3.51 \(\int \frac{\sinh ^4(c+d x)}{(a+b \sinh ^2(c+d x))^3} \, dx\)

Optimal. Leaf size=124 \[ \frac{\tanh ^3(c+d x)}{4 d (a-b) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}-\frac{3 \tanh (c+d x)}{8 d (a-b)^2 \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 \sqrt{a} d (a-b)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.123065, antiderivative size = 124, 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 = {3187, 288, 208} \[ \frac{\tanh ^3(c+d x)}{4 d (a-b) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}-\frac{3 \tanh (c+d x)}{8 d (a-b)^2 \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 \sqrt{a} d (a-b)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

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

Rule 3187

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

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 208

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

Rubi steps

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

Mathematica [A]  time = 1.3691, size = 104, normalized size = 0.84 \[ \frac{\frac{3 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} (a-b)^{5/2}}+\frac{\sinh (2 (c+d x)) ((2 a-5 b) \cosh (2 (c+d x))-8 a+5 b)}{(a-b)^2 (2 a+b \cosh (2 (c+d x))-b)^2}}{8 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

((3*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)^(5/2)) + ((-8*a + 5*b + (2*a - 5*b)*Cosh[2*
(c + d*x)])*Sinh[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 = 0.052, size = 768, normalized size = 6.2 \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)^4/(a+b*sinh(d*x+c)^2)^3,x)

[Out]

-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/(a^2-2*a*b+b^2)*tan
h(1/2*d*x+1/2*c)^7+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
^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^5*a-5/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)^5*b+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^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3*a-5/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)^3*b-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/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)+3/8/d
/(a^2-2*a*b+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))-3/8/d/(a^2-2*a*b+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))*b-3/8/d/(a^2-2*a*b+b^2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arc
tan(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*a*b+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))*b

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^4/(a+b*sinh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.94201, size = 11786, normalized size = 95.05 \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)^4/(a+b*sinh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[-1/16*(4*(8*a^4*b - 24*a^3*b^2 + 21*a^2*b^3 - 5*a*b^4)*cosh(d*x + c)^6 + 24*(8*a^4*b - 24*a^3*b^2 + 21*a^2*b^
3 - 5*a*b^4)*cosh(d*x + c)*sinh(d*x + c)^5 + 4*(8*a^4*b - 24*a^3*b^2 + 21*a^2*b^3 - 5*a*b^4)*sinh(d*x + c)^6 +
8*a^3*b^2 - 28*a^2*b^3 + 20*a*b^4 + 4*(16*a^5 - 72*a^4*b + 102*a^3*b^2 - 61*a^2*b^3 + 15*a*b^4)*cosh(d*x + c)
^4 + 4*(16*a^5 - 72*a^4*b + 102*a^3*b^2 - 61*a^2*b^3 + 15*a*b^4 + 15*(8*a^4*b - 24*a^3*b^2 + 21*a^2*b^3 - 5*a*
b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*(5*(8*a^4*b - 24*a^3*b^2 + 21*a^2*b^3 - 5*a*b^4)*cosh(d*x + c)^3 +
(16*a^5 - 72*a^4*b + 102*a^3*b^2 - 61*a^2*b^3 + 15*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(8*a^4*b - 40*a^3
*b^2 + 47*a^2*b^3 - 15*a*b^4)*cosh(d*x + c)^2 + 4*(8*a^4*b - 40*a^3*b^2 + 47*a^2*b^3 - 15*a*b^4 + 15*(8*a^4*b
- 24*a^3*b^2 + 21*a^2*b^3 - 5*a*b^4)*cosh(d*x + c)^4 + 6*(16*a^5 - 72*a^4*b + 102*a^3*b^2 - 61*a^2*b^3 + 15*a*
b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 3*(b^4*cosh(d*x + c)^8 + 8*b^4*cosh(d*x + c)*sinh(d*x + c)^7 + b^4*sin
h(d*x + c)^8 + 4*(2*a*b^3 - b^4)*cosh(d*x + c)^6 + 4*(7*b^4*cosh(d*x + c)^2 + 2*a*b^3 - b^4)*sinh(d*x + c)^6 +
8*(7*b^4*cosh(d*x + c)^3 + 3*(2*a*b^3 - b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(8*a^2*b^2 - 8*a*b^3 + 3*b^4)
*cosh(d*x + c)^4 + 2*(35*b^4*cosh(d*x + c)^4 + 8*a^2*b^2 - 8*a*b^3 + 3*b^4 + 30*(2*a*b^3 - b^4)*cosh(d*x + c)^
2)*sinh(d*x + c)^4 + b^4 + 8*(7*b^4*cosh(d*x + c)^5 + 10*(2*a*b^3 - b^4)*cosh(d*x + c)^3 + (8*a^2*b^2 - 8*a*b^
3 + 3*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(2*a*b^3 - b^4)*cosh(d*x + c)^2 + 4*(7*b^4*cosh(d*x + c)^6 + 15*
(2*a*b^3 - b^4)*cosh(d*x + c)^4 + 2*a*b^3 - b^4 + 3*(8*a^2*b^2 - 8*a*b^3 + 3*b^4)*cosh(d*x + c)^2)*sinh(d*x +
c)^2 + 8*(b^4*cosh(d*x + c)^7 + 3*(2*a*b^3 - b^4)*cosh(d*x + c)^5 + (8*a^2*b^2 - 8*a*b^3 + 3*b^4)*cosh(d*x + c
)^3 + (2*a*b^3 - b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a^2 - a*b)*log((b^2*cosh(d*x + c)^4 + 4*b^2*cosh(d*x
+ c)*sinh(d*x + c)^3 + b^2*sinh(d*x + c)^4 + 2*(2*a*b - b^2)*cosh(d*x + c)^2 + 2*(3*b^2*cosh(d*x + c)^2 + 2*a*
b - b^2)*sinh(d*x + c)^2 + 8*a^2 - 8*a*b + b^2 + 4*(b^2*cosh(d*x + c)^3 + (2*a*b - b^2)*cosh(d*x + c))*sinh(d*
x + c) - 4*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + 2*a - b)*sqrt(a^2 - a*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)) + 8*(3*(8*a^4*b - 24*a^3*b^2 + 21*a^2*b^3 - 5*a*b^4)*cosh(d*x + c)^5 + 2*(16*a^5 - 72*a^4*b + 102*a^3
*b^2 - 61*a^2*b^3 + 15*a*b^4)*cosh(d*x + c)^3 + (8*a^4*b - 40*a^3*b^2 + 47*a^2*b^3 - 15*a*b^4)*cosh(d*x + c))*
sinh(d*x + c))/((a^4*b^4 - 3*a^3*b^5 + 3*a^2*b^6 - a*b^7)*d*cosh(d*x + c)^8 + 8*(a^4*b^4 - 3*a^3*b^5 + 3*a^2*b
^6 - a*b^7)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^4*b^4 - 3*a^3*b^5 + 3*a^2*b^6 - a*b^7)*d*sinh(d*x + c)^8 + 4*
(2*a^5*b^3 - 7*a^4*b^4 + 9*a^3*b^5 - 5*a^2*b^6 + a*b^7)*d*cosh(d*x + c)^6 + 4*(7*(a^4*b^4 - 3*a^3*b^5 + 3*a^2*
b^6 - a*b^7)*d*cosh(d*x + c)^2 + (2*a^5*b^3 - 7*a^4*b^4 + 9*a^3*b^5 - 5*a^2*b^6 + a*b^7)*d)*sinh(d*x + c)^6 +
2*(8*a^6*b^2 - 32*a^5*b^3 + 51*a^4*b^4 - 41*a^3*b^5 + 17*a^2*b^6 - 3*a*b^7)*d*cosh(d*x + c)^4 + 8*(7*(a^4*b^4
- 3*a^3*b^5 + 3*a^2*b^6 - a*b^7)*d*cosh(d*x + c)^3 + 3*(2*a^5*b^3 - 7*a^4*b^4 + 9*a^3*b^5 - 5*a^2*b^6 + a*b^7)
*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^4*b^4 - 3*a^3*b^5 + 3*a^2*b^6 - a*b^7)*d*cosh(d*x + c)^4 + 30*(2*
a^5*b^3 - 7*a^4*b^4 + 9*a^3*b^5 - 5*a^2*b^6 + a*b^7)*d*cosh(d*x + c)^2 + (8*a^6*b^2 - 32*a^5*b^3 + 51*a^4*b^4
- 41*a^3*b^5 + 17*a^2*b^6 - 3*a*b^7)*d)*sinh(d*x + c)^4 + 4*(2*a^5*b^3 - 7*a^4*b^4 + 9*a^3*b^5 - 5*a^2*b^6 + a
*b^7)*d*cosh(d*x + c)^2 + 8*(7*(a^4*b^4 - 3*a^3*b^5 + 3*a^2*b^6 - a*b^7)*d*cosh(d*x + c)^5 + 10*(2*a^5*b^3 - 7
*a^4*b^4 + 9*a^3*b^5 - 5*a^2*b^6 + a*b^7)*d*cosh(d*x + c)^3 + (8*a^6*b^2 - 32*a^5*b^3 + 51*a^4*b^4 - 41*a^3*b^
5 + 17*a^2*b^6 - 3*a*b^7)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^4*b^4 - 3*a^3*b^5 + 3*a^2*b^6 - a*b^7)*d*
cosh(d*x + c)^6 + 15*(2*a^5*b^3 - 7*a^4*b^4 + 9*a^3*b^5 - 5*a^2*b^6 + a*b^7)*d*cosh(d*x + c)^4 + 3*(8*a^6*b^2
- 32*a^5*b^3 + 51*a^4*b^4 - 41*a^3*b^5 + 17*a^2*b^6 - 3*a*b^7)*d*cosh(d*x + c)^2 + (2*a^5*b^3 - 7*a^4*b^4 + 9*
a^3*b^5 - 5*a^2*b^6 + a*b^7)*d)*sinh(d*x + c)^2 + (a^4*b^4 - 3*a^3*b^5 + 3*a^2*b^6 - a*b^7)*d + 8*((a^4*b^4 -
3*a^3*b^5 + 3*a^2*b^6 - a*b^7)*d*cosh(d*x + c)^7 + 3*(2*a^5*b^3 - 7*a^4*b^4 + 9*a^3*b^5 - 5*a^2*b^6 + a*b^7)*d
*cosh(d*x + c)^5 + (8*a^6*b^2 - 32*a^5*b^3 + 51*a^4*b^4 - 41*a^3*b^5 + 17*a^2*b^6 - 3*a*b^7)*d*cosh(d*x + c)^3
+ (2*a^5*b^3 - 7*a^4*b^4 + 9*a^3*b^5 - 5*a^2*b^6 + a*b^7)*d*cosh(d*x + c))*sinh(d*x + c)), -1/8*(2*(8*a^4*b -
24*a^3*b^2 + 21*a^2*b^3 - 5*a*b^4)*cosh(d*x + c)^6 + 12*(8*a^4*b - 24*a^3*b^2 + 21*a^2*b^3 - 5*a*b^4)*cosh(d*
x + c)*sinh(d*x + c)^5 + 2*(8*a^4*b - 24*a^3*b^2 + 21*a^2*b^3 - 5*a*b^4)*sinh(d*x + c)^6 + 4*a^3*b^2 - 14*a^2*
b^3 + 10*a*b^4 + 2*(16*a^5 - 72*a^4*b + 102*a^3*b^2 - 61*a^2*b^3 + 15*a*b^4)*cosh(d*x + c)^4 + 2*(16*a^5 - 72*
a^4*b + 102*a^3*b^2 - 61*a^2*b^3 + 15*a*b^4 + 15*(8*a^4*b - 24*a^3*b^2 + 21*a^2*b^3 - 5*a*b^4)*cosh(d*x + c)^2
)*sinh(d*x + c)^4 + 8*(5*(8*a^4*b - 24*a^3*b^2 + 21*a^2*b^3 - 5*a*b^4)*cosh(d*x + c)^3 + (16*a^5 - 72*a^4*b +
102*a^3*b^2 - 61*a^2*b^3 + 15*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(8*a^4*b - 40*a^3*b^2 + 47*a^2*b^3 - 1
5*a*b^4)*cosh(d*x + c)^2 + 2*(8*a^4*b - 40*a^3*b^2 + 47*a^2*b^3 - 15*a*b^4 + 15*(8*a^4*b - 24*a^3*b^2 + 21*a^2
*b^3 - 5*a*b^4)*cosh(d*x + c)^4 + 6*(16*a^5 - 72*a^4*b + 102*a^3*b^2 - 61*a^2*b^3 + 15*a*b^4)*cosh(d*x + c)^2)
*sinh(d*x + c)^2 + 3*(b^4*cosh(d*x + c)^8 + 8*b^4*cosh(d*x + c)*sinh(d*x + c)^7 + b^4*sinh(d*x + c)^8 + 4*(2*a
*b^3 - b^4)*cosh(d*x + c)^6 + 4*(7*b^4*cosh(d*x + c)^2 + 2*a*b^3 - b^4)*sinh(d*x + c)^6 + 8*(7*b^4*cosh(d*x +
c)^3 + 3*(2*a*b^3 - b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(8*a^2*b^2 - 8*a*b^3 + 3*b^4)*cosh(d*x + c)^4 + 2*
(35*b^4*cosh(d*x + c)^4 + 8*a^2*b^2 - 8*a*b^3 + 3*b^4 + 30*(2*a*b^3 - b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 +
b^4 + 8*(7*b^4*cosh(d*x + c)^5 + 10*(2*a*b^3 - b^4)*cosh(d*x + c)^3 + (8*a^2*b^2 - 8*a*b^3 + 3*b^4)*cosh(d*x +
c))*sinh(d*x + c)^3 + 4*(2*a*b^3 - b^4)*cosh(d*x + c)^2 + 4*(7*b^4*cosh(d*x + c)^6 + 15*(2*a*b^3 - b^4)*cosh(
d*x + c)^4 + 2*a*b^3 - b^4 + 3*(8*a^2*b^2 - 8*a*b^3 + 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*(b^4*cosh(d*
x + c)^7 + 3*(2*a*b^3 - b^4)*cosh(d*x + c)^5 + (8*a^2*b^2 - 8*a*b^3 + 3*b^4)*cosh(d*x + c)^3 + (2*a*b^3 - b^4)
*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a^2 + a*b)*arctan(-1/2*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x +
c) + b*sinh(d*x + c)^2 + 2*a - b)*sqrt(-a^2 + a*b)/(a^2 - a*b)) + 4*(3*(8*a^4*b - 24*a^3*b^2 + 21*a^2*b^3 - 5*
a*b^4)*cosh(d*x + c)^5 + 2*(16*a^5 - 72*a^4*b + 102*a^3*b^2 - 61*a^2*b^3 + 15*a*b^4)*cosh(d*x + c)^3 + (8*a^4*
b - 40*a^3*b^2 + 47*a^2*b^3 - 15*a*b^4)*cosh(d*x + c))*sinh(d*x + c))/((a^4*b^4 - 3*a^3*b^5 + 3*a^2*b^6 - a*b^
7)*d*cosh(d*x + c)^8 + 8*(a^4*b^4 - 3*a^3*b^5 + 3*a^2*b^6 - a*b^7)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^4*b^4
- 3*a^3*b^5 + 3*a^2*b^6 - a*b^7)*d*sinh(d*x + c)^8 + 4*(2*a^5*b^3 - 7*a^4*b^4 + 9*a^3*b^5 - 5*a^2*b^6 + a*b^7)
*d*cosh(d*x + c)^6 + 4*(7*(a^4*b^4 - 3*a^3*b^5 + 3*a^2*b^6 - a*b^7)*d*cosh(d*x + c)^2 + (2*a^5*b^3 - 7*a^4*b^4
+ 9*a^3*b^5 - 5*a^2*b^6 + a*b^7)*d)*sinh(d*x + c)^6 + 2*(8*a^6*b^2 - 32*a^5*b^3 + 51*a^4*b^4 - 41*a^3*b^5 + 1
7*a^2*b^6 - 3*a*b^7)*d*cosh(d*x + c)^4 + 8*(7*(a^4*b^4 - 3*a^3*b^5 + 3*a^2*b^6 - a*b^7)*d*cosh(d*x + c)^3 + 3*
(2*a^5*b^3 - 7*a^4*b^4 + 9*a^3*b^5 - 5*a^2*b^6 + a*b^7)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^4*b^4 - 3*
a^3*b^5 + 3*a^2*b^6 - a*b^7)*d*cosh(d*x + c)^4 + 30*(2*a^5*b^3 - 7*a^4*b^4 + 9*a^3*b^5 - 5*a^2*b^6 + a*b^7)*d*
cosh(d*x + c)^2 + (8*a^6*b^2 - 32*a^5*b^3 + 51*a^4*b^4 - 41*a^3*b^5 + 17*a^2*b^6 - 3*a*b^7)*d)*sinh(d*x + c)^4
+ 4*(2*a^5*b^3 - 7*a^4*b^4 + 9*a^3*b^5 - 5*a^2*b^6 + a*b^7)*d*cosh(d*x + c)^2 + 8*(7*(a^4*b^4 - 3*a^3*b^5 + 3
*a^2*b^6 - a*b^7)*d*cosh(d*x + c)^5 + 10*(2*a^5*b^3 - 7*a^4*b^4 + 9*a^3*b^5 - 5*a^2*b^6 + a*b^7)*d*cosh(d*x +
c)^3 + (8*a^6*b^2 - 32*a^5*b^3 + 51*a^4*b^4 - 41*a^3*b^5 + 17*a^2*b^6 - 3*a*b^7)*d*cosh(d*x + c))*sinh(d*x + c
)^3 + 4*(7*(a^4*b^4 - 3*a^3*b^5 + 3*a^2*b^6 - a*b^7)*d*cosh(d*x + c)^6 + 15*(2*a^5*b^3 - 7*a^4*b^4 + 9*a^3*b^5
- 5*a^2*b^6 + a*b^7)*d*cosh(d*x + c)^4 + 3*(8*a^6*b^2 - 32*a^5*b^3 + 51*a^4*b^4 - 41*a^3*b^5 + 17*a^2*b^6 - 3
*a*b^7)*d*cosh(d*x + c)^2 + (2*a^5*b^3 - 7*a^4*b^4 + 9*a^3*b^5 - 5*a^2*b^6 + a*b^7)*d)*sinh(d*x + c)^2 + (a^4*
b^4 - 3*a^3*b^5 + 3*a^2*b^6 - a*b^7)*d + 8*((a^4*b^4 - 3*a^3*b^5 + 3*a^2*b^6 - a*b^7)*d*cosh(d*x + c)^7 + 3*(2
*a^5*b^3 - 7*a^4*b^4 + 9*a^3*b^5 - 5*a^2*b^6 + a*b^7)*d*cosh(d*x + c)^5 + (8*a^6*b^2 - 32*a^5*b^3 + 51*a^4*b^4
- 41*a^3*b^5 + 17*a^2*b^6 - 3*a*b^7)*d*cosh(d*x + c)^3 + (2*a^5*b^3 - 7*a^4*b^4 + 9*a^3*b^5 - 5*a^2*b^6 + a*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)**4/(a+b*sinh(d*x+c)**2)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.56499, size = 386, normalized size = 3.11 \begin{align*} \frac{3 \, \arctan \left (\frac{b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt{-a^{2} + a b}}\right )}{8 \,{\left (a^{2} d - 2 \, a b d + b^{2} d\right )} \sqrt{-a^{2} + a b}} - \frac{8 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 16 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 5 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 56 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 46 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 15 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 32 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 15 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b^{2} - 5 \, b^{3}}{4 \,{\left (a^{2} b^{2} d - 2 \, a b^{3} d + b^{4} d\right )}{\left (b e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + b\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^4/(a+b*sinh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

3/8*arctan(1/2*(b*e^(2*d*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/((a^2*d - 2*a*b*d + b^2*d)*sqrt(-a^2 + a*b)) -
1/4*(8*a^2*b*e^(6*d*x + 6*c) - 16*a*b^2*e^(6*d*x + 6*c) + 5*b^3*e^(6*d*x + 6*c) + 16*a^3*e^(4*d*x + 4*c) - 56*
a^2*b*e^(4*d*x + 4*c) + 46*a*b^2*e^(4*d*x + 4*c) - 15*b^3*e^(4*d*x + 4*c) + 8*a^2*b*e^(2*d*x + 2*c) - 32*a*b^2
*e^(2*d*x + 2*c) + 15*b^3*e^(2*d*x + 2*c) + 2*a*b^2 - 5*b^3)/((a^2*b^2*d - 2*a*b^3*d + b^4*d)*(b*e^(4*d*x + 4*
c) + 4*a*e^(2*d*x + 2*c) - 2*b*e^(2*d*x + 2*c) + b)^2)