3.3.41 \(\int \frac {\sinh ^9(c+d x)}{(a-b \sinh ^4(c+d x))^2} \, dx\) [241]

Optimal. Leaf size=235 \[ -\frac {\sqrt {a} \left (5 \sqrt {a}-6 \sqrt {b}\right ) \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 \left (\sqrt {a}-\sqrt {b}\right )^{3/2} b^{9/4} d}-\frac {\sqrt {a} \left (5 \sqrt {a}+6 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 \left (\sqrt {a}+\sqrt {b}\right )^{3/2} b^{9/4} d}+\frac {\cosh (c+d x)}{b^2 d}+\frac {a \cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{4 (a-b) b^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )} \]

[Out]

cosh(d*x+c)/b^2/d+1/4*a*cosh(d*x+c)*(a+b-b*cosh(d*x+c)^2)/(a-b)/b^2/d/(a-b+2*b*cosh(d*x+c)^2-b*cosh(d*x+c)^4)-
1/8*arctan(b^(1/4)*cosh(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))*a^(1/2)*(5*a^(1/2)-6*b^(1/2))/b^(9/4)/d/(a^(1/2)-b^(1/
2))^(3/2)-1/8*arctanh(b^(1/4)*cosh(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))*a^(1/2)*(5*a^(1/2)+6*b^(1/2))/b^(9/4)/d/(a^
(1/2)+b^(1/2))^(3/2)

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Rubi [A]
time = 0.37, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3294, 1219, 1690, 1180, 211, 214} \begin {gather*} -\frac {\sqrt {a} \left (5 \sqrt {a}-6 \sqrt {b}\right ) \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 b^{9/4} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}-\frac {\sqrt {a} \left (5 \sqrt {a}+6 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 b^{9/4} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}+\frac {a \cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{4 b^2 d (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}+\frac {\cosh (c+d x)}{b^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^9/(a - b*Sinh[c + d*x]^4)^2,x]

[Out]

-1/8*(Sqrt[a]*(5*Sqrt[a] - 6*Sqrt[b])*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/((Sqrt[a] - Sqr
t[b])^(3/2)*b^(9/4)*d) - (Sqrt[a]*(5*Sqrt[a] + 6*Sqrt[b])*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[
b]]])/(8*(Sqrt[a] + Sqrt[b])^(3/2)*b^(9/4)*d) + Cosh[c + d*x]/(b^2*d) + (a*Cosh[c + d*x]*(a + b - b*Cosh[c + d
*x]^2))/(4*(a - b)*b^2*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4))

Rule 211

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

Rule 214

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

Rule 1180

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 1219

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 1690

Int[(Pq_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^2 + c*x^4), x], x
] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1

Rule 3294

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]

Rubi steps

\begin {align*} \int \frac {\sinh ^9(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^4}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {a \cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{4 (a-b) b^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {2 a \left (a+\frac {a^2}{b}-4 b\right )-2 a (7 a-8 b) x^2+8 a (a-b) x^4}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{8 a (a-b) b d}\\ &=\frac {a \cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{4 (a-b) b^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}-\frac {\text {Subst}\left (\int \left (-\frac {8 a (a-b)}{b}+\frac {2 \left (a^2 (5 a-7 b)+a^2 b x^2\right )}{b \left (a-b+2 b x^2-b x^4\right )}\right ) \, dx,x,\cosh (c+d x)\right )}{8 a (a-b) b d}\\ &=\frac {\cosh (c+d x)}{b^2 d}+\frac {a \cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{4 (a-b) b^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {a^2 (5 a-7 b)+a^2 b x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{4 a (a-b) b^2 d}\\ &=\frac {\cosh (c+d x)}{b^2 d}+\frac {a \cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{4 (a-b) b^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}+\frac {\left (\sqrt {a} \left (5 \sqrt {a}-6 \sqrt {b}\right )\right ) \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{8 \left (\sqrt {a}-\sqrt {b}\right ) b^{3/2} d}-\frac {\left (\sqrt {a} \left (5 a+\sqrt {a} \sqrt {b}-6 b\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{8 (a-b) b^{3/2} d}\\ &=-\frac {\sqrt {a} \left (5 \sqrt {a}-6 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 \left (\sqrt {a}-\sqrt {b}\right )^{3/2} b^{9/4} d}-\frac {\sqrt {a} \left (5 \sqrt {a}+6 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 \left (\sqrt {a}+\sqrt {b}\right )^{3/2} b^{9/4} d}+\frac {\cosh (c+d x)}{b^2 d}+\frac {a \cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{4 (a-b) b^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.70, size = 615, normalized size = 2.62 \begin {gather*} \frac {32 \cosh (c+d x)+\frac {32 a \cosh (c+d x) (2 a+b-b \cosh (2 (c+d x)))}{(a-b) (8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x)))}+\frac {a \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-b c-b d x-2 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )-20 a c \text {$\#$1}^2+27 b c \text {$\#$1}^2-20 a d x \text {$\#$1}^2+27 b d x \text {$\#$1}^2-40 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+54 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+20 a c \text {$\#$1}^4-27 b c \text {$\#$1}^4+20 a d x \text {$\#$1}^4-27 b d x \text {$\#$1}^4+40 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-54 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+b c \text {$\#$1}^6+b d x \text {$\#$1}^6+2 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{a-b}}{32 b^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^9/(a - b*Sinh[c + d*x]^4)^2,x]

[Out]

(32*Cosh[c + d*x] + (32*a*Cosh[c + d*x]*(2*a + b - b*Cosh[2*(c + d*x)]))/((a - b)*(8*a - 3*b + 4*b*Cosh[2*(c +
d*x)] - b*Cosh[4*(c + d*x)])) + (a*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-(b*c
) - b*d*x - 2*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 20
*a*c*#1^2 + 27*b*c*#1^2 - 20*a*d*x*#1^2 + 27*b*d*x*#1^2 - 40*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Co
sh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + 54*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 + 20*a*c*#1^4 - 27*b*c*#1^4 + 20*a*d*x*#1^4 - 27*b*d*x*#1^4 + 40*a*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 - 54*b*Log[-Cosh[(
c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + b*c*#1^6 + b*d*x*#1^6 +
2*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)/(-(b*#1) -
8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(a - b))/(32*b^2*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(372\) vs. \(2(185)=370\).
time = 5.93, size = 373, normalized size = 1.59

method	result	size
derivativedivides	\(\frac {-\frac {2 a \left (\frac {\frac {\left (-2 b +a \right ) \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a -4 b}-\frac {\left (3 a -8 b \right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a -b \right )}+\frac {\left (3 a +2 b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a -4 b}-\frac {a}{4 \left (a -b \right )}}{a \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {a \left (-\frac {\left (\sqrt {a b}+5 a -6 b \right ) \arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {-\sqrt {a b}\, a -a b}}+\frac {\left (-\sqrt {a b}+5 a -6 b \right ) \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {\sqrt {a b}\, a -a b}}\right )}{4 a -4 b}\right )}{b^{2}}-\frac {1}{b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {1}{b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\)	\(373\)
default	\(\frac {-\frac {2 a \left (\frac {\frac {\left (-2 b +a \right ) \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a -4 b}-\frac {\left (3 a -8 b \right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a -b \right )}+\frac {\left (3 a +2 b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a -4 b}-\frac {a}{4 \left (a -b \right )}}{a \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {a \left (-\frac {\left (\sqrt {a b}+5 a -6 b \right ) \arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {-\sqrt {a b}\, a -a b}}+\frac {\left (-\sqrt {a b}+5 a -6 b \right ) \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {\sqrt {a b}\, a -a b}}\right )}{4 a -4 b}\right )}{b^{2}}-\frac {1}{b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {1}{b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\)	\(373\)
risch	\(\frac {{\mathrm e}^{d x +c}}{2 b^{2} d}+\frac {{\mathrm e}^{-d x -c}}{2 b^{2} d}+\frac {a \,{\mathrm e}^{d x +c} \left (-b \,{\mathrm e}^{6 d x +6 c}+4 a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}+b \,{\mathrm e}^{2 d x +2 c}-b \right )}{2 b^{2} \left (a -b \right ) d \left (-b \,{\mathrm e}^{8 d x +8 c}+4 b \,{\mathrm e}^{6 d x +6 c}+16 a \,{\mathrm e}^{4 d x +4 c}-6 b \,{\mathrm e}^{4 d x +4 c}+4 b \,{\mathrm e}^{2 d x +2 c}-b \right )}+\left (\munderset {\textit {\_R} =\RootOf \left (\left (65536 a^{3} b^{9} d^{4}-196608 a^{2} b^{10} d^{4}+196608 a \,b^{11} d^{4}-65536 b^{12} d^{4}\right ) \textit {\_Z}^{4}+\left (7680 a^{3} b^{5} d^{2}-24064 a^{2} b^{6} d^{2}+18432 a \,b^{7} d^{2}\right ) \textit {\_Z}^{2}-625 a^{4}+1800 a^{3} b -1296 a^{2} b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\left (-\frac {32768 a^{4} b^{7} d^{3}}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}+\frac {147456 a^{3} b^{8} d^{3}}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}-\frac {245760 a^{2} b^{9} d^{3}}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}+\frac {180224 a \,b^{10} d^{3}}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}-\frac {49152 b^{11} d^{3}}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}\right ) \textit {\_R}^{3}+\left (-\frac {4000 a^{5} b^{2} d}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}+\frac {14720 a^{4} b^{3} d}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}-\frac {14240 a^{3} b^{4} d}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}-\frac {2880 a^{2} b^{5} d}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}+\frac {6912 a \,b^{6} d}{625 a^{5}-2625 a^{4} b +3684 a^{3} b^{2}-1728 a^{2} b^{3}}\right ) \textit {\_R} \right ) {\mathrm e}^{d x +c}+1\right )\right )\)	\(710\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^9/(a-b*sinh(d*x+c)^4)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^9/(a-b*sinh(d*x+c)^4)^2,x, algorithm="maxima")

[Out]

1/2*(a*b - b^2 + (a*b*e^(10*c) - b^2*e^(10*c))*e^(10*d*x) - (2*a*b*e^(8*c) - 3*b^2*e^(8*c))*e^(8*d*x) - (20*a^
2*e^(6*c) - 17*a*b*e^(6*c) + 2*b^2*e^(6*c))*e^(6*d*x) - (20*a^2*e^(4*c) - 17*a*b*e^(4*c) + 2*b^2*e^(4*c))*e^(4
*d*x) - (2*a*b*e^(2*c) - 3*b^2*e^(2*c))*e^(2*d*x))/((a*b^3*d*e^(9*c) - b^4*d*e^(9*c))*e^(9*d*x) - 4*(a*b^3*d*e
^(7*c) - b^4*d*e^(7*c))*e^(7*d*x) - 2*(8*a^2*b^2*d*e^(5*c) - 11*a*b^3*d*e^(5*c) + 3*b^4*d*e^(5*c))*e^(5*d*x) -
4*(a*b^3*d*e^(3*c) - b^4*d*e^(3*c))*e^(3*d*x) + (a*b^3*d*e^c - b^4*d*e^c)*e^(d*x)) + 1/512*integrate(256*(a*b
*e^(7*d*x + 7*c) - a*b*e^(d*x + c) + (20*a^2*e^(5*c) - 27*a*b*e^(5*c))*e^(5*d*x) - (20*a^2*e^(3*c) - 27*a*b*e^
(3*c))*e^(3*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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 7664 vs. \(2 (187) = 374\).
time = 0.58, size = 7664, normalized size = 32.61 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^9/(a-b*sinh(d*x+c)^4)^2,x, algorithm="fricas")

[Out]

1/16*(8*(a*b - b^2)*cosh(d*x + c)^10 + 80*(a*b - b^2)*cosh(d*x + c)*sinh(d*x + c)^9 + 8*(a*b - b^2)*sinh(d*x +
c)^10 - 8*(2*a*b - 3*b^2)*cosh(d*x + c)^8 + 8*(45*(a*b - b^2)*cosh(d*x + c)^2 - 2*a*b + 3*b^2)*sinh(d*x + c)^
8 + 64*(15*(a*b - b^2)*cosh(d*x + c)^3 - (2*a*b - 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 8*(20*a^2 - 17*a*b +
2*b^2)*cosh(d*x + c)^6 + 8*(210*(a*b - b^2)*cosh(d*x + c)^4 - 28*(2*a*b - 3*b^2)*cosh(d*x + c)^2 - 20*a^2 + 1
7*a*b - 2*b^2)*sinh(d*x + c)^6 + 16*(126*(a*b - b^2)*cosh(d*x + c)^5 - 28*(2*a*b - 3*b^2)*cosh(d*x + c)^3 - 3*
(20*a^2 - 17*a*b + 2*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 8*(20*a^2 - 17*a*b + 2*b^2)*cosh(d*x + c)^4 + 8*(21
0*(a*b - b^2)*cosh(d*x + c)^6 - 70*(2*a*b - 3*b^2)*cosh(d*x + c)^4 - 15*(20*a^2 - 17*a*b + 2*b^2)*cosh(d*x + c
)^2 - 20*a^2 + 17*a*b - 2*b^2)*sinh(d*x + c)^4 + 32*(30*(a*b - b^2)*cosh(d*x + c)^7 - 14*(2*a*b - 3*b^2)*cosh(
d*x + c)^5 - 5*(20*a^2 - 17*a*b + 2*b^2)*cosh(d*x + c)^3 - (20*a^2 - 17*a*b + 2*b^2)*cosh(d*x + c))*sinh(d*x +
c)^3 - 8*(2*a*b - 3*b^2)*cosh(d*x + c)^2 + 8*(45*(a*b - b^2)*cosh(d*x + c)^8 - 28*(2*a*b - 3*b^2)*cosh(d*x +
c)^6 - 15*(20*a^2 - 17*a*b + 2*b^2)*cosh(d*x + c)^4 - 6*(20*a^2 - 17*a*b + 2*b^2)*cosh(d*x + c)^2 - 2*a*b + 3*
b^2)*sinh(d*x + c)^2 + ((a*b^3 - b^4)*d*cosh(d*x + c)^9 + 9*(a*b^3 - b^4)*d*cosh(d*x + c)*sinh(d*x + c)^8 + (a
*b^3 - b^4)*d*sinh(d*x + c)^9 - 4*(a*b^3 - b^4)*d*cosh(d*x + c)^7 + 4*(9*(a*b^3 - b^4)*d*cosh(d*x + c)^2 - (a*
b^3 - b^4)*d)*sinh(d*x + c)^7 - 2*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cosh(d*x + c)^5 + 28*(3*(a*b^3 - b^4)*d*cos
h(d*x + c)^3 - (a*b^3 - b^4)*d*cosh(d*x + c))*sinh(d*x + c)^6 + 2*(63*(a*b^3 - b^4)*d*cosh(d*x + c)^4 - 42*(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)^5 - 4*(a*b^3 - b^4)*d*cosh(d*x
+ c)^3 + 2*(63*(a*b^3 - b^4)*d*cosh(d*x + c)^5 - 70*(a*b^3 - b^4)*d*cosh(d*x + c)^3 - 5*(8*a^2*b^2 - 11*a*b^3
+ 3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(21*(a*b^3 - b^4)*d*cosh(d*x + c)^6 - 35*(a*b^3 - b^4)*d*cosh(d*
x + c)^4 - 5*(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)^3 + (a*b^3 - b^
4)*d*cosh(d*x + c) + 4*(9*(a*b^3 - b^4)*d*cosh(d*x + c)^7 - 21*(a*b^3 - b^4)*d*cosh(d*x + c)^5 - 5*(8*a^2*b^2
- 11*a*b^3 + 3*b^4)*d*cosh(d*x + c)^3 - 3*(a*b^3 - b^4)*d*cosh(d*x + c))*sinh(d*x + c)^2 + (9*(a*b^3 - b^4)*d*
cosh(d*x + c)^8 - 28*(a*b^3 - b^4)*d*cosh(d*x + c)^6 - 10*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cosh(d*x + c)^4 - 1
2*(a*b^3 - b^4)*d*cosh(d*x + c)^2 + (a*b^3 - b^4)*d)*sinh(d*x + c))*sqrt(-((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^
7)*d^2*sqrt((625*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4)/((a^6*b^9 - 6*a^5*b^10 + 15*a^
4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) + 15*a^3 - 47*a^2*b + 36*a*b^2)/((a^3*b^4 - 3*a^2*
b^5 + 3*a*b^6 - b^7)*d^2))*log(-625*a^5 + 2625*a^4*b - 3684*a^3*b^2 + 1728*a^2*b^3 - (625*a^5 - 2625*a^4*b + 3
684*a^3*b^2 - 1728*a^2*b^3)*cosh(d*x + c)^2 - 2*(625*a^5 - 2625*a^4*b + 3684*a^3*b^2 - 1728*a^2*b^3)*cosh(d*x
+ c)*sinh(d*x + c) - (625*a^5 - 2625*a^4*b + 3684*a^3*b^2 - 1728*a^2*b^3)*sinh(d*x + c)^2 + 2*((125*a^5*b^2 -
520*a^4*b^3 + 723*a^3*b^4 - 336*a^2*b^5)*d*cosh(d*x + c) + (125*a^5*b^2 - 520*a^4*b^3 + 723*a^3*b^4 - 336*a^2*
b^5)*d*sinh(d*x + c) - 2*((2*a^4*b^7 - 9*a^3*b^8 + 15*a^2*b^9 - 11*a*b^10 + 3*b^11)*d^3*cosh(d*x + c) + (2*a^4
*b^7 - 9*a^3*b^8 + 15*a^2*b^9 - 11*a*b^10 + 3*b^11)*d^3*sinh(d*x + c))*sqrt((625*a^7 - 3450*a^6*b + 7161*a^5*b
^2 - 6624*a^4*b^3 + 2304*a^3*b^4)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^14
+ b^15)*d^4)))*sqrt(-((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((625*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 66
24*a^4*b^3 + 2304*a^3*b^4)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^14 + b^15)
*d^4)) + 15*a^3 - 47*a^2*b + 36*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)^9 + 9*(a*b^3 - b^4)*d*cosh(d*x + c)*sinh(d*x + c)^8 + (a*b^3 - b^4)*d*sinh(d*x + c)^9 - 4*(a*b^3 - b^4
)*d*cosh(d*x + c)^7 + 4*(9*(a*b^3 - b^4)*d*cosh(d*x + c)^2 - (a*b^3 - b^4)*d)*sinh(d*x + c)^7 - 2*(8*a^2*b^2 -
11*a*b^3 + 3*b^4)*d*cosh(d*x + c)^5 + 28*(3*(a*b^3 - b^4)*d*cosh(d*x + c)^3 - (a*b^3 - b^4)*d*cosh(d*x + c))*
sinh(d*x + c)^6 + 2*(63*(a*b^3 - b^4)*d*cosh(d*x + c)^4 - 42*(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)^5 - 4*(a*b^3 - b^4)*d*cosh(d*x + c)^3 + 2*(63*(a*b^3 - b^4)*d*cosh(d*x + c)^5
- 70*(a*b^3 - b^4)*d*cosh(d*x + c)^3 - 5*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^4 + 4*
(21*(a*b^3 - b^4)*d*cosh(d*x + c)^6 - 35*(a*b^3 - b^4)*d*cosh(d*x + c)^4 - 5*(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)^3 + (a*b^3 - b^4)*d*cosh(d*x + c) + 4*(9*(a*b^3 - b^4)*d*cosh
(d*x + c)^7 - 21*(a*b^3 - b^4)*d*cosh(d*x + c)^5 - 5*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cosh(d*x + c)^3 - 3*(a*b
^3 - b^4)*d*cosh(d*x + c))*sinh(d*x + c)^2 + (9*(a*b^3 - b^4)*d*cosh(d*x + c)^8 - 28*(a*b^3 - b^4)*d*cosh(d*x
+ c)^6 - 10*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*co...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)**9/(a-b*sinh(d*x+c)**4)**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1082 vs. \(2 (187) = 374\).
time = 0.88, size = 1082, normalized size = 4.60 \begin {gather*} -\frac {\frac {{\left ({\left (4 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{2} + 5 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b\right )} {\left (a b^{2} - b^{3}\right )}^{2} {\left | b \right |} + {\left (20 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{4} b^{2} - 23 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{3} b^{3} - 32 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b^{4} + 35 \, \sqrt {-b^{2} + \sqrt {a b} b} a b^{5}\right )} {\left | -a b^{2} + b^{3} \right |} {\left | b \right |} - {\left (20 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{4} b^{4} - 39 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{3} b^{5} - 12 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b^{6} + 61 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b^{7} - 30 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} b^{8}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a b^{3} - b^{4} + \sqrt {{\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} {\left (a b^{3} - b^{4}\right )} + {\left (a b^{3} - b^{4}\right )}^{2}}}{a b^{3} - b^{4}}}}\right )}{{\left (4 \, a^{4} b^{6} - 7 \, a^{3} b^{7} - 3 \, a^{2} b^{8} + 11 \, a b^{9} - 5 \, b^{10}\right )} {\left | -a b^{2} + b^{3} \right |}} - \frac {{\left ({\left (4 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{2} + 5 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a b\right )} {\left (a b^{2} - b^{3}\right )}^{2} {\left | b \right |} - {\left (20 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{4} b^{2} - 23 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{3} b^{3} - 32 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b^{4} + 35 \, \sqrt {-b^{2} - \sqrt {a b} b} a b^{5}\right )} {\left | -a b^{2} + b^{3} \right |} {\left | b \right |} - {\left (20 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{4} b^{4} - 39 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{3} b^{5} - 12 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b^{6} + 61 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a b^{7} - 30 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} b^{8}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a b^{3} - b^{4} - \sqrt {{\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} {\left (a b^{3} - b^{4}\right )} + {\left (a b^{3} - b^{4}\right )}^{2}}}{a b^{3} - b^{4}}}}\right )}{{\left (4 \, a^{4} b^{6} - 7 \, a^{3} b^{7} - 3 \, a^{2} b^{8} + 11 \, a b^{9} - 5 \, b^{10}\right )} {\left | -a b^{2} + b^{3} \right |}} - \frac {4 \, {\left (a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 4 \, a^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 4 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}\right )}}{{\left (b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4} - 8 \, b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 16 \, a + 16 \, b\right )} {\left (a b^{2} - b^{3}\right )}} - \frac {4 \, {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{b^{2}}}{8 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^9/(a-b*sinh(d*x+c)^4)^2,x, algorithm="giac")

[Out]

-1/8*(((4*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^2 + 5*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a*b)*(a*b^2 - b^3)^2*a
bs(b) + (20*sqrt(-b^2 + sqrt(a*b)*b)*a^4*b^2 - 23*sqrt(-b^2 + sqrt(a*b)*b)*a^3*b^3 - 32*sqrt(-b^2 + sqrt(a*b)*
b)*a^2*b^4 + 35*sqrt(-b^2 + sqrt(a*b)*b)*a*b^5)*abs(-a*b^2 + b^3)*abs(b) - (20*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)
*b)*a^4*b^4 - 39*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^3*b^5 - 12*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^2*b^6 +
61*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a*b^7 - 30*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*b^8)*abs(b))*arctan(1/2*(e
^(d*x + c) + e^(-d*x - c))/sqrt(-(a*b^3 - b^4 + sqrt((a^2*b^2 - 2*a*b^3 + b^4)*(a*b^3 - b^4) + (a*b^3 - b^4)^2
))/(a*b^3 - b^4)))/((4*a^4*b^6 - 7*a^3*b^7 - 3*a^2*b^8 + 11*a*b^9 - 5*b^10)*abs(-a*b^2 + b^3)) - ((4*sqrt(a*b)
*sqrt(-b^2 - sqrt(a*b)*b)*a^2 + 5*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a*b)*(a*b^2 - b^3)^2*abs(b) - (20*sqrt(-b
^2 - sqrt(a*b)*b)*a^4*b^2 - 23*sqrt(-b^2 - sqrt(a*b)*b)*a^3*b^3 - 32*sqrt(-b^2 - sqrt(a*b)*b)*a^2*b^4 + 35*sqr
t(-b^2 - sqrt(a*b)*b)*a*b^5)*abs(-a*b^2 + b^3)*abs(b) - (20*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^4*b^4 - 39*sq
rt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^3*b^5 - 12*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^2*b^6 + 61*sqrt(a*b)*sqrt(-
b^2 - sqrt(a*b)*b)*a*b^7 - 30*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*b^8)*abs(b))*arctan(1/2*(e^(d*x + c) + e^(-d*
x - c))/sqrt(-(a*b^3 - b^4 - sqrt((a^2*b^2 - 2*a*b^3 + b^4)*(a*b^3 - b^4) + (a*b^3 - b^4)^2))/(a*b^3 - b^4)))/
((4*a^4*b^6 - 7*a^3*b^7 - 3*a^2*b^8 + 11*a*b^9 - 5*b^10)*abs(-a*b^2 + b^3)) - 4*(a*b*(e^(d*x + c) + e^(-d*x -
c))^3 - 4*a^2*(e^(d*x + c) + e^(-d*x - c)) - 4*a*b*(e^(d*x + c) + e^(-d*x - c)))/((b*(e^(d*x + c) + e^(-d*x -
c))^4 - 8*b*(e^(d*x + c) + e^(-d*x - c))^2 - 16*a + 16*b)*(a*b^2 - b^3)) - 4*(e^(d*x + c) + e^(-d*x - c))/b^2)
/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^9}{{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^9/(a - b*sinh(c + d*x)^4)^2,x)

[Out]

int(sinh(c + d*x)^9/(a - b*sinh(c + d*x)^4)^2, x)

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