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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.361405, antiderivative size = 325, normalized size of antiderivative = 1., 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 = {3215, 1238, 207, 1178, 1166, 205, 208} \[ -\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^2 d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{8 a^{3/2} d \left (\sqrt{a}-\sqrt{b}\right )^{3/2}}+\frac{\sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^2 d \sqrt{\sqrt{a}+\sqrt{b}}}+\frac{\sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cosh (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{8 a^{3/2} d \left (\sqrt{a}+\sqrt{b}\right )^{3/2}}-\frac{\tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac{b \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{4 a d (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^(1/4)*ArcTan[(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)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^2*Sqrt[Sqrt[a] - Sqrt[b]]*d) - ArcTanh[C
osh[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 3215

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]

Rule 1238

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 207

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

Rule 1178

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 1166

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 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{\text{csch}(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx &=-\frac{\operatorname{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{\operatorname{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{\operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\cosh (c+d x)\right )}{a^2 d}-\frac{\operatorname{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{\operatorname{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{\operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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*}

Mathematica [C]  time = 0.837766, size = 761, normalized size = 2.34 \[ \frac{-\frac{b \text{RootSum}\left [-16 \text{$\#$1}^4 a+\text{$\#$1}^8 b-4 \text{$\#$1}^6 b+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\& ,\frac{10 \text{$\#$1}^6 a \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \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 )\right )-38 \text{$\#$1}^4 a \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \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 )\right )+38 \text{$\#$1}^2 a \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \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 )\right )+5 \text{$\#$1}^6 a c-19 \text{$\#$1}^4 a c+19 \text{$\#$1}^2 a c+5 \text{$\#$1}^6 a d x-19 \text{$\#$1}^4 a d x+19 \text{$\#$1}^2 a d x-8 \text{$\#$1}^6 b \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \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 )\right )+24 \text{$\#$1}^4 b \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \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 )\right )-24 \text{$\#$1}^2 b \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \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 )\right )-4 \text{$\#$1}^6 b c+12 \text{$\#$1}^4 b c-12 \text{$\#$1}^2 b c-4 \text{$\#$1}^6 b d x+12 \text{$\#$1}^4 b d x-12 \text{$\#$1}^2 b d x-10 a \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \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 )\right )+8 b \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \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 )\right )-5 a c-5 a d x+4 b c+4 b d x}{-8 \text{$\#$1}^3 a+\text{$\#$1}^7 b-3 \text{$\#$1}^5 b+3 \text{$\#$1}^3 b-\text{$\#$1} b}\& \right ]}{a-b}+\frac{16 a b (\cosh (3 (c+d x))-5 \cosh (c+d x))}{(a-b) (8 a+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))-3 b)}+32 \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{32 a^2 d} \]

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 [B]  time = 0.096, size = 966, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

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)

[Out]

-1/2/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4
-4*tanh(1/2*d*x+1/2*c)^2*a+a)/(a-b)/a*tanh(1/2*d*x+1/2*c)^6*b-3/2/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/
2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)/a/(a-b)*tanh(1/2*d*
x+1/2*c)^4*b+4/d*b^2/a^2/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tan
h(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)/(a-b)*tanh(1/2*d*x+1/2*c)^4+5/2/d/(tanh(1/2*d*x+1/2*c)^8*a-4*t
anh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)/a/(a-
b)*tanh(1/2*d*x+1/2*c)^2*b-1/2/d*b/a/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^
4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)/(a-b)-5/8/d/(a-b)/a/(-a*b-(a*b)^(1/2)*a)^(1/2)*arc
tan(1/4*(-2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)+2*a)/(-a*b-(a*b)^(1/2)*a)^(1/2))*(a*b)^(1/2)+1/2/d*b/a^2/(a-
b)/(-a*b-(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(-2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)+2*a)/(-a*b-(a*b)^(1/2)*a)^(
1/2))*(a*b)^(1/2)+1/8/d/(a-b)/a/(-a*b-(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(-2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2
)+2*a)/(-a*b-(a*b)^(1/2)*a)^(1/2))*b-5/8/d/(a-b)/a/(-a*b+(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(2*tanh(1/2*d*x+1/2*c
)^2*a+4*(a*b)^(1/2)-2*a)/(-a*b+(a*b)^(1/2)*a)^(1/2))*(a*b)^(1/2)+1/2/d*b/a^2/(a-b)/(-a*b+(a*b)^(1/2)*a)^(1/2)*
arctan(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)-2*a)/(-a*b+(a*b)^(1/2)*a)^(1/2))*(a*b)^(1/2)-1/8/d/(a-b)/a
/(-a*b+(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)-2*a)/(-a*b+(a*b)^(1/2)*a)^(1/2
))*b+1/d/a^2*ln(tanh(1/2*d*x+1/2*c))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{b e^{\left (7 \, d x + 7 \, c\right )} - 5 \, b e^{\left (5 \, d x + 5 \, c\right )} - 5 \, b e^{\left (3 \, d x + 3 \, c\right )} + b e^{\left (d x + c\right )}}{2 \,{\left (a^{2} b d - a b^{2} d +{\left (a^{2} b d e^{\left (8 \, c\right )} - a b^{2} d e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} - 4 \,{\left (a^{2} b d e^{\left (6 \, c\right )} - a b^{2} d e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} - 2 \,{\left (8 \, a^{3} d e^{\left (4 \, c\right )} - 11 \, a^{2} b d e^{\left (4 \, c\right )} + 3 \, a b^{2} d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} - 4 \,{\left (a^{2} b d e^{\left (2 \, c\right )} - a b^{2} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}} - \frac{\log \left ({\left (e^{\left (d x + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{a^{2} d} + \frac{\log \left ({\left (e^{\left (d x + c\right )} - 1\right )} e^{\left (-c\right )}\right )}{a^{2} d} - 2 \, \int \frac{{\left (5 \, a b e^{\left (7 \, c\right )} - 4 \, b^{2} e^{\left (7 \, c\right )}\right )} e^{\left (7 \, d x\right )} -{\left (19 \, a b e^{\left (5 \, c\right )} - 12 \, b^{2} e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} +{\left (19 \, a b e^{\left (3 \, c\right )} - 12 \, b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} -{\left (5 \, a b e^{c} - 4 \, b^{2} e^{c}\right )} e^{\left (d x\right )}}{4 \,{\left (a^{3} b - a^{2} b^{2} +{\left (a^{3} b e^{\left (8 \, c\right )} - a^{2} b^{2} e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} - 4 \,{\left (a^{3} b e^{\left (6 \, c\right )} - a^{2} b^{2} e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} - 2 \,{\left (8 \, a^{4} e^{\left (4 \, c\right )} - 11 \, a^{3} b e^{\left (4 \, c\right )} + 3 \, a^{2} b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} - 4 \,{\left (a^{3} b e^{\left (2 \, c\right )} - a^{2} b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \end{align*}

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]  time = 4.22233, size = 17785, normalized size = 54.72 \begin{align*} \text{result too large to display} \end{align*}

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^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*co
sh(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 - 1
1*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 - 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))*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^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*c
osh(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 - 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 - 112
5*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*si
nh(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 - 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)))*sqr
t(((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))) + 16*((a*b - b^2)*cosh(d*x + c)^8 + 8*(a*b - b^2)*c
osh(d*x + c)*sinh(d*x + c)^7 + (a*b - b^2)*sinh(d*x + c)^8 - 4*(a*b - b^2)*cosh(d*x + c)^6 + 4*(7*(a*b - b^2)*
cosh(d*x + c)^2 - a*b + b^2)*sinh(d*x + c)^6 + 8*(7*(a*b - b^2)*cosh(d*x + c)^3 - 3*(a*b - b^2)*cosh(d*x + c))
*sinh(d*x + c)^5 - 2*(8*a^2 - 11*a*b + 3*b^2)*cosh(d*x + c)^4 + 2*(35*(a*b - b^2)*cosh(d*x + c)^4 - 30*(a*b -
b^2)*cosh(d*x + c)^2 - 8*a^2 + 11*a*b - 3*b^2)*sinh(d*x + c)^4 + 8*(7*(a*b - b^2)*cosh(d*x + c)^5 - 10*(a*b -
b^2)*cosh(d*x + c)^3 - (8*a^2 - 11*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*(a*b - b^2)*cosh(d*x + c)^2
+ 4*(7*(a*b - b^2)*cosh(d*x + c)^6 - 15*(a*b - b^2)*cosh(d*x + c)^4 - 3*(8*a^2 - 11*a*b + 3*b^2)*cosh(d*x + c
)^2 - a*b + b^2)*sinh(d*x + c)^2 + a*b - b^2 + 8*((a*b - b^2)*cosh(d*x + c)^7 - 3*(a*b - b^2)*cosh(d*x + c)^5
- (8*a^2 - 11*a*b + 3*b^2)*cosh(d*x + c)^3 - (a*b - b^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sin
h(d*x + c) + 1) - 16*((a*b - b^2)*cosh(d*x + c)^8 + 8*(a*b - b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (a*b - b^2)*
sinh(d*x + c)^8 - 4*(a*b - b^2)*cosh(d*x + c)^6 + 4*(7*(a*b - b^2)*cosh(d*x + c)^2 - a*b + b^2)*sinh(d*x + c)^
6 + 8*(7*(a*b - b^2)*cosh(d*x + c)^3 - 3*(a*b - b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(8*a^2 - 11*a*b + 3*b^
2)*cosh(d*x + c)^4 + 2*(35*(a*b - b^2)*cosh(d*x + c)^4 - 30*(a*b - b^2)*cosh(d*x + c)^2 - 8*a^2 + 11*a*b - 3*b
^2)*sinh(d*x + c)^4 + 8*(7*(a*b - b^2)*cosh(d*x + c)^5 - 10*(a*b - b^2)*cosh(d*x + c)^3 - (8*a^2 - 11*a*b + 3*
b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*(a*b - b^2)*cosh(d*x + c)^2 + 4*(7*(a*b - b^2)*cosh(d*x + c)^6 - 15*(a
*b - b^2)*cosh(d*x + c)^4 - 3*(8*a^2 - 11*a*b + 3*b^2)*cosh(d*x + c)^2 - a*b + b^2)*sinh(d*x + c)^2 + a*b - b^
2 + 8*((a*b - b^2)*cosh(d*x + c)^7 - 3*(a*b - b^2)*cosh(d*x + c)^5 - (8*a^2 - 11*a*b + 3*b^2)*cosh(d*x + c)^3
- (a*b - b^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 8*(7*a*b*cosh(d*x + c)^6
- 25*a*b*cosh(d*x + c)^4 - 15*a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c))/((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))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

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]

Exception raised: NotImplementedError