3.3.48 \(\int \frac {\sinh ^6(c+d x)}{(a-b \sinh ^4(c+d x))^2} \, dx\) [248]

Optimal. Leaf size=233 \[ \frac {\left (2 \sqrt {a}-3 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} b^{3/2} d}-\frac {\left (2 \sqrt {a}+3 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} b^{3/2} d}+\frac {\tanh (c+d x)}{4 (a-b) b d}+\frac {\text {sech}^2(c+d x) \tanh ^3(c+d x)}{4 b d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )} \]

[Out]

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

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Rubi [A]
time = 0.23, antiderivative size = 233, 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, 1134, 1293, 1180, 214} \begin {gather*} \frac {\left (2 \sqrt {a}-3 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a} b^{3/2} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}-\frac {\left (2 \sqrt {a}+3 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a} b^{3/2} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}+\frac {\tanh (c+d x)}{4 b d (a-b)}+\frac {\tanh ^3(c+d x) \text {sech}^2(c+d x)}{4 b d \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^6/(a - b*Sinh[c + d*x]^4)^2,x]

[Out]

((2*Sqrt[a] - 3*Sqrt[b])*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(8*a^(1/4)*(Sqrt[a] - Sqrt[
b])^(3/2)*b^(3/2)*d) - ((2*Sqrt[a] + 3*Sqrt[b])*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(8*a
^(1/4)*(Sqrt[a] + Sqrt[b])^(3/2)*b^(3/2)*d) + Tanh[c + d*x]/(4*(a - b)*b*d) + (Sech[c + d*x]^2*Tanh[c + d*x]^3
)/(4*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 1134

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

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 1293

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

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 )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {x^6}{\left (a-2 a x^2+(a-b) x^4\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {sech}^2(c+d x) \tanh ^3(c+d x)}{4 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 {x^2 \left (6 a-2 a x^2\right )}{a-2 a x^2+(a-b) x^4} \, dx,x,\tanh (c+d x)\right )}{8 a b d}\\ &=\frac {\tanh (c+d x)}{4 (a-b) b d}+\frac {\text {sech}^2(c+d x) \tanh ^3(c+d x)}{4 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 {-2 a^2-2 a (a-3 b) x^2}{a-2 a x^2+(a-b) x^4} \, dx,x,\tanh (c+d x)\right )}{8 a (a-b) b d}\\ &=\frac {\tanh (c+d x)}{4 (a-b) b d}+\frac {\text {sech}^2(c+d x) \tanh ^3(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 (a-\frac {2 \sqrt {a} (a-2 b)}{\sqrt {b}}-3 b\right ) \text {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 {\left (a+\frac {2 \sqrt {a} (a-2 b)}{\sqrt {b}}-3 b\right ) \text {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 {\left (2 \sqrt {a}-3 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} b^{3/2} d}-\frac {\left (2 \sqrt {a}+3 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} b^{3/2} d}+\frac {\tanh (c+d x)}{4 (a-b) b d}+\frac {\text {sech}^2(c+d x) \tanh ^3(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 = 1.84, size = 238, normalized size = 1.02 \begin {gather*} \frac {\frac {\sqrt {b} \left (-2 a+\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 )}{\sqrt {-a+\sqrt {a} \sqrt {b}}}+\frac {\sqrt {b} \left (-2 a-\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 )}{\sqrt {a+\sqrt {a} \sqrt {b}}}-\frac {4 b (-2 a-b+b \cosh (2 (c+d x))) \sinh (2 (c+d x))}{8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))}}{8 (a-b) b^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^6/(a - b*Sinh[c + d*x]^4)^2,x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.95, size = 305, normalized size = 1.31

method	result	size
derivativedivides	\(\frac {-\frac {128 \left (-\frac {a \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256 b \left (a -b \right )}+\frac {\left (a +4 b \right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256 b \left (a -b \right )}+\frac {\left (a +4 b \right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256 b \left (a -b \right )}-\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{256 b \left (a -b \right )}\right )}{a \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}-\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (-a \,\textit {\_R}^{6}+\left (-5 a +12 b \right ) \textit {\_R}^{4}+\left (5 a -12 b \right ) \textit {\_R}^{2}+a \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}}{16 b \left (a -b \right )}}{d}\)	\(305\)
default	\(\frac {-\frac {128 \left (-\frac {a \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256 b \left (a -b \right )}+\frac {\left (a +4 b \right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256 b \left (a -b \right )}+\frac {\left (a +4 b \right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256 b \left (a -b \right )}-\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{256 b \left (a -b \right )}\right )}{a \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}-\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (-a \,\textit {\_R}^{6}+\left (-5 a +12 b \right ) \textit {\_R}^{4}+\left (5 a -12 b \right ) \textit {\_R}^{2}+a \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}}{16 b \left (a -b \right )}}{d}\)	\(305\)
risch	\(\frac {2 a \,{\mathrm e}^{6 d x +6 c}-b \,{\mathrm e}^{6 d x +6 c}-8 a \,{\mathrm e}^{4 d x +4 c}+3 b \,{\mathrm e}^{4 d x +4 c}-2 a \,{\mathrm e}^{2 d x +2 c}-3 b \,{\mathrm e}^{2 d x +2 c}+b}{2 b d \left (a -b \right ) \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^{4} b^{6} d^{4}-196608 a^{3} b^{7} d^{4}+196608 a^{2} b^{8} d^{4}-65536 a \,b^{9} d^{4}\right ) \textit {\_Z}^{4}+\left (-2048 a^{3} d^{2} b^{3}+7680 a^{2} b^{4} d^{2}-7680 a \,b^{5} d^{2}\right ) \textit {\_Z}^{2}+16 a^{2}-72 a b +81 b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (-\frac {8192 a^{5} b^{5} d^{3}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}+\frac {49152 a^{4} b^{6} d^{3}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}-\frac {98304 a^{3} b^{7} d^{3}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}+\frac {81920 a^{2} b^{8} d^{3}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}-\frac {24576 a \,b^{9} d^{3}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}\right ) \textit {\_R}^{3}+\left (-\frac {2048 d^{2} b^{3} a^{5}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}+\frac {10752 d^{2} b^{4} a^{4}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}-\frac {19968 d^{2} b^{5} a^{3}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}+\frac {15872 d^{2} b^{6} a^{2}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}-\frac {4608 d^{2} b^{7} a}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}\right ) \textit {\_R}^{2}+\left (\frac {128 a^{4} b^{2} d}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}-\frac {1184 a^{3} b^{3} d}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}+\frac {3136 a^{2} b^{4} d}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}-\frac {2592 a \,b^{5} d}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}\right ) \textit {\_R} +\frac {32 a^{4}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}-\frac {192 a^{3} b}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}+\frac {370 a^{2} b^{2}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}-\frac {189 a \,b^{3}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}-\frac {81 b^{4}}{20 a^{2} b^{2}-81 a \,b^{3}+81 b^{4}}\right )\right )\)	\(873\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^6/(a-b*sinh(d*x+c)^4)^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(-128*(-1/256*a/b/(a-b)*tanh(1/2*d*x+1/2*c)^7+1/256/b*(a+4*b)/(a-b)*tanh(1/2*d*x+1/2*c)^5+1/256/b*(a+4*b)/
(a-b)*tanh(1/2*d*x+1/2*c)^3-1/256*a/b/(a-b)*tanh(1/2*d*x+1/2*c))/(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)-1/16/b/(a-b)*sum((-a*_
R^6+(-5*a+12*b)*_R^4+(5*a-12*b)*_R^2+a)/(_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)^2,x, algorithm="maxima")

[Out]

-1/2*((2*a*e^(6*c) - b*e^(6*c))*e^(6*d*x) - (8*a*e^(4*c) - 3*b*e^(4*c))*e^(4*d*x) - (2*a*e^(2*c) + 3*b*e^(2*c)
)*e^(2*d*x) + b)/(a*b^2*d - b^3*d + (a*b^2*d*e^(8*c) - b^3*d*e^(8*c))*e^(8*d*x) - 4*(a*b^2*d*e^(6*c) - b^3*d*e
^(6*c))*e^(6*d*x) - 2*(8*a^2*b*d*e^(4*c) - 11*a*b^2*d*e^(4*c) + 3*b^3*d*e^(4*c))*e^(4*d*x) - 4*(a*b^2*d*e^(2*c
) - b^3*d*e^(2*c))*e^(2*d*x)) + 1/64*integrate(64*((2*a*e^(6*c) - 3*b*e^(6*c))*e^(6*d*x) + (2*a*e^(2*c) - 3*b*
e^(2*c))*e^(2*d*x) + 6*b*e^(4*d*x + 4*c))/(a*b^2 - b^3 + (a*b^2*e^(8*c) - b^3*e^(8*c))*e^(8*d*x) - 4*(a*b^2*e^
(6*c) - b^3*e^(6*c))*e^(6*d*x) - 2*(8*a^2*b*e^(4*c) - 11*a*b^2*e^(4*c) + 3*b^3*e^(4*c))*e^(4*d*x) - 4*(a*b^2*e
^(2*c) - b^3*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. 6045 vs. \(2 (181) = 362\).
time = 0.63, size = 6045, normalized size = 25.94 \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)^2,x, algorithm="fricas")

[Out]

-1/16*(8*(2*a - b)*cosh(d*x + c)^6 + 48*(2*a - b)*cosh(d*x + c)*sinh(d*x + c)^5 + 8*(2*a - b)*sinh(d*x + c)^6
- 8*(8*a - 3*b)*cosh(d*x + c)^4 + 8*(15*(2*a - b)*cosh(d*x + c)^2 - 8*a + 3*b)*sinh(d*x + c)^4 + 32*(5*(2*a -
b)*cosh(d*x + c)^3 - (8*a - 3*b)*cosh(d*x + c))*sinh(d*x + c)^3 - 8*(2*a + 3*b)*cosh(d*x + c)^2 + 8*(15*(2*a -
b)*cosh(d*x + c)^4 - 6*(8*a - 3*b)*cosh(d*x + c)^2 - 2*a - 3*b)*sinh(d*x + c)^2 - ((a*b^2 - b^3)*d*cosh(d*x +
c)^8 + 8*(a*b^2 - b^3)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a*b^2 - b^3)*d*sinh(d*x + c)^8 - 4*(a*b^2 - b^3)*d*
cosh(d*x + c)^6 + 4*(7*(a*b^2 - b^3)*d*cosh(d*x + c)^2 - (a*b^2 - b^3)*d)*sinh(d*x + c)^6 - 2*(8*a^2*b - 11*a*
b^2 + 3*b^3)*d*cosh(d*x + c)^4 + 8*(7*(a*b^2 - b^3)*d*cosh(d*x + c)^3 - 3*(a*b^2 - b^3)*d*cosh(d*x + c))*sinh(
d*x + c)^5 + 2*(35*(a*b^2 - b^3)*d*cosh(d*x + c)^4 - 30*(a*b^2 - b^3)*d*cosh(d*x + c)^2 - (8*a^2*b - 11*a*b^2
+ 3*b^3)*d)*sinh(d*x + c)^4 - 4*(a*b^2 - b^3)*d*cosh(d*x + c)^2 + 8*(7*(a*b^2 - b^3)*d*cosh(d*x + c)^5 - 10*(a
*b^2 - b^3)*d*cosh(d*x + c)^3 - (8*a^2*b - 11*a*b^2 + 3*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a*b^2 -
b^3)*d*cosh(d*x + c)^6 - 15*(a*b^2 - b^3)*d*cosh(d*x + c)^4 - 3*(8*a^2*b - 11*a*b^2 + 3*b^3)*d*cosh(d*x + c)^2
- (a*b^2 - b^3)*d)*sinh(d*x + c)^2 + (a*b^2 - b^3)*d + 8*((a*b^2 - b^3)*d*cosh(d*x + c)^7 - 3*(a*b^2 - b^3)*d
*cosh(d*x + c)^5 - (8*a^2*b - 11*a*b^2 + 3*b^3)*d*cosh(d*x + c)^3 - (a*b^2 - b^3)*d*cosh(d*x + c))*sinh(d*x +
c))*sqrt(-((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 1
5*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) - 4*a^2 + 15*a*b - 15*b^2)/((a^3*b^3 - 3*a^2*b^
4 + 3*a*b^5 - b^6)*d^2))*log(2*(4*a^5*b - 21*a^4*b^2 + 39*a^3*b^3 - 31*a^2*b^4 + 9*a*b^5)*d^2*sqrt((25*a^2 - 9
0*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) + (20*
a^2 - 81*a*b + 81*b^2)*cosh(d*x + c)^2 + 2*(20*a^2 - 81*a*b + 81*b^2)*cosh(d*x + c)*sinh(d*x + c) + (20*a^2 -
81*a*b + 81*b^2)*sinh(d*x + c)^2 - 20*a^2 + 81*a*b - 81*b^2 + 2*((a^5*b^3 - 6*a^4*b^4 + 12*a^3*b^5 - 10*a^2*b^
6 + 3*a*b^7)*d^3*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7
- 6*a^2*b^8 + a*b^9)*d^4)) - 2*(5*a^3*b - 19*a^2*b^2 + 18*a*b^3)*d)*sqrt(-((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^
6)*d^2*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^
8 + a*b^9)*d^4)) - 4*a^2 + 15*a*b - 15*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2))) + ((a*b^2 - b^3)*d*c
osh(d*x + c)^8 + 8*(a*b^2 - b^3)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a*b^2 - b^3)*d*sinh(d*x + c)^8 - 4*(a*b^2
- b^3)*d*cosh(d*x + c)^6 + 4*(7*(a*b^2 - b^3)*d*cosh(d*x + c)^2 - (a*b^2 - b^3)*d)*sinh(d*x + c)^6 - 2*(8*a^2*
b - 11*a*b^2 + 3*b^3)*d*cosh(d*x + c)^4 + 8*(7*(a*b^2 - b^3)*d*cosh(d*x + c)^3 - 3*(a*b^2 - b^3)*d*cosh(d*x +
c))*sinh(d*x + c)^5 + 2*(35*(a*b^2 - b^3)*d*cosh(d*x + c)^4 - 30*(a*b^2 - b^3)*d*cosh(d*x + c)^2 - (8*a^2*b -
11*a*b^2 + 3*b^3)*d)*sinh(d*x + c)^4 - 4*(a*b^2 - b^3)*d*cosh(d*x + c)^2 + 8*(7*(a*b^2 - b^3)*d*cosh(d*x + c)^
5 - 10*(a*b^2 - b^3)*d*cosh(d*x + c)^3 - (8*a^2*b - 11*a*b^2 + 3*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*
(a*b^2 - b^3)*d*cosh(d*x + c)^6 - 15*(a*b^2 - b^3)*d*cosh(d*x + c)^4 - 3*(8*a^2*b - 11*a*b^2 + 3*b^3)*d*cosh(d
*x + c)^2 - (a*b^2 - b^3)*d)*sinh(d*x + c)^2 + (a*b^2 - b^3)*d + 8*((a*b^2 - b^3)*d*cosh(d*x + c)^7 - 3*(a*b^2
- b^3)*d*cosh(d*x + c)^5 - (8*a^2*b - 11*a*b^2 + 3*b^3)*d*cosh(d*x + c)^3 - (a*b^2 - b^3)*d*cosh(d*x + c))*si
nh(d*x + c))*sqrt(-((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^
6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) - 4*a^2 + 15*a*b - 15*b^2)/((a^3*b^3 -
3*a^2*b^4 + 3*a*b^5 - b^6)*d^2))*log(2*(4*a^5*b - 21*a^4*b^2 + 39*a^3*b^3 - 31*a^2*b^4 + 9*a*b^5)*d^2*sqrt((2
5*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4
)) + (20*a^2 - 81*a*b + 81*b^2)*cosh(d*x + c)^2 + 2*(20*a^2 - 81*a*b + 81*b^2)*cosh(d*x + c)*sinh(d*x + c) + (
20*a^2 - 81*a*b + 81*b^2)*sinh(d*x + c)^2 - 20*a^2 + 81*a*b - 81*b^2 - 2*((a^5*b^3 - 6*a^4*b^4 + 12*a^3*b^5 -
10*a^2*b^6 + 3*a*b^7)*d^3*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15
*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) - 2*(5*a^3*b - 19*a^2*b^2 + 18*a*b^3)*d)*sqrt(-((a^3*b^3 - 3*a^2*b^4 + 3*a
*b^5 - b^6)*d^2*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 -
6*a^2*b^8 + a*b^9)*d^4)) - 4*a^2 + 15*a*b - 15*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2))) + ((a*b^2 -
b^3)*d*cosh(d*x + c)^8 + 8*(a*b^2 - b^3)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a*b^2 - b^3)*d*sinh(d*x + c)^8 -
4*(a*b^2 - b^3)*d*cosh(d*x + c)^6 + 4*(7*(a*b^2 - b^3)*d*cosh(d*x + c)^2 - (a*b^2 - b^3)*d)*sinh(d*x + c)^6 -
2*(8*a^2*b - 11*a*b^2 + 3*b^3)*d*cosh(d*x + c)^4 + 8*(7*(a*b^2 - b^3)*d*cosh(d*x + c)^3 - 3*(a*b^2 - b^3)*d*co
sh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a*b^2 - b...

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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)**2,x)

[Out]

Timed out

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Giac [A]
time = 1.10, size = 153, normalized size = 0.66 \begin {gather*} -\frac {2 \, a e^{\left (6 \, d x + 6 \, c\right )} - b e^{\left (6 \, d x + 6 \, c\right )} - 8 \, a e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b e^{\left (4 \, d x + 4 \, c\right )} - 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + b}{2 \, {\left (a b - b^{2}\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 )} 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)^2,x, algorithm="giac")

[Out]

-1/2*(2*a*e^(6*d*x + 6*c) - b*e^(6*d*x + 6*c) - 8*a*e^(4*d*x + 4*c) + 3*b*e^(4*d*x + 4*c) - 2*a*e^(2*d*x + 2*c
) - 3*b*e^(2*d*x + 2*c) + b)/((a*b - b^2)*(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)*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 )}^2} \,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)^2,x)

[Out]

int(sinh(c + d*x)^6/(a - b*sinh(c + d*x)^4)^2, x)

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