3.3.46 \(\int \frac {\text {csch}(c+d x)}{(a-b \sinh ^4(c+d x))^2} \, dx\) [246]

Optimal. Leaf size=325 \[ -\frac {\sqrt [4]{b} \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}-\frac {\sqrt [4]{b} \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a^2 d}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 a (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )} \]

[Out]

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

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Rubi [A]
time = 0.31, antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {3294, 1252, 213, 1192, 1180, 211, 214} \begin {gather*} -\frac {\sqrt [4]{b} \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{3/2} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{3/2} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-\frac {\sqrt [4]{b} \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^2 d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^2 d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac {b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 a 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[Csch[c + d*x]/(a - b*Sinh[c + d*x]^4)^2,x]

[Out]

-1/8*(b^(1/4)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(a^(3/2)*(Sqrt[a] - Sqrt[b])^(3/2)*d) -
(b^(1/4)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^2*Sqrt[Sqrt[a] - Sqrt[b]]*d) - ArcTanh
[Cosh[c + d*x]]/(a^2*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*a^(3/2)*(Sqrt[
a] + Sqrt[b])^(3/2)*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^2*Sqrt[Sqrt[a
] + Sqrt[b]]*d) - (b*Cosh[c + d*x]*(2 - Cosh[c + d*x]^2))/(4*a*(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 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 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 1252

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d
+ e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && ((Intege
rQ[p] && IntegerQ[q]) || IGtQ[p, 0] || IGtQ[q, 0])

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}(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \left (-\frac {1}{a^2 \left (-1+x^2\right )}+\frac {b-b x^2}{a \left (a-b+2 b x^2-b x^4\right )^2}+\frac {b-b x^2}{a^2 \left (a-b+2 b x^2-b x^4\right )}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\cosh (c+d x)\right )}{a^2 d}-\frac {\text {Subst}\left (\int \frac {b-b x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{a^2 d}-\frac {\text {Subst}\left (\int \frac {b-b x^2}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cosh (c+d x)\right )}{a d}\\ &=-\frac {\tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac {b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 a (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+2 a b^2 x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{8 a^2 (a-b) b d}+\frac {b \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{2 a^2 d}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{2 a^2 d}\\ &=-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a^2 d}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 a (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}+\frac {b \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{8 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right ) d}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{8 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right ) d}\\ &=-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a^2 d}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^2 \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 a (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 = 0.63, size = 761, normalized size = 2.34 \begin {gather*} \frac {\frac {16 a b (-5 \cosh (c+d x)+\cosh (3 (c+d x)))}{(a-b) (8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x)))}+32 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )-\frac {b \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-5 a c+4 b c-5 a d x+4 b d x-10 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+8 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+19 a c \text {$\#$1}^2-12 b c \text {$\#$1}^2+19 a d x \text {$\#$1}^2-12 b d x \text {$\#$1}^2+38 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-24 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-19 a c \text {$\#$1}^4+12 b c \text {$\#$1}^4-19 a d x \text {$\#$1}^4+12 b d x \text {$\#$1}^4-38 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+24 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+5 a c \text {$\#$1}^6-4 b c \text {$\#$1}^6+5 a d x \text {$\#$1}^6-4 b d x \text {$\#$1}^6+10 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6-8 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{a-b}}{32 a^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((16*a*b*(-5*Cosh[c + d*x] + Cosh[3*(c + d*x)]))/((a - b)*(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d
*x)])) + 32*Log[Tanh[(c + d*x)/2]] - (b*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-
5*a*c + 4*b*c - 5*a*d*x + 4*b*d*x - 10*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - S
inh[(c + d*x)/2]*#1] + 8*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/
2]*#1] + 19*a*c*#1^2 - 12*b*c*#1^2 + 19*a*d*x*#1^2 - 12*b*d*x*#1^2 + 38*a*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 - 24*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] +
Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 19*a*c*#1^4 + 12*b*c*#1^4 - 19*a*d*x*#1^4 + 12*b*d*x*#1^4
- 38*a*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 + 24*b*
Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + 5*a*c*#1^6 -
4*b*c*#1^6 + 5*a*d*x*#1^6 - 4*b*d*x*#1^6 + 10*a*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 - 8*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sin
h[(c + d*x)/2]*#1]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(a - b))/(32*a^2*d)

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Maple [A]
time = 2.80, size = 364, normalized size = 1.12

method	result	size
derivativedivides	\(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}+\frac {16 b \left (\frac {-\frac {a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 \left (a -b \right )}-\frac {\left (3 a -8 b \right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 \left (a -b \right )}+\frac {5 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 \left (a -b \right )}-\frac {a}{32 \left (a -b \right )}}{a \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {a \left (-\frac {\left (5 \sqrt {a b}\, a -4 \sqrt {a b}\, b -a b \right ) \arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{4 a b \sqrt {-\sqrt {a b}\, a -a b}}+\frac {\left (-5 \sqrt {a b}\, a +4 \sqrt {a b}\, b -a b \right ) \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{4 a b \sqrt {\sqrt {a b}\, a -a b}}\right )}{32 a -32 b}\right )}{a^{2}}}{d}\)	\(364\)
default	\(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}+\frac {16 b \left (\frac {-\frac {a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 \left (a -b \right )}-\frac {\left (3 a -8 b \right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 \left (a -b \right )}+\frac {5 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 \left (a -b \right )}-\frac {a}{32 \left (a -b \right )}}{a \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {a \left (-\frac {\left (5 \sqrt {a b}\, a -4 \sqrt {a b}\, b -a b \right ) \arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{4 a b \sqrt {-\sqrt {a b}\, a -a b}}+\frac {\left (-5 \sqrt {a b}\, a +4 \sqrt {a b}\, b -a b \right ) \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{4 a b \sqrt {\sqrt {a b}\, a -a b}}\right )}{32 a -32 b}\right )}{a^{2}}}{d}\)	\(364\)
risch	\(\frac {b \,{\mathrm e}^{d x +c} \left ({\mathrm e}^{6 d x +6 c}-5 \,{\mathrm e}^{4 d x +4 c}-5 \,{\mathrm e}^{2 d x +2 c}+1\right )}{2 a \left (a -b \right ) d \left (-b \,{\mathrm e}^{8 d x +8 c}+4 b \,{\mathrm e}^{6 d x +6 c}+16 a \,{\mathrm e}^{4 d x +4 c}-6 b \,{\mathrm e}^{4 d x +4 c}+4 b \,{\mathrm e}^{2 d x +2 c}-b \right )}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{2} d}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{a^{2} d}+2 \left (\munderset {\textit {\_R} =\RootOf \left (\left (1048576 a^{11} d^{4}-3145728 a^{10} b \,d^{4}+3145728 a^{9} b^{2} d^{4}-1048576 a^{8} b^{3} d^{4}\right ) \textit {\_Z}^{4}+\left (71680 a^{6} b \,d^{2}-96256 a^{5} b^{2} d^{2}+32768 a^{4} b^{3} d^{2}\right ) \textit {\_Z}^{2}-625 a^{2} b +800 a \,b^{2}-256 b^{3}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\left (\frac {327680 a^{10} d^{3}}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}-\frac {1179648 a^{9} d^{3} b}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}+\frac {1572864 a^{8} d^{3} b^{2}}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}-\frac {917504 a^{7} d^{3} b^{3}}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}+\frac {196608 a^{6} b^{4} d^{3}}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}\right ) \textit {\_R}^{3}+\left (\frac {20800 a^{5} d b}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}-\frac {39296 a^{4} d \,b^{2}}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}+\frac {24640 a^{3} b^{3} d}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}-\frac {5120 a^{2} b^{4} d}{625 a^{3} b -1125 a^{2} b^{2}+664 a \,b^{3}-128 b^{4}}\right ) \textit {\_R} \right ) {\mathrm e}^{d x +c}+1\right )\right )\)	\(626\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/a^2*ln(tanh(1/2*d*x+1/2*c))+16*b/a^2*((-1/32*a/(a-b)*tanh(1/2*d*x+1/2*c)^6-1/32*(3*a-8*b)/(a-b)*tanh(1/
2*d*x+1/2*c)^4+5/32*a/(a-b)*tanh(1/2*d*x+1/2*c)^2-1/32*a/(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)+1/32/(a-b)*a*(-1/4*(5*(
a*b)^(1/2)*a-4*(a*b)^(1/2)*b-a*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*(-5*(a*b)^(1/2)*a+4*(a*b)^(1/2)*b-a*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(csch(d*x+c)/(a-b*sinh(d*x+c)^4)^2,x, algorithm="maxima")

[Out]

-1/2*(b*e^(7*d*x + 7*c) - 5*b*e^(5*d*x + 5*c) - 5*b*e^(3*d*x + 3*c) + b*e^(d*x + c))/(a^2*b*d - a*b^2*d + (a^2
*b*d*e^(8*c) - a*b^2*d*e^(8*c))*e^(8*d*x) - 4*(a^2*b*d*e^(6*c) - a*b^2*d*e^(6*c))*e^(6*d*x) - 2*(8*a^3*d*e^(4*
c) - 11*a^2*b*d*e^(4*c) + 3*a*b^2*d*e^(4*c))*e^(4*d*x) - 4*(a^2*b*d*e^(2*c) - a*b^2*d*e^(2*c))*e^(2*d*x)) - lo
g((e^(d*x + c) + 1)*e^(-c))/(a^2*d) + log((e^(d*x + c) - 1)*e^(-c))/(a^2*d) - 2*integrate(1/4*((5*a*b*e^(7*c)
- 4*b^2*e^(7*c))*e^(7*d*x) - (19*a*b*e^(5*c) - 12*b^2*e^(5*c))*e^(5*d*x) + (19*a*b*e^(3*c) - 12*b^2*e^(3*c))*e
^(3*d*x) - (5*a*b*e^c - 4*b^2*e^c)*e^(d*x))/(a^3*b - a^2*b^2 + (a^3*b*e^(8*c) - a^2*b^2*e^(8*c))*e^(8*d*x) - 4
*(a^3*b*e^(6*c) - a^2*b^2*e^(6*c))*e^(6*d*x) - 2*(8*a^4*e^(4*c) - 11*a^3*b*e^(4*c) + 3*a^2*b^2*e^(4*c))*e^(4*d
*x) - 4*(a^3*b*e^(2*c) - a^2*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. 7793 vs. \(2 (244) = 488\).
time = 0.68, size = 7793, normalized size = 23.98 \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)/(a-b*sinh(d*x+c)^4)^2,x, algorithm="fricas")

[Out]

-1/16*(8*a*b*cosh(d*x + c)^7 + 56*a*b*cosh(d*x + c)*sinh(d*x + c)^6 + 8*a*b*sinh(d*x + c)^7 - 40*a*b*cosh(d*x
+ c)^5 + 8*(21*a*b*cosh(d*x + c)^2 - 5*a*b)*sinh(d*x + c)^5 - 40*a*b*cosh(d*x + c)^3 + 40*(7*a*b*cosh(d*x + c)
^3 - 5*a*b*cosh(d*x + c))*sinh(d*x + c)^4 + 40*(7*a*b*cosh(d*x + c)^4 - 10*a*b*cosh(d*x + c)^2 - a*b)*sinh(d*x
+ c)^3 + 8*a*b*cosh(d*x + c) + 8*(21*a*b*cosh(d*x + c)^5 - 50*a*b*cosh(d*x + c)^3 - 15*a*b*cosh(d*x + c))*sin
h(d*x + c)^2 + ((a^3*b - a^2*b^2)*d*cosh(d*x + c)^8 + 8*(a^3*b - a^2*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a
^3*b - a^2*b^2)*d*sinh(d*x + c)^8 - 4*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^6 + 4*(7*(a^3*b - a^2*b^2)*d*cosh(d*x
+ c)^2 - (a^3*b - a^2*b^2)*d)*sinh(d*x + c)^6 - 2*(8*a^4 - 11*a^3*b + 3*a^2*b^2)*d*cosh(d*x + c)^4 + 8*(7*(a^3
*b - a^2*b^2)*d*cosh(d*x + c)^3 - 3*(a^3*b - a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^3*b - a^2*b^
2)*d*cosh(d*x + c)^4 - 30*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^2 - (8*a^4 - 11*a^3*b + 3*a^2*b^2)*d)*sinh(d*x + c
)^4 - 4*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^2 + 8*(7*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^5 - 10*(a^3*b - a^2*b^2)*
d*cosh(d*x + c)^3 - (8*a^4 - 11*a^3*b + 3*a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^3*b - a^2*b^2)*d
*cosh(d*x + c)^6 - 15*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^4 - 3*(8*a^4 - 11*a^3*b + 3*a^2*b^2)*d*cosh(d*x + c)^2
- (a^3*b - a^2*b^2)*d)*sinh(d*x + c)^2 + (a^3*b - a^2*b^2)*d + 8*((a^3*b - a^2*b^2)*d*cosh(d*x + c)^7 - 3*(a^
3*b - a^2*b^2)*d*cosh(d*x + c)^5 - (8*a^4 - 11*a^3*b + 3*a^2*b^2)*d*cosh(d*x + c)^3 - (a^3*b - a^2*b^2)*d*cosh
(d*x + c))*sinh(d*x + c))*sqrt(-((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((625*a^4*b - 1450*a^3*b^2 + 12
41*a^2*b^3 - 464*a*b^4 + 64*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*
b^6)*d^4)) + 35*a^2*b - 47*a*b^2 + 16*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))*log(-625*a^3*b + 1125*
a^2*b^2 - 664*a*b^3 + 128*b^4 - (625*a^3*b - 1125*a^2*b^2 + 664*a*b^3 - 128*b^4)*cosh(d*x + c)^2 - 2*(625*a^3*
b - 1125*a^2*b^2 + 664*a*b^3 - 128*b^4)*cosh(d*x + c)*sinh(d*x + c) - (625*a^3*b - 1125*a^2*b^2 + 664*a*b^3 -
128*b^4)*sinh(d*x + c)^2 + 2*(2*(75*a^5*b - 137*a^4*b^2 + 82*a^3*b^3 - 16*a^2*b^4)*d*cosh(d*x + c) + 2*(75*a^5
*b - 137*a^4*b^2 + 82*a^3*b^3 - 16*a^2*b^4)*d*sinh(d*x + c) - ((5*a^10 - 18*a^9*b + 24*a^8*b^2 - 14*a^7*b^3 +
3*a^6*b^4)*d^3*cosh(d*x + c) + (5*a^10 - 18*a^9*b + 24*a^8*b^2 - 14*a^7*b^3 + 3*a^6*b^4)*d^3*sinh(d*x + c))*sq
rt((625*a^4*b - 1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^
3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)))*sqrt(-((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((625*a^4*b
- 1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4
- 6*a^8*b^5 + a^7*b^6)*d^4)) + 35*a^2*b - 47*a*b^2 + 16*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))) -
((a^3*b - a^2*b^2)*d*cosh(d*x + c)^8 + 8*(a^3*b - a^2*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^3*b - a^2*b^2)
*d*sinh(d*x + c)^8 - 4*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^6 + 4*(7*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^2 - (a^3*b
- a^2*b^2)*d)*sinh(d*x + c)^6 - 2*(8*a^4 - 11*a^3*b + 3*a^2*b^2)*d*cosh(d*x + c)^4 + 8*(7*(a^3*b - a^2*b^2)*d
*cosh(d*x + c)^3 - 3*(a^3*b - a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^3*b - a^2*b^2)*d*cosh(d*x +
c)^4 - 30*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^2 - (8*a^4 - 11*a^3*b + 3*a^2*b^2)*d)*sinh(d*x + c)^4 - 4*(a^3*b
- a^2*b^2)*d*cosh(d*x + c)^2 + 8*(7*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^5 - 10*(a^3*b - a^2*b^2)*d*cosh(d*x + c)
^3 - (8*a^4 - 11*a^3*b + 3*a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^
6 - 15*(a^3*b - a^2*b^2)*d*cosh(d*x + c)^4 - 3*(8*a^4 - 11*a^3*b + 3*a^2*b^2)*d*cosh(d*x + c)^2 - (a^3*b - a^2
*b^2)*d)*sinh(d*x + c)^2 + (a^3*b - a^2*b^2)*d + 8*((a^3*b - a^2*b^2)*d*cosh(d*x + c)^7 - 3*(a^3*b - a^2*b^2)*
d*cosh(d*x + c)^5 - (8*a^4 - 11*a^3*b + 3*a^2*b^2)*d*cosh(d*x + c)^3 - (a^3*b - a^2*b^2)*d*cosh(d*x + c))*sinh
(d*x + c))*sqrt(-((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((625*a^4*b - 1450*a^3*b^2 + 1241*a^2*b^3 - 46
4*a*b^4 + 64*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 35
*a^2*b - 47*a*b^2 + 16*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))*log(-625*a^3*b + 1125*a^2*b^2 - 664*a
*b^3 + 128*b^4 - (625*a^3*b - 1125*a^2*b^2 + 664*a*b^3 - 128*b^4)*cosh(d*x + c)^2 - 2*(625*a^3*b - 1125*a^2*b^
2 + 664*a*b^3 - 128*b^4)*cosh(d*x + c)*sinh(d*x + c) - (625*a^3*b - 1125*a^2*b^2 + 664*a*b^3 - 128*b^4)*sinh(d
*x + c)^2 - 2*(2*(75*a^5*b - 137*a^4*b^2 + 82*a^3*b^3 - 16*a^2*b^4)*d*cosh(d*x + c) + 2*(75*a^5*b - 137*a^4*b^
2 + 82*a^3*b^3 - 16*a^2*b^4)*d*sinh(d*x + c) - ((5*a^10 - 18*a^9*b + 24*a^8*b^2 - 14*a^7*b^3 + 3*a^6*b^4)*d^3*
cosh(d*x + c) + (5*a^10 - 18*a^9*b + 24*a^8*b^2 - 14*a^7*b^3 + 3*a^6*b^4)*d^3*sinh(d*x + c))*sqrt((625*a^4*b -
1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4
- 6*a^8*b^5 + a^7*b^6)*d^4)))*sqrt(-((a^7 - 3*a...

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)/(a-b*sinh(d*x+c)**4)**2,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4373 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1116 vs. \(2 (244) = 488\).
time = 0.55, size = 1116, normalized size = 3.43 \begin {gather*} \frac {\frac {{\left ({\left (20 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{2} + 9 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b - 20 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} b^{2}\right )} {\left (a^{3} - a^{2} b\right )}^{2} {\left | b \right |} - 2 \, {\left (12 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{5} b - 5 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{4} b^{2} - 17 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{3} b^{3} + 10 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b^{4}\right )} {\left | -a^{3} + a^{2} b \right |} {\left | b \right |} + {\left (4 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{7} b - 3 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{6} b^{2} - 6 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{5} b^{3} + 5 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{4} 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^{3} b - a^{2} b^{2} + \sqrt {{\left (a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right )} {\left (a^{3} b - a^{2} b^{2}\right )} + {\left (a^{3} b - a^{2} b^{2}\right )}^{2}}}{a^{3} b - a^{2} b^{2}}}}\right )}{{\left (4 \, a^{8} b^{2} - 7 \, a^{7} b^{3} - 3 \, a^{6} b^{4} + 11 \, a^{5} b^{5} - 5 \, a^{4} b^{6}\right )} {\left | -a^{3} + a^{2} b \right |}} - \frac {{\left ({\left (20 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{2} + 9 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a b - 20 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} b^{2}\right )} {\left (a^{3} - a^{2} b\right )}^{2} {\left | b \right |} + 2 \, {\left (12 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{5} b - 5 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{4} b^{2} - 17 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{3} b^{3} + 10 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b^{4}\right )} {\left | -a^{3} + a^{2} b \right |} {\left | b \right |} + {\left (4 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{7} b - 3 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{6} b^{2} - 6 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{5} b^{3} + 5 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{4} 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^{3} b - a^{2} b^{2} - \sqrt {{\left (a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right )} {\left (a^{3} b - a^{2} b^{2}\right )} + {\left (a^{3} b - a^{2} b^{2}\right )}^{2}}}{a^{3} b - a^{2} b^{2}}}}\right )}{{\left (4 \, a^{8} b^{2} - 7 \, a^{7} b^{3} - 3 \, a^{6} b^{4} + 11 \, a^{5} b^{5} - 5 \, a^{4} b^{6}\right )} {\left | -a^{3} + a^{2} b \right |}} - \frac {4 \, {\left (b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 8 \, b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}\right )}}{{\left (b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4} - 8 \, b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 16 \, a + 16 \, b\right )} {\left (a^{2} - a b\right )}} - \frac {4 \, \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right )}{a^{2}} + \frac {4 \, \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right )}{a^{2}}}{8 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)/(a-b*sinh(d*x+c)^4)^2,x, algorithm="giac")

[Out]

1/8*(((20*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^2 + 9*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a*b - 20*sqrt(a*b)*sqr
t(-b^2 + sqrt(a*b)*b)*b^2)*(a^3 - a^2*b)^2*abs(b) - 2*(12*sqrt(-b^2 + sqrt(a*b)*b)*a^5*b - 5*sqrt(-b^2 + sqrt(
a*b)*b)*a^4*b^2 - 17*sqrt(-b^2 + sqrt(a*b)*b)*a^3*b^3 + 10*sqrt(-b^2 + sqrt(a*b)*b)*a^2*b^4)*abs(-a^3 + a^2*b)
*abs(b) + (4*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^7*b - 3*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^6*b^2 - 6*sqrt(
a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^5*b^3 + 5*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^4*b^4)*abs(b))*arctan(1/2*(e^(d
*x + c) + e^(-d*x - c))/sqrt(-(a^3*b - a^2*b^2 + sqrt((a^4 - 2*a^3*b + a^2*b^2)*(a^3*b - a^2*b^2) + (a^3*b - a
^2*b^2)^2))/(a^3*b - a^2*b^2)))/((4*a^8*b^2 - 7*a^7*b^3 - 3*a^6*b^4 + 11*a^5*b^5 - 5*a^4*b^6)*abs(-a^3 + a^2*b
)) - ((20*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^2 + 9*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a*b - 20*sqrt(a*b)*sqr
t(-b^2 - sqrt(a*b)*b)*b^2)*(a^3 - a^2*b)^2*abs(b) + 2*(12*sqrt(-b^2 - sqrt(a*b)*b)*a^5*b - 5*sqrt(-b^2 - sqrt(
a*b)*b)*a^4*b^2 - 17*sqrt(-b^2 - sqrt(a*b)*b)*a^3*b^3 + 10*sqrt(-b^2 - sqrt(a*b)*b)*a^2*b^4)*abs(-a^3 + a^2*b)
*abs(b) + (4*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^7*b - 3*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^6*b^2 - 6*sqrt(
a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^5*b^3 + 5*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^4*b^4)*abs(b))*arctan(1/2*(e^(d
*x + c) + e^(-d*x - c))/sqrt(-(a^3*b - a^2*b^2 - sqrt((a^4 - 2*a^3*b + a^2*b^2)*(a^3*b - a^2*b^2) + (a^3*b - a
^2*b^2)^2))/(a^3*b - a^2*b^2)))/((4*a^8*b^2 - 7*a^7*b^3 - 3*a^6*b^4 + 11*a^5*b^5 - 5*a^4*b^6)*abs(-a^3 + a^2*b
)) - 4*(b*(e^(d*x + c) + e^(-d*x - c))^3 - 8*b*(e^(d*x + c) + e^(-d*x - c)))/((b*(e^(d*x + c) + e^(-d*x - c))^
4 - 8*b*(e^(d*x + c) + e^(-d*x - c))^2 - 16*a + 16*b)*(a^2 - a*b)) - 4*log(e^(d*x + c) + e^(-d*x - c) + 2)/a^2
+ 4*log(e^(d*x + c) + e^(-d*x - c) - 2)/a^2)/d

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

[Out]

int(1/(sinh(c + d*x)*(a - b*sinh(c + d*x)^4)^2), x)

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