3.247 \(\int \frac {\sinh ^8(c+d x)}{(a-b \sinh ^4(c+d x))^2} \, dx\)

Optimal. Leaf size=320 \[ \frac {\sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 b^{3/2} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}-\frac {\sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 b^{3/2} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-\frac {\sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 b^2 d \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 b^2 d \sqrt {\sqrt {a}+\sqrt {b}}}+\frac {\tanh ^5(c+d x)}{4 b d \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac {\tanh (c+d x)}{4 b d (a-b)}+\frac {x}{b^2} \]

[Out]

x/b^2+1/8*a^(1/4)*arctanh((a^(1/2)-b^(1/2))^(1/2)*tanh(d*x+c)/a^(1/4))/b^(3/2)/d/(a^(1/2)-b^(1/2))^(3/2)-1/8*a
^(1/4)*arctanh((a^(1/2)+b^(1/2))^(1/2)*tanh(d*x+c)/a^(1/4))/b^(3/2)/d/(a^(1/2)+b^(1/2))^(3/2)-1/2*a^(1/4)*arct
anh((a^(1/2)-b^(1/2))^(1/2)*tanh(d*x+c)/a^(1/4))/b^2/d/(a^(1/2)-b^(1/2))^(1/2)-1/2*a^(1/4)*arctanh((a^(1/2)+b^
(1/2))^(1/2)*tanh(d*x+c)/a^(1/4))/b^2/d/(a^(1/2)+b^(1/2))^(1/2)-1/4*tanh(d*x+c)/(a-b)/b/d+1/4*tanh(d*x+c)^5/b/
d/(a-2*a*tanh(d*x+c)^2+(a-b)*tanh(d*x+c)^4)

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Rubi [A]  time = 0.46, antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3217, 1313, 1275, 12, 1122, 1166, 208, 1287, 207} \[ \frac {\sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 b^{3/2} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}-\frac {\sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 b^2 d \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 b^{3/2} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-\frac {\sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 b^2 d \sqrt {\sqrt {a}+\sqrt {b}}}+\frac {\tanh ^5(c+d x)}{4 b d \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac {\tanh (c+d x)}{4 b d (a-b)}+\frac {x}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^8/(a - b*Sinh[c + d*x]^4)^2,x]

[Out]

x/b^2 - (a^(1/4)*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^2*d) +
(a^(1/4)*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(8*(Sqrt[a] - Sqrt[b])^(3/2)*b^(3/2)*d) -
(a^(1/4)*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^2*d) - (a^(1/4
)*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(8*(Sqrt[a] + Sqrt[b])^(3/2)*b^(3/2)*d) - Tanh[c +
d*x]/(4*(a - b)*b*d) + Tanh[c + d*x]^5/(4*b*d*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4))

Rule 12

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

Rule 207

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

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]

Rule 1122

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(d^3*(d*x)^(m - 3)*(a + b*
x^2 + c*x^4)^(p + 1))/(c*(m + 4*p + 1)), x] - Dist[d^4/(c*(m + 4*p + 1)), Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b
*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && Gt
Q[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

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 1275

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[(f*
(f*x)^(m - 1)*(a + b*x^2 + c*x^4)^(p + 1)*(b*d - 2*a*e - (b*e - 2*c*d)*x^2))/(2*(p + 1)*(b^2 - 4*a*c)), x] - D
ist[f^2/(2*(p + 1)*(b^2 - 4*a*c)), Int[(f*x)^(m - 2)*(a + b*x^2 + c*x^4)^(p + 1)*Simp[(m - 1)*(b*d - 2*a*e) -
(4*p + 4 + m + 1)*(b*e - 2*c*d)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[
p, -1] && GtQ[m, 1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1287

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[Ex
pandIntegrand[((f*x)^m*(d + e*x^2)^q)/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^
2 - 4*a*c, 0] && IntegerQ[q] && IntegerQ[m]

Rule 1313

Int[(((f_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_))/((d_.) + (e_.)*(x_)^2), x_Symbol] :> -Dist
[f^4/(c*d^2 - b*d*e + a*e^2), Int[(f*x)^(m - 4)*(a*d + (b*d - a*e)*x^2)*(a + b*x^2 + c*x^4)^p, x], x] + Dist[(
d^2*f^4)/(c*d^2 - b*d*e + a*e^2), Int[((f*x)^(m - 4)*(a + b*x^2 + c*x^4)^(p + 1))/(d + e*x^2), x], x] /; FreeQ
[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 2]

Rule 3217

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p)/(1 + ff^2
*x^2)^(m/2 + 2*p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rubi steps

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

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Mathematica [A]  time = 4.78, size = 262, normalized size = 0.82 \[ \frac {-\frac {\sqrt {a} \left (4 \sqrt {a}+5 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {\sqrt {a} \sqrt {b}+a}}\right )}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {\sqrt {a} \sqrt {b}+a}}+\frac {\sqrt {a} \left (4 \sqrt {a}-5 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {\sqrt {a} \sqrt {b}-a}}\right )}{\left (\sqrt {a}-\sqrt {b}\right ) \sqrt {\sqrt {a} \sqrt {b}-a}}+\frac {2 a b (\sinh (4 (c+d x))-6 \sinh (2 (c+d x)))}{(a-b) (8 a+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))-3 b)}+8 (c+d x)}{8 b^2 d} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sinh[c + d*x]^8/(a - b*Sinh[c + d*x]^4)^2,x]

[Out]

(8*(c + d*x) + (Sqrt[a]*(4*Sqrt[a] - 5*Sqrt[b])*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*S
qrt[b]]])/((Sqrt[a] - Sqrt[b])*Sqrt[-a + Sqrt[a]*Sqrt[b]]) - (Sqrt[a]*(4*Sqrt[a] + 5*Sqrt[b])*ArcTanh[((Sqrt[a
] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] + Sqrt[b])*Sqrt[a + Sqrt[a]*Sqrt[b]]) + (2*a
*b*(-6*Sinh[2*(c + d*x)] + Sinh[4*(c + d*x)]))/((a - b)*(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x
)])))/(8*b^2*d)

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fricas [B]  time = 1.41, size = 6944, normalized size = 21.70 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^8/(a-b*sinh(d*x+c)^4)^2,x, algorithm="fricas")

[Out]

1/16*(16*(a*b - b^2)*d*x*cosh(d*x + c)^8 + 128*(a*b - b^2)*d*x*cosh(d*x + c)*sinh(d*x + c)^7 + 16*(a*b - b^2)*
d*x*sinh(d*x + c)^8 - 8*(8*(a*b - b^2)*d*x - a*b)*cosh(d*x + c)^6 + 8*(56*(a*b - b^2)*d*x*cosh(d*x + c)^2 - 8*
(a*b - b^2)*d*x + a*b)*sinh(d*x + c)^6 + 16*(56*(a*b - b^2)*d*x*cosh(d*x + c)^3 - 3*(8*(a*b - b^2)*d*x - a*b)*
cosh(d*x + c))*sinh(d*x + c)^5 - 8*(4*(8*a^2 - 11*a*b + 3*b^2)*d*x + 8*a^2 - 3*a*b)*cosh(d*x + c)^4 + 8*(140*(
a*b - b^2)*d*x*cosh(d*x + c)^4 - 4*(8*a^2 - 11*a*b + 3*b^2)*d*x - 15*(8*(a*b - b^2)*d*x - a*b)*cosh(d*x + c)^2
- 8*a^2 + 3*a*b)*sinh(d*x + c)^4 + 32*(28*(a*b - b^2)*d*x*cosh(d*x + c)^5 - 5*(8*(a*b - b^2)*d*x - a*b)*cosh(
d*x + c)^3 - (4*(8*a^2 - 11*a*b + 3*b^2)*d*x + 8*a^2 - 3*a*b)*cosh(d*x + c))*sinh(d*x + c)^3 + 16*(a*b - b^2)*
d*x - 8*(8*(a*b - b^2)*d*x + 5*a*b)*cosh(d*x + c)^2 + 8*(56*(a*b - b^2)*d*x*cosh(d*x + c)^6 - 15*(8*(a*b - b^2
)*d*x - a*b)*cosh(d*x + c)^4 - 8*(a*b - b^2)*d*x - 6*(4*(8*a^2 - 11*a*b + 3*b^2)*d*x + 8*a^2 - 3*a*b)*cosh(d*x
+ c)^2 - 5*a*b)*sinh(d*x + c)^2 + ((a*b^3 - b^4)*d*cosh(d*x + c)^8 + 8*(a*b^3 - b^4)*d*cosh(d*x + c)*sinh(d*x
+ c)^7 + (a*b^3 - b^4)*d*sinh(d*x + c)^8 - 4*(a*b^3 - b^4)*d*cosh(d*x + c)^6 + 4*(7*(a*b^3 - b^4)*d*cosh(d*x
+ c)^2 - (a*b^3 - b^4)*d)*sinh(d*x + c)^6 - 2*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cosh(d*x + c)^4 + 8*(7*(a*b^3 -
b^4)*d*cosh(d*x + c)^3 - 3*(a*b^3 - b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a*b^3 - b^4)*d*cosh(d*x +
c)^4 - 30*(a*b^3 - b^4)*d*cosh(d*x + c)^2 - (8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d)*sinh(d*x + c)^4 - 4*(a*b^3 - b^4
)*d*cosh(d*x + c)^2 + 8*(7*(a*b^3 - b^4)*d*cosh(d*x + c)^5 - 10*(a*b^3 - b^4)*d*cosh(d*x + c)^3 - (8*a^2*b^2 -
11*a*b^3 + 3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a*b^3 - b^4)*d*cosh(d*x + c)^6 - 15*(a*b^3 - b^4)*
d*cosh(d*x + c)^4 - 3*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cosh(d*x + c)^2 - (a*b^3 - b^4)*d)*sinh(d*x + c)^2 + (a
*b^3 - b^4)*d + 8*((a*b^3 - b^4)*d*cosh(d*x + c)^7 - 3*(a*b^3 - b^4)*d*cosh(d*x + c)^5 - (8*a^2*b^2 - 11*a*b^3
+ 3*b^4)*d*cosh(d*x + c)^3 - (a*b^3 - b^4)*d*cosh(d*x + c))*sinh(d*x + c))*sqrt(-((a^3*b^4 - 3*a^2*b^5 + 3*a*
b^6 - b^7)*d^2*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*
a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) - 16*a^3 + 47*a^2*b - 35*a*b^2)/((a^3*b^4 - 3*a^2
*b^5 + 3*a*b^6 - b^7)*d^2))*log(2*(16*a^4*b^3 - 73*a^3*b^4 + 123*a^2*b^5 - 91*a*b^6 + 25*b^7)*d^2*sqrt((64*a^5
- 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*
a^2*b^11 - 6*a*b^12 + b^13)*d^4)) + 128*a^3 - 664*a^2*b + 1125*a*b^2 - 625*b^3 - (128*a^3 - 664*a^2*b + 1125*a
*b^2 - 625*b^3)*cosh(d*x + c)^2 - 2*(128*a^3 - 664*a^2*b + 1125*a*b^2 - 625*b^3)*cosh(d*x + c)*sinh(d*x + c) -
(128*a^3 - 664*a^2*b + 1125*a*b^2 - 625*b^3)*sinh(d*x + c)^2 + 2*(2*(2*a^4*b^5 - 9*a^3*b^6 + 15*a^2*b^7 - 11*
a*b^8 + 3*b^9)*d^3*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 +
15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) + (24*a^3*b^2 - 127*a^2*b^3 + 220*a*b^4 - 125
*b^5)*d)*sqrt(-((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b
^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) - 16*a
^3 + 47*a^2*b - 35*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))) - ((a*b^3 - b^4)*d*cosh(d*x + c)^8 + 8
*(a*b^3 - b^4)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a*b^3 - b^4)*d*sinh(d*x + c)^8 - 4*(a*b^3 - b^4)*d*cosh(d*x
+ c)^6 + 4*(7*(a*b^3 - b^4)*d*cosh(d*x + c)^2 - (a*b^3 - b^4)*d)*sinh(d*x + c)^6 - 2*(8*a^2*b^2 - 11*a*b^3 + 3
*b^4)*d*cosh(d*x + c)^4 + 8*(7*(a*b^3 - b^4)*d*cosh(d*x + c)^3 - 3*(a*b^3 - b^4)*d*cosh(d*x + c))*sinh(d*x + c
)^5 + 2*(35*(a*b^3 - b^4)*d*cosh(d*x + c)^4 - 30*(a*b^3 - b^4)*d*cosh(d*x + c)^2 - (8*a^2*b^2 - 11*a*b^3 + 3*b
^4)*d)*sinh(d*x + c)^4 - 4*(a*b^3 - b^4)*d*cosh(d*x + c)^2 + 8*(7*(a*b^3 - b^4)*d*cosh(d*x + c)^5 - 10*(a*b^3
- b^4)*d*cosh(d*x + c)^3 - (8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a*b^3 - b^4
)*d*cosh(d*x + c)^6 - 15*(a*b^3 - b^4)*d*cosh(d*x + c)^4 - 3*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cosh(d*x + c)^2
- (a*b^3 - b^4)*d)*sinh(d*x + c)^2 + (a*b^3 - b^4)*d + 8*((a*b^3 - b^4)*d*cosh(d*x + c)^7 - 3*(a*b^3 - b^4)*d*
cosh(d*x + c)^5 - (8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cosh(d*x + c)^3 - (a*b^3 - b^4)*d*cosh(d*x + c))*sinh(d*x +
c))*sqrt(-((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 +
625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) - 16*a^3 +
47*a^2*b - 35*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))*log(2*(16*a^4*b^3 - 73*a^3*b^4 + 123*a^2*b^
5 - 91*a*b^6 + 25*b^7)*d^2*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a
^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) + 128*a^3 - 664*a^2*b + 1125*a*b^2 -
625*b^3 - (128*a^3 - 664*a^2*b + 1125*a*b^2 - 625*b^3)*cosh(d*x + c)^2 - 2*(128*a^3 - 664*a^2*b + 1125*a*b^2 -
625*b^3)*cosh(d*x + c)*sinh(d*x + c) - (128*a^3 - 664*a^2*b + 1125*a*b^2 - 625*b^3)*sinh(d*x + c)^2 - 2*(2*(2
*a^4*b^5 - 9*a^3*b^6 + 15*a^2*b^7 - 11*a*b^8 + 3*b^9)*d^3*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b
^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) + (24*
a^3*b^2 - 127*a^2*b^3 + 220*a*b^4 - 125*b^5)*d)*sqrt(-((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((64*a^5
- 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a
^2*b^11 - 6*a*b^12 + b^13)*d^4)) - 16*a^3 + 47*a^2*b - 35*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2)))
- ((a*b^3 - b^4)*d*cosh(d*x + c)^8 + 8*(a*b^3 - b^4)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a*b^3 - b^4)*d*sinh(d
*x + c)^8 - 4*(a*b^3 - b^4)*d*cosh(d*x + c)^6 + 4*(7*(a*b^3 - b^4)*d*cosh(d*x + c)^2 - (a*b^3 - b^4)*d)*sinh(d
*x + c)^6 - 2*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cosh(d*x + c)^4 + 8*(7*(a*b^3 - b^4)*d*cosh(d*x + c)^3 - 3*(a*b
^3 - b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a*b^3 - b^4)*d*cosh(d*x + c)^4 - 30*(a*b^3 - b^4)*d*cosh(d
*x + c)^2 - (8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d)*sinh(d*x + c)^4 - 4*(a*b^3 - b^4)*d*cosh(d*x + c)^2 + 8*(7*(a*b^
3 - b^4)*d*cosh(d*x + c)^5 - 10*(a*b^3 - b^4)*d*cosh(d*x + c)^3 - (8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cosh(d*x +
c))*sinh(d*x + c)^3 + 4*(7*(a*b^3 - b^4)*d*cosh(d*x + c)^6 - 15*(a*b^3 - b^4)*d*cosh(d*x + c)^4 - 3*(8*a^2*b^2
- 11*a*b^3 + 3*b^4)*d*cosh(d*x + c)^2 - (a*b^3 - b^4)*d)*sinh(d*x + c)^2 + (a*b^3 - b^4)*d + 8*((a*b^3 - b^4)
*d*cosh(d*x + c)^7 - 3*(a*b^3 - b^4)*d*cosh(d*x + c)^5 - (8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cosh(d*x + c)^3 - (a
*b^3 - b^4)*d*cosh(d*x + c))*sinh(d*x + c))*sqrt(((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((64*a^5 - 464
*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^
11 - 6*a*b^12 + b^13)*d^4)) + 16*a^3 - 47*a^2*b + 35*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))*log(-
2*(16*a^4*b^3 - 73*a^3*b^4 + 123*a^2*b^5 - 91*a*b^6 + 25*b^7)*d^2*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 14
50*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)
) + 128*a^3 - 664*a^2*b + 1125*a*b^2 - 625*b^3 - (128*a^3 - 664*a^2*b + 1125*a*b^2 - 625*b^3)*cosh(d*x + c)^2
- 2*(128*a^3 - 664*a^2*b + 1125*a*b^2 - 625*b^3)*cosh(d*x + c)*sinh(d*x + c) - (128*a^3 - 664*a^2*b + 1125*a*b
^2 - 625*b^3)*sinh(d*x + c)^2 + 2*(2*(2*a^4*b^5 - 9*a^3*b^6 + 15*a^2*b^7 - 11*a*b^8 + 3*b^9)*d^3*sqrt((64*a^5
- 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a
^2*b^11 - 6*a*b^12 + b^13)*d^4)) - (24*a^3*b^2 - 127*a^2*b^3 + 220*a*b^4 - 125*b^5)*d)*sqrt(((a^3*b^4 - 3*a^2*
b^5 + 3*a*b^6 - b^7)*d^2*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5
*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) + 16*a^3 - 47*a^2*b + 35*a*b^2)/((a^3*b
^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))) + ((a*b^3 - b^4)*d*cosh(d*x + c)^8 + 8*(a*b^3 - b^4)*d*cosh(d*x + c)*si
nh(d*x + c)^7 + (a*b^3 - b^4)*d*sinh(d*x + c)^8 - 4*(a*b^3 - b^4)*d*cosh(d*x + c)^6 + 4*(7*(a*b^3 - b^4)*d*cos
h(d*x + c)^2 - (a*b^3 - b^4)*d)*sinh(d*x + c)^6 - 2*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cosh(d*x + c)^4 + 8*(7*(a
*b^3 - b^4)*d*cosh(d*x + c)^3 - 3*(a*b^3 - b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a*b^3 - b^4)*d*cosh(
d*x + c)^4 - 30*(a*b^3 - b^4)*d*cosh(d*x + c)^2 - (8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d)*sinh(d*x + c)^4 - 4*(a*b^3
- b^4)*d*cosh(d*x + c)^2 + 8*(7*(a*b^3 - b^4)*d*cosh(d*x + c)^5 - 10*(a*b^3 - b^4)*d*cosh(d*x + c)^3 - (8*a^2
*b^2 - 11*a*b^3 + 3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a*b^3 - b^4)*d*cosh(d*x + c)^6 - 15*(a*b^3 -
b^4)*d*cosh(d*x + c)^4 - 3*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cosh(d*x + c)^2 - (a*b^3 - b^4)*d)*sinh(d*x + c)^
2 + (a*b^3 - b^4)*d + 8*((a*b^3 - b^4)*d*cosh(d*x + c)^7 - 3*(a*b^3 - b^4)*d*cosh(d*x + c)^5 - (8*a^2*b^2 - 11
*a*b^3 + 3*b^4)*d*cosh(d*x + c)^3 - (a*b^3 - b^4)*d*cosh(d*x + c))*sinh(d*x + c))*sqrt(((a^3*b^4 - 3*a^2*b^5 +
3*a*b^6 - b^7)*d^2*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8
+ 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) + 16*a^3 - 47*a^2*b + 35*a*b^2)/((a^3*b^4 -
3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))*log(-2*(16*a^4*b^3 - 73*a^3*b^4 + 123*a^2*b^5 - 91*a*b^6 + 25*b^7)*d^2*sqrt((
64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10
+ 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) + 128*a^3 - 664*a^2*b + 1125*a*b^2 - 625*b^3 - (128*a^3 - 664*a^2*b +
1125*a*b^2 - 625*b^3)*cosh(d*x + c)^2 - 2*(128*a^3 - 664*a^2*b + 1125*a*b^2 - 625*b^3)*cosh(d*x + c)*sinh(d*x
+ c) - (128*a^3 - 664*a^2*b + 1125*a*b^2 - 625*b^3)*sinh(d*x + c)^2 - 2*(2*(2*a^4*b^5 - 9*a^3*b^6 + 15*a^2*b^7
- 11*a*b^8 + 3*b^9)*d^3*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5
*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) - (24*a^3*b^2 - 127*a^2*b^3 + 220*a*b^4
- 125*b^5)*d)*sqrt(((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*
a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) +
16*a^3 - 47*a^2*b + 35*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))) + 8*a*b + 16*(8*(a*b - b^2)*d*x*c
osh(d*x + c)^7 - 3*(8*(a*b - b^2)*d*x - a*b)*cosh(d*x + c)^5 - 2*(4*(8*a^2 - 11*a*b + 3*b^2)*d*x + 8*a^2 - 3*a
*b)*cosh(d*x + c)^3 - (8*(a*b - b^2)*d*x + 5*a*b)*cosh(d*x + c))*sinh(d*x + c))/((a*b^3 - b^4)*d*cosh(d*x + c)
^8 + 8*(a*b^3 - b^4)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a*b^3 - b^4)*d*sinh(d*x + c)^8 - 4*(a*b^3 - b^4)*d*cos
h(d*x + c)^6 + 4*(7*(a*b^3 - b^4)*d*cosh(d*x + c)^2 - (a*b^3 - b^4)*d)*sinh(d*x + c)^6 - 2*(8*a^2*b^2 - 11*a*b
^3 + 3*b^4)*d*cosh(d*x + c)^4 + 8*(7*(a*b^3 - b^4)*d*cosh(d*x + c)^3 - 3*(a*b^3 - b^4)*d*cosh(d*x + c))*sinh(d
*x + c)^5 + 2*(35*(a*b^3 - b^4)*d*cosh(d*x + c)^4 - 30*(a*b^3 - b^4)*d*cosh(d*x + c)^2 - (8*a^2*b^2 - 11*a*b^3
+ 3*b^4)*d)*sinh(d*x + c)^4 - 4*(a*b^3 - b^4)*d*cosh(d*x + c)^2 + 8*(7*(a*b^3 - b^4)*d*cosh(d*x + c)^5 - 10*(
a*b^3 - b^4)*d*cosh(d*x + c)^3 - (8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a*b^3
- b^4)*d*cosh(d*x + c)^6 - 15*(a*b^3 - b^4)*d*cosh(d*x + c)^4 - 3*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cosh(d*x +
c)^2 - (a*b^3 - b^4)*d)*sinh(d*x + c)^2 + (a*b^3 - b^4)*d + 8*((a*b^3 - b^4)*d*cosh(d*x + c)^7 - 3*(a*b^3 - b
^4)*d*cosh(d*x + c)^5 - (8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cosh(d*x + c)^3 - (a*b^3 - b^4)*d*cosh(d*x + c))*sinh
(d*x + c))

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giac [A]  time = 1.98, size = 149, normalized size = 0.47 \[ \frac {\frac {a b e^{\left (6 \, d x + 6 \, c\right )} - 8 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 5 \, a b e^{\left (2 \, d x + 2 \, c\right )} + a b}{{\left (a b^{2} - b^{3}\right )} {\left (b e^{\left (8 \, d x + 8 \, c\right )} - 4 \, b e^{\left (6 \, d x + 6 \, c\right )} - 16 \, a e^{\left (4 \, d x + 4 \, c\right )} + 6 \, b e^{\left (4 \, d x + 4 \, c\right )} - 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + b\right )}} + \frac {2 \, {\left (d x + c\right )}}{b^{2}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^8/(a-b*sinh(d*x+c)^4)^2,x, algorithm="giac")

[Out]

1/2*((a*b*e^(6*d*x + 6*c) - 8*a^2*e^(4*d*x + 4*c) + 3*a*b*e^(4*d*x + 4*c) - 5*a*b*e^(2*d*x + 2*c) + a*b)/((a*b
^2 - b^3)*(b*e^(8*d*x + 8*c) - 4*b*e^(6*d*x + 6*c) - 16*a*e^(4*d*x + 4*c) + 6*b*e^(4*d*x + 4*c) - 4*b*e^(2*d*x
+ 2*c) + b)) + 2*(d*x + c)/b^2)/d

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maple [C]  time = 0.11, size = 574, normalized size = 1.79 \[ -\frac {a \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d b \left (\left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -4 \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +6 \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a \right ) \left (a -b \right )}+\frac {5 a \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d b \left (\left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -4 \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +6 \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a \right ) \left (a -b \right )}+\frac {5 a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d b \left (\left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -4 \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +6 \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a \right ) \left (a -b \right )}-\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d b \left (\left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -4 \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +6 \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a \right ) \left (a -b \right )}+\frac {a \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\left (4 a -5 b \right ) \textit {\_R}^{6}+\left (-12 a +19 b \right ) \textit {\_R}^{4}+\left (12 a -19 b \right ) \textit {\_R}^{2}-4 a +5 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}\right )}{16 d \,b^{2} \left (a -b \right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d \,b^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^8/(a-b*sinh(d*x+c)^4)^2,x)

[Out]

-1/2/d*a/b/(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)/(a-b)*tanh(1/2*d*x+1/2*c)^7+5/2/d*a/b/(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)/(a-b)*tanh(1/2*
d*x+1/2*c)^5+5/2/d*a/b/(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)/(a-b)*tanh(1/2*d*x+1/2*c)^3-1/2/d*a/b/(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)/(a-
b)*tanh(1/2*d*x+1/2*c)+1/16/d*a/b^2/(a-b)*sum(((4*a-5*b)*_R^6+(-12*a+19*b)*_R^4+(12*a-19*b)*_R^2-4*a+5*b)/(_R^
7*a-3*_R^5*a+3*_R^3*a-8*_R^3*b-_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R=RootOf(a*_Z^8-4*a*_Z^6+(6*a-16*b)*_Z^4-4*a*
_Z^2+a))-1/d/b^2*ln(tanh(1/2*d*x+1/2*c)-1)+1/d/b^2*ln(tanh(1/2*d*x+1/2*c)+1)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (a b d e^{\left (8 \, c\right )} - b^{2} d e^{\left (8 \, c\right )}\right )} x e^{\left (8 \, d x\right )} + a b + 2 \, {\left (a b d - b^{2} d\right )} x + {\left (a b e^{\left (6 \, c\right )} - 8 \, {\left (a b d e^{\left (6 \, c\right )} - b^{2} d e^{\left (6 \, c\right )}\right )} x\right )} e^{\left (6 \, d x\right )} - {\left (8 \, a^{2} e^{\left (4 \, c\right )} - 3 \, a b e^{\left (4 \, c\right )} + 4 \, {\left (8 \, a^{2} d e^{\left (4 \, c\right )} - 11 \, a b d e^{\left (4 \, c\right )} + 3 \, b^{2} d e^{\left (4 \, c\right )}\right )} x\right )} e^{\left (4 \, d x\right )} - {\left (5 \, a b e^{\left (2 \, c\right )} + 8 \, {\left (a b d e^{\left (2 \, c\right )} - b^{2} d e^{\left (2 \, c\right )}\right )} x\right )} e^{\left (2 \, d x\right )}}{2 \, {\left (a b^{3} d - b^{4} d + {\left (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 (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^{2} b^{2} d e^{\left (4 \, c\right )} - 11 \, 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 (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}{256} \, \int \frac {256 \, {\left (a b e^{\left (6 \, d x + 6 \, c\right )} + a b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (8 \, a^{2} e^{\left (4 \, c\right )} - 11 \, a b e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )}\right )}}{a b^{3} - b^{4} + {\left (a b^{3} e^{\left (8 \, c\right )} - b^{4} e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} - 4 \, {\left (a b^{3} e^{\left (6 \, c\right )} - b^{4} e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} - 2 \, {\left (8 \, a^{2} b^{2} e^{\left (4 \, c\right )} - 11 \, a b^{3} e^{\left (4 \, c\right )} + 3 \, b^{4} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} - 4 \, {\left (a b^{3} e^{\left (2 \, c\right )} - b^{4} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^8/(a-b*sinh(d*x+c)^4)^2,x, algorithm="maxima")

[Out]

1/2*(2*(a*b*d*e^(8*c) - b^2*d*e^(8*c))*x*e^(8*d*x) + a*b + 2*(a*b*d - b^2*d)*x + (a*b*e^(6*c) - 8*(a*b*d*e^(6*
c) - b^2*d*e^(6*c))*x)*e^(6*d*x) - (8*a^2*e^(4*c) - 3*a*b*e^(4*c) + 4*(8*a^2*d*e^(4*c) - 11*a*b*d*e^(4*c) + 3*
b^2*d*e^(4*c))*x)*e^(4*d*x) - (5*a*b*e^(2*c) + 8*(a*b*d*e^(2*c) - b^2*d*e^(2*c))*x)*e^(2*d*x))/(a*b^3*d - b^4*
d + (a*b^3*d*e^(8*c) - b^4*d*e^(8*c))*e^(8*d*x) - 4*(a*b^3*d*e^(6*c) - b^4*d*e^(6*c))*e^(6*d*x) - 2*(8*a^2*b^2
*d*e^(4*c) - 11*a*b^3*d*e^(4*c) + 3*b^4*d*e^(4*c))*e^(4*d*x) - 4*(a*b^3*d*e^(2*c) - b^4*d*e^(2*c))*e^(2*d*x))
+ 1/256*integrate(256*(a*b*e^(6*d*x + 6*c) + a*b*e^(2*d*x + 2*c) + 2*(8*a^2*e^(4*c) - 11*a*b*e^(4*c))*e^(4*d*x
))/(a*b^3 - b^4 + (a*b^3*e^(8*c) - b^4*e^(8*c))*e^(8*d*x) - 4*(a*b^3*e^(6*c) - b^4*e^(6*c))*e^(6*d*x) - 2*(8*a
^2*b^2*e^(4*c) - 11*a*b^3*e^(4*c) + 3*b^4*e^(4*c))*e^(4*d*x) - 4*(a*b^3*e^(2*c) - b^4*e^(2*c))*e^(2*d*x)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^8}{{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^8/(a - b*sinh(c + d*x)^4)^2,x)

[Out]

int(sinh(c + d*x)^8/(a - b*sinh(c + d*x)^4)^2, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)**8/(a-b*sinh(d*x+c)**4)**2,x)

[Out]

Timed out

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