Integrand size = 24, antiderivative size = 357 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=-\frac {3 \sqrt {b} \left (20 a-34 \sqrt {a} \sqrt {b}+15 b\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{13/4} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} d}+\frac {3 \sqrt {b} \left (20 a+34 \sqrt {a} \sqrt {b}+15 b\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{13/4} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} d}-\frac {\coth (c+d x)}{a^3 d}+\frac {b^2 \tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 a^2 (a-b)^3 d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )^2}+\frac {b \tanh (c+d x) \left (2 a^2 (9 a-17 b)-(a-b) \left (18 a^2+15 a b-13 b^2\right ) \tanh ^2(c+d x)\right )}{32 a^3 (a-b)^3 d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )} \] Output:
-3/64*b^(1/2)*(20*a-34*a^(1/2)*b^(1/2)+15*b)*arctanh((a^(1/2)-b^(1/2))^(1/ 2)*tanh(d*x+c)/a^(1/4))/a^(13/4)/(a^(1/2)-b^(1/2))^(5/2)/d+3/64*b^(1/2)*(2 0*a+34*a^(1/2)*b^(1/2)+15*b)*arctanh((a^(1/2)+b^(1/2))^(1/2)*tanh(d*x+c)/a ^(1/4))/a^(13/4)/(a^(1/2)+b^(1/2))^(5/2)/d-coth(d*x+c)/a^3/d+1/8*b^2*tanh( d*x+c)*(a*(a+3*b)-(a^2+6*a*b+b^2)*tanh(d*x+c)^2)/a^2/(a-b)^3/d/(a-2*a*tanh (d*x+c)^2+(a-b)*tanh(d*x+c)^4)^2+1/32*b*tanh(d*x+c)*(2*a^2*(9*a-17*b)-(a-b )*(18*a^2+15*a*b-13*b^2)*tanh(d*x+c)^2)/a^3/(a-b)^3/d/(a-2*a*tanh(d*x+c)^2 +(a-b)*tanh(d*x+c)^4)
Time = 7.93 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.00 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\frac {\frac {3 \sqrt {b} \left (20 a-34 \sqrt {a} \sqrt {b}+15 b\right ) \arctan \left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\left (\sqrt {a}-\sqrt {b}\right )^2 \sqrt {-a+\sqrt {a} \sqrt {b}}}+\frac {3 \sqrt {b} \left (20 a+34 \sqrt {a} \sqrt {b}+15 b\right ) \text {arctanh}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\left (\sqrt {a}+\sqrt {b}\right )^2 \sqrt {a+\sqrt {a} \sqrt {b}}}-64 \coth (c+d x)+\frac {4 b \left (28 a^2+3 a b-13 b^2+b (-19 a+13 b) \cosh (2 (c+d x))\right ) \sinh (2 (c+d x))}{(a-b)^2 (8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x)))}+\frac {128 a b (2 a+b-b \cosh (2 (c+d x))) \sinh (2 (c+d x))}{(a-b) (-8 a+3 b-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x)))^2}}{64 a^3 d} \] Input:
Integrate[Csch[c + d*x]^2/(a - b*Sinh[c + d*x]^4)^3,x]
Output:
((3*Sqrt[b]*(20*a - 34*Sqrt[a]*Sqrt[b] + 15*b)*ArcTan[((Sqrt[a] - Sqrt[b]) *Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] - Sqrt[b])^2*Sqrt[- a + Sqrt[a]*Sqrt[b]]) + (3*Sqrt[b]*(20*a + 34*Sqrt[a]*Sqrt[b] + 15*b)*ArcT anh[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/((Sqrt [a] + Sqrt[b])^2*Sqrt[a + Sqrt[a]*Sqrt[b]]) - 64*Coth[c + d*x] + (4*b*(28* a^2 + 3*a*b - 13*b^2 + b*(-19*a + 13*b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x )])/((a - b)^2*(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)])) + (128*a*b*(2*a + b - b*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a - b)*(-8 *a + 3*b - 4*b*Cosh[2*(c + d*x)] + b*Cosh[4*(c + d*x)])^2))/(64*a^3*d)
Time = 1.68 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3042, 25, 3696, 1673, 27, 2198, 27, 2195, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\text {csch}^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\sin (i c+i d x)^2 \left (a-b \sin (i c+i d x)^4\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\sin (i c+i d x)^2 \left (a-b \sin (i c+i d x)^4\right )^3}dx\) |
| \(\Big \downarrow \) 3696 |
| \(\displaystyle \frac {\int \frac {\coth ^2(c+d x) \left (1-\tanh ^2(c+d x)\right )^6}{\left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1673 |
| \(\displaystyle \frac {\frac {b^2 \tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 a^2 (a-b)^3 \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}-\frac {\int -\frac {2 \coth ^2(c+d x) \left (\frac {8 a^2 b \tanh ^8(c+d x)}{a-b}-\frac {16 a^2 (2 a-3 b) b \tanh ^6(c+d x)}{(a-b)^2}+\frac {b \left (48 a^4-136 b a^3+115 b^2 a^2-30 b^3 a-5 b^4\right ) \tanh ^4(c+d x)}{(a-b)^3}-\frac {a b \left (32 a^3-96 b a^2+97 b^2 a-29 b^3\right ) \tanh ^2(c+d x)}{(a-b)^3}+8 a b\right )}{\left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{16 a^2 b}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\coth ^2(c+d x) \left (\frac {8 a^2 b \tanh ^8(c+d x)}{a-b}-\frac {16 a^2 (2 a-3 b) b \tanh ^6(c+d x)}{(a-b)^2}+\frac {b \left (48 a^4-136 b a^3+115 b^2 a^2-30 b^3 a-5 b^4\right ) \tanh ^4(c+d x)}{(a-b)^3}-\frac {a b \left (32 a^3-96 b a^2+97 b^2 a-29 b^3\right ) \tanh ^2(c+d x)}{(a-b)^3}+8 a b\right )}{\left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{8 a^2 b}+\frac {b^2 \tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 a^2 (a-b)^3 \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}}{d}\) |
| \(\Big \downarrow \) 2198 |
| \(\displaystyle \frac {\frac {\frac {b^2 \tanh (c+d x) \left (\frac {2 a^2 (9 a-17 b)}{(a-b)^3}-\frac {\left (18 a^2+15 a b-13 b^2\right ) \tanh ^2(c+d x)}{(a-b)^2}\right )}{4 a \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac {\int -\frac {2 \coth ^2(c+d x) \left (\frac {a b^2 \left (32 a^3-18 b a^2-15 b^2 a+13 b^3\right ) \tanh ^4(c+d x)}{(a-b)^2}-\frac {2 a^2 b^2 \left (32 a^2-55 b a+26 b^2\right ) \tanh ^2(c+d x)}{(a-b)^2}+32 a^2 b^2\right )}{(a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a}d\tanh (c+d x)}{8 a^2 b}}{8 a^2 b}+\frac {b^2 \tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 a^2 (a-b)^3 \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\coth ^2(c+d x) \left (\frac {a b^2 \left (32 a^3-18 b a^2-15 b^2 a+13 b^3\right ) \tanh ^4(c+d x)}{(a-b)^2}-\frac {2 a^2 b^2 \left (32 a^2-55 b a+26 b^2\right ) \tanh ^2(c+d x)}{(a-b)^2}+32 a^2 b^2\right )}{(a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a}d\tanh (c+d x)}{4 a^2 b}+\frac {b^2 \tanh (c+d x) \left (\frac {2 a^2 (9 a-17 b)}{(a-b)^3}-\frac {\left (18 a^2+15 a b-13 b^2\right ) \tanh ^2(c+d x)}{(a-b)^2}\right )}{4 a \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}}{8 a^2 b}+\frac {b^2 \tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 a^2 (a-b)^3 \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}}{d}\) |
| \(\Big \downarrow \) 2195 |
| \(\displaystyle \frac {\frac {\frac {\int \left (\frac {3 a \left (\left (26 a^2-37 b a+15 b^2\right ) \tanh ^2(c+d x)-2 a (3 a-2 b)\right ) b^3}{(a-b)^2 \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}+32 a \coth ^2(c+d x) b^2\right )d\tanh (c+d x)}{4 a^2 b}+\frac {b^2 \tanh (c+d x) \left (\frac {2 a^2 (9 a-17 b)}{(a-b)^3}-\frac {\left (18 a^2+15 a b-13 b^2\right ) \tanh ^2(c+d x)}{(a-b)^2}\right )}{4 a \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}}{8 a^2 b}+\frac {b^2 \tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 a^2 (a-b)^3 \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {b^2 \tanh (c+d x) \left (a (a+3 b)-\left (a^2+6 a b+b^2\right ) \tanh ^2(c+d x)\right )}{8 a^2 (a-b)^3 \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}+\frac {\frac {b^2 \tanh (c+d x) \left (\frac {2 a^2 (9 a-17 b)}{(a-b)^3}-\frac {\left (18 a^2+15 a b-13 b^2\right ) \tanh ^2(c+d x)}{(a-b)^2}\right )}{4 a \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}+\frac {-\frac {3 a^{3/4} b^{5/2} \left (-34 \sqrt {a} \sqrt {b}+20 a+15 b\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}+\frac {3 a^{3/4} b^{5/2} \left (34 \sqrt {a} \sqrt {b}+20 a+15 b\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}-32 a b^2 \coth (c+d x)}{4 a^2 b}}{8 a^2 b}}{d}\) |
Input:
Int[Csch[c + d*x]^2/(a - b*Sinh[c + d*x]^4)^3,x]
Output:
((b^2*Tanh[c + d*x]*(a*(a + 3*b) - (a^2 + 6*a*b + b^2)*Tanh[c + d*x]^2))/( 8*a^2*(a - b)^3*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4)^2) + ( ((-3*a^(3/4)*b^(5/2)*(20*a - 34*Sqrt[a]*Sqrt[b] + 15*b)*ArcTanh[(Sqrt[Sqrt [a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(2*(Sqrt[a] - Sqrt[b])^(5/2)) + (3 *a^(3/4)*b^(5/2)*(20*a + 34*Sqrt[a]*Sqrt[b] + 15*b)*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(2*(Sqrt[a] + Sqrt[b])^(5/2)) - 32*a*b ^2*Coth[c + d*x])/(4*a^2*b) + (b^2*Tanh[c + d*x]*((2*a^2*(9*a - 17*b))/(a - b)^3 - ((18*a^2 + 15*a*b - 13*b^2)*Tanh[c + d*x]^2)/(a - b)^2))/(4*a*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4)))/(8*a^2*b))/d
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^ 4)^(p_), x_Symbol] :> With[{f = Coeff[PolynomialRemainder[x^m*(d + e*x^2)^q , a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[x^m*(d + e*x^ 2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a *b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[x^m*(a + b*x^2 + c*x^4)^(p + 1)*Simp[ExpandToSum[(2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x])/x^m + (b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5 ) - a*b*g)/x^m + c*(4*p + 7)*(b*f - 2*a*g)*x^(2 - m), x], x], x], x]] /; Fr eeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] && ILtQ[m/2, 0]
Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d*x)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]
Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{Qx = PolynomialQuotient[x^m*Pq, a + b*x^2 + c*x^4, x], d = Coeff[Pol ynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Polynomial Remainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4) ^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[x^m*(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[(2*a*(p + 1)*(b^2 - 4*a*c)*Qx)/x^m + (b^2*d*(2* p + 3) - 2*a*c*d*(4*p + 5) - a*b*e)/x^m + c*(4*p + 7)*(b*d - 2*a*e)*x^(2 - m), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && GtQ[Expon[Pq, x ^2], 1] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && ILtQ[m/2, 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff^(m + 1 )/f Subst[Int[x^m*((a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2) ^(m/2 + 2*p + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] & & IntegerQ[m/2] && IntegerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 13.99 (sec) , antiderivative size = 608, normalized size of antiderivative = 1.70
| method | result | size |
| derivativedivides | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{3}}-\frac {1}{2 a^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {8 b \left (\frac {-\frac {3 a^{2} \left (3 a -2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (45 a^{2}+16 a b -34 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{64 a^{2}-128 a b +64 b^{2}}-\frac {a \left (81 a^{2}-28 a b -38 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (45 a^{3}-50 a^{2} b -612 b^{2} a +416 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{64 a^{2}-128 a b +64 b^{2}}+\frac {\left (45 a^{3}-50 a^{2} b -612 b^{2} a +416 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{64 a^{2}-128 a b +64 b^{2}}-\frac {a \left (81 a^{2}-28 a b -38 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (45 a^{2}+16 a b -34 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{64 a^{2}-128 a b +64 b^{2}}-\frac {3 a^{2} \left (3 a -2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{64 \left (a^{2}-2 a b +b^{2}\right )}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a -4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a +6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -16 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (a \left (-3 a +2 b \right ) \textit {\_R}^{6}+\left (49 a^{2}-72 a b +30 b^{2}\right ) \textit {\_R}^{4}+\left (-49 a^{2}+72 a b -30 b^{2}\right ) \textit {\_R}^{2}+3 a^{2}-2 a b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}\right )}{512 \left (a^{2}-2 a b +b^{2}\right )}\right )}{a^{3}}}{d}\) | \(608\) |
| default | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{3}}-\frac {1}{2 a^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {8 b \left (\frac {-\frac {3 a^{2} \left (3 a -2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (45 a^{2}+16 a b -34 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{64 a^{2}-128 a b +64 b^{2}}-\frac {a \left (81 a^{2}-28 a b -38 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (45 a^{3}-50 a^{2} b -612 b^{2} a +416 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{64 a^{2}-128 a b +64 b^{2}}+\frac {\left (45 a^{3}-50 a^{2} b -612 b^{2} a +416 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{64 a^{2}-128 a b +64 b^{2}}-\frac {a \left (81 a^{2}-28 a b -38 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{64 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (45 a^{2}+16 a b -34 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{64 a^{2}-128 a b +64 b^{2}}-\frac {3 a^{2} \left (3 a -2 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{64 \left (a^{2}-2 a b +b^{2}\right )}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a -4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a +6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -16 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (a \left (-3 a +2 b \right ) \textit {\_R}^{6}+\left (49 a^{2}-72 a b +30 b^{2}\right ) \textit {\_R}^{4}+\left (-49 a^{2}+72 a b -30 b^{2}\right ) \textit {\_R}^{2}+3 a^{2}-2 a b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}\right )}{512 \left (a^{2}-2 a b +b^{2}\right )}\right )}{a^{3}}}{d}\) | \(608\) |
| risch | \(\text {Expression too large to display}\) | \(2723\) |
Input:
int(csch(d*x+c)^2/(a-b*sinh(d*x+c)^4)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(-1/2/a^3*tanh(1/2*d*x+1/2*c)-1/2/a^3/tanh(1/2*d*x+1/2*c)-8*b/a^3*((-3 /64*a^2*(3*a-2*b)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)+1/64*(45*a^2+16*a*b- 34*b^2)*a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3-1/64*a*(81*a^2-28*a*b-38*b ^2)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^5+1/64*(45*a^3-50*a^2*b-612*a*b^2+ 416*b^3)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^7+1/64*(45*a^3-50*a^2*b-612*a *b^2+416*b^3)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^9-1/64*a*(81*a^2-28*a*b- 38*b^2)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^11+1/64*(45*a^2+16*a*b-34*b^2) *a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^13-3/64*a^2*(3*a-2*b)/(a^2-2*a*b+b^ 2)*tanh(1/2*d*x+1/2*c)^15)/(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)^2+3/512/(a^2-2*a*b+b^2)*sum((a*(-3*a+2*b)*_R^6+(49*a^2-72*a*b+ 30*b^2)*_R^4+(-49*a^2+72*a*b-30*b^2)*_R^2+3*a^2-2*a*b)/(_R^7*a-3*_R^5*a+3* _R^3*a-8*_R^3*b-_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R=RootOf(a*_Z^8-4*a*_Z^6 +(6*a-16*b)*_Z^4-4*a*_Z^2+a))))
Leaf count of result is larger than twice the leaf count of optimal. 28429 vs. \(2 (304) = 608\).
Time = 1.38 (sec) , antiderivative size = 28429, normalized size of antiderivative = 79.63 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)^2/(a-b*sinh(d*x+c)^4)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\text {csch}^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\text {Timed out} \] Input:
integrate(csch(d*x+c)**2/(a-b*sinh(d*x+c)**4)**3,x)
Output:
Timed out
\[ \int \frac {\text {csch}^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\int { -\frac {\operatorname {csch}\left (d x + c\right )^{2}}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \] Input:
integrate(csch(d*x+c)^2/(a-b*sinh(d*x+c)^4)^3,x, algorithm="maxima")
Output:
1/16*(32*a^2*b^2 - 83*a*b^3 + 45*b^4 + 3*(20*a^2*b^2*e^(16*c) - 33*a*b^3*e ^(16*c) + 15*b^4*e^(16*c))*e^(16*d*x) - 12*(43*a^2*b^2*e^(14*c) - 68*a*b^3 *e^(14*c) + 30*b^4*e^(14*c))*e^(14*d*x) - 4*(400*a^3*b*e^(12*c) - 1137*a^2 *b^2*e^(12*c) + 1031*a*b^3*e^(12*c) - 315*b^4*e^(12*c))*e^(12*d*x) + 12*(5 92*a^3*b*e^(10*c) - 1237*a^2*b^2*e^(10*c) + 886*a*b^3*e^(10*c) - 210*b^4*e ^(10*c))*e^(10*d*x) + 2*(4096*a^4*e^(8*c) - 12192*a^3*b*e^(8*c) + 13634*a^ 2*b^2*e^(8*c) - 7113*a*b^3*e^(8*c) + 1575*b^4*e^(8*c))*e^(8*d*x) + 4*(880* a^3*b*e^(6*c) - 2855*a^2*b^2*e^(6*c) + 2512*a*b^3*e^(6*c) - 630*b^4*e^(6*c ))*e^(6*d*x) - 4*(256*a^3*b*e^(4*c) - 823*a^2*b^2*e^(4*c) + 903*a*b^3*e^(4 *c) - 315*b^4*e^(4*c))*e^(4*d*x) - 12*(19*a^2*b^2*e^(2*c) - 54*a*b^3*e^(2* c) + 30*b^4*e^(2*c))*e^(2*d*x))/(a^5*b^2*d - 2*a^4*b^3*d + a^3*b^4*d - (a^ 5*b^2*d*e^(18*c) - 2*a^4*b^3*d*e^(18*c) + a^3*b^4*d*e^(18*c))*e^(18*d*x) + 9*(a^5*b^2*d*e^(16*c) - 2*a^4*b^3*d*e^(16*c) + a^3*b^4*d*e^(16*c))*e^(16* d*x) + 4*(8*a^6*b*d*e^(14*c) - 25*a^5*b^2*d*e^(14*c) + 26*a^4*b^3*d*e^(14* c) - 9*a^3*b^4*d*e^(14*c))*e^(14*d*x) - 4*(40*a^6*b*d*e^(12*c) - 101*a^5*b ^2*d*e^(12*c) + 82*a^4*b^3*d*e^(12*c) - 21*a^3*b^4*d*e^(12*c))*e^(12*d*x) - 2*(128*a^7*d*e^(10*c) - 416*a^6*b*d*e^(10*c) + 511*a^5*b^2*d*e^(10*c) - 286*a^4*b^3*d*e^(10*c) + 63*a^3*b^4*d*e^(10*c))*e^(10*d*x) + 2*(128*a^7*d* e^(8*c) - 416*a^6*b*d*e^(8*c) + 511*a^5*b^2*d*e^(8*c) - 286*a^4*b^3*d*e^(8 *c) + 63*a^3*b^4*d*e^(8*c))*e^(8*d*x) + 4*(40*a^6*b*d*e^(6*c) - 101*a^5...
\[ \int \frac {\text {csch}^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\int { -\frac {\operatorname {csch}\left (d x + c\right )^{2}}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{3}} \,d x } \] Input:
integrate(csch(d*x+c)^2/(a-b*sinh(d*x+c)^4)^3,x, algorithm="giac")
Output:
sage0*x
Timed out. \[ \int \frac {\text {csch}^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\int \frac {1}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^3} \,d x \] Input:
int(1/(sinh(c + d*x)^2*(a - b*sinh(c + d*x)^4)^3),x)
Output:
int(1/(sinh(c + d*x)^2*(a - b*sinh(c + d*x)^4)^3), x)
\[ \int \frac {\text {csch}^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx=\text {too large to display} \] Input:
int(csch(d*x+c)^2/(a-b*sinh(d*x+c)^4)^3,x)
Output:
(963049744316906864640*e**(22*c + 18*d*x)*int(e**(4*d*x)/(9732096*e**(28*c + 28*d*x)*a**4*b**3 + 9236480*e**(28*c + 28*d*x)*a**3*b**4 + 254720*e**(2 8*c + 28*d*x)*a**2*b**5 + 127920*e**(28*c + 28*d*x)*a*b**6 + 8505*e**(28*c + 28*d*x)*b**7 - 136249344*e**(26*c + 26*d*x)*a**4*b**3 - 129310720*e**(2 6*c + 26*d*x)*a**3*b**4 - 3566080*e**(26*c + 26*d*x)*a**2*b**5 - 1790880*e **(26*c + 26*d*x)*a*b**6 - 119070*e**(26*c + 26*d*x)*b**7 - 467140608*e**( 24*c + 24*d*x)*a**5*b**2 + 442269696*e**(24*c + 24*d*x)*a**4*b**3 + 828293 120*e**(24*c + 24*d*x)*a**3*b**4 + 17039360*e**(24*c + 24*d*x)*a**2*b**5 + 11232480*e**(24*c + 24*d*x)*a*b**6 + 773955*e**(24*c + 24*d*x)*b**7 + 467 1406080*e**(22*c + 22*d*x)*a**5*b**2 + 891027456*e**(22*c + 22*d*x)*a**4*b **3 - 3239813120*e**(22*c + 22*d*x)*a**3*b**4 - 31316480*e**(22*c + 22*d*x )*a**2*b**5 - 42480480*e**(22*c + 22*d*x)*a*b**6 - 3095820*e**(22*c + 22*d *x)*b**7 + 7474249728*e**(20*c + 20*d*x)*a**6*b - 13927710720*e**(20*c + 2 0*d*x)*a**5*b**2 - 10013343744*e**(20*c + 20*d*x)*a**4*b**3 + 8793763840*e **(20*c + 20*d*x)*a**3*b**4 - 14800640*e**(20*c + 20*d*x)*a**2*b**5 + 1096 77120*e**(20*c + 20*d*x)*a*b**6 + 8513505*e**(20*c + 20*d*x)*b**7 - 448454 98368*e**(18*c + 18*d*x)*a**6*b + 13495173120*e**(18*c + 18*d*x)*a**5*b**2 + 32544718848*e**(18*c + 18*d*x)*a**4*b**3 - 17613701120*e**(18*c + 18*d* x)*a**3*b**4 + 187678720*e**(18*c + 18*d*x)*a**2*b**5 - 207107040*e**(18*c + 18*d*x)*a*b**6 - 17027010*e**(18*c + 18*d*x)*b**7 - 39862665216*e**(...