### 3.860 $$\int \frac{1}{(a+b \cosh (c+d x) \sinh (c+d x))^3} \, dx$$

Optimal. Leaf size=143 $-\frac{4 \left (8 a^2-b^2\right ) \tanh ^{-1}\left (\frac{b-2 a \tanh (c+d x)}{\sqrt{4 a^2+b^2}}\right )}{d \left (4 a^2+b^2\right )^{5/2}}-\frac{12 a b \cosh (2 c+2 d x)}{d \left (4 a^2+b^2\right )^2 (2 a+b \sinh (2 c+2 d x))}-\frac{2 b \cosh (2 c+2 d x)}{d \left (4 a^2+b^2\right ) (2 a+b \sinh (2 c+2 d x))^2}$

[Out]

(-4*(8*a^2 - b^2)*ArcTanh[(b - 2*a*Tanh[c + d*x])/Sqrt[4*a^2 + b^2]])/((4*a^2 + b^2)^(5/2)*d) - (2*b*Cosh[2*c
+ 2*d*x])/((4*a^2 + b^2)*d*(2*a + b*Sinh[2*c + 2*d*x])^2) - (12*a*b*Cosh[2*c + 2*d*x])/((4*a^2 + b^2)^2*d*(2*a
+ b*Sinh[2*c + 2*d*x]))

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Rubi [A]  time = 0.171853, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.389, Rules used = {2666, 2664, 2754, 12, 2660, 618, 204} $-\frac{4 \left (8 a^2-b^2\right ) \tanh ^{-1}\left (\frac{b-2 a \tanh (c+d x)}{\sqrt{4 a^2+b^2}}\right )}{d \left (4 a^2+b^2\right )^{5/2}}-\frac{12 a b \cosh (2 c+2 d x)}{d \left (4 a^2+b^2\right )^2 (2 a+b \sinh (2 c+2 d x))}-\frac{2 b \cosh (2 c+2 d x)}{d \left (4 a^2+b^2\right ) (2 a+b \sinh (2 c+2 d x))^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-4*(8*a^2 - b^2)*ArcTanh[(b - 2*a*Tanh[c + d*x])/Sqrt[4*a^2 + b^2]])/((4*a^2 + b^2)^(5/2)*d) - (2*b*Cosh[2*c
+ 2*d*x])/((4*a^2 + b^2)*d*(2*a + b*Sinh[2*c + 2*d*x])^2) - (12*a*b*Cosh[2*c + 2*d*x])/((4*a^2 + b^2)^2*d*(2*a
+ b*Sinh[2*c + 2*d*x]))

Rule 2666

Int[((a_) + cos[(c_.) + (d_.)*(x_)]*(b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Int[(a + (b*Sin[2*c + 2*
d*x])/2)^n, x] /; FreeQ[{a, b, c, d, n}, x]

Rule 2664

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n +
1))/(d*(n + 1)*(a^2 - b^2)), x] + Dist[1/((n + 1)*(a^2 - b^2)), Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n + 1
) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integer
Q[2*n]

Rule 2754

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[((
b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(a^2 - b^2
)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + 2)*Sin[e + f*x], x], x], x] /
; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{(a+b \cosh (c+d x) \sinh (c+d x))^3} \, dx &=\int \frac{1}{\left (a+\frac{1}{2} b \sinh (2 c+2 d x)\right )^3} \, dx\\ &=-\frac{2 b \cosh (2 c+2 d x)}{\left (4 a^2+b^2\right ) d (2 a+b \sinh (2 c+2 d x))^2}-\frac{2 \int \frac{-2 a+\frac{1}{2} b \sinh (2 c+2 d x)}{\left (a+\frac{1}{2} b \sinh (2 c+2 d x)\right )^2} \, dx}{4 a^2+b^2}\\ &=-\frac{2 b \cosh (2 c+2 d x)}{\left (4 a^2+b^2\right ) d (2 a+b \sinh (2 c+2 d x))^2}-\frac{12 a b \cosh (2 c+2 d x)}{\left (4 a^2+b^2\right )^2 d (2 a+b \sinh (2 c+2 d x))}+\frac{8 \int \frac{2 a^2-\frac{b^2}{4}}{a+\frac{1}{2} b \sinh (2 c+2 d x)} \, dx}{\left (4 a^2+b^2\right )^2}\\ &=-\frac{2 b \cosh (2 c+2 d x)}{\left (4 a^2+b^2\right ) d (2 a+b \sinh (2 c+2 d x))^2}-\frac{12 a b \cosh (2 c+2 d x)}{\left (4 a^2+b^2\right )^2 d (2 a+b \sinh (2 c+2 d x))}+\frac{\left (2 \left (8 a^2-b^2\right )\right ) \int \frac{1}{a+\frac{1}{2} b \sinh (2 c+2 d x)} \, dx}{\left (4 a^2+b^2\right )^2}\\ &=-\frac{2 b \cosh (2 c+2 d x)}{\left (4 a^2+b^2\right ) d (2 a+b \sinh (2 c+2 d x))^2}-\frac{12 a b \cosh (2 c+2 d x)}{\left (4 a^2+b^2\right )^2 d (2 a+b \sinh (2 c+2 d x))}-\frac{\left (2 i \left (8 a^2-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a-i b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (2 i c+2 i d x)\right )\right )}{\left (4 a^2+b^2\right )^2 d}\\ &=-\frac{2 b \cosh (2 c+2 d x)}{\left (4 a^2+b^2\right ) d (2 a+b \sinh (2 c+2 d x))^2}-\frac{12 a b \cosh (2 c+2 d x)}{\left (4 a^2+b^2\right )^2 d (2 a+b \sinh (2 c+2 d x))}+\frac{\left (4 i \left (8 a^2-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-4 a^2-b^2-x^2} \, dx,x,-i b+2 a \tan \left (\frac{1}{2} (2 i c+2 i d x)\right )\right )}{\left (4 a^2+b^2\right )^2 d}\\ &=-\frac{4 \left (8 a^2-b^2\right ) \tanh ^{-1}\left (\frac{b-2 a \tanh (c+d x)}{\sqrt{4 a^2+b^2}}\right )}{\left (4 a^2+b^2\right )^{5/2} d}-\frac{2 b \cosh (2 c+2 d x)}{\left (4 a^2+b^2\right ) d (2 a+b \sinh (2 c+2 d x))^2}-\frac{12 a b \cosh (2 c+2 d x)}{\left (4 a^2+b^2\right )^2 d (2 a+b \sinh (2 c+2 d x))}\\ \end{align*}

Mathematica [A]  time = 0.640178, size = 121, normalized size = 0.85 $\frac{2 \left (\frac{2 \left (8 a^2-b^2\right ) \tan ^{-1}\left (\frac{b-2 a \tanh (c+d x)}{\sqrt{-4 a^2-b^2}}\right )}{\sqrt{-4 a^2-b^2}}-\frac{b \cosh (2 (c+d x)) \left (16 a^2+6 a b \sinh (2 (c+d x))+b^2\right )}{(2 a+b \sinh (2 (c+d x)))^2}\right )}{d \left (4 a^2+b^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*((2*(8*a^2 - b^2)*ArcTan[(b - 2*a*Tanh[c + d*x])/Sqrt[-4*a^2 - b^2]])/Sqrt[-4*a^2 - b^2] - (b*Cosh[2*(c + d
*x)]*(16*a^2 + b^2 + 6*a*b*Sinh[2*(c + d*x)]))/(2*a + b*Sinh[2*(c + d*x)])^2))/((4*a^2 + b^2)^2*d)

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Maple [B]  time = 0.158, size = 2082, normalized size = 14.6 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*cosh(d*x+c)*sinh(d*x+c))^3,x)

[Out]

20/d/(tanh(1/2*d*x+1/2*c)^4*a+2*b*tanh(1/2*d*x+1/2*c)^3-2*tanh(1/2*d*x+1/2*c)^2*a+2*b*tanh(1/2*d*x+1/2*c)+a)^2
*a*b^2/(16*a^4+8*a^2*b^2+b^4)*tanh(1/2*d*x+1/2*c)^7+2/d/(tanh(1/2*d*x+1/2*c)^4*a+2*b*tanh(1/2*d*x+1/2*c)^3-2*t
anh(1/2*d*x+1/2*c)^2*a+2*b*tanh(1/2*d*x+1/2*c)+a)^2/a*b^4/(16*a^4+8*a^2*b^2+b^4)*tanh(1/2*d*x+1/2*c)^7-64/d/(t
anh(1/2*d*x+1/2*c)^4*a+2*b*tanh(1/2*d*x+1/2*c)^3-2*tanh(1/2*d*x+1/2*c)^2*a+2*b*tanh(1/2*d*x+1/2*c)+a)^2*b*a^2/
(16*a^4+8*a^2*b^2+b^4)*tanh(1/2*d*x+1/2*c)^6+28/d/(tanh(1/2*d*x+1/2*c)^4*a+2*b*tanh(1/2*d*x+1/2*c)^3-2*tanh(1/
2*d*x+1/2*c)^2*a+2*b*tanh(1/2*d*x+1/2*c)+a)^2*b^3/(16*a^4+8*a^2*b^2+b^4)*tanh(1/2*d*x+1/2*c)^6+2/d/(tanh(1/2*d
*x+1/2*c)^4*a+2*b*tanh(1/2*d*x+1/2*c)^3-2*tanh(1/2*d*x+1/2*c)^2*a+2*b*tanh(1/2*d*x+1/2*c)+a)^2*b^5/a^2/(16*a^4
+8*a^2*b^2+b^4)*tanh(1/2*d*x+1/2*c)^6-116/d/(tanh(1/2*d*x+1/2*c)^4*a+2*b*tanh(1/2*d*x+1/2*c)^3-2*tanh(1/2*d*x+
1/2*c)^2*a+2*b*tanh(1/2*d*x+1/2*c)+a)^2*a*b^2/(16*a^4+8*a^2*b^2+b^4)*tanh(1/2*d*x+1/2*c)^5-2/d/(tanh(1/2*d*x+1
/2*c)^4*a+2*b*tanh(1/2*d*x+1/2*c)^3-2*tanh(1/2*d*x+1/2*c)^2*a+2*b*tanh(1/2*d*x+1/2*c)+a)^2/a*b^4/(16*a^4+8*a^2
*b^2+b^4)*tanh(1/2*d*x+1/2*c)^5+128/d/(tanh(1/2*d*x+1/2*c)^4*a+2*b*tanh(1/2*d*x+1/2*c)^3-2*tanh(1/2*d*x+1/2*c)
^2*a+2*b*tanh(1/2*d*x+1/2*c)+a)^2*a^2*b/(16*a^4+8*a^2*b^2+b^4)*tanh(1/2*d*x+1/2*c)^4+72/d/(tanh(1/2*d*x+1/2*c)
^4*a+2*b*tanh(1/2*d*x+1/2*c)^3-2*tanh(1/2*d*x+1/2*c)^2*a+2*b*tanh(1/2*d*x+1/2*c)+a)^2*b^3/(16*a^4+8*a^2*b^2+b^
4)*tanh(1/2*d*x+1/2*c)^4+4/d/(tanh(1/2*d*x+1/2*c)^4*a+2*b*tanh(1/2*d*x+1/2*c)^3-2*tanh(1/2*d*x+1/2*c)^2*a+2*b*
tanh(1/2*d*x+1/2*c)+a)^2/a^2*b^5/(16*a^4+8*a^2*b^2+b^4)*tanh(1/2*d*x+1/2*c)^4-116/d/(tanh(1/2*d*x+1/2*c)^4*a+2
*b*tanh(1/2*d*x+1/2*c)^3-2*tanh(1/2*d*x+1/2*c)^2*a+2*b*tanh(1/2*d*x+1/2*c)+a)^2*a*b^2/(16*a^4+8*a^2*b^2+b^4)*t
anh(1/2*d*x+1/2*c)^3-2/d/(tanh(1/2*d*x+1/2*c)^4*a+2*b*tanh(1/2*d*x+1/2*c)^3-2*tanh(1/2*d*x+1/2*c)^2*a+2*b*tanh
(1/2*d*x+1/2*c)+a)^2/a*b^4/(16*a^4+8*a^2*b^2+b^4)*tanh(1/2*d*x+1/2*c)^3-64/d/(tanh(1/2*d*x+1/2*c)^4*a+2*b*tanh
(1/2*d*x+1/2*c)^3-2*tanh(1/2*d*x+1/2*c)^2*a+2*b*tanh(1/2*d*x+1/2*c)+a)^2*b*a^2/(16*a^4+8*a^2*b^2+b^4)*tanh(1/2
*d*x+1/2*c)^2+28/d/(tanh(1/2*d*x+1/2*c)^4*a+2*b*tanh(1/2*d*x+1/2*c)^3-2*tanh(1/2*d*x+1/2*c)^2*a+2*b*tanh(1/2*d
*x+1/2*c)+a)^2*b^3/(16*a^4+8*a^2*b^2+b^4)*tanh(1/2*d*x+1/2*c)^2+2/d/(tanh(1/2*d*x+1/2*c)^4*a+2*b*tanh(1/2*d*x+
1/2*c)^3-2*tanh(1/2*d*x+1/2*c)^2*a+2*b*tanh(1/2*d*x+1/2*c)+a)^2*b^5/a^2/(16*a^4+8*a^2*b^2+b^4)*tanh(1/2*d*x+1/
2*c)^2+20/d/(tanh(1/2*d*x+1/2*c)^4*a+2*b*tanh(1/2*d*x+1/2*c)^3-2*tanh(1/2*d*x+1/2*c)^2*a+2*b*tanh(1/2*d*x+1/2*
c)+a)^2*a*b^2/(16*a^4+8*a^2*b^2+b^4)*tanh(1/2*d*x+1/2*c)+2/d/(tanh(1/2*d*x+1/2*c)^4*a+2*b*tanh(1/2*d*x+1/2*c)^
3-2*tanh(1/2*d*x+1/2*c)^2*a+2*b*tanh(1/2*d*x+1/2*c)+a)^2/a*b^4/(16*a^4+8*a^2*b^2+b^4)*tanh(1/2*d*x+1/2*c)+64/d
/(16*a^4+8*a^2*b^2+b^4)*a^4/(4*a^2+b^2)^(3/2)*ln(tanh(1/2*d*x+1/2*c)^2*a+(4*a^2+b^2)^(1/2)*tanh(1/2*d*x+1/2*c)
+b*tanh(1/2*d*x+1/2*c)+a)+8/d/(16*a^4+8*a^2*b^2+b^4)*a^2/(4*a^2+b^2)^(3/2)*ln(tanh(1/2*d*x+1/2*c)^2*a+(4*a^2+b
^2)^(1/2)*tanh(1/2*d*x+1/2*c)+b*tanh(1/2*d*x+1/2*c)+a)*b^2-2/d/(16*a^4+8*a^2*b^2+b^4)/(4*a^2+b^2)^(3/2)*ln(tan
h(1/2*d*x+1/2*c)^2*a+(4*a^2+b^2)^(1/2)*tanh(1/2*d*x+1/2*c)+b*tanh(1/2*d*x+1/2*c)+a)*b^4-16/d/(16*a^4+8*a^2*b^2
+b^4)*a^2/(4*a^2+b^2)^(1/2)*ln(-tanh(1/2*d*x+1/2*c)^2*a+(4*a^2+b^2)^(1/2)*tanh(1/2*d*x+1/2*c)-b*tanh(1/2*d*x+1
/2*c)-a)+2/d/(16*a^4+8*a^2*b^2+b^4)/(4*a^2+b^2)^(1/2)*ln(-tanh(1/2*d*x+1/2*c)^2*a+(4*a^2+b^2)^(1/2)*tanh(1/2*d
*x+1/2*c)-b*tanh(1/2*d*x+1/2*c)-a)*b^2

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.07796, size = 5688, normalized size = 39.78 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(d*x+c)*sinh(d*x+c))^3,x, algorithm="fricas")

[Out]

2*(2*(32*a^4*b + 4*a^2*b^3 - b^5)*cosh(d*x + c)^6 + 12*(32*a^4*b + 4*a^2*b^3 - b^5)*cosh(d*x + c)*sinh(d*x + c
)^5 + 2*(32*a^4*b + 4*a^2*b^3 - b^5)*sinh(d*x + c)^6 + 48*a^3*b^2 + 12*a*b^4 + 12*(32*a^5 + 4*a^3*b^2 - a*b^4)
*cosh(d*x + c)^4 + 6*(64*a^5 + 8*a^3*b^2 - 2*a*b^4 + 5*(32*a^4*b + 4*a^2*b^3 - b^5)*cosh(d*x + c)^2)*sinh(d*x
+ c)^4 + 8*(5*(32*a^4*b + 4*a^2*b^3 - b^5)*cosh(d*x + c)^3 + 6*(32*a^5 + 4*a^3*b^2 - a*b^4)*cosh(d*x + c))*sin
h(d*x + c)^3 - 2*(160*a^4*b + 44*a^2*b^3 + b^5)*cosh(d*x + c)^2 - 2*(160*a^4*b + 44*a^2*b^3 + b^5 - 15*(32*a^4
*b + 4*a^2*b^3 - b^5)*cosh(d*x + c)^4 - 36*(32*a^5 + 4*a^3*b^2 - a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((8
*a^2*b^2 - b^4)*cosh(d*x + c)^8 + 8*(8*a^2*b^2 - b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (8*a^2*b^2 - b^4)*sinh(d
*x + c)^8 + 8*(8*a^3*b - a*b^3)*cosh(d*x + c)^6 + 4*(16*a^3*b - 2*a*b^3 + 7*(8*a^2*b^2 - b^4)*cosh(d*x + c)^2)
*sinh(d*x + c)^6 + 8*(7*(8*a^2*b^2 - b^4)*cosh(d*x + c)^3 + 6*(8*a^3*b - a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5
+ 2*(64*a^4 - 16*a^2*b^2 + b^4)*cosh(d*x + c)^4 + 2*(35*(8*a^2*b^2 - b^4)*cosh(d*x + c)^4 + 64*a^4 - 16*a^2*b
^2 + b^4 + 60*(8*a^3*b - a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*a^2*b^2 - b^4 + 8*(7*(8*a^2*b^2 - b^4)*co
sh(d*x + c)^5 + 20*(8*a^3*b - a*b^3)*cosh(d*x + c)^3 + (64*a^4 - 16*a^2*b^2 + b^4)*cosh(d*x + c))*sinh(d*x + c
)^3 - 8*(8*a^3*b - a*b^3)*cosh(d*x + c)^2 + 4*(7*(8*a^2*b^2 - b^4)*cosh(d*x + c)^6 + 30*(8*a^3*b - a*b^3)*cosh
(d*x + c)^4 - 16*a^3*b + 2*a*b^3 + 3*(64*a^4 - 16*a^2*b^2 + b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((8*a^2*
b^2 - b^4)*cosh(d*x + c)^7 + 6*(8*a^3*b - a*b^3)*cosh(d*x + c)^5 + (64*a^4 - 16*a^2*b^2 + b^4)*cosh(d*x + c)^3
- 2*(8*a^3*b - a*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(4*a^2 + b^2)*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 + 4*a*b*cosh(d*x + c)^2 + 2*(3*b^2*cosh(d*x + c)^2 + 2*a*b)*sinh
(d*x + c)^2 + 8*a^2 + b^2 + 4*(b^2*cosh(d*x + c)^3 + 2*a*b*cosh(d*x + c))*sinh(d*x + c) + 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)*sqrt(4*a^2 + b^2))/(b*cosh(d*x + c)^4 + 4*b*cosh
(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 + 4*a*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 + 2*a)*sinh(d*x +
c)^2 + 4*(b*cosh(d*x + c)^3 + 2*a*cosh(d*x + c))*sinh(d*x + c) - b)) + 4*(3*(32*a^4*b + 4*a^2*b^3 - b^5)*cosh
(d*x + c)^5 + 12*(32*a^5 + 4*a^3*b^2 - a*b^4)*cosh(d*x + c)^3 - (160*a^4*b + 44*a^2*b^3 + b^5)*cosh(d*x + c))*
sinh(d*x + c))/((64*a^6*b^2 + 48*a^4*b^4 + 12*a^2*b^6 + b^8)*d*cosh(d*x + c)^8 + 8*(64*a^6*b^2 + 48*a^4*b^4 +
12*a^2*b^6 + b^8)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (64*a^6*b^2 + 48*a^4*b^4 + 12*a^2*b^6 + b^8)*d*sinh(d*x +
c)^8 + 8*(64*a^7*b + 48*a^5*b^3 + 12*a^3*b^5 + a*b^7)*d*cosh(d*x + c)^6 + 4*(7*(64*a^6*b^2 + 48*a^4*b^4 + 12*a
^2*b^6 + b^8)*d*cosh(d*x + c)^2 + 2*(64*a^7*b + 48*a^5*b^3 + 12*a^3*b^5 + a*b^7)*d)*sinh(d*x + c)^6 + 2*(512*a
^8 + 320*a^6*b^2 + 48*a^4*b^4 - 4*a^2*b^6 - b^8)*d*cosh(d*x + c)^4 + 8*(7*(64*a^6*b^2 + 48*a^4*b^4 + 12*a^2*b^
6 + b^8)*d*cosh(d*x + c)^3 + 6*(64*a^7*b + 48*a^5*b^3 + 12*a^3*b^5 + a*b^7)*d*cosh(d*x + c))*sinh(d*x + c)^5 +
2*(35*(64*a^6*b^2 + 48*a^4*b^4 + 12*a^2*b^6 + b^8)*d*cosh(d*x + c)^4 + 60*(64*a^7*b + 48*a^5*b^3 + 12*a^3*b^5
+ a*b^7)*d*cosh(d*x + c)^2 + (512*a^8 + 320*a^6*b^2 + 48*a^4*b^4 - 4*a^2*b^6 - b^8)*d)*sinh(d*x + c)^4 - 8*(6
4*a^7*b + 48*a^5*b^3 + 12*a^3*b^5 + a*b^7)*d*cosh(d*x + c)^2 + 8*(7*(64*a^6*b^2 + 48*a^4*b^4 + 12*a^2*b^6 + b^
8)*d*cosh(d*x + c)^5 + 20*(64*a^7*b + 48*a^5*b^3 + 12*a^3*b^5 + a*b^7)*d*cosh(d*x + c)^3 + (512*a^8 + 320*a^6*
b^2 + 48*a^4*b^4 - 4*a^2*b^6 - b^8)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(64*a^6*b^2 + 48*a^4*b^4 + 12*a^2*
b^6 + b^8)*d*cosh(d*x + c)^6 + 30*(64*a^7*b + 48*a^5*b^3 + 12*a^3*b^5 + a*b^7)*d*cosh(d*x + c)^4 + 3*(512*a^8
+ 320*a^6*b^2 + 48*a^4*b^4 - 4*a^2*b^6 - b^8)*d*cosh(d*x + c)^2 - 2*(64*a^7*b + 48*a^5*b^3 + 12*a^3*b^5 + a*b^
7)*d)*sinh(d*x + c)^2 + (64*a^6*b^2 + 48*a^4*b^4 + 12*a^2*b^6 + b^8)*d + 8*((64*a^6*b^2 + 48*a^4*b^4 + 12*a^2*
b^6 + b^8)*d*cosh(d*x + c)^7 + 6*(64*a^7*b + 48*a^5*b^3 + 12*a^3*b^5 + a*b^7)*d*cosh(d*x + c)^5 + (512*a^8 + 3
20*a^6*b^2 + 48*a^4*b^4 - 4*a^2*b^6 - b^8)*d*cosh(d*x + c)^3 - 2*(64*a^7*b + 48*a^5*b^3 + 12*a^3*b^5 + 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(1/(a+b*cosh(d*x+c)*sinh(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.57034, size = 351, normalized size = 2.45 \begin{align*} -\frac{2 \,{\left (8 \, a^{2} - b^{2}\right )} \log \left (\frac{{\left | -2 \, b e^{\left (2 \, d x + 2 \, c\right )} - 4 \, a - 2 \, \sqrt{4 \, a^{2} + b^{2}} \right |}}{{\left | -2 \, b e^{\left (2 \, d x + 2 \, c\right )} - 4 \, a + 2 \, \sqrt{4 \, a^{2} + b^{2}} \right |}}\right )}{{\left (16 \, a^{4} d + 8 \, a^{2} b^{2} d + b^{4} d\right )} \sqrt{4 \, a^{2} + b^{2}}} + \frac{4 \,{\left (8 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 48 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 6 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 40 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 6 \, a b^{2}\right )}}{{\left (16 \, a^{4} d + 8 \, a^{2} b^{2} 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 )} - b\right )}^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(d*x+c)*sinh(d*x+c))^3,x, algorithm="giac")

[Out]

-2*(8*a^2 - b^2)*log(abs(-2*b*e^(2*d*x + 2*c) - 4*a - 2*sqrt(4*a^2 + b^2))/abs(-2*b*e^(2*d*x + 2*c) - 4*a + 2*
sqrt(4*a^2 + b^2)))/((16*a^4*d + 8*a^2*b^2*d + b^4*d)*sqrt(4*a^2 + b^2)) + 4*(8*a^2*b*e^(6*d*x + 6*c) - b^3*e^
(6*d*x + 6*c) + 48*a^3*e^(4*d*x + 4*c) - 6*a*b^2*e^(4*d*x + 4*c) - 40*a^2*b*e^(2*d*x + 2*c) - b^3*e^(2*d*x + 2
*c) + 6*a*b^2)/((16*a^4*d + 8*a^2*b^2*d + b^4*d)*(b*e^(4*d*x + 4*c) + 4*a*e^(2*d*x + 2*c) - b)^2)