*a^3*b*d*e^(6*c) - 5*a^2*b^2*d*e^(6*c) + 4*a*b^3*d*e^(6*c) - b^4*d*e^(6*c))*e^(6*d*x) + 2*(8*a^4*
d*e^(4*c) - 24*a^3*b*d*e^(4*c) + 27*a^2*b^2*d*e^(4*c) - 14*a*b^3*d*e^(4*c) + 3*b^4*d*e^(4*c))*e^(4*d*x) + 4*(2
*a^3*b*d*e^(2*c) - 5*a^2*b^2*d*e^(2*c) + 4*a*b^3*d*e^(2*c) - b^4*d*e^(2*c))*e^(2*d*x)) + 1/2*integrate(3/2*(e^
(3*d*x + 3*c) - e^(d*x + c))/(a^2*b - 2*a*b^2 + b^3 + (a^2*b*e^(4*c) - 2*a*b^2*e^(4*c) + b^3*e^(4*c))*e^(4*d*x
) + 2*(2*a^3*e^(2*c) - 5*a^2*b*e^(2*c) + 4*a*b^2*e^(2*c) - b^3*e^(2*c))*e^(2*d*x)), x)

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Fricas [B]  time = 2.86219, size = 12081, normalized size = 102.38 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(12*(a*b^2 - b^3)*cosh(d*x + c)^7 + 84*(a*b^2 - b^3)*cosh(d*x + c)*sinh(d*x + c)^6 + 12*(a*b^2 - b^3)*si
nh(d*x + c)^7 + 4*(20*a^2*b - 31*a*b^2 + 11*b^3)*cosh(d*x + c)^5 + 4*(20*a^2*b - 31*a*b^2 + 11*b^3 + 63*(a*b^2
- b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 20*(21*(a*b^2 - b^3)*cosh(d*x + c)^3 + (20*a^2*b - 31*a*b^2 + 11*b^
3)*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(20*a^2*b - 31*a*b^2 + 11*b^3)*cosh(d*x + c)^3 + 4*(105*(a*b^2 - b^3)*co
sh(d*x + c)^4 + 20*a^2*b - 31*a*b^2 + 11*b^3 + 10*(20*a^2*b - 31*a*b^2 + 11*b^3)*cosh(d*x + c)^2)*sinh(d*x + c
)^3 + 4*(63*(a*b^2 - b^3)*cosh(d*x + c)^5 + 10*(20*a^2*b - 31*a*b^2 + 11*b^3)*cosh(d*x + c)^3 + 3*(20*a^2*b -
31*a*b^2 + 11*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 - 3*(b^2*cosh(d*x + c)^8 + 8*b^2*cosh(d*x + c)*sinh(d*x + c)
^7 + b^2*sinh(d*x + c)^8 + 4*(2*a*b - b^2)*cosh(d*x + c)^6 + 4*(7*b^2*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x
+ c)^6 + 8*(7*b^2*cosh(d*x + c)^3 + 3*(2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(8*a^2 - 8*a*b + 3*b^2)
*cosh(d*x + c)^4 + 2*(35*b^2*cosh(d*x + c)^4 + 30*(2*a*b - b^2)*cosh(d*x + c)^2 + 8*a^2 - 8*a*b + 3*b^2)*sinh(
d*x + c)^4 + 8*(7*b^2*cosh(d*x + c)^5 + 10*(2*a*b - b^2)*cosh(d*x + c)^3 + (8*a^2 - 8*a*b + 3*b^2)*cosh(d*x +
c))*sinh(d*x + c)^3 + 4*(2*a*b - b^2)*cosh(d*x + c)^2 + 4*(7*b^2*cosh(d*x + c)^6 + 15*(2*a*b - b^2)*cosh(d*x +
c)^4 + 3*(8*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x + c)^2 + b^2 + 8*(b^2*cosh(d*x + c)^
7 + 3*(2*a*b - b^2)*cosh(d*x + c)^5 + (8*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c)^3 + (2*a*b - b^2)*cosh(d*x + c))*s
inh(d*x + c))*sqrt(-a*b + b^2)*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 - 3*b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 - 2*a + 3*b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 -
(2*a - 3*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^2) + b)/(b*cosh(d*x + c)^4 + 4*b*c
osh(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)) + 12*(a*b^2 - b^3)*
cosh(d*x + c) + 4*(21*(a*b^2 - b^3)*cosh(d*x + c)^6 + 5*(20*a^2*b - 31*a*b^2 + 11*b^3)*cosh(d*x + c)^4 + 3*a*b
^2 - 3*b^3 + 3*(20*a^2*b - 31*a*b^2 + 11*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5
- b^6)*d*cosh(d*x + c)^8 + 8*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^3*b^3
- 3*a^2*b^4 + 3*a*b^5 - b^6)*d*sinh(d*x + c)^8 + 4*(2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4 - 5*a*b^5 + b^6)*d*cosh(
d*x + c)^6 + 4*(7*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cosh(d*x + c)^2 + (2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4
- 5*a*b^5 + b^6)*d)*sinh(d*x + c)^6 + 2*(8*a^5*b - 32*a^4*b^2 + 51*a^3*b^3 - 41*a^2*b^4 + 17*a*b^5 - 3*b^6)*d
*cosh(d*x + c)^4 + 8*(7*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cosh(d*x + c)^3 + 3*(2*a^4*b^2 - 7*a^3*b^3 + 9
*a^2*b^4 - 5*a*b^5 + b^6)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cos
h(d*x + c)^4 + 30*(2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4 - 5*a*b^5 + b^6)*d*cosh(d*x + c)^2 + (8*a^5*b - 32*a^4*b^
2 + 51*a^3*b^3 - 41*a^2*b^4 + 17*a*b^5 - 3*b^6)*d)*sinh(d*x + c)^4 + 4*(2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4 - 5*
a*b^5 + b^6)*d*cosh(d*x + c)^2 + 8*(7*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cosh(d*x + c)^5 + 10*(2*a^4*b^2
- 7*a^3*b^3 + 9*a^2*b^4 - 5*a*b^5 + b^6)*d*cosh(d*x + c)^3 + (8*a^5*b - 32*a^4*b^2 + 51*a^3*b^3 - 41*a^2*b^4 +
17*a*b^5 - 3*b^6)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cosh(d*x +
c)^6 + 15*(2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4 - 5*a*b^5 + b^6)*d*cosh(d*x + c)^4 + 3*(8*a^5*b - 32*a^4*b^2 + 51
*a^3*b^3 - 41*a^2*b^4 + 17*a*b^5 - 3*b^6)*d*cosh(d*x + c)^2 + (2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4 - 5*a*b^5 + b
^6)*d)*sinh(d*x + c)^2 + (a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d + 8*((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*
cosh(d*x + c)^7 + 3*(2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4 - 5*a*b^5 + b^6)*d*cosh(d*x + c)^5 + (8*a^5*b - 32*a^4*
b^2 + 51*a^3*b^3 - 41*a^2*b^4 + 17*a*b^5 - 3*b^6)*d*cosh(d*x + c)^3 + (2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4 - 5*a
*b^5 + b^6)*d*cosh(d*x + c))*sinh(d*x + c)), 1/8*(6*(a*b^2 - b^3)*cosh(d*x + c)^7 + 42*(a*b^2 - b^3)*cosh(d*x
+ c)*sinh(d*x + c)^6 + 6*(a*b^2 - b^3)*sinh(d*x + c)^7 + 2*(20*a^2*b - 31*a*b^2 + 11*b^3)*cosh(d*x + c)^5 + 2*
(20*a^2*b - 31*a*b^2 + 11*b^3 + 63*(a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 10*(21*(a*b^2 - b^3)*cosh(
d*x + c)^3 + (20*a^2*b - 31*a*b^2 + 11*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 2*(20*a^2*b - 31*a*b^2 + 11*b^3)*
cosh(d*x + c)^3 + 2*(105*(a*b^2 - b^3)*cosh(d*x + c)^4 + 20*a^2*b - 31*a*b^2 + 11*b^3 + 10*(20*a^2*b - 31*a*b^
2 + 11*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(63*(a*b^2 - b^3)*cosh(d*x + c)^5 + 10*(20*a^2*b - 31*a*b^2 +
11*b^3)*cosh(d*x + c)^3 + 3*(20*a^2*b - 31*a*b^2 + 11*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + 3*(b^2*cosh(d*x +
c)^8 + 8*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + b^2*sinh(d*x + c)^8 + 4*(2*a*b - b^2)*cosh(d*x + c)^6 + 4*(7*b^2
*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x + c)^6 + 8*(7*b^2*cosh(d*x + c)^3 + 3*(2*a*b - b^2)*cosh(d*x + c))*si
nh(d*x + c)^5 + 2*(8*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c)^4 + 2*(35*b^2*cosh(d*x + c)^4 + 30*(2*a*b - b^2)*cosh(
d*x + c)^2 + 8*a^2 - 8*a*b + 3*b^2)*sinh(d*x + c)^4 + 8*(7*b^2*cosh(d*x + c)^5 + 10*(2*a*b - b^2)*cosh(d*x + c
)^3 + (8*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(2*a*b - b^2)*cosh(d*x + c)^2 + 4*(7*b^2*cosh
(d*x + c)^6 + 15*(2*a*b - b^2)*cosh(d*x + c)^4 + 3*(8*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh
(d*x + c)^2 + b^2 + 8*(b^2*cosh(d*x + c)^7 + 3*(2*a*b - b^2)*cosh(d*x + c)^5 + (8*a^2 - 8*a*b + 3*b^2)*cosh(d*
x + c)^3 + (2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a*b - b^2)*arctan(-1/2*(b*cosh(d*x + c)^3 + 3*b*co
sh(d*x + c)*sinh(d*x + c)^2 + b*sinh(d*x + c)^3 + (4*a - 3*b)*cosh(d*x + c) + (3*b*cosh(d*x + c)^2 + 4*a - 3*b
)*sinh(d*x + c))/sqrt(a*b - b^2)) - 3*(b^2*cosh(d*x + c)^8 + 8*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + b^2*sinh(d*
x + c)^8 + 4*(2*a*b - b^2)*cosh(d*x + c)^6 + 4*(7*b^2*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x + c)^6 + 8*(7*b^
2*cosh(d*x + c)^3 + 3*(2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(8*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c)^4
+ 2*(35*b^2*cosh(d*x + c)^4 + 30*(2*a*b - b^2)*cosh(d*x + c)^2 + 8*a^2 - 8*a*b + 3*b^2)*sinh(d*x + c)^4 + 8*(
7*b^2*cosh(d*x + c)^5 + 10*(2*a*b - b^2)*cosh(d*x + c)^3 + (8*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c
)^3 + 4*(2*a*b - b^2)*cosh(d*x + c)^2 + 4*(7*b^2*cosh(d*x + c)^6 + 15*(2*a*b - b^2)*cosh(d*x + c)^4 + 3*(8*a^2
- 8*a*b + 3*b^2)*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x + c)^2 + b^2 + 8*(b^2*cosh(d*x + c)^7 + 3*(2*a*b - b
^2)*cosh(d*x + c)^5 + (8*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c)^3 + (2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c))*sq
rt(a*b - b^2)*arctan(-1/2*sqrt(a*b - b^2)*(cosh(d*x + c) + sinh(d*x + c))/(a - b)) + 6*(a*b^2 - b^3)*cosh(d*x
+ c) + 2*(21*(a*b^2 - b^3)*cosh(d*x + c)^6 + 5*(20*a^2*b - 31*a*b^2 + 11*b^3)*cosh(d*x + c)^4 + 3*a*b^2 - 3*b^
3 + 3*(20*a^2*b - 31*a*b^2 + 11*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*
cosh(d*x + c)^8 + 8*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^3*b^3 - 3*a^2*b
^4 + 3*a*b^5 - b^6)*d*sinh(d*x + c)^8 + 4*(2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4 - 5*a*b^5 + b^6)*d*cosh(d*x + c)^
6 + 4*(7*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cosh(d*x + c)^2 + (2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4 - 5*a*b^
5 + b^6)*d)*sinh(d*x + c)^6 + 2*(8*a^5*b - 32*a^4*b^2 + 51*a^3*b^3 - 41*a^2*b^4 + 17*a*b^5 - 3*b^6)*d*cosh(d*x
+ c)^4 + 8*(7*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cosh(d*x + c)^3 + 3*(2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4
- 5*a*b^5 + b^6)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cosh(d*x + c
)^4 + 30*(2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4 - 5*a*b^5 + b^6)*d*cosh(d*x + c)^2 + (8*a^5*b - 32*a^4*b^2 + 51*a^
3*b^3 - 41*a^2*b^4 + 17*a*b^5 - 3*b^6)*d)*sinh(d*x + c)^4 + 4*(2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4 - 5*a*b^5 + b
^6)*d*cosh(d*x + c)^2 + 8*(7*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cosh(d*x + c)^5 + 10*(2*a^4*b^2 - 7*a^3*b
^3 + 9*a^2*b^4 - 5*a*b^5 + b^6)*d*cosh(d*x + c)^3 + (8*a^5*b - 32*a^4*b^2 + 51*a^3*b^3 - 41*a^2*b^4 + 17*a*b^5
- 3*b^6)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cosh(d*x + c)^6 + 15
*(2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4 - 5*a*b^5 + b^6)*d*cosh(d*x + c)^4 + 3*(8*a^5*b - 32*a^4*b^2 + 51*a^3*b^3
- 41*a^2*b^4 + 17*a*b^5 - 3*b^6)*d*cosh(d*x + c)^2 + (2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4 - 5*a*b^5 + b^6)*d)*si
nh(d*x + c)^2 + (a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d + 8*((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cosh(d*x
+ c)^7 + 3*(2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4 - 5*a*b^5 + b^6)*d*cosh(d*x + c)^5 + (8*a^5*b - 32*a^4*b^2 + 51*
a^3*b^3 - 41*a^2*b^4 + 17*a*b^5 - 3*b^6)*d*cosh(d*x + c)^3 + (2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4 - 5*a*b^5 + b^
6)*d*cosh(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(sinh(d*x+c)/(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(sinh(d*x+c)/(a+b*sinh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError