3.3.60 \(\int \frac {\sinh ^6(c+d x)}{(a-b \sinh ^4(c+d x))^3} \, dx\) [260]

Optimal. Leaf size=345 \[ \frac {\left (4 a-10 \sqrt {a} \sqrt {b}+3 b\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{5/4} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{3/2} d}-\frac {\left (4 a+10 \sqrt {a} \sqrt {b}+3 b\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{5/4} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{3/2} d}+\frac {\tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 (a-b)^3 d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )^2}+\frac {\tanh (c+d x) \left (\frac {2 a \left (a^2-a b-8 b^2\right )}{(a-b)^3}-\frac {\left (2 a^2+15 a b+3 b^2\right ) \tanh ^2(c+d x)}{(a-b)^2}\right )}{32 a b d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )} \]

[Out]

1/64*arctanh((a^(1/2)-b^(1/2))^(1/2)*tanh(d*x+c)/a^(1/4))*(4*a+3*b-10*a^(1/2)*b^(1/2))/a^(5/4)/b^(3/2)/d/(a^(1
/2)-b^(1/2))^(5/2)-1/64*arctanh((a^(1/2)+b^(1/2))^(1/2)*tanh(d*x+c)/a^(1/4))*(4*a+3*b+10*a^(1/2)*b^(1/2))/a^(5
/4)/b^(3/2)/d/(a^(1/2)+b^(1/2))^(5/2)+1/8*tanh(d*x+c)*(a*(a+3*b)-(a^2+6*a*b+b^2)*tanh(d*x+c)^2)/(a-b)^3/d/(a-2
*a*tanh(d*x+c)^2+(a-b)*tanh(d*x+c)^4)^2+1/32*tanh(d*x+c)*(2*a*(a^2-a*b-8*b^2)/(a-b)^3-(2*a^2+15*a*b+3*b^2)*tan
h(d*x+c)^2/(a-b)^2)/a/b/d/(a-2*a*tanh(d*x+c)^2+(a-b)*tanh(d*x+c)^4)

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Rubi [A]
time = 0.52, antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3296, 1347, 1692, 1180, 214} \begin {gather*} \frac {\left (-10 \sqrt {a} \sqrt {b}+4 a+3 b\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{5/4} b^{3/2} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}-\frac {\left (10 \sqrt {a} \sqrt {b}+4 a+3 b\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{5/4} b^{3/2} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}+\frac {\tanh (c+d x) \left (\frac {2 a \left (a^2-a b-8 b^2\right )}{(a-b)^3}-\frac {\left (2 a^2+15 a b+3 b^2\right ) \tanh ^2(c+d x)}{(a-b)^2}\right )}{32 a b d \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}+\frac {\tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 d (a-b)^3 \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^6/(a - b*Sinh[c + d*x]^4)^3,x]

[Out]

((4*a - 10*Sqrt[a]*Sqrt[b] + 3*b)*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(64*a^(5/4)*(Sqrt[
a] - Sqrt[b])^(5/2)*b^(3/2)*d) - ((4*a + 10*Sqrt[a]*Sqrt[b] + 3*b)*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d
*x])/a^(1/4)])/(64*a^(5/4)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(3/2)*d) + (Tanh[c + d*x]*(a*(a + 3*b) - (a^2 + 6*a*b +
b^2)*Tanh[c + d*x]^2))/(8*(a - b)^3*d*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4)^2) + (Tanh[c + d*x]
*((2*a*(a^2 - a*b - 8*b^2))/(a - b)^3 - ((2*a^2 + 15*a*b + 3*b^2)*Tanh[c + d*x]^2)/(a - b)^2))/(32*a*b*d*(a -
2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4))

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 1347

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{f = Coe
ff[PolynomialRemainder[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[x^m*(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)*Simp[ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[x^m*(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], x]] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] && IGtQ[m/2, 0]

Rule 1692

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

Rule 3296

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 ^6(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {x^6 \left (1-x^2\right )^2}{\left (a-2 a x^2+(a-b) x^4\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 (a-b)^3 d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {\frac {2 a^3 b (a+3 b)}{(a-b)^3}+\frac {2 a^2 b \left (5 a^2+6 a b-3 b^2\right ) x^2}{(a-b)^3}-\frac {32 a^2 b^2 x^4}{(a-b)^2}-\frac {16 a^2 b x^6}{a-b}}{\left (a-2 a x^2+(a-b) x^4\right )^2} \, dx,x,\tanh (c+d x)\right )}{16 a^2 b d}\\ &=\frac {\tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 (a-b)^3 d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )^2}+\frac {\tanh (c+d x) \left (\frac {2 a \left (a^2-a b-8 b^2\right )}{(a-b)^3}-\frac {\left (2 a^2+15 a b+3 b^2\right ) \tanh ^2(c+d x)}{(a-b)^2}\right )}{32 a b d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {-\frac {8 a^4 b (a+2 b)}{(a-b)^2}-\frac {4 a^3 b \left (2 a^2-17 a b+3 b^2\right ) x^2}{(a-b)^2}}{a-2 a x^2+(a-b) x^4} \, dx,x,\tanh (c+d x)\right )}{128 a^4 b^2 d}\\ &=\frac {\tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 (a-b)^3 d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )^2}+\frac {\tanh (c+d x) \left (\frac {2 a \left (a^2-a b-8 b^2\right )}{(a-b)^3}-\frac {\left (2 a^2+15 a b+3 b^2\right ) \tanh ^2(c+d x)}{(a-b)^2}\right )}{32 a b d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )}-\frac {\left (\left (\sqrt {a}+\sqrt {b}\right ) \left (4 a-10 \sqrt {a} \sqrt {b}+3 b\right )\right ) \text {Subst}\left (\int \frac {1}{-a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{64 a \left (\sqrt {a}-\sqrt {b}\right )^2 b^{3/2} d}+\frac {\left (-\frac {2 a^3 b \left (2 a^2-17 a b+3 b^2\right )}{(a-b)^2}-\frac {-\frac {16 a^4 b (a+2 b)}{a-b}-\frac {8 a^4 b \left (2 a^2-17 a b+3 b^2\right )}{(a-b)^2}}{4 \sqrt {a} \sqrt {b}}\right ) \text {Subst}\left (\int \frac {1}{-a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{128 a^4 b^2 d}\\ &=\frac {\left (4 a-10 \sqrt {a} \sqrt {b}+3 b\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{5/4} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{3/2} d}-\frac {\left (4 a+10 \sqrt {a} \sqrt {b}+3 b\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{5/4} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{3/2} d}+\frac {\tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 (a-b)^3 d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )^2}+\frac {\tanh (c+d x) \left (\frac {2 a \left (a^2-a b-8 b^2\right )}{(a-b)^3}-\frac {\left (2 a^2+15 a b+3 b^2\right ) \tanh ^2(c+d x)}{(a-b)^2}\right )}{32 a 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 = 2.33, size = 351, normalized size = 1.02 \begin {gather*} -\frac {\frac {\left (\sqrt {a}+\sqrt {b}\right )^2 \left (4 a-10 \sqrt {a} \sqrt {b}+3 b\right ) \text {ArcTan}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{a \sqrt {-a+\sqrt {a} \sqrt {b}} b^{3/2}}+\frac {\left (\sqrt {a}-\sqrt {b}\right )^2 \left (4 a+10 \sqrt {a} \sqrt {b}+3 b\right ) \tanh ^{-1}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{a \sqrt {a+\sqrt {a} \sqrt {b}} b^{3/2}}+\frac {4 \left (4 a^2-19 a b-3 b^2+3 b (a+b) \cosh (2 (c+d x))\right ) \sinh (2 (c+d x))}{a b (8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x)))}-\frac {128 (a-b) (2 a+b-b \cosh (2 (c+d x))) \sinh (2 (c+d x))}{b (-8 a+3 b-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x)))^2}}{64 (a-b)^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^6/(a - b*Sinh[c + d*x]^4)^3,x]

[Out]

-1/64*(((Sqrt[a] + Sqrt[b])^2*(4*a - 10*Sqrt[a]*Sqrt[b] + 3*b)*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt
[-a + Sqrt[a]*Sqrt[b]]])/(a*Sqrt[-a + Sqrt[a]*Sqrt[b]]*b^(3/2)) + ((Sqrt[a] - Sqrt[b])^2*(4*a + 10*Sqrt[a]*Sqr
t[b] + 3*b)*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(a*Sqrt[a + Sqrt[a]*Sqrt[b
]]*b^(3/2)) + (4*(4*a^2 - 19*a*b - 3*b^2 + 3*b*(a + b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/(a*b*(8*a - 3*b +
4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)])) - (128*(a - b)*(2*a + b - b*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)
])/(b*(-8*a + 3*b - 4*b*Cosh[2*(c + d*x)] + b*Cosh[4*(c + d*x)])^2))/((a - b)^2*d)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.80, size = 594, normalized size = 1.72

method	result	size
derivativedivides	\(\frac {-\frac {128 \left (-\frac {\left (2 b +a \right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{1024 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (5 a^{2}+24 a b -2 b^{2}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (9 a^{2}+76 a b -70 b^{2}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (5 a^{3}+54 a^{2} b -164 a \,b^{2}-96 b^{3}\right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024 a b \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (5 a^{3}+54 a^{2} b -164 a \,b^{2}-96 b^{3}\right ) \left (\tanh ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024 a b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (9 a^{2}+76 a b -70 b^{2}\right ) \left (\tanh ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (5 a^{2}+24 a b -2 b^{2}\right ) \left (\tanh ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (2 b +a \right ) a \left (\tanh ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024 b \left (a^{2}-2 a b +b^{2}\right )}\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 {\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 (a \left (-2 b -a \right ) \textit {\_R}^{6}+\left (-5 a^{2}+32 a b -6 b^{2}\right ) \textit {\_R}^{4}+\left (5 a^{2}-32 a b +6 b^{2}\right ) \textit {\_R}^{2}+a^{2}+2 a 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}}{64 a b \left (a^{2}-2 a b +b^{2}\right )}}{d}\)	\(594\)
default	\(\frac {-\frac {128 \left (-\frac {\left (2 b +a \right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{1024 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (5 a^{2}+24 a b -2 b^{2}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (9 a^{2}+76 a b -70 b^{2}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (5 a^{3}+54 a^{2} b -164 a \,b^{2}-96 b^{3}\right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024 a b \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (5 a^{3}+54 a^{2} b -164 a \,b^{2}-96 b^{3}\right ) \left (\tanh ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024 a b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (9 a^{2}+76 a b -70 b^{2}\right ) \left (\tanh ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024 b \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (5 a^{2}+24 a b -2 b^{2}\right ) \left (\tanh ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (2 b +a \right ) a \left (\tanh ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024 b \left (a^{2}-2 a b +b^{2}\right )}\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 {\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 (a \left (-2 b -a \right ) \textit {\_R}^{6}+\left (-5 a^{2}+32 a b -6 b^{2}\right ) \textit {\_R}^{4}+\left (5 a^{2}-32 a b +6 b^{2}\right ) \textit {\_R}^{2}+a^{2}+2 a 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}}{64 a b \left (a^{2}-2 a b +b^{2}\right )}}{d}\)	\(594\)
risch	\(\text {Expression too large to display}\)	\(2359\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^6/(a-b*sinh(d*x+c)^4)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-128*(-1/1024*(2*b+a)*a/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)+1/1024*(5*a^2+24*a*b-2*b^2)/b/(a^2-2*a*b+b^
2)*tanh(1/2*d*x+1/2*c)^3-1/1024/b*(9*a^2+76*a*b-70*b^2)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^5+1/1024*(5*a^3+54
*a^2*b-164*a*b^2-96*b^3)/a/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^7+1/1024*(5*a^3+54*a^2*b-164*a*b^2-96*b^3)/a/
b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^9-1/1024/b*(9*a^2+76*a*b-70*b^2)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^11+
1/1024*(5*a^2+24*a*b-2*b^2)/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^13-1/1024*(2*b+a)*a/b/(a^2-2*a*b+b^2)*tanh(1
/2*d*x+1/2*c)^15)/(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)^2-1/64/a/b/(a^2-2*a*b+b^2)*sum((a*(-2*b-a)*_R^6+(-5*a^2+32*a*b-6*b^2)
*_R^4+(5*a^2-32*a*b+6*b^2)*_R^2+a^2+2*a*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)))

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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)^6/(a-b*sinh(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

-1/16*(3*a*b^2 + 3*b^3 - (4*a^2*b*e^(14*c) - 13*a*b^2*e^(14*c) + 3*b^3*e^(14*c))*e^(14*d*x) + 3*(8*a^2*b*e^(12
*c) - 33*a*b^2*e^(12*c) + 7*b^3*e^(12*c))*e^(12*d*x) - (64*a^3*e^(10*c) + 68*a^2*b*e^(10*c) - 225*a*b^2*e^(10*
c) + 63*b^3*e^(10*c))*e^(10*d*x) + 3*(128*a^3*e^(8*c) + 32*a^2*b*e^(8*c) - 61*a*b^2*e^(8*c) + 35*b^3*e^(8*c))*
e^(8*d*x) + (64*a^3*e^(6*c) + 452*a^2*b*e^(6*c) - 9*a*b^2*e^(6*c) - 105*b^3*e^(6*c))*e^(6*d*x) - 3*(40*a^2*b*e
^(4*c) - 29*a*b^2*e^(4*c) - 21*b^3*e^(4*c))*e^(4*d*x) + (4*a^2*b*e^(2*c) - 37*a*b^2*e^(2*c) - 21*b^3*e^(2*c))*
e^(2*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*(1
6*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) + 22*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)) + 1/64*integrate(8*((4*a^2*e^(6*c) - 13*a*b*e^(6*c) + 3*
b^2*e^(6*c))*e^(6*d*x) + 6*(7*a*b*e^(4*c) - b^2*e^(4*c))*e^(4*d*x) + (4*a^2*e^(2*c) - 13*a*b*e^(2*c) + 3*b^2*e
^(2*c))*e^(2*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. 22729 vs. \(2 (292) = 584\).
time = 1.20, size = 22729, normalized size = 65.88 \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)^6/(a-b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

1/128*(8*(4*a^2*b - 13*a*b^2 + 3*b^3)*cosh(d*x + c)^14 + 112*(4*a^2*b - 13*a*b^2 + 3*b^3)*cosh(d*x + c)*sinh(d
*x + c)^13 + 8*(4*a^2*b - 13*a*b^2 + 3*b^3)*sinh(d*x + c)^14 - 24*(8*a^2*b - 33*a*b^2 + 7*b^3)*cosh(d*x + c)^1
2 - 8*(24*a^2*b - 99*a*b^2 + 21*b^3 - 91*(4*a^2*b - 13*a*b^2 + 3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^12 + 32*(
91*(4*a^2*b - 13*a*b^2 + 3*b^3)*cosh(d*x + c)^3 - 9*(8*a^2*b - 33*a*b^2 + 7*b^3)*cosh(d*x + c))*sinh(d*x + c)^
11 + 8*(64*a^3 + 68*a^2*b - 225*a*b^2 + 63*b^3)*cosh(d*x + c)^10 + 8*(1001*(4*a^2*b - 13*a*b^2 + 3*b^3)*cosh(d
*x + c)^4 + 64*a^3 + 68*a^2*b - 225*a*b^2 + 63*b^3 - 198*(8*a^2*b - 33*a*b^2 + 7*b^3)*cosh(d*x + c)^2)*sinh(d*
x + c)^10 + 16*(1001*(4*a^2*b - 13*a*b^2 + 3*b^3)*cosh(d*x + c)^5 - 330*(8*a^2*b - 33*a*b^2 + 7*b^3)*cosh(d*x
+ c)^3 + 5*(64*a^3 + 68*a^2*b - 225*a*b^2 + 63*b^3)*cosh(d*x + c))*sinh(d*x + c)^9 - 24*(128*a^3 + 32*a^2*b -
61*a*b^2 + 35*b^3)*cosh(d*x + c)^8 + 24*(1001*(4*a^2*b - 13*a*b^2 + 3*b^3)*cosh(d*x + c)^6 - 495*(8*a^2*b - 33
*a*b^2 + 7 ...

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

[Out]

Timed out

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Giac [A]
time = 1.27, size = 451, normalized size = 1.31 \begin {gather*} \frac {4 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} - 13 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} + 3 \, b^{3} e^{\left (14 \, d x + 14 \, c\right )} - 24 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 99 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} - 21 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 64 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 68 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} - 225 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 63 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} - 384 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} - 96 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 183 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 105 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 64 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 452 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 9 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 105 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 120 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 87 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 63 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 4 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 37 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 21 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 3 \, a b^{2} - 3 \, b^{3}}{16 \, {\left (a^{3} b - 2 \, a^{2} b^{2} + a 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 )}^{2} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^6/(a-b*sinh(d*x+c)^4)^3,x, algorithm="giac")

[Out]

1/16*(4*a^2*b*e^(14*d*x + 14*c) - 13*a*b^2*e^(14*d*x + 14*c) + 3*b^3*e^(14*d*x + 14*c) - 24*a^2*b*e^(12*d*x +
12*c) + 99*a*b^2*e^(12*d*x + 12*c) - 21*b^3*e^(12*d*x + 12*c) + 64*a^3*e^(10*d*x + 10*c) + 68*a^2*b*e^(10*d*x
+ 10*c) - 225*a*b^2*e^(10*d*x + 10*c) + 63*b^3*e^(10*d*x + 10*c) - 384*a^3*e^(8*d*x + 8*c) - 96*a^2*b*e^(8*d*x
+ 8*c) + 183*a*b^2*e^(8*d*x + 8*c) - 105*b^3*e^(8*d*x + 8*c) - 64*a^3*e^(6*d*x + 6*c) - 452*a^2*b*e^(6*d*x +
6*c) + 9*a*b^2*e^(6*d*x + 6*c) + 105*b^3*e^(6*d*x + 6*c) + 120*a^2*b*e^(4*d*x + 4*c) - 87*a*b^2*e^(4*d*x + 4*c
) - 63*b^3*e^(4*d*x + 4*c) - 4*a^2*b*e^(2*d*x + 2*c) + 37*a*b^2*e^(2*d*x + 2*c) + 21*b^3*e^(2*d*x + 2*c) - 3*a
*b^2 - 3*b^3)/((a^3*b - 2*a^2*b^2 + a*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)

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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 )}^6}{{\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)^6/(a - b*sinh(c + d*x)^4)^3,x)

[Out]

int(sinh(c + d*x)^6/(a - b*sinh(c + d*x)^4)^3, x)

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