3.3.55 \(\int \frac {\sinh ^5(c+d x)}{(a-b \sinh ^4(c+d x))^3} \, dx\) [255]

Optimal. Leaf size=313 \[ -\frac {\left (3 a-10 \sqrt {a} \sqrt {b}+4 b\right ) \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{5/4} d}-\frac {\left (3 a+10 \sqrt {a} \sqrt {b}+4 b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{5/4} d}+\frac {\cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac {\cosh (c+d x) \left (a^2-11 a b-2 b^2+2 b (2 a+b) \cosh ^2(c+d x)\right )}{32 a (a-b)^2 b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )} \]

[Out]

1/8*cosh(d*x+c)*(a+b-b*cosh(d*x+c)^2)/(a-b)/b/d/(a-b+2*b*cosh(d*x+c)^2-b*cosh(d*x+c)^4)^2-1/32*cosh(d*x+c)*(a^
2-11*a*b-2*b^2+2*b*(2*a+b)*cosh(d*x+c)^2)/a/(a-b)^2/b/d/(a-b+2*b*cosh(d*x+c)^2-b*cosh(d*x+c)^4)-1/64*arctan(b^
(1/4)*cosh(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))*(3*a+4*b-10*a^(1/2)*b^(1/2))/a^(3/2)/b^(5/4)/d/(a^(1/2)-b^(1/2))^(5
/2)-1/64*arctanh(b^(1/4)*cosh(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))*(3*a+4*b+10*a^(1/2)*b^(1/2))/a^(3/2)/b^(5/4)/d/(
a^(1/2)+b^(1/2))^(5/2)

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Rubi [A]
time = 0.39, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3294, 1219, 1192, 1180, 211, 214} \begin {gather*} -\frac {\left (-10 \sqrt {a} \sqrt {b}+3 a+4 b\right ) \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{3/2} b^{5/4} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}-\frac {\left (10 \sqrt {a} \sqrt {b}+3 a+4 b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{3/2} b^{5/4} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}-\frac {\cosh (c+d x) \left (a^2+2 b (2 a+b) \cosh ^2(c+d x)-11 a b-2 b^2\right )}{32 a b d (a-b)^2 \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}+\frac {\cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{8 b d (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^5/(a - b*Sinh[c + d*x]^4)^3,x]

[Out]

-1/64*((3*a - 10*Sqrt[a]*Sqrt[b] + 4*b)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(a^(3/2)*(Sqr
t[a] - Sqrt[b])^(5/2)*b^(5/4)*d) - ((3*a + 10*Sqrt[a]*Sqrt[b] + 4*b)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt
[a] + Sqrt[b]]])/(64*a^(3/2)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(5/4)*d) + (Cosh[c + d*x]*(a + b - b*Cosh[c + d*x]^2)
)/(8*(a - b)*b*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4)^2) - (Cosh[c + d*x]*(a^2 - 11*a*b - 2*b^2 +
2*b*(2*a + b)*Cosh[c + d*x]^2))/(32*a*(a - b)^2*b*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 1192

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*(a*b*e - d*(b^2 - 2*a
*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), 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] && LtQ[p, -1] && IntegerQ[2*p]

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 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 ^5(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{\left (a-b+2 b x^2-b x^4\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {2 a (a-7 b)+10 a b x^2}{\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 {\cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{8 (a-b) b 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 (a^2-11 a b-2 b^2+2 b (2 a+b) \cosh ^2(c+d x)\right )}{32 a (a-b)^2 b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {-4 a b \left (3 a^2-17 a b+2 b^2\right )-8 a b^2 (2 a+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 {\cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{8 (a-b) b 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 (a^2-11 a b-2 b^2+2 b (2 a+b) \cosh ^2(c+d x)\right )}{32 a (a-b)^2 b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}+\frac {\left (3 a-10 \sqrt {a} \sqrt {b}+4 b\right ) \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{64 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^2 \sqrt {b} d}-\frac {\left (3 a+10 \sqrt {a} \sqrt {b}+4 b\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{64 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^2 \sqrt {b} d}\\ &=-\frac {\left (3 a-10 \sqrt {a} \sqrt {b}+4 b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{5/4} d}-\frac {\left (3 a+10 \sqrt {a} \sqrt {b}+4 b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{5/4} d}+\frac {\cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{8 (a-b) b 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 (a^2-11 a b-2 b^2+2 b (2 a+b) \cosh ^2(c+d x)\right )}{32 a (a-b)^2 b 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 = 1.36, size = 1019, normalized size = 3.26 \begin {gather*} -\frac {\frac {32 \cosh (c+d x) \left (a^2-9 a b-b^2+b (2 a+b) \cosh (2 (c+d x))\right )}{a (8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x)))}-\frac {512 (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}+\frac {\text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {2 a b c+b^2 c+2 a b d x+b^2 d x+4 a b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+2 b^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+6 a^2 c \text {$\#$1}^2-32 a b c \text {$\#$1}^2+5 b^2 c \text {$\#$1}^2+6 a^2 d x \text {$\#$1}^2-32 a b d x \text {$\#$1}^2+5 b^2 d x \text {$\#$1}^2+12 a^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-64 a b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+10 b^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-6 a^2 c \text {$\#$1}^4+32 a b c \text {$\#$1}^4-5 b^2 c \text {$\#$1}^4-6 a^2 d x \text {$\#$1}^4+32 a b d x \text {$\#$1}^4-5 b^2 d x \text {$\#$1}^4-12 a^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+64 a b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-10 b^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-2 a b c \text {$\#$1}^6-b^2 c \text {$\#$1}^6-2 a b d x \text {$\#$1}^6-b^2 d x \text {$\#$1}^6-4 a b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6-2 b^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{a}}{128 (a-b)^2 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^5/(a - b*Sinh[c + d*x]^4)^3,x]

[Out]

-1/128*((32*Cosh[c + d*x]*(a^2 - 9*a*b - b^2 + b*(2*a + b)*Cosh[2*(c + d*x)]))/(a*(8*a - 3*b + 4*b*Cosh[2*(c +
d*x)] - b*Cosh[4*(c + d*x)])) - (512*(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 + b^2*c + 2*a*b*d*x + 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] + 2*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sin
h[(c + d*x)/2]*#1] + 6*a^2*c*#1^2 - 32*a*b*c*#1^2 + 5*b^2*c*#1^2 + 6*a^2*d*x*#1^2 - 32*a*b*d*x*#1^2 + 5*b^2*d*
x*#1^2 + 12*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
- 64*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 + 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]*#1^2 - 6*a^2*c*#
1^4 + 32*a*b*c*#1^4 - 5*b^2*c*#1^4 - 6*a^2*d*x*#1^4 + 32*a*b*d*x*#1^4 - 5*b^2*d*x*#1^4 - 12*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]*#1^4 + 64*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]*#1^4 - 10*b^2*Log[-Cosh[(c + d*x)/2] - S
inh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 2*a*b*c*#1^6 - b^2*c*#1^6 - 2*a*b*d*x*#
1^6 - 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 - 2*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) & ]/a)/((a - b)^2*b*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(649\) vs. \(2(261)=522\).
time = 10.68, size = 650, normalized size = 2.08

method	result	size
derivativedivides	\(\frac {\frac {-\frac {\left (3 a^{2}-13 a b +4 b^{2}\right ) \left (\tanh ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (7 a^{2}-33 a b +8 b^{2}\right ) \left (\tanh ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (63 a^{3}-225 a^{2} b +68 a \,b^{2}+64 b^{3}\right ) \left (\tanh ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right ) a}+\frac {3 \left (35 a^{3}-61 a^{2} b +32 a \,b^{2}+128 b^{3}\right ) \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (105 a^{3}+9 a^{2} b -452 a \,b^{2}-64 b^{3}\right ) \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a b \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (21 a^{2}+29 a b -40 b^{2}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (21 a^{2}+37 a b -4 b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 a \left (a +b \right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}}{\left (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 \right )^{2}}+\frac {\frac {\left (4 \sqrt {a b}\, a +2 \sqrt {a b}\, b -3 a^{2}+13 a b -4 b^{2}\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 (-4 \sqrt {a b}\, a -2 \sqrt {a b}\, b -3 a^{2}+13 a b -4 b^{2}\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}}}{16 b \left (a^{2}-2 a b +b^{2}\right )}}{d}\)	\(650\)
default	\(\frac {\frac {-\frac {\left (3 a^{2}-13 a b +4 b^{2}\right ) \left (\tanh ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (7 a^{2}-33 a b +8 b^{2}\right ) \left (\tanh ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (63 a^{3}-225 a^{2} b +68 a \,b^{2}+64 b^{3}\right ) \left (\tanh ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right ) a}+\frac {3 \left (35 a^{3}-61 a^{2} b +32 a \,b^{2}+128 b^{3}\right ) \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (105 a^{3}+9 a^{2} b -452 a \,b^{2}-64 b^{3}\right ) \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a b \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 \left (21 a^{2}+29 a b -40 b^{2}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (21 a^{2}+37 a b -4 b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 a \left (a +b \right )}{16 b \left (a^{2}-2 a b +b^{2}\right )}}{\left (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 \right )^{2}}+\frac {\frac {\left (4 \sqrt {a b}\, a +2 \sqrt {a b}\, b -3 a^{2}+13 a b -4 b^{2}\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 (-4 \sqrt {a b}\, a -2 \sqrt {a b}\, b -3 a^{2}+13 a b -4 b^{2}\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}}}{16 b \left (a^{2}-2 a b +b^{2}\right )}}{d}\)	\(650\)
risch	\(\text {Expression too large to display}\)	\(1427\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^5/(a-b*sinh(d*x+c)^4)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(32*(-1/512*(3*a^2-13*a*b+4*b^2)/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^14+3/512/b*(7*a^2-33*a*b+8*b^2)/(a^
2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^12-1/512/b*(63*a^3-225*a^2*b+68*a*b^2+64*b^3)/(a^2-2*a*b+b^2)/a*tanh(1/2*d*x+
1/2*c)^10+3/512*(35*a^3-61*a^2*b+32*a*b^2+128*b^3)/a/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^8-1/512/a*(105*a^3+
9*a^2*b-452*a*b^2-64*b^3)/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^6+3/512*(21*a^2+29*a*b-40*b^2)/b/(a^2-2*a*b+b^
2)*tanh(1/2*d*x+1/2*c)^4-1/512*(21*a^2+37*a*b-4*b^2)/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^2+3/512*a*(a+b)/b/(
a^2-2*a*b+b^2))/(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)^2+1/16/b/(a^2-2*a*b+b^2)*(1/4*(4*(a*b)^(1/2)*a+2*(a*b)^(1/2)*b-3*a^2+13
*a*b-4*b^2)/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*(-4*(a*b)^(1/2)*a-2*(a*b)^(1/2)*b-3*a^2+13*a*b-4*b^2)/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))))

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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)^5/(a-b*sinh(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

1/8*((2*a*b^2*e^(15*c) + b^3*e^(15*c))*e^(15*d*x) + (2*a^2*b*e^(13*c) - 24*a*b^2*e^(13*c) - 5*b^3*e^(13*c))*e^
(13*d*x) - (70*a^2*b*e^(11*c) - 76*a*b^2*e^(11*c) - 9*b^3*e^(11*c))*e^(11*d*x) + (96*a^3*e^(9*c) + 164*a^2*b*e
^(9*c) - 54*a*b^2*e^(9*c) - 5*b^3*e^(9*c))*e^(9*d*x) + (96*a^3*e^(7*c) + 164*a^2*b*e^(7*c) - 54*a*b^2*e^(7*c)
- 5*b^3*e^(7*c))*e^(7*d*x) - (70*a^2*b*e^(5*c) - 76*a*b^2*e^(5*c) - 9*b^3*e^(5*c))*e^(5*d*x) + (2*a^2*b*e^(3*c
) - 24*a*b^2*e^(3*c) - 5*b^3*e^(3*c))*e^(3*d*x) + (2*a*b^2*e^c + b^3*e^c)*e^(d*x))/(a^3*b^3*d - 2*a^2*b^4*d +
a*b^5*d + (a^3*b^3*d*e^(16*c) - 2*a^2*b^4*d*e^(16*c) + a*b^5*d*e^(16*c))*e^(16*d*x) - 8*(a^3*b^3*d*e^(14*c) -
2*a^2*b^4*d*e^(14*c) + a*b^5*d*e^(14*c))*e^(14*d*x) - 4*(8*a^4*b^2*d*e^(12*c) - 23*a^3*b^3*d*e^(12*c) + 22*a^2
*b^4*d*e^(12*c) - 7*a*b^5*d*e^(12*c))*e^(12*d*x) + 8*(16*a^4*b^2*d*e^(10*c) - 39*a^3*b^3*d*e^(10*c) + 30*a^2*b
^4*d*e^(10*c) - 7*a*b^5*d*e^(10*c))*e^(10*d*x) + 2*(128*a^5*b*d*e^(8*c) - 352*a^4*b^2*d*e^(8*c) + 355*a^3*b^3*
d*e^(8*c) - 166*a^2*b^4*d*e^(8*c) + 35*a*b^5*d*e^(8*c))*e^(8*d*x) + 8*(16*a^4*b^2*d*e^(6*c) - 39*a^3*b^3*d*e^(
6*c) + 30*a^2*b^4*d*e^(6*c) - 7*a*b^5*d*e^(6*c))*e^(6*d*x) - 4*(8*a^4*b^2*d*e^(4*c) - 23*a^3*b^3*d*e^(4*c) + 2
2*a^2*b^4*d*e^(4*c) - 7*a*b^5*d*e^(4*c))*e^(4*d*x) - 8*(a^3*b^3*d*e^(2*c) - 2*a^2*b^4*d*e^(2*c) + a*b^5*d*e^(2
*c))*e^(2*d*x)) + 1/32*integrate(4*((2*a*b*e^(7*c) + b^2*e^(7*c))*e^(7*d*x) + (6*a^2*e^(5*c) - 32*a*b*e^(5*c)
+ 5*b^2*e^(5*c))*e^(5*d*x) - (6*a^2*e^(3*c) - 32*a*b*e^(3*c) + 5*b^2*e^(3*c))*e^(3*d*x) - (2*a*b*e^c + b^2*e^c
)*e^(d*x))/(a^3*b^2 - 2*a^2*b^3 + a*b^4 + (a^3*b^2*e^(8*c) - 2*a^2*b^3*e^(8*c) + a*b^4*e^(8*c))*e^(8*d*x) - 4*
(a^3*b^2*e^(6*c) - 2*a^2*b^3*e^(6*c) + a*b^4*e^(6*c))*e^(6*d*x) - 2*(8*a^4*b*e^(4*c) - 19*a^3*b^2*e^(4*c) + 14
*a^2*b^3*e^(4*c) - 3*a*b^4*e^(4*c))*e^(4*d*x) - 4*(a^3*b^2*e^(2*c) - 2*a^2*b^3*e^(2*c) + a*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. 22506 vs. \(2 (262) = 524\).
time = 0.83, size = 22506, normalized size = 71.90 \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)^5/(a-b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

1/128*(16*(2*a*b^2 + b^3)*cosh(d*x + c)^15 + 240*(2*a*b^2 + b^3)*cosh(d*x + c)*sinh(d*x + c)^14 + 16*(2*a*b^2
+ b^3)*sinh(d*x + c)^15 + 16*(2*a^2*b - 24*a*b^2 - 5*b^3)*cosh(d*x + c)^13 + 16*(2*a^2*b - 24*a*b^2 - 5*b^3 +
105*(2*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^13 + 208*(35*(2*a*b^2 + b^3)*cosh(d*x + c)^3 + (2*a^2*b - 2
4*a*b^2 - 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^12 - 16*(70*a^2*b - 76*a*b^2 - 9*b^3)*cosh(d*x + c)^11 + 16*(136
5*(2*a*b^2 + b^3)*cosh(d*x + c)^4 - 70*a^2*b + 76*a*b^2 + 9*b^3 + 78*(2*a^2*b - 24*a*b^2 - 5*b^3)*cosh(d*x + c
)^2)*sinh(d*x + c)^11 + 176*(273*(2*a*b^2 + b^3)*cosh(d*x + c)^5 + 26*(2*a^2*b - 24*a*b^2 - 5*b^3)*cosh(d*x +
c)^3 - (70*a^2*b - 76*a*b^2 - 9*b^3)*cosh(d*x + c))*sinh(d*x + c)^10 + 16*(96*a^3 + 164*a^2*b - 54*a*b^2 - 5*b
^3)*cosh(d*x + c)^9 + 16*(5005*(2*a*b^2 + b^3)*cosh(d*x + c)^6 + 715*(2*a^2*b - 24*a*b^2 - 5*b^3)*cosh(d*x + c
)^4 + 96*a^3 + 164*a^2*b - 54*a*b^2 - 5*b^3 - 55*(70*a^2*b - 76*a*b^2 - 9*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^
9 + 48*(21 ...

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1797 vs. \(2 (262) = 524\).
time = 0.93, size = 1797, normalized size = 5.74 \begin {gather*} \frac {\frac {{\left (2 \, {\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )}^{2} {\left (8 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{2} + 14 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a b + 5 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} b^{2}\right )} {\left | b \right |} - {\left (12 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{6} b - 77 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{5} b^{2} + 41 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{4} b^{3} + 111 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{3} b^{4} - 97 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b^{5} + 10 \, \sqrt {-b^{2} - \sqrt {a b} b} a b^{6}\right )} {\left | a^{3} b - 2 \, a^{2} b^{2} + a b^{3} \right |} {\left | b \right |} - {\left (12 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{8} b^{2} - 85 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{7} b^{3} + 171 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{6} b^{4} - 54 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{5} b^{5} - 214 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{4} b^{6} + 279 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{3} b^{7} - 129 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b^{8} + 20 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a b^{9}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + \sqrt {{\left (a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right )} {\left (a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right )} + {\left (a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right )}^{2}}}{a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}}}}\right )}{{\left (4 \, a^{8} b^{4} - 15 \, a^{7} b^{5} + 15 \, a^{6} b^{6} + 10 \, a^{5} b^{7} - 30 \, a^{4} b^{8} + 21 \, a^{3} b^{9} - 5 \, a^{2} b^{10}\right )} {\left | a^{3} b - 2 \, a^{2} b^{2} + a b^{3} \right |}} + \frac {{\left (2 \, {\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )}^{2} {\left (8 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{2} + 14 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b + 5 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} b^{2}\right )} {\left | b \right |} - {\left (12 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{6} b - 77 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{5} b^{2} + 41 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{4} b^{3} + 111 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{3} b^{4} - 97 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b^{5} + 10 \, \sqrt {-b^{2} + \sqrt {a b} b} a b^{6}\right )} {\left | a^{3} b - 2 \, a^{2} b^{2} + a b^{3} \right |} {\left | b \right |} - {\left (12 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{8} b^{2} - 85 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{7} b^{3} + 171 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{6} b^{4} - 54 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{5} b^{5} - 214 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{4} b^{6} + 279 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{3} b^{7} - 129 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b^{8} + 20 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b^{9}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} - \sqrt {{\left (a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right )} {\left (a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right )} + {\left (a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right )}^{2}}}{a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}}}}\right )}{{\left (4 \, a^{8} b^{4} - 15 \, a^{7} b^{5} + 15 \, a^{6} b^{6} + 10 \, a^{5} b^{7} - 30 \, a^{4} b^{8} + 21 \, a^{3} b^{9} - 5 \, a^{2} b^{10}\right )} {\left | a^{3} b - 2 \, a^{2} b^{2} + a b^{3} \right |}} + \frac {8 \, {\left (2 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} + b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} + 2 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 38 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 12 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 80 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 224 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 48 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 96 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 384 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 416 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 64 \, b^{3} {\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 )}^{2} {\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )}}}{64 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^5/(a-b*sinh(d*x+c)^4)^3,x, algorithm="giac")

[Out]

1/64*((2*(a^3*b - 2*a^2*b^2 + a*b^3)^2*(8*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^2 + 14*sqrt(a*b)*sqrt(-b^2 - sq
rt(a*b)*b)*a*b + 5*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*b^2)*abs(b) - (12*sqrt(-b^2 - sqrt(a*b)*b)*a^6*b - 77*sq
rt(-b^2 - sqrt(a*b)*b)*a^5*b^2 + 41*sqrt(-b^2 - sqrt(a*b)*b)*a^4*b^3 + 111*sqrt(-b^2 - sqrt(a*b)*b)*a^3*b^4 -
97*sqrt(-b^2 - sqrt(a*b)*b)*a^2*b^5 + 10*sqrt(-b^2 - sqrt(a*b)*b)*a*b^6)*abs(a^3*b - 2*a^2*b^2 + a*b^3)*abs(b)
- (12*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^8*b^2 - 85*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^7*b^3 + 171*sqrt(a
*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^6*b^4 - 54*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^5*b^5 - 214*sqrt(a*b)*sqrt(-b^2
- sqrt(a*b)*b)*a^4*b^6 + 279*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^3*b^7 - 129*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)
*b)*a^2*b^8 + 20*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a*b^9)*abs(b))*arctan(1/2*(e^(d*x + c) + e^(-d*x - c))/sqr
t(-(a^3*b^2 - 2*a^2*b^3 + a*b^4 + sqrt((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*(a^3*b^2 - 2*a^2*b^3 + a*b^4) +
(a^3*b^2 - 2*a^2*b^3 + a*b^4)^2))/(a^3*b^2 - 2*a^2*b^3 + a*b^4)))/((4*a^8*b^4 - 15*a^7*b^5 + 15*a^6*b^6 + 10*
a^5*b^7 - 30*a^4*b^8 + 21*a^3*b^9 - 5*a^2*b^10)*abs(a^3*b - 2*a^2*b^2 + a*b^3)) + (2*(a^3*b - 2*a^2*b^2 + a*b^
3)^2*(8*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^2 + 14*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a*b + 5*sqrt(a*b)*sqrt(
-b^2 + sqrt(a*b)*b)*b^2)*abs(b) - (12*sqrt(-b^2 + sqrt(a*b)*b)*a^6*b - 77*sqrt(-b^2 + sqrt(a*b)*b)*a^5*b^2 + 4
1*sqrt(-b^2 + sqrt(a*b)*b)*a^4*b^3 + 111*sqrt(-b^2 + sqrt(a*b)*b)*a^3*b^4 - 97*sqrt(-b^2 + sqrt(a*b)*b)*a^2*b^
5 + 10*sqrt(-b^2 + sqrt(a*b)*b)*a*b^6)*abs(a^3*b - 2*a^2*b^2 + a*b^3)*abs(b) - (12*sqrt(a*b)*sqrt(-b^2 + sqrt(
a*b)*b)*a^8*b^2 - 85*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^7*b^3 + 171*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^6*b
^4 - 54*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^5*b^5 - 214*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^4*b^6 + 279*sqrt
(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^3*b^7 - 129*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^2*b^8 + 20*sqrt(a*b)*sqrt(-b
^2 + sqrt(a*b)*b)*a*b^9)*abs(b))*arctan(1/2*(e^(d*x + c) + e^(-d*x - c))/sqrt(-(a^3*b^2 - 2*a^2*b^3 + a*b^4 -
sqrt((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*(a^3*b^2 - 2*a^2*b^3 + a*b^4) + (a^3*b^2 - 2*a^2*b^3 + a*b^4)^2))
/(a^3*b^2 - 2*a^2*b^3 + a*b^4)))/((4*a^8*b^4 - 15*a^7*b^5 + 15*a^6*b^6 + 10*a^5*b^7 - 30*a^4*b^8 + 21*a^3*b^9
- 5*a^2*b^10)*abs(a^3*b - 2*a^2*b^2 + a*b^3)) + 8*(2*a*b^2*(e^(d*x + c) + e^(-d*x - c))^7 + b^3*(e^(d*x + c) +
e^(-d*x - c))^7 + 2*a^2*b*(e^(d*x + c) + e^(-d*x - c))^5 - 38*a*b^2*(e^(d*x + c) + e^(-d*x - c))^5 - 12*b^3*(
e^(d*x + c) + e^(-d*x - c))^5 - 80*a^2*b*(e^(d*x + c) + e^(-d*x - c))^3 + 224*a*b^2*(e^(d*x + c) + e^(-d*x - c
))^3 + 48*b^3*(e^(d*x + c) + e^(-d*x - c))^3 + 96*a^3*(e^(d*x + c) + e^(-d*x - c)) + 384*a^2*b*(e^(d*x + c) +
e^(-d*x - c)) - 416*a*b^2*(e^(d*x + c) + e^(-d*x - c)) - 64*b^3*(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)^2*(a^3*b - 2*a^2*b^2 + a*b^3)))/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 )}^5}{{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^5/(a - b*sinh(c + d*x)^4)^3,x)

[Out]

int(sinh(c + d*x)^5/(a - b*sinh(c + d*x)^4)^3, x)

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