3.3.44 \(\int \frac {\sinh ^3(c+d x)}{(a-b \sinh ^4(c+d x))^2} \, dx\) [244]

Optimal. Leaf size=186 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 \sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} b^{3/4} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 \sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} b^{3/4} d}-\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )} \]

[Out]

-1/4*cosh(d*x+c)*(2-cosh(d*x+c)^2)/(a-b)/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))/b^(3/4)/d/a^(1/2)/(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))/b^(3/4)/d/a^(1/2)/(a^(1/2)+b^(1/2))^(3/2)

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Rubi [A]
time = 0.16, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3294, 1192, 1180, 211, 214} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 \sqrt {a} b^{3/4} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 \sqrt {a} b^{3/4} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 d (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^3/(a - b*Sinh[c + d*x]^4)^2,x]

[Out]

-1/8*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]]/(Sqrt[a]*(Sqrt[a] - Sqrt[b])^(3/2)*b^(3/4)*d) + A
rcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]]/(8*Sqrt[a]*(Sqrt[a] + Sqrt[b])^(3/2)*b^(3/4)*d) - (Cos
h[c + d*x]*(2 - Cosh[c + d*x]^2))/(4*(a - 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 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 ^3(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx &=-\frac {\text {Subst}\left (\int \frac {1-x^2}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 (a-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+2 a b x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{8 a (a-b) b d}\\ &=-\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 (a-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 {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{8 \sqrt {a} \left (\sqrt {a}-\sqrt {b}\right ) d}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{8 \sqrt {a} \left (\sqrt {a}+\sqrt {b}\right ) d}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 \sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} b^{3/4} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 \sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} b^{3/4} d}-\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 (a-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 = 0.35, size = 422, normalized size = 2.27 \begin {gather*} -\frac {\frac {16 (-5 \cosh (c+d x)+\cosh (3 (c+d x)))}{-8 a+3 b-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x))}+\text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-c-d x-2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+7 c \text {$\#$1}^2+7 d x \text {$\#$1}^2+14 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-7 c \text {$\#$1}^4-7 d x \text {$\#$1}^4-14 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+c \text {$\#$1}^6+d x \text {$\#$1}^6+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{32 (a-b) d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^3/(a - b*Sinh[c + d*x]^4)^2,x]

[Out]

-1/32*((16*(-5*Cosh[c + d*x] + Cosh[3*(c + d*x)]))/(-8*a + 3*b - 4*b*Cosh[2*(c + d*x)] + b*Cosh[4*(c + d*x)])
+ RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-c - d*x - 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] + 7*c*#1^2 + 7*d*x*#1^2 + 14*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 - 7*c*#1^4 - 7*d*x*#1^4 - 14*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 + c*#1^6 + d*x*#1^
6 + 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 - b)*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(321\) vs. \(2(144)=288\).
time = 4.10, size = 322, normalized size = 1.73

method	result	size
derivativedivides	\(\frac {\frac {-\frac {\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a -b \right )}-\frac {\left (3 a -8 b \right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a \left (a -b \right )}+\frac {5 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right )}-\frac {1}{2 \left (a -b \right )}}{a \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {a \left (-\frac {\left (\sqrt {a b}-b \right ) \arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{4 a b \sqrt {-\sqrt {a b}\, a -a b}}+\frac {\left (-\sqrt {a b}-b \right ) \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{4 a b \sqrt {\sqrt {a b}\, a -a b}}\right )}{2 a -2 b}}{d}\)	\(322\)
default	\(\frac {\frac {-\frac {\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a -b \right )}-\frac {\left (3 a -8 b \right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a \left (a -b \right )}+\frac {5 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right )}-\frac {1}{2 \left (a -b \right )}}{a \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {a \left (-\frac {\left (\sqrt {a b}-b \right ) \arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{4 a b \sqrt {-\sqrt {a b}\, a -a b}}+\frac {\left (-\sqrt {a b}-b \right ) \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{4 a b \sqrt {\sqrt {a b}\, a -a b}}\right )}{2 a -2 b}}{d}\)	\(322\)
risch	\(\frac {{\mathrm e}^{d x +c} \left ({\mathrm e}^{6 d x +6 c}-5 \,{\mathrm e}^{4 d x +4 c}-5 \,{\mathrm e}^{2 d x +2 c}+1\right )}{2 \left (a -b \right ) d \left (-b \,{\mathrm e}^{8 d x +8 c}+4 b \,{\mathrm e}^{6 d x +6 c}+16 a \,{\mathrm e}^{4 d x +4 c}-6 b \,{\mathrm e}^{4 d x +4 c}+4 b \,{\mathrm e}^{2 d x +2 c}-b \right )}+\left (\munderset {\textit {\_R} =\RootOf \left (-1+\left (65536 a^{5} b^{3} d^{4}-196608 a^{4} b^{4} d^{4}+196608 a^{3} b^{5} d^{4}-65536 a^{2} b^{6} d^{4}\right ) \textit {\_Z}^{4}+\left (1536 a^{2} b^{2} d^{2}+512 a \,b^{3} d^{2}\right ) \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\left (\frac {8192 a^{5} b^{2} d^{3}}{a +3 b}-\frac {16384 a^{4} b^{3} d^{3}}{a +3 b}+\frac {16384 a^{2} b^{5} d^{3}}{a +3 b}-\frac {8192 a \,b^{6} d^{3}}{a +3 b}\right ) \textit {\_R}^{3}+\left (\frac {160 a^{2} b d}{a +3 b}+\frac {320 a \,b^{2} d}{a +3 b}+\frac {32 b^{3} d}{a +3 b}\right ) \textit {\_R} \right ) {\mathrm e}^{d x +c}+1\right )\right )\)	\(340\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^3/(a-b*sinh(d*x+c)^4)^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(8*(-1/16/(a-b)*tanh(1/2*d*x+1/2*c)^6-1/16*(3*a-8*b)/a/(a-b)*tanh(1/2*d*x+1/2*c)^4+5/16/(a-b)*tanh(1/2*d*x
+1/2*c)^2-1/16/(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)+1/2/(a-b)*a*(-1/4*((a*b)^(1/2)-b)/a/b/(-(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)-b)/a/b
/((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)^3/(a-b*sinh(d*x+c)^4)^2,x, algorithm="maxima")

[Out]

-1/2*(e^(7*d*x + 7*c) - 5*e^(5*d*x + 5*c) - 5*e^(3*d*x + 3*c) + e^(d*x + c))/(a*b*d - b^2*d + (a*b*d*e^(8*c) -
b^2*d*e^(8*c))*e^(8*d*x) - 4*(a*b*d*e^(6*c) - b^2*d*e^(6*c))*e^(6*d*x) - 2*(8*a^2*d*e^(4*c) - 11*a*b*d*e^(4*c
) + 3*b^2*d*e^(4*c))*e^(4*d*x) - 4*(a*b*d*e^(2*c) - b^2*d*e^(2*c))*e^(2*d*x)) - 1/8*integrate(4*(e^(7*d*x + 7*
c) - 7*e^(5*d*x + 5*c) + 7*e^(3*d*x + 3*c) - e^(d*x + c))/(a*b - b^2 + (a*b*e^(8*c) - b^2*e^(8*c))*e^(8*d*x) -
4*(a*b*e^(6*c) - b^2*e^(6*c))*e^(6*d*x) - 2*(8*a^2*e^(4*c) - 11*a*b*e^(4*c) + 3*b^2*e^(4*c))*e^(4*d*x) - 4*(a
*b*e^(2*c) - b^2*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. 5238 vs. \(2 (141) = 282\).
time = 0.49, size = 5238, normalized size = 28.16 \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)^3/(a-b*sinh(d*x+c)^4)^2,x, algorithm="fricas")

[Out]

-1/16*(8*cosh(d*x + c)^7 + 56*cosh(d*x + c)*sinh(d*x + c)^6 + 8*sinh(d*x + c)^7 + 8*(21*cosh(d*x + c)^2 - 5)*s
inh(d*x + c)^5 - 40*cosh(d*x + c)^5 + 40*(7*cosh(d*x + c)^3 - 5*cosh(d*x + c))*sinh(d*x + c)^4 + 40*(7*cosh(d*
x + c)^4 - 10*cosh(d*x + c)^2 - 1)*sinh(d*x + c)^3 - 40*cosh(d*x + c)^3 + 8*(21*cosh(d*x + c)^5 - 50*cosh(d*x
+ c)^3 - 15*cosh(d*x + c))*sinh(d*x + c)^2 - ((a*b - b^2)*d*cosh(d*x + c)^8 + 8*(a*b - b^2)*d*cosh(d*x + c)*si
nh(d*x + c)^7 + (a*b - b^2)*d*sinh(d*x + c)^8 - 4*(a*b - b^2)*d*cosh(d*x + c)^6 + 4*(7*(a*b - b^2)*d*cosh(d*x
+ c)^2 - (a*b - b^2)*d)*sinh(d*x + c)^6 - 2*(8*a^2 - 11*a*b + 3*b^2)*d*cosh(d*x + c)^4 + 8*(7*(a*b - b^2)*d*co
sh(d*x + c)^3 - 3*(a*b - b^2)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a*b - b^2)*d*cosh(d*x + c)^4 - 30*(a*b
- b^2)*d*cosh(d*x + c)^2 - (8*a^2 - 11*a*b + 3*b^2)*d)*sinh(d*x + c)^4 - 4*(a*b - b^2)*d*cosh(d*x + c)^2 + 8*
(7*(a*b - b^2)*d*cosh(d*x + c)^5 - 10*(a*b - b^2)*d*cosh(d*x + c)^3 - (8*a^2 - 11*a*b + 3*b^2)*d*cosh(d*x + c)
)*sinh(d*x + c)^3 + 4*(7*(a*b - b^2)*d*cosh(d*x + c)^6 - 15*(a*b - b^2)*d*cosh(d*x + c)^4 - 3*(8*a^2 - 11*a*b
+ 3*b^2)*d*cosh(d*x + c)^2 - (a*b - b^2)*d)*sinh(d*x + c)^2 + (a*b - b^2)*d + 8*((a*b - b^2)*d*cosh(d*x + c)^7
- 3*(a*b - b^2)*d*cosh(d*x + c)^5 - (8*a^2 - 11*a*b + 3*b^2)*d*cosh(d*x + c)^3 - (a*b - b^2)*d*cosh(d*x + c))
*sinh(d*x + c))*sqrt(-((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((a^2 + 6*a*b + 9*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)) + 3*a + b)/((a^4*b - 3*a^3*b^2 + 3*a^2
*b^3 - a*b^4)*d^2))*log((a + 3*b)*cosh(d*x + c)^2 + 2*(a + 3*b)*cosh(d*x + c)*sinh(d*x + c) + (a + 3*b)*sinh(d
*x + c)^2 + 2*(2*(a^2*b + 3*a*b^2)*d*cosh(d*x + c) + 2*(a^2*b + 3*a*b^2)*d*sinh(d*x + c) - ((a^5*b^2 - 2*a^4*b
^3 + 2*a^2*b^5 - a*b^6)*d^3*cosh(d*x + c) + (a^5*b^2 - 2*a^4*b^3 + 2*a^2*b^5 - a*b^6)*d^3*sinh(d*x + c))*sqrt(
(a^2 + 6*a*b + 9*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)))
*sqrt(-((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((a^2 + 6*a*b + 9*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)) + 3*a + b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d
^2)) + a + 3*b) + ((a*b - b^2)*d*cosh(d*x + c)^8 + 8*(a*b - b^2)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a*b - b^2)
*d*sinh(d*x + c)^8 - 4*(a*b - b^2)*d*cosh(d*x + c)^6 + 4*(7*(a*b - b^2)*d*cosh(d*x + c)^2 - (a*b - b^2)*d)*sin
h(d*x + c)^6 - 2*(8*a^2 - 11*a*b + 3*b^2)*d*cosh(d*x + c)^4 + 8*(7*(a*b - b^2)*d*cosh(d*x + c)^3 - 3*(a*b - b^
2)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a*b - b^2)*d*cosh(d*x + c)^4 - 30*(a*b - b^2)*d*cosh(d*x + c)^2 -
(8*a^2 - 11*a*b + 3*b^2)*d)*sinh(d*x + c)^4 - 4*(a*b - b^2)*d*cosh(d*x + c)^2 + 8*(7*(a*b - b^2)*d*cosh(d*x +
c)^5 - 10*(a*b - b^2)*d*cosh(d*x + c)^3 - (8*a^2 - 11*a*b + 3*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a
*b - b^2)*d*cosh(d*x + c)^6 - 15*(a*b - b^2)*d*cosh(d*x + c)^4 - 3*(8*a^2 - 11*a*b + 3*b^2)*d*cosh(d*x + c)^2
- (a*b - b^2)*d)*sinh(d*x + c)^2 + (a*b - b^2)*d + 8*((a*b - b^2)*d*cosh(d*x + c)^7 - 3*(a*b - b^2)*d*cosh(d*x
+ c)^5 - (8*a^2 - 11*a*b + 3*b^2)*d*cosh(d*x + c)^3 - (a*b - b^2)*d*cosh(d*x + c))*sinh(d*x + c))*sqrt(-((a^4
*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((a^2 + 6*a*b + 9*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)) + 3*a + b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2))*log((a
+ 3*b)*cosh(d*x + c)^2 + 2*(a + 3*b)*cosh(d*x + c)*sinh(d*x + c) + (a + 3*b)*sinh(d*x + c)^2 - 2*(2*(a^2*b + 3
*a*b^2)*d*cosh(d*x + c) + 2*(a^2*b + 3*a*b^2)*d*sinh(d*x + c) - ((a^5*b^2 - 2*a^4*b^3 + 2*a^2*b^5 - a*b^6)*d^3
*cosh(d*x + c) + (a^5*b^2 - 2*a^4*b^3 + 2*a^2*b^5 - a*b^6)*d^3*sinh(d*x + c))*sqrt((a^2 + 6*a*b + 9*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)))*sqrt(-((a^4*b - 3*a^3*b^2
+ 3*a^2*b^3 - a*b^4)*d^2*sqrt((a^2 + 6*a*b + 9*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)) + 3*a + b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2)) + a + 3*b) - ((a*b - b
^2)*d*cosh(d*x + c)^8 + 8*(a*b - b^2)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a*b - b^2)*d*sinh(d*x + c)^8 - 4*(a*b
- b^2)*d*cosh(d*x + c)^6 + 4*(7*(a*b - b^2)*d*cosh(d*x + c)^2 - (a*b - b^2)*d)*sinh(d*x + c)^6 - 2*(8*a^2 - 1
1*a*b + 3*b^2)*d*cosh(d*x + c)^4 + 8*(7*(a*b - b^2)*d*cosh(d*x + c)^3 - 3*(a*b - b^2)*d*cosh(d*x + c))*sinh(d*
x + c)^5 + 2*(35*(a*b - b^2)*d*cosh(d*x + c)^4 - 30*(a*b - b^2)*d*cosh(d*x + c)^2 - (8*a^2 - 11*a*b + 3*b^2)*d
)*sinh(d*x + c)^4 - 4*(a*b - b^2)*d*cosh(d*x + c)^2 + 8*(7*(a*b - b^2)*d*cosh(d*x + c)^5 - 10*(a*b - b^2)*d*co
sh(d*x + c)^3 - (8*a^2 - 11*a*b + 3*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a*b - b^2)*d*cosh(d*x + c)^6
- 15*(a*b - b^2)*d*cosh(d*x + c)^4 - 3*(8*a^2 - 11*a*b + 3*b^2)*d*cosh(d*x + c)^2 - (a*b - b^2)*d)*sinh(d*x +
c)^2 + (a*b - b^2)*d + 8*((a*b - b^2)*d*cosh(d*x + c)^7 - 3*(a*b - b^2)*d*cosh(d*x + c)^5 - (8*a^2 - 11*a*b +
3*b^2)*d*cosh(d*x + c)^3 - (a*b - b^2)*d*cosh(...

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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)**3/(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. 861 vs. \(2 (141) = 282\).
time = 0.63, size = 861, normalized size = 4.63 \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\right )}^{2} {\left | b \right |} - 2 \, {\left (4 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{3} b + \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b^{2} - 5 \, \sqrt {-b^{2} + \sqrt {a b} b} a b^{3}\right )} {\left | -a + b \right |} {\left | b \right |} + {\left (4 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{3} b - 3 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b^{2} - 6 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b^{3} + 5 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} b^{4}\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 - b^{2} + \sqrt {{\left (a^{2} - 2 \, a b + b^{2}\right )} {\left (a b - b^{2}\right )} + {\left (a b - b^{2}\right )}^{2}}}{a b - b^{2}}}}\right )}{{\left (4 \, a^{5} b^{3} - 7 \, a^{4} b^{4} - 3 \, a^{3} b^{5} + 11 \, a^{2} b^{6} - 5 \, a b^{7}\right )} {\left | -a + b \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\right )}^{2} {\left | b \right |} + 2 \, {\left (4 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{3} b + \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b^{2} - 5 \, \sqrt {-b^{2} - \sqrt {a b} b} a b^{3}\right )} {\left | -a + b \right |} {\left | b \right |} + {\left (4 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{3} b - 3 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b^{2} - 6 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a b^{3} + 5 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} b^{4}\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 - b^{2} - \sqrt {{\left (a^{2} - 2 \, a b + b^{2}\right )} {\left (a b - b^{2}\right )} + {\left (a b - b^{2}\right )}^{2}}}{a b - b^{2}}}}\right )}{{\left (4 \, a^{5} b^{3} - 7 \, a^{4} b^{4} - 3 \, a^{3} b^{5} + 11 \, a^{2} b^{6} - 5 \, a b^{7}\right )} {\left | -a + b \right |}} - \frac {4 \, {\left ({\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 8 \, e^{\left (d x + c\right )} - 8 \, e^{\left (-d x - c\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\right )}}}{8 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^3/(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*abs(b) -
2*(4*sqrt(-b^2 + sqrt(a*b)*b)*a^3*b + sqrt(-b^2 + sqrt(a*b)*b)*a^2*b^2 - 5*sqrt(-b^2 + sqrt(a*b)*b)*a*b^3)*ab
s(-a + b)*abs(b) + (4*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^3*b - 3*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^2*b^2
- 6*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a*b^3 + 5*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*b^4)*abs(b))*arctan(1/2*(e
^(d*x + c) + e^(-d*x - c))/sqrt(-(a*b - b^2 + sqrt((a^2 - 2*a*b + b^2)*(a*b - b^2) + (a*b - b^2)^2))/(a*b - b^
2)))/((4*a^5*b^3 - 7*a^4*b^4 - 3*a^3*b^5 + 11*a^2*b^6 - 5*a*b^7)*abs(-a + b)) - ((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*abs(b) + 2*(4*sqrt(-b^2 - sqrt(a*b)*b)*a^3*
b + sqrt(-b^2 - sqrt(a*b)*b)*a^2*b^2 - 5*sqrt(-b^2 - sqrt(a*b)*b)*a*b^3)*abs(-a + b)*abs(b) + (4*sqrt(a*b)*sqr
t(-b^2 - sqrt(a*b)*b)*a^3*b - 3*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^2*b^2 - 6*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)
*b)*a*b^3 + 5*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*b^4)*abs(b))*arctan(1/2*(e^(d*x + c) + e^(-d*x - c))/sqrt(-(a
*b - b^2 - sqrt((a^2 - 2*a*b + b^2)*(a*b - b^2) + (a*b - b^2)^2))/(a*b - b^2)))/((4*a^5*b^3 - 7*a^4*b^4 - 3*a^
3*b^5 + 11*a^2*b^6 - 5*a*b^7)*abs(-a + b)) - 4*((e^(d*x + c) + e^(-d*x - c))^3 - 8*e^(d*x + c) - 8*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)))/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^3}{{\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)^3/(a - b*sinh(c + d*x)^4)^2,x)

[Out]

int(sinh(c + d*x)^3/(a - b*sinh(c + d*x)^4)^2, x)

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