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

Optimal. Leaf size=135 \[ \frac{(a-4 b) \tan ^{-1}\left (\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right )}{8 b^{3/2} d (a-b)^{5/2}}+\frac{(a-4 b) \cosh (c+d x)}{8 b d (a-b)^2 \left (a+b \cosh ^2(c+d x)-b\right )}-\frac{a \cosh (c+d x)}{4 b d (a-b) \left (a+b \cosh ^2(c+d x)-b\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.141105, antiderivative size = 135, 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 = {3186, 385, 199, 205} \[ \frac{(a-4 b) \tan ^{-1}\left (\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right )}{8 b^{3/2} d (a-b)^{5/2}}+\frac{(a-4 b) \cosh (c+d x)}{8 b d (a-b)^2 \left (a+b \cosh ^2(c+d x)-b\right )}-\frac{a \cosh (c+d x)}{4 b d (a-b) \left (a+b \cosh ^2(c+d x)-b\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [C]  time = 1.22242, size = 170, normalized size = 1.26 \[ \frac{\frac{2 \sqrt{b} \cosh (c+d x) \left (-2 a^2+b (a-4 b) \cosh (2 (c+d x))-5 a b+4 b^2\right )}{(a-b)^2 (2 a+b \cosh (2 (c+d x))-b)^2}+\frac{(a-4 b) \left (\tan ^{-1}\left (\frac{\sqrt{b}-i \sqrt{a} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a-b}}\right )+\tan ^{-1}\left (\frac{\sqrt{b}+i \sqrt{a} \tanh \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a-b}}\right )\right )}{(a-b)^{5/2}}}{8 b^{3/2} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.042, size = 961, normalized size = 7.1 \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)^3/(a+b*sinh(d*x+c)^2)^3,x)

[Out]

1/4/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/b/(a^2-2*a*b+b^2)*
tanh(1/2*d*x+1/2*c)^6-1/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)^6-3/4/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/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4+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/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4+2/d/(tanh(1/2*d*x+1/2*c)^4*a-2*t
anh(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)^4-4/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*b^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c
)^4+3/4/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/b/(a^2-2*a*b+b^2)*
tanh(1/2*d*x+1/2*c)^2*a^2+1/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-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*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^2-1/4/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/b/(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/(a^2-2*a*b+b^2)+1/8/d/b/(a^2-2*a*b+b^2)/(a*b-b^2)^(1/2)*arctan(1/4*(2*ta
nh(1/2*d*x+1/2*c)^2*a-2*a+4*b)/(a*b-b^2)^(1/2))*a-1/2/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 (a b e^{\left (7 \, c\right )} - 4 \, b^{2} e^{\left (7 \, c\right )}\right )} e^{\left (7 \, d x\right )} -{\left (4 \, a^{2} e^{\left (5 \, c\right )} + 9 \, a b e^{\left (5 \, c\right )} - 4 \, b^{2} e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} -{\left (4 \, a^{2} e^{\left (3 \, c\right )} + 9 \, a b e^{\left (3 \, c\right )} - 4 \, b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} +{\left (a b e^{c} - 4 \, b^{2} e^{c}\right )} e^{\left (d x\right )}}{4 \,{\left (a^{2} b^{3} d - 2 \, a b^{4} d + b^{5} d +{\left (a^{2} b^{3} d e^{\left (8 \, c\right )} - 2 \, a b^{4} d e^{\left (8 \, c\right )} + b^{5} d e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} + 4 \,{\left (2 \, a^{3} b^{2} d e^{\left (6 \, c\right )} - 5 \, a^{2} b^{3} d e^{\left (6 \, c\right )} + 4 \, a b^{4} d e^{\left (6 \, c\right )} - b^{5} 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 )} - 24 \, a^{3} b^{2} d e^{\left (4 \, c\right )} + 27 \, a^{2} b^{3} d e^{\left (4 \, c\right )} - 14 \, a b^{4} d e^{\left (4 \, c\right )} + 3 \, b^{5} 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 )} - 5 \, a^{2} b^{3} d e^{\left (2 \, c\right )} + 4 \, a b^{4} d e^{\left (2 \, c\right )} - b^{5} 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 )} - 4 \, b e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} -{\left (a e^{c} - 4 \, b e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{2} b^{2} - 2 \, a b^{3} + b^{4} +{\left (a^{2} b^{2} e^{\left (4 \, c\right )} - 2 \, a b^{3} e^{\left (4 \, c\right )} + b^{4} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (2 \, a^{3} b e^{\left (2 \, c\right )} - 5 \, a^{2} b^{2} e^{\left (2 \, c\right )} + 4 \, 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(sinh(d*x+c)^3/(a+b*sinh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

1/4*((a*b*e^(7*c) - 4*b^2*e^(7*c))*e^(7*d*x) - (4*a^2*e^(5*c) + 9*a*b*e^(5*c) - 4*b^2*e^(5*c))*e^(5*d*x) - (4*
a^2*e^(3*c) + 9*a*b*e^(3*c) - 4*b^2*e^(3*c))*e^(3*d*x) + (a*b*e^c - 4*b^2*e^c)*e^(d*x))/(a^2*b^3*d - 2*a*b^4*d
+ b^5*d + (a^2*b^3*d*e^(8*c) - 2*a*b^4*d*e^(8*c) + b^5*d*e^(8*c))*e^(8*d*x) + 4*(2*a^3*b^2*d*e^(6*c) - 5*a^2*
b^3*d*e^(6*c) + 4*a*b^4*d*e^(6*c) - b^5*d*e^(6*c))*e^(6*d*x) + 2*(8*a^4*b*d*e^(4*c) - 24*a^3*b^2*d*e^(4*c) + 2
7*a^2*b^3*d*e^(4*c) - 14*a*b^4*d*e^(4*c) + 3*b^5*d*e^(4*c))*e^(4*d*x) + 4*(2*a^3*b^2*d*e^(2*c) - 5*a^2*b^3*d*e
^(2*c) + 4*a*b^4*d*e^(2*c) - b^5*d*e^(2*c))*e^(2*d*x)) + 1/8*integrate(2*((a*e^(3*c) - 4*b*e^(3*c))*e^(3*d*x)
- (a*e^c - 4*b*e^c)*e^(d*x))/(a^2*b^2 - 2*a*b^3 + b^4 + (a^2*b^2*e^(4*c) - 2*a*b^3*e^(4*c) + b^4*e^(4*c))*e^(4
*d*x) + 2*(2*a^3*b*e^(2*c) - 5*a^2*b^2*e^(2*c) + 4*a*b^3*e^(2*c) - b^4*e^(2*c))*e^(2*d*x)), x)

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Fricas [B]  time = 2.96323, size = 14074, normalized size = 104.25 \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)^3/(a+b*sinh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[1/16*(4*(a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(d*x + c)^7 + 28*(a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(d*x + c)*sinh(d*x +
c)^6 + 4*(a^2*b^2 - 5*a*b^3 + 4*b^4)*sinh(d*x + c)^7 - 4*(4*a^3*b + 5*a^2*b^2 - 13*a*b^3 + 4*b^4)*cosh(d*x +
c)^5 - 4*(4*a^3*b + 5*a^2*b^2 - 13*a*b^3 + 4*b^4 - 21*(a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(d*x + c)^2)*sinh(d*x +
c)^5 + 20*(7*(a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(d*x + c)^3 - (4*a^3*b + 5*a^2*b^2 - 13*a*b^3 + 4*b^4)*cosh(d*x +
c))*sinh(d*x + c)^4 - 4*(4*a^3*b + 5*a^2*b^2 - 13*a*b^3 + 4*b^4)*cosh(d*x + c)^3 + 4*(35*(a^2*b^2 - 5*a*b^3 +
4*b^4)*cosh(d*x + c)^4 - 4*a^3*b - 5*a^2*b^2 + 13*a*b^3 - 4*b^4 - 10*(4*a^3*b + 5*a^2*b^2 - 13*a*b^3 + 4*b^4)
*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 4*(21*(a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(d*x + c)^5 - 10*(4*a^3*b + 5*a^2*b^
2 - 13*a*b^3 + 4*b^4)*cosh(d*x + c)^3 - 3*(4*a^3*b + 5*a^2*b^2 - 13*a*b^3 + 4*b^4)*cosh(d*x + c))*sinh(d*x + c
)^2 + ((a*b^2 - 4*b^3)*cosh(d*x + c)^8 + 8*(a*b^2 - 4*b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (a*b^2 - 4*b^3)*sin
h(d*x + c)^8 + 4*(2*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^6 + 4*(2*a^2*b - 9*a*b^2 + 4*b^3 + 7*(a*b^2 - 4*b^3
)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a*b^2 - 4*b^3)*cosh(d*x + c)^3 + 3*(2*a^2*b - 9*a*b^2 + 4*b^3)*cosh
(d*x + c))*sinh(d*x + c)^5 + 2*(8*a^3 - 40*a^2*b + 35*a*b^2 - 12*b^3)*cosh(d*x + c)^4 + 2*(35*(a*b^2 - 4*b^3)*
cosh(d*x + c)^4 + 8*a^3 - 40*a^2*b + 35*a*b^2 - 12*b^3 + 30*(2*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^2)*sinh(
d*x + c)^4 + 8*(7*(a*b^2 - 4*b^3)*cosh(d*x + c)^5 + 10*(2*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^3 + (8*a^3 -
40*a^2*b + 35*a*b^2 - 12*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + a*b^2 - 4*b^3 + 4*(2*a^2*b - 9*a*b^2 + 4*b^3)*c
osh(d*x + c)^2 + 4*(7*(a*b^2 - 4*b^3)*cosh(d*x + c)^6 + 15*(2*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^4 + 2*a^2
*b - 9*a*b^2 + 4*b^3 + 3*(8*a^3 - 40*a^2*b + 35*a*b^2 - 12*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a*b^2 -
4*b^3)*cosh(d*x + c)^7 + 3*(2*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^5 + (8*a^3 - 40*a^2*b + 35*a*b^2 - 12*b^
3)*cosh(d*x + c)^3 + (2*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c))*sinh(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*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 - 5*a*b^3 + 4*b^4)*cosh(d*x + c) + 4*(7*(a^2*b^2 -
5*a*b^3 + 4*b^4)*cosh(d*x + c)^6 - 5*(4*a^3*b + 5*a^2*b^2 - 13*a*b^3 + 4*b^4)*cosh(d*x + c)^4 + a^2*b^2 - 5*a*
b^3 + 4*b^4 - 3*(4*a^3*b + 5*a^2*b^2 - 13*a*b^3 + 4*b^4)*cosh(d*x + c)^2)*sinh(d*x + c))/((a^3*b^4 - 3*a^2*b^5
+ 3*a*b^6 - b^7)*d*cosh(d*x + c)^8 + 8*(a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d*cosh(d*x + c)*sinh(d*x + c)^7
+ (a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d*sinh(d*x + c)^8 + 4*(2*a^4*b^3 - 7*a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b
^7)*d*cosh(d*x + c)^6 + 4*(7*(a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d*cosh(d*x + c)^2 + (2*a^4*b^3 - 7*a^3*b^4
+ 9*a^2*b^5 - 5*a*b^6 + b^7)*d)*sinh(d*x + c)^6 + 2*(8*a^5*b^2 - 32*a^4*b^3 + 51*a^3*b^4 - 41*a^2*b^5 + 17*a*b
^6 - 3*b^7)*d*cosh(d*x + c)^4 + 8*(7*(a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d*cosh(d*x + c)^3 + 3*(2*a^4*b^3 -
7*a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^3*b^4 - 3*a^2*b^5 + 3*a*b^6
- b^7)*d*cosh(d*x + c)^4 + 30*(2*a^4*b^3 - 7*a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7)*d*cosh(d*x + c)^2 + (8*a^5*
b^2 - 32*a^4*b^3 + 51*a^3*b^4 - 41*a^2*b^5 + 17*a*b^6 - 3*b^7)*d)*sinh(d*x + c)^4 + 4*(2*a^4*b^3 - 7*a^3*b^4 +
9*a^2*b^5 - 5*a*b^6 + b^7)*d*cosh(d*x + c)^2 + 8*(7*(a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d*cosh(d*x + c)^5 +
10*(2*a^4*b^3 - 7*a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7)*d*cosh(d*x + c)^3 + (8*a^5*b^2 - 32*a^4*b^3 + 51*a^3*b
^4 - 41*a^2*b^5 + 17*a*b^6 - 3*b^7)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b
^7)*d*cosh(d*x + c)^6 + 15*(2*a^4*b^3 - 7*a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7)*d*cosh(d*x + c)^4 + 3*(8*a^5*b^
2 - 32*a^4*b^3 + 51*a^3*b^4 - 41*a^2*b^5 + 17*a*b^6 - 3*b^7)*d*cosh(d*x + c)^2 + (2*a^4*b^3 - 7*a^3*b^4 + 9*a^
2*b^5 - 5*a*b^6 + b^7)*d)*sinh(d*x + c)^2 + (a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d + 8*((a^3*b^4 - 3*a^2*b^5
+ 3*a*b^6 - b^7)*d*cosh(d*x + c)^7 + 3*(2*a^4*b^3 - 7*a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7)*d*cosh(d*x + c)^5 +
(8*a^5*b^2 - 32*a^4*b^3 + 51*a^3*b^4 - 41*a^2*b^5 + 17*a*b^6 - 3*b^7)*d*cosh(d*x + c)^3 + (2*a^4*b^3 - 7*a^3*
b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7)*d*cosh(d*x + c))*sinh(d*x + c)), 1/8*(2*(a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(d*x
+ c)^7 + 14*(a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(d*x + c)*sinh(d*x + c)^6 + 2*(a^2*b^2 - 5*a*b^3 + 4*b^4)*sinh(d*x
+ c)^7 - 2*(4*a^3*b + 5*a^2*b^2 - 13*a*b^3 + 4*b^4)*cosh(d*x + c)^5 - 2*(4*a^3*b + 5*a^2*b^2 - 13*a*b^3 + 4*b
^4 - 21*(a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 10*(7*(a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(
d*x + c)^3 - (4*a^3*b + 5*a^2*b^2 - 13*a*b^3 + 4*b^4)*cosh(d*x + c))*sinh(d*x + c)^4 - 2*(4*a^3*b + 5*a^2*b^2
- 13*a*b^3 + 4*b^4)*cosh(d*x + c)^3 + 2*(35*(a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(d*x + c)^4 - 4*a^3*b - 5*a^2*b^2
+ 13*a*b^3 - 4*b^4 - 10*(4*a^3*b + 5*a^2*b^2 - 13*a*b^3 + 4*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(21*(a^2
*b^2 - 5*a*b^3 + 4*b^4)*cosh(d*x + c)^5 - 10*(4*a^3*b + 5*a^2*b^2 - 13*a*b^3 + 4*b^4)*cosh(d*x + c)^3 - 3*(4*a
^3*b + 5*a^2*b^2 - 13*a*b^3 + 4*b^4)*cosh(d*x + c))*sinh(d*x + c)^2 + ((a*b^2 - 4*b^3)*cosh(d*x + c)^8 + 8*(a*
b^2 - 4*b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (a*b^2 - 4*b^3)*sinh(d*x + c)^8 + 4*(2*a^2*b - 9*a*b^2 + 4*b^3)*c
osh(d*x + c)^6 + 4*(2*a^2*b - 9*a*b^2 + 4*b^3 + 7*(a*b^2 - 4*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a*b
^2 - 4*b^3)*cosh(d*x + c)^3 + 3*(2*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(8*a^3 - 40*a^2
*b + 35*a*b^2 - 12*b^3)*cosh(d*x + c)^4 + 2*(35*(a*b^2 - 4*b^3)*cosh(d*x + c)^4 + 8*a^3 - 40*a^2*b + 35*a*b^2
- 12*b^3 + 30*(2*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(a*b^2 - 4*b^3)*cosh(d*x + c
)^5 + 10*(2*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^3 + (8*a^3 - 40*a^2*b + 35*a*b^2 - 12*b^3)*cosh(d*x + c))*s
inh(d*x + c)^3 + a*b^2 - 4*b^3 + 4*(2*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^2 + 4*(7*(a*b^2 - 4*b^3)*cosh(d*x
+ c)^6 + 15*(2*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^4 + 2*a^2*b - 9*a*b^2 + 4*b^3 + 3*(8*a^3 - 40*a^2*b + 3
5*a*b^2 - 12*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a*b^2 - 4*b^3)*cosh(d*x + c)^7 + 3*(2*a^2*b - 9*a*b^2
+ 4*b^3)*cosh(d*x + c)^5 + (8*a^3 - 40*a^2*b + 35*a*b^2 - 12*b^3)*cosh(d*x + c)^3 + (2*a^2*b - 9*a*b^2 + 4*b^
3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a*b - b^2)*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 - 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)) - ((a*b^2 - 4*b^3)*cosh(d*x + c)^8 + 8*(a*b^2 - 4*b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (a*b^2 - 4*
b^3)*sinh(d*x + c)^8 + 4*(2*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^6 + 4*(2*a^2*b - 9*a*b^2 + 4*b^3 + 7*(a*b^2
- 4*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a*b^2 - 4*b^3)*cosh(d*x + c)^3 + 3*(2*a^2*b - 9*a*b^2 + 4*b
^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(8*a^3 - 40*a^2*b + 35*a*b^2 - 12*b^3)*cosh(d*x + c)^4 + 2*(35*(a*b^2 -
4*b^3)*cosh(d*x + c)^4 + 8*a^3 - 40*a^2*b + 35*a*b^2 - 12*b^3 + 30*(2*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^
2)*sinh(d*x + c)^4 + 8*(7*(a*b^2 - 4*b^3)*cosh(d*x + c)^5 + 10*(2*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^3 + (
8*a^3 - 40*a^2*b + 35*a*b^2 - 12*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + a*b^2 - 4*b^3 + 4*(2*a^2*b - 9*a*b^2 +
4*b^3)*cosh(d*x + c)^2 + 4*(7*(a*b^2 - 4*b^3)*cosh(d*x + c)^6 + 15*(2*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^4
+ 2*a^2*b - 9*a*b^2 + 4*b^3 + 3*(8*a^3 - 40*a^2*b + 35*a*b^2 - 12*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*(
(a*b^2 - 4*b^3)*cosh(d*x + c)^7 + 3*(2*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^5 + (8*a^3 - 40*a^2*b + 35*a*b^2
- 12*b^3)*cosh(d*x + c)^3 + (2*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a*b - b^2)*arctan(
-1/2*sqrt(a*b - b^2)*(cosh(d*x + c) + sinh(d*x + c))/(a - b)) + 2*(a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(d*x + c) +
2*(7*(a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(d*x + c)^6 - 5*(4*a^3*b + 5*a^2*b^2 - 13*a*b^3 + 4*b^4)*cosh(d*x + c)^4
+ a^2*b^2 - 5*a*b^3 + 4*b^4 - 3*(4*a^3*b + 5*a^2*b^2 - 13*a*b^3 + 4*b^4)*cosh(d*x + c)^2)*sinh(d*x + c))/((a^3
*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d*cosh(d*x + c)^8 + 8*(a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d*cosh(d*x + c)*
sinh(d*x + c)^7 + (a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d*sinh(d*x + c)^8 + 4*(2*a^4*b^3 - 7*a^3*b^4 + 9*a^2*b
^5 - 5*a*b^6 + b^7)*d*cosh(d*x + c)^6 + 4*(7*(a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d*cosh(d*x + c)^2 + (2*a^4*
b^3 - 7*a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7)*d)*sinh(d*x + c)^6 + 2*(8*a^5*b^2 - 32*a^4*b^3 + 51*a^3*b^4 - 41*
a^2*b^5 + 17*a*b^6 - 3*b^7)*d*cosh(d*x + c)^4 + 8*(7*(a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d*cosh(d*x + c)^3 +
3*(2*a^4*b^3 - 7*a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^3*b^4 - 3*a
^2*b^5 + 3*a*b^6 - b^7)*d*cosh(d*x + c)^4 + 30*(2*a^4*b^3 - 7*a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7)*d*cosh(d*x
+ c)^2 + (8*a^5*b^2 - 32*a^4*b^3 + 51*a^3*b^4 - 41*a^2*b^5 + 17*a*b^6 - 3*b^7)*d)*sinh(d*x + c)^4 + 4*(2*a^4*b
^3 - 7*a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7)*d*cosh(d*x + c)^2 + 8*(7*(a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d*c
osh(d*x + c)^5 + 10*(2*a^4*b^3 - 7*a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7)*d*cosh(d*x + c)^3 + (8*a^5*b^2 - 32*a^
4*b^3 + 51*a^3*b^4 - 41*a^2*b^5 + 17*a*b^6 - 3*b^7)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^3*b^4 - 3*a^2*b
^5 + 3*a*b^6 - b^7)*d*cosh(d*x + c)^6 + 15*(2*a^4*b^3 - 7*a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7)*d*cosh(d*x + c)
^4 + 3*(8*a^5*b^2 - 32*a^4*b^3 + 51*a^3*b^4 - 41*a^2*b^5 + 17*a*b^6 - 3*b^7)*d*cosh(d*x + c)^2 + (2*a^4*b^3 -
7*a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7)*d)*sinh(d*x + c)^2 + (a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d + 8*((a^3*
b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d*cosh(d*x + c)^7 + 3*(2*a^4*b^3 - 7*a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7)*d*c
osh(d*x + c)^5 + (8*a^5*b^2 - 32*a^4*b^3 + 51*a^3*b^4 - 41*a^2*b^5 + 17*a*b^6 - 3*b^7)*d*cosh(d*x + c)^3 + (2*
a^4*b^3 - 7*a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7)*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)**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(sinh(d*x+c)^3/(a+b*sinh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError