3.253 \(\int \frac{\sinh ^9(c+d x)}{(a-b \sinh ^4(c+d x))^3} \, dx\)

Optimal. Leaf size=315 \[ -\frac{\cosh (c+d x) \left (9 a^2-2 b (2 a-5 b) \cosh ^2(c+d x)-11 a b-10 b^2\right )}{32 b^2 d (a-b)^2 \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}+\frac{a \cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{8 b^2 d (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}+\frac{\left (-14 \sqrt{a} \sqrt{b}+5 a+12 b\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{64 \sqrt{a} b^{9/4} d \left (\sqrt{a}-\sqrt{b}\right )^{5/2}}+\frac{\left (14 \sqrt{a} \sqrt{b}+5 a+12 b\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{64 \sqrt{a} b^{9/4} d \left (\sqrt{a}+\sqrt{b}\right )^{5/2}} \]

[Out]

((5*a - 14*Sqrt[a]*Sqrt[b] + 12*b)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*Sqrt[a]*(Sqrt[
a] - Sqrt[b])^(5/2)*b^(9/4)*d) + ((5*a + 14*Sqrt[a]*Sqrt[b] + 12*b)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[
a] + Sqrt[b]]])/(64*Sqrt[a]*(Sqrt[a] + Sqrt[b])^(5/2)*b^(9/4)*d) + (a*Cosh[c + d*x]*(a + b - b*Cosh[c + d*x]^2
))/(8*(a - b)*b^2*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4)^2) - (Cosh[c + d*x]*(9*a^2 - 11*a*b - 10
*b^2 - 2*(2*a - 5*b)*b*Cosh[c + d*x]^2))/(32*(a - b)^2*b^2*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4)
)

________________________________________________________________________________________

Rubi [A]  time = 0.577358, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3215, 1205, 1678, 1166, 205, 208} \[ -\frac{\cosh (c+d x) \left (9 a^2-2 b (2 a-5 b) \cosh ^2(c+d x)-11 a b-10 b^2\right )}{32 b^2 d (a-b)^2 \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}+\frac{a \cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{8 b^2 d (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2}+\frac{\left (-14 \sqrt{a} \sqrt{b}+5 a+12 b\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{64 \sqrt{a} b^{9/4} d \left (\sqrt{a}-\sqrt{b}\right )^{5/2}}+\frac{\left (14 \sqrt{a} \sqrt{b}+5 a+12 b\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{64 \sqrt{a} b^{9/4} d \left (\sqrt{a}+\sqrt{b}\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^9/(a - b*Sinh[c + d*x]^4)^3,x]

[Out]

((5*a - 14*Sqrt[a]*Sqrt[b] + 12*b)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*Sqrt[a]*(Sqrt[
a] - Sqrt[b])^(5/2)*b^(9/4)*d) + ((5*a + 14*Sqrt[a]*Sqrt[b] + 12*b)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[
a] + Sqrt[b]]])/(64*Sqrt[a]*(Sqrt[a] + Sqrt[b])^(5/2)*b^(9/4)*d) + (a*Cosh[c + d*x]*(a + b - b*Cosh[c + d*x]^2
))/(8*(a - b)*b^2*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4)^2) - (Cosh[c + d*x]*(9*a^2 - 11*a*b - 10
*b^2 - 2*(2*a - 5*b)*b*Cosh[c + d*x]^2))/(32*(a - b)^2*b^2*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4)
)

Rule 3215

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(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 - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 1205

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{f = Coeff[Polynom
ialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x
^2 + c*x^4, x], x, 2]}, Simp[(x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2))/(
2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToS
um[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c
*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)*x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*
a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1] && LtQ[p, -1]

Rule 1678

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Pq, a +
b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[(x*(a + b*x^2
+ c*x^4)^(p + 1)*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*
a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuot
ient[Pq, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4*p + 7)*(b*d - 2*a*e)*x^2,
x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

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]

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 ^9(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^4}{\left (a-b+2 b x^2-b x^4\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac{a \cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{8 (a-b) b^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{\frac{2 a \left (a^2+a b-8 b^2\right )}{b}-2 a (11 a-16 b) x^2+16 a (a-b) x^4}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cosh (c+d x)\right )}{16 a (a-b) b d}\\ &=\frac{a \cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{8 (a-b) b^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac{\cosh (c+d x) \left (9 a^2-11 a b-10 b^2-2 (2 a-5 b) b \cosh ^2(c+d x)\right )}{32 (a-b)^2 b^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{4 a^2 \left (5 a^2-15 a b+22 b^2\right )+8 a^2 (2 a-5 b) b x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{128 a^2 (a-b)^2 b^2 d}\\ &=\frac{a \cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{8 (a-b) b^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac{\cosh (c+d x) \left (9 a^2-11 a b-10 b^2-2 (2 a-5 b) b \cosh ^2(c+d x)\right )}{32 (a-b)^2 b^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}-\frac{\left (5 a-14 \sqrt{a} \sqrt{b}+12 b\right ) \operatorname{Subst}\left (\int \frac{1}{-\sqrt{a} \sqrt{b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{64 \sqrt{a} \left (\sqrt{a}-\sqrt{b}\right )^2 b^{3/2} d}+\frac{\left (5 a+14 \sqrt{a} \sqrt{b}+12 b\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a} \sqrt{b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{64 \sqrt{a} \left (\sqrt{a}+\sqrt{b}\right )^2 b^{3/2} d}\\ &=\frac{\left (5 a-14 \sqrt{a} \sqrt{b}+12 b\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{64 \sqrt{a} \left (\sqrt{a}-\sqrt{b}\right )^{5/2} b^{9/4} d}+\frac{\left (5 a+14 \sqrt{a} \sqrt{b}+12 b\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{64 \sqrt{a} \left (\sqrt{a}+\sqrt{b}\right )^{5/2} b^{9/4} d}+\frac{a \cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{8 (a-b) b^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac{\cosh (c+d x) \left (9 a^2-11 a b-10 b^2-2 (2 a-5 b) b \cosh ^2(c+d x)\right )}{32 (a-b)^2 b^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}\\ \end{align*}

Mathematica [C]  time = 1.68906, size = 1021, normalized size = 3.24 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^9/(a - b*Sinh[c + d*x]^4)^3,x]

[Out]

((32*Cosh[c + d*x]*(-9*a^2 + 13*a*b + 5*b^2 + (2*a - 5*b)*b*Cosh[2*(c + d*x)]))/(8*a - 3*b + 4*b*Cosh[2*(c + d
*x)] - b*Cosh[4*(c + d*x)]) + (512*a*(a - b)*Cosh[c + d*x]*(2*a + b - b*Cosh[2*(c + d*x)]))/(-8*a + 3*b - 4*b*
Cosh[2*(c + d*x)] + b*Cosh[4*(c + d*x)])^2 - RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 &
, (-2*a*b*c + 5*b^2*c - 2*a*b*d*x + 5*b^2*d*x - 4*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c +
d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 10*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1
- Sinh[(c + d*x)/2]*#1] - 10*a^2*c*#1^2 + 28*a*b*c*#1^2 - 39*b^2*c*#1^2 - 10*a^2*d*x*#1^2 + 28*a*b*d*x*#1^2 -
39*b^2*d*x*#1^2 - 20*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]
*#1]*#1^2 + 56*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#
1^2 - 78*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 +
10*a^2*c*#1^4 - 28*a*b*c*#1^4 + 39*b^2*c*#1^4 + 10*a^2*d*x*#1^4 - 28*a*b*d*x*#1^4 + 39*b^2*d*x*#1^4 + 20*a^2*L
og[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 56*a*b*Log[-Co
sh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + 78*b^2*Log[-Cosh[(c
+ d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + 2*a*b*c*#1^6 - 5*b^2*c*#1^
6 + 2*a*b*d*x*#1^6 - 5*b^2*d*x*#1^6 + 4*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1
- Sinh[(c + d*x)/2]*#1]*#1^6 - 10*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh
[(c + d*x)/2]*#1]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(128*(a - b)^2*b^2*d)

________________________________________________________________________________________

Maple [B]  time = 0.099, size = 3542, normalized size = 11.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^9/(a-b*sinh(d*x+c)^4)^3,x)

[Out]

-1/16/d/b^2/(a^2-2*a*b+b^2)*a/(-a*b-(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(-2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)+
2*a)/(-a*b-(a*b)^(1/2)*a)^(1/2))*(a*b)^(1/2)-77/16/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh
(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*a^2/b/(a^2-2*a*b+b^2)*tanh(1/2*d
*x+1/2*c)^2-11/16/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2
*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*a^2/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^14-35/16/d/(tanh(1/2*d*
x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2
*c)^2*a+a)^2*a^3/b^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^12+85/16/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/
2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*a^2/b/(a^2-2*a*b+
b^2)*tanh(1/2*d*x+1/2*c)^12+105/16/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^
4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/b^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^10*a^3-1
1/64/d/b/(a^2-2*a*b+b^2)*a/(-a*b+(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)-2*a)
/(-a*b+(a*b)^(1/2)*a)^(1/2))-5/64/d/b^2/(a^2-2*a*b+b^2)*a^2/(-a*b-(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(-2*tanh(1/2
*d*x+1/2*c)^2*a+4*(a*b)^(1/2)+2*a)/(-a*b-(a*b)^(1/2)*a)^(1/2))+11/64/d/b/(a^2-2*a*b+b^2)*a/(-a*b-(a*b)^(1/2)*a
)^(1/2)*arctan(1/4*(-2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)+2*a)/(-a*b-(a*b)^(1/2)*a)^(1/2))+35/16/d/(tanh(1/
2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x
+1/2*c)^2*a+a)^2*a^3/b^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^2-175/16/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*
x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/b^2/(a^2-2*a*
b+b^2)*tanh(1/2*d*x+1/2*c)^8*a^3+5/32/d/b/(a^2-2*a*b+b^2)/(-a*b+(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(2*tanh(1/2*d*
x+1/2*c)^2*a+4*(a*b)^(1/2)-2*a)/(-a*b+(a*b)^(1/2)*a)^(1/2))*(a*b)^(1/2)+5/32/d/b/(a^2-2*a*b+b^2)/(-a*b-(a*b)^(
1/2)*a)^(1/2)*arctan(1/4*(-2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)+2*a)/(-a*b-(a*b)^(1/2)*a)^(1/2))*(a*b)^(1/2
)+5/16/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)
^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*a^3/b^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^14+865/16/d/(tanh(1/2*d*x+1/2*c)
^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+
a)^2/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^8*a^2+175/16/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6
*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/b^2/(a^2-2*a*b+b^2)*tanh(1/
2*d*x+1/2*c)^6*a^3-849/16/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*
tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^6*a^2-105/16/d/(tan
h(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2
*d*x+1/2*c)^2*a+a)^2*a^3/b^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4+383/16/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/
2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*a^2/b/(a^
2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4-407/16/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+
1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^10*
a^2+5/64/d/b^2/(a^2-2*a*b+b^2)*a^2/(-a*b+(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1
/2)-2*a)/(-a*b+(a*b)^(1/2)*a)^(1/2))-1/16/d/b^2/(a^2-2*a*b+b^2)*a/(-a*b+(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(2*tan
h(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)-2*a)/(-a*b+(a*b)^(1/2)*a)^(1/2))*(a*b)^(1/2)+88/d/(tanh(1/2*d*x+1/2*c)^8*a-
4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*
b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^8+20/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x
+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^6-
5/16/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4
-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*a^3/b^2/(a^2-2*a*b+b^2)+11/16/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)
^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*a^2/b/(a^2-2*a*b+b^2)
-106/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4
-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^8*a+189/4/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tan
h(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/(a^2-
2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^6*a-31/2/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/
2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*a/(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)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*t
anh(1/2*d*x+1/2*c)^2*a+a)^2*a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^2+3/4/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*
d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*a/(a^2-2*a*
b+b^2)*tanh(1/2*d*x+1/2*c)^14-13/2/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^
4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^12+163/4/d
/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tan
h(1/2*d*x+1/2*c)^2*a+a)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^10*a-20/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*
x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*b/(a^2-2*a*b+
b^2)*tanh(1/2*d*x+1/2*c)^10-3/16/d/(a^2-2*a*b+b^2)/(-a*b-(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(-2*tanh(1/2*d*x+1/2*
c)^2*a+4*(a*b)^(1/2)+2*a)/(-a*b-(a*b)^(1/2)*a)^(1/2))+3/16/d/(a^2-2*a*b+b^2)/(-a*b+(a*b)^(1/2)*a)^(1/2)*arctan
(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)-2*a)/(-a*b+(a*b)^(1/2)*a)^(1/2))

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Maxima [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)^9/(a-b*sinh(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

Timed out

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Fricas [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)^9/(a-b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

Timed out

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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)**9/(a-b*sinh(d*x+c)**4)**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^9/(a-b*sinh(d*x+c)^4)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError