3.3.34 \(\int \frac {\text {csch}^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx\) [234]

Optimal. Leaf size=184 \[ \frac {b^{3/4} \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^{3/2} \sqrt {\sqrt {a}-\sqrt {b}} d}+\frac {\tanh ^{-1}(\cosh (c+d x))}{2 a d}+\frac {b^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^{3/2} \sqrt {\sqrt {a}+\sqrt {b}} d}+\frac {1}{4 a d (1-\cosh (c+d x))}-\frac {1}{4 a d (1+\cosh (c+d x))} \]

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3294, 1184, 213, 1107, 211, 214} \begin {gather*} \frac {b^{3/4} \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^{3/2} d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {b^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^{3/2} d \sqrt {\sqrt {a}+\sqrt {b}}}+\frac {1}{4 a d (1-\cosh (c+d x))}-\frac {1}{4 a d (\cosh (c+d x)+1)}+\frac {\tanh ^{-1}(\cosh (c+d x))}{2 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

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 1107

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/
2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && PosQ[b^2 - 4*a*c]

Rule 1184

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x
^2)^q/(a + b*x^2 + c*x^4), 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] && IntegerQ[q]

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

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.28, size = 265, normalized size = 1.44 \begin {gather*} -\frac {\text {csch}^2\left (\frac {1}{2} (c+d x)\right )+4 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+4 b \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-c \text {$\#$1}-d x \text {$\#$1}-2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}+c \text {$\#$1}^3+d x \text {$\#$1}^3+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^3}{-b-8 a \text {$\#$1}^2+3 b \text {$\#$1}^2-3 b \text {$\#$1}^4+b \text {$\#$1}^6}\&\right ]+\text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(Csch[(c + d*x)/2]^2 + 4*Log[Tanh[(c + d*x)/2]] + 4*b*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#
1^6 + b*#1^8 & , (-(c*#1) - d*x*#1 - 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]*#1 + c*#1^3 + d*x*#1^3 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#
1 - Sinh[(c + d*x)/2]*#1]*#1^3)/(-b - 8*a*#1^2 + 3*b*#1^2 - 3*b*#1^4 + b*#1^6) & ] + Sech[(c + d*x)/2]^2)/(a*d
)

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Maple [A]
time = 2.16, size = 181, normalized size = 0.98

method	result	size
derivativedivides	\(\frac {\frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}+2 b \left (\frac {\arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {\sqrt {a b}\, a -a b}}-\frac {\arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {-\sqrt {a b}\, a -a b}}\right )-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{d}\)	\(181\)
default	\(\frac {\frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}+2 b \left (\frac {\arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {\sqrt {a b}\, a -a b}}-\frac {\arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {-\sqrt {a b}\, a -a b}}\right )-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{d}\)	\(181\)
risch	\(-\frac {{\mathrm e}^{d x +c} \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d a}+\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d a}+8 \left (\munderset {\textit {\_R} =\RootOf \left (\left (1048576 a^{7} d^{4}-1048576 a^{6} b \,d^{4}\right ) \textit {\_Z}^{4}+2048 a^{3} b^{2} d^{2} \textit {\_Z}^{2}-b^{3}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\left (\frac {65536 d^{3} a^{5}}{b^{2}}-\frac {65536 d^{3} a^{4}}{b}\right ) \textit {\_R}^{3}+\left (\frac {64 a^{2} d}{b}+64 a d \right ) \textit {\_R} \right ) {\mathrm e}^{d x +c}+1\right )\right )\)	\(187\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/8*tanh(1/2*d*x+1/2*c)^2/a+2*b*(1/4/a/((a*b)^(1/2)*a-a*b)^(1/2)*arctan(1/4*(2*a*tanh(1/2*d*x+1/2*c)^2+4*
(a*b)^(1/2)-2*a)/((a*b)^(1/2)*a-a*b)^(1/2))-1/4/a/(-(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)))-1/8/a/tanh(1/2*d*x+1/2*c)^2-1/2/a*ln(tanh(1/2*d*x+1/2*c)
))

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

[Out]

-(e^(3*d*x + 3*c) + e^(d*x + c))/(a*d*e^(4*d*x + 4*c) - 2*a*d*e^(2*d*x + 2*c) + a*d) + 1/2*log((e^(d*x + c) +
1)*e^(-c))/(a*d) - 1/2*log((e^(d*x + c) - 1)*e^(-c))/(a*d) - 8*integrate((b*e^(5*d*x + 5*c) - b*e^(3*d*x + 3*c
))/(a*b*e^(8*d*x + 8*c) - 4*a*b*e^(6*d*x + 6*c) - 4*a*b*e^(2*d*x + 2*c) + a*b - 2*(8*a^2*e^(4*c) - 3*a*b*e^(4*
c))*e^(4*d*x)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1954 vs. \(2 (136) = 272\).
time = 0.44, size = 1954, normalized size = 10.62 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*cosh(d*x + c)^3 + 12*cosh(d*x + c)*sinh(d*x + c)^2 + 4*sinh(d*x + c)^3 - (a*d*cosh(d*x + c)^4 + 4*a*d*
cosh(d*x + c)*sinh(d*x + c)^3 + a*d*sinh(d*x + c)^4 - 2*a*d*cosh(d*x + c)^2 + 2*(3*a*d*cosh(d*x + c)^2 - a*d)*
sinh(d*x + c)^2 + a*d + 4*(a*d*cosh(d*x + c)^3 - a*d*cosh(d*x + c))*sinh(d*x + c))*sqrt(-((a^4 - a^3*b)*d^2*sq
rt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2))*log(b^2*cosh(d*x + c)^2 + 2*b^2*cosh(d*x +
c)*sinh(d*x + c) + b^2*sinh(d*x + c)^2 + b^2 + 2*(a^2*b*d*cosh(d*x + c) + a^2*b*d*sinh(d*x + c) - ((a^5 - a^4
*b)*d^3*cosh(d*x + c) + (a^5 - a^4*b)*d^3*sinh(d*x + c))*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)))*sqrt(-((a^
4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2))) + (a*d*cosh(d*x + c)^4 +
4*a*d*cosh(d*x + c)*sinh(d*x + c)^3 + a*d*sinh(d*x + c)^4 - 2*a*d*cosh(d*x + c)^2 + 2*(3*a*d*cosh(d*x + c)^2
- a*d)*sinh(d*x + c)^2 + a*d + 4*(a*d*cosh(d*x + c)^3 - a*d*cosh(d*x + c))*sinh(d*x + c))*sqrt(-((a^4 - a^3*b)
*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2))*log(b^2*cosh(d*x + c)^2 + 2*b^2*cos
h(d*x + c)*sinh(d*x + c) + b^2*sinh(d*x + c)^2 + b^2 - 2*(a^2*b*d*cosh(d*x + c) + a^2*b*d*sinh(d*x + c) - ((a^
5 - a^4*b)*d^3*cosh(d*x + c) + (a^5 - a^4*b)*d^3*sinh(d*x + c))*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)))*sqr
t(-((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2))) - (a*d*cosh(d*x +
c)^4 + 4*a*d*cosh(d*x + c)*sinh(d*x + c)^3 + a*d*sinh(d*x + c)^4 - 2*a*d*cosh(d*x + c)^2 + 2*(3*a*d*cosh(d*x
+ c)^2 - a*d)*sinh(d*x + c)^2 + a*d + 4*(a*d*cosh(d*x + c)^3 - a*d*cosh(d*x + c))*sinh(d*x + c))*sqrt(((a^4 -
a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2))*log(b^2*cosh(d*x + c)^2 + 2*b
^2*cosh(d*x + c)*sinh(d*x + c) + b^2*sinh(d*x + c)^2 + b^2 + 2*(a^2*b*d*cosh(d*x + c) + a^2*b*d*sinh(d*x + c)
+ ((a^5 - a^4*b)*d^3*cosh(d*x + c) + (a^5 - a^4*b)*d^3*sinh(d*x + c))*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)
))*sqrt(((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2))) + (a*d*cosh(
d*x + c)^4 + 4*a*d*cosh(d*x + c)*sinh(d*x + c)^3 + a*d*sinh(d*x + c)^4 - 2*a*d*cosh(d*x + c)^2 + 2*(3*a*d*cosh
(d*x + c)^2 - a*d)*sinh(d*x + c)^2 + a*d + 4*(a*d*cosh(d*x + c)^3 - a*d*cosh(d*x + c))*sinh(d*x + c))*sqrt(((a
^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2))*log(b^2*cosh(d*x + c)^2
+ 2*b^2*cosh(d*x + c)*sinh(d*x + c) + b^2*sinh(d*x + c)^2 + b^2 - 2*(a^2*b*d*cosh(d*x + c) + a^2*b*d*sinh(d*x
+ c) + ((a^5 - a^4*b)*d^3*cosh(d*x + c) + (a^5 - a^4*b)*d^3*sinh(d*x + c))*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)
*d^4)))*sqrt(((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2))) - 2*(co
sh(d*x + c)^4 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4 + 2*(3*cosh(d*x + c)^2 - 1)*sinh(d*x + c)^2
- 2*cosh(d*x + c)^2 + 4*(cosh(d*x + c)^3 - cosh(d*x + c))*sinh(d*x + c) + 1)*log(cosh(d*x + c) + sinh(d*x + c)
+ 1) + 2*(cosh(d*x + c)^4 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4 + 2*(3*cosh(d*x + c)^2 - 1)*sin
h(d*x + c)^2 - 2*cosh(d*x + c)^2 + 4*(cosh(d*x + c)^3 - cosh(d*x + c))*sinh(d*x + c) + 1)*log(cosh(d*x + c) +
sinh(d*x + c) - 1) + 4*(3*cosh(d*x + c)^2 + 1)*sinh(d*x + c) + 4*cosh(d*x + c))/(a*d*cosh(d*x + c)^4 + 4*a*d*c
osh(d*x + c)*sinh(d*x + c)^3 + a*d*sinh(d*x + c)^4 - 2*a*d*cosh(d*x + c)^2 + 2*(3*a*d*cosh(d*x + c)^2 - a*d)*s
inh(d*x + c)^2 + a*d + 4*(a*d*cosh(d*x + c)^3 - a*d*cosh(d*x + c))*sinh(d*x + c))

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 467 vs. \(2 (136) = 272\).
time = 0.51, size = 467, normalized size = 2.54 \begin {gather*} \frac {\frac {2 \, {\left ({\left (4 \, \sqrt {-b^{2} - \sqrt {a b} b} a b + 5 \, \sqrt {-b^{2} - \sqrt {a b} b} b^{2}\right )} {\left | a \right |} {\left | b \right |} - {\left (4 \, \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 |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a b + \sqrt {a^{2} b^{2} + {\left (a^{2} - a b\right )} a b}}{a b}}}\right )}{{\left (4 \, a^{3} b^{2} + a^{2} b^{3} - 5 \, a b^{4}\right )} {\left | a \right |}} + \frac {2 \, {\left ({\left (4 \, \sqrt {-b^{2} + \sqrt {a b} b} a b + 5 \, \sqrt {-b^{2} + \sqrt {a b} b} b^{2}\right )} {\left | a \right |} {\left | b \right |} + {\left (4 \, \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 |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a b - \sqrt {a^{2} b^{2} + {\left (a^{2} - a b\right )} a b}}{a b}}}\right )}{{\left (4 \, a^{3} b^{2} + a^{2} b^{3} - 5 \, a b^{4}\right )} {\left | a \right |}} + \frac {\log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right )}{a} - \frac {\log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right )}{a} - \frac {4 \, {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4\right )} a}}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*((4*sqrt(-b^2 - sqrt(a*b)*b)*a*b + 5*sqrt(-b^2 - sqrt(a*b)*b)*b^2)*abs(a)*abs(b) - (4*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))*arctan(1/2*(e^(d*x + c) + e^(-d*x -
c))/sqrt(-(a*b + sqrt(a^2*b^2 + (a^2 - a*b)*a*b))/(a*b)))/((4*a^3*b^2 + a^2*b^3 - 5*a*b^4)*abs(a)) + 2*((4*sqr
t(-b^2 + sqrt(a*b)*b)*a*b + 5*sqrt(-b^2 + sqrt(a*b)*b)*b^2)*abs(a)*abs(b) + (4*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))*arctan(1/2*(e^(d*x + c) + e^(-d*x - c))/sqrt(-(a*b
- sqrt(a^2*b^2 + (a^2 - a*b)*a*b))/(a*b)))/((4*a^3*b^2 + a^2*b^3 - 5*a*b^4)*abs(a)) + log(e^(d*x + c) + e^(-d
*x - c) + 2)/a - log(e^(d*x + c) + e^(-d*x - c) - 2)/a - 4*(e^(d*x + c) + e^(-d*x - c))/(((e^(d*x + c) + e^(-d
*x - c))^2 - 4)*a))/d

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Mupad [B]
time = 12.21, size = 1517, normalized size = 8.24 \begin {gather*} \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (256\,a^6\,\sqrt {-a^2\,d^2}+b^6\,\sqrt {-a^2\,d^2}+96\,a^2\,b^4\,\sqrt {-a^2\,d^2}-288\,a^3\,b^3\,\sqrt {-a^2\,d^2}+512\,a^4\,b^2\,\sqrt {-a^2\,d^2}-16\,a\,b^5\,\sqrt {-a^2\,d^2}-512\,a^5\,b\,\sqrt {-a^2\,d^2}\right )}{256\,d\,a^7-512\,d\,a^6\,b+512\,d\,a^5\,b^2-288\,d\,a^4\,b^3+96\,d\,a^3\,b^4-16\,d\,a^2\,b^5+d\,a\,b^6}\right )}{\sqrt {-a^2\,d^2}}-\ln \left (\frac {\left (\frac {\left (\frac {8589934592\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (8\,a^2-7\,a\,b+3\,b^2\right )\,\sqrt {-\frac {\sqrt {a^7\,b^3}+a^3\,b^2}{a^6\,d^2\,\left (a-b\right )}}}{b^5\,{\left (a-b\right )}^2}-\frac {4294967296\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (12\,a^3-7\,a^2\,b+2\,a\,b^2+b^3\right )}{a^2\,b^4\,{\left (a-b\right )}^3}\right )\,\sqrt {-\frac {\sqrt {a^7\,b^3}+a^3\,b^2}{a^6\,d^2\,\left (a-b\right )}}}{4}-\frac {4294967296\,d\,{\mathrm {e}}^{c+d\,x}\,\left (2\,a^2-2\,a\,b+b^2\right )}{a^3\,b^4\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {\sqrt {a^7\,b^3}+a^3\,b^2}{a^6\,d^2\,\left (a-b\right )}}}{4}+\frac {268435456\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (4\,a^3-a\,b^2+b^3\right )}{a^5\,b^3\,{\left (a-b\right )}^3}\right )\,\sqrt {-\frac {\sqrt {a^7\,b^3}+a^3\,b^2}{16\,\left (a^7\,d^2-a^6\,b\,d^2\right )}}+\ln \left (\frac {268435456\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (4\,a^3-a\,b^2+b^3\right )}{a^5\,b^3\,{\left (a-b\right )}^3}-\frac {\left (\frac {\left (\frac {8589934592\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (8\,a^2-7\,a\,b+3\,b^2\right )\,\sqrt {-\frac {\sqrt {a^7\,b^3}+a^3\,b^2}{a^6\,d^2\,\left (a-b\right )}}}{b^5\,{\left (a-b\right )}^2}+\frac {4294967296\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (12\,a^3-7\,a^2\,b+2\,a\,b^2+b^3\right )}{a^2\,b^4\,{\left (a-b\right )}^3}\right )\,\sqrt {-\frac {\sqrt {a^7\,b^3}+a^3\,b^2}{a^6\,d^2\,\left (a-b\right )}}}{4}-\frac {4294967296\,d\,{\mathrm {e}}^{c+d\,x}\,\left (2\,a^2-2\,a\,b+b^2\right )}{a^3\,b^4\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {\sqrt {a^7\,b^3}+a^3\,b^2}{a^6\,d^2\,\left (a-b\right )}}}{4}\right )\,\sqrt {-\frac {\sqrt {a^7\,b^3}+a^3\,b^2}{16\,\left (a^7\,d^2-a^6\,b\,d^2\right )}}-\ln \left (\frac {\left (\frac {\left (\frac {8589934592\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (8\,a^2-7\,a\,b+3\,b^2\right )\,\sqrt {\frac {\sqrt {a^7\,b^3}-a^3\,b^2}{a^6\,d^2\,\left (a-b\right )}}}{b^5\,{\left (a-b\right )}^2}-\frac {4294967296\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (12\,a^3-7\,a^2\,b+2\,a\,b^2+b^3\right )}{a^2\,b^4\,{\left (a-b\right )}^3}\right )\,\sqrt {\frac {\sqrt {a^7\,b^3}-a^3\,b^2}{a^6\,d^2\,\left (a-b\right )}}}{4}-\frac {4294967296\,d\,{\mathrm {e}}^{c+d\,x}\,\left (2\,a^2-2\,a\,b+b^2\right )}{a^3\,b^4\,{\left (a-b\right )}^2}\right )\,\sqrt {\frac {\sqrt {a^7\,b^3}-a^3\,b^2}{a^6\,d^2\,\left (a-b\right )}}}{4}+\frac {268435456\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (4\,a^3-a\,b^2+b^3\right )}{a^5\,b^3\,{\left (a-b\right )}^3}\right )\,\sqrt {\frac {\sqrt {a^7\,b^3}-a^3\,b^2}{16\,\left (a^7\,d^2-a^6\,b\,d^2\right )}}+\ln \left (\frac {268435456\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (4\,a^3-a\,b^2+b^3\right )}{a^5\,b^3\,{\left (a-b\right )}^3}-\frac {\left (\frac {\left (\frac {8589934592\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (8\,a^2-7\,a\,b+3\,b^2\right )\,\sqrt {\frac {\sqrt {a^7\,b^3}-a^3\,b^2}{a^6\,d^2\,\left (a-b\right )}}}{b^5\,{\left (a-b\right )}^2}+\frac {4294967296\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (12\,a^3-7\,a^2\,b+2\,a\,b^2+b^3\right )}{a^2\,b^4\,{\left (a-b\right )}^3}\right )\,\sqrt {\frac {\sqrt {a^7\,b^3}-a^3\,b^2}{a^6\,d^2\,\left (a-b\right )}}}{4}-\frac {4294967296\,d\,{\mathrm {e}}^{c+d\,x}\,\left (2\,a^2-2\,a\,b+b^2\right )}{a^3\,b^4\,{\left (a-b\right )}^2}\right )\,\sqrt {\frac {\sqrt {a^7\,b^3}-a^3\,b^2}{a^6\,d^2\,\left (a-b\right )}}}{4}\right )\,\sqrt {\frac {\sqrt {a^7\,b^3}-a^3\,b^2}{16\,\left (a^7\,d^2-a^6\,b\,d^2\right )}}-\frac {{\mathrm {e}}^{c+d\,x}}{a\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}}{a\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)^3*(a - b*sinh(c + d*x)^4)),x)

[Out]

atan((exp(d*x)*exp(c)*(256*a^6*(-a^2*d^2)^(1/2) + b^6*(-a^2*d^2)^(1/2) + 96*a^2*b^4*(-a^2*d^2)^(1/2) - 288*a^3
*b^3*(-a^2*d^2)^(1/2) + 512*a^4*b^2*(-a^2*d^2)^(1/2) - 16*a*b^5*(-a^2*d^2)^(1/2) - 512*a^5*b*(-a^2*d^2)^(1/2))
)/(256*a^7*d - 16*a^2*b^5*d + 96*a^3*b^4*d - 288*a^4*b^3*d + 512*a^5*b^2*d + a*b^6*d - 512*a^6*b*d))/(-a^2*d^2
)^(1/2) - log((((((8589934592*d^3*exp(c + d*x)*(8*a^2 - 7*a*b + 3*b^2)*(-((a^7*b^3)^(1/2) + a^3*b^2)/(a^6*d^2*
(a - b)))^(1/2))/(b^5*(a - b)^2) - (4294967296*d^2*(exp(2*c + 2*d*x) + 1)*(2*a*b^2 - 7*a^2*b + 12*a^3 + b^3))/
(a^2*b^4*(a - b)^3))*(-((a^7*b^3)^(1/2) + a^3*b^2)/(a^6*d^2*(a - b)))^(1/2))/4 - (4294967296*d*exp(c + d*x)*(2
*a^2 - 2*a*b + b^2))/(a^3*b^4*(a - b)^2))*(-((a^7*b^3)^(1/2) + a^3*b^2)/(a^6*d^2*(a - b)))^(1/2))/4 + (2684354
56*(exp(2*c + 2*d*x) + 1)*(4*a^3 - a*b^2 + b^3))/(a^5*b^3*(a - b)^3))*(-((a^7*b^3)^(1/2) + a^3*b^2)/(16*(a^7*d
^2 - a^6*b*d^2)))^(1/2) + log((268435456*(exp(2*c + 2*d*x) + 1)*(4*a^3 - a*b^2 + b^3))/(a^5*b^3*(a - b)^3) - (
((((8589934592*d^3*exp(c + d*x)*(8*a^2 - 7*a*b + 3*b^2)*(-((a^7*b^3)^(1/2) + a^3*b^2)/(a^6*d^2*(a - b)))^(1/2)
)/(b^5*(a - b)^2) + (4294967296*d^2*(exp(2*c + 2*d*x) + 1)*(2*a*b^2 - 7*a^2*b + 12*a^3 + b^3))/(a^2*b^4*(a - b
)^3))*(-((a^7*b^3)^(1/2) + a^3*b^2)/(a^6*d^2*(a - b)))^(1/2))/4 - (4294967296*d*exp(c + d*x)*(2*a^2 - 2*a*b +
b^2))/(a^3*b^4*(a - b)^2))*(-((a^7*b^3)^(1/2) + a^3*b^2)/(a^6*d^2*(a - b)))^(1/2))/4)*(-((a^7*b^3)^(1/2) + a^3
*b^2)/(16*(a^7*d^2 - a^6*b*d^2)))^(1/2) - log((((((8589934592*d^3*exp(c + d*x)*(8*a^2 - 7*a*b + 3*b^2)*(((a^7*
b^3)^(1/2) - a^3*b^2)/(a^6*d^2*(a - b)))^(1/2))/(b^5*(a - b)^2) - (4294967296*d^2*(exp(2*c + 2*d*x) + 1)*(2*a*
b^2 - 7*a^2*b + 12*a^3 + b^3))/(a^2*b^4*(a - b)^3))*(((a^7*b^3)^(1/2) - a^3*b^2)/(a^6*d^2*(a - b)))^(1/2))/4 -
(4294967296*d*exp(c + d*x)*(2*a^2 - 2*a*b + b^2))/(a^3*b^4*(a - b)^2))*(((a^7*b^3)^(1/2) - a^3*b^2)/(a^6*d^2*
(a - b)))^(1/2))/4 + (268435456*(exp(2*c + 2*d*x) + 1)*(4*a^3 - a*b^2 + b^3))/(a^5*b^3*(a - b)^3))*(((a^7*b^3)
^(1/2) - a^3*b^2)/(16*(a^7*d^2 - a^6*b*d^2)))^(1/2) + log((268435456*(exp(2*c + 2*d*x) + 1)*(4*a^3 - a*b^2 + b
^3))/(a^5*b^3*(a - b)^3) - (((((8589934592*d^3*exp(c + d*x)*(8*a^2 - 7*a*b + 3*b^2)*(((a^7*b^3)^(1/2) - a^3*b^
2)/(a^6*d^2*(a - b)))^(1/2))/(b^5*(a - b)^2) + (4294967296*d^2*(exp(2*c + 2*d*x) + 1)*(2*a*b^2 - 7*a^2*b + 12*
a^3 + b^3))/(a^2*b^4*(a - b)^3))*(((a^7*b^3)^(1/2) - a^3*b^2)/(a^6*d^2*(a - b)))^(1/2))/4 - (4294967296*d*exp(
c + d*x)*(2*a^2 - 2*a*b + b^2))/(a^3*b^4*(a - b)^2))*(((a^7*b^3)^(1/2) - a^3*b^2)/(a^6*d^2*(a - b)))^(1/2))/4)
*(((a^7*b^3)^(1/2) - a^3*b^2)/(16*(a^7*d^2 - a^6*b*d^2)))^(1/2) - exp(c + d*x)/(a*d*(exp(2*c + 2*d*x) - 1)) -
(2*exp(c + d*x))/(a*d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1))

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