Integrand size = 23, antiderivative size = 248 \[ \int \text {csch}^{20}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {(a+b)^3 \coth (c+d x)}{d}-\frac {(a+b)^2 (3 a+b) \coth ^3(c+d x)}{d}+\frac {3 (a+b) \left (12 a^2+9 a b+b^2\right ) \coth ^5(c+d x)}{5 d}-\frac {\left (84 a^3+105 a^2 b+30 a b^2+b^3\right ) \coth ^7(c+d x)}{7 d}+\frac {a \left (42 a^2+35 a b+5 b^2\right ) \coth ^9(c+d x)}{3 d}-\frac {3 a \left (42 a^2+21 a b+b^2\right ) \coth ^{11}(c+d x)}{11 d}+\frac {21 a^2 (4 a+b) \coth ^{13}(c+d x)}{13 d}-\frac {a^2 (12 a+b) \coth ^{15}(c+d x)}{5 d}+\frac {9 a^3 \coth ^{17}(c+d x)}{17 d}-\frac {a^3 \coth ^{19}(c+d x)}{19 d} \] Output:
(a+b)^3*coth(d*x+c)/d-(a+b)^2*(3*a+b)*coth(d*x+c)^3/d+3/5*(a+b)*(12*a^2+9* a*b+b^2)*coth(d*x+c)^5/d-1/7*(84*a^3+105*a^2*b+30*a*b^2+b^3)*coth(d*x+c)^7 /d+1/3*a*(42*a^2+35*a*b+5*b^2)*coth(d*x+c)^9/d-3/11*a*(42*a^2+21*a*b+b^2)* coth(d*x+c)^11/d+21/13*a^2*(4*a+b)*coth(d*x+c)^13/d-1/5*a^2*(12*a+b)*coth( d*x+c)^15/d+9/17*a^3*coth(d*x+c)^17/d-1/19*a^3*coth(d*x+c)^19/d
Leaf count is larger than twice the leaf count of optimal. \(548\) vs. \(2(248)=496\).
Time = 7.35 (sec) , antiderivative size = 548, normalized size of antiderivative = 2.21 \[ \int \text {csch}^{20}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {\left (-7945986048 a^3 \cosh (c+d x)-8939234304 a^2 b \cosh (c+d x)-6518191680 a b^2 \cosh (c+d x)-1792502712 b^3 \cosh (c+d x)+6501261312 a^3 \cosh (3 (c+d x))+18149354496 a^2 b \cosh (3 (c+d x))+14814072000 a b^2 \cosh (3 (c+d x))+4260103848 b^3 \cosh (3 (c+d x))-4334174208 a^3 \cosh (5 (c+d x))-14582690304 a^2 b \cosh (5 (c+d x))-14221509120 a b^2 \cosh (5 (c+d x))-4440518082 b^3 \cosh (5 (c+d x))+2333786112 a^3 \cosh (7 (c+d x))+7852217856 a^2 b \cosh (7 (c+d x))+8803791360 a b^2 \cosh (7 (c+d x))+3047642598 b^3 \cosh (7 (c+d x))-1000194048 a^3 \cosh (9 (c+d x))-3365236224 a^2 b \cosh (9 (c+d x))-3906077760 a b^2 \cosh (9 (c+d x))-1489040982 b^3 \cosh (9 (c+d x))+333398016 a^3 \cosh (11 (c+d x))+1121745408 a^2 b \cosh (11 (c+d x))+1302025920 a b^2 \cosh (11 (c+d x))+527386002 b^3 \cosh (11 (c+d x))-83349504 a^3 \cosh (13 (c+d x))-280436352 a^2 b \cosh (13 (c+d x))-325506480 a b^2 \cosh (13 (c+d x))-134271423 b^3 \cosh (13 (c+d x))+14708736 a^3 \cosh (15 (c+d x))+49488768 a^2 b \cosh (15 (c+d x))+57442320 a b^2 \cosh (15 (c+d x))+23694957 b^3 \cosh (15 (c+d x))-1634304 a^3 \cosh (17 (c+d x))-5498752 a^2 b \cosh (17 (c+d x))-6382480 a b^2 \cosh (17 (c+d x))-2632773 b^3 \cosh (17 (c+d x))+86016 a^3 \cosh (19 (c+d x))+289408 a^2 b \cosh (19 (c+d x))+335920 a b^2 \cosh (19 (c+d x))+138567 b^3 \cosh (19 (c+d x))\right ) \text {csch}^{19}(c+d x)}{79459860480 d} \] Input:
Integrate[Csch[c + d*x]^20*(a + b*Sinh[c + d*x]^4)^3,x]
Output:
((-7945986048*a^3*Cosh[c + d*x] - 8939234304*a^2*b*Cosh[c + d*x] - 6518191 680*a*b^2*Cosh[c + d*x] - 1792502712*b^3*Cosh[c + d*x] + 6501261312*a^3*Co sh[3*(c + d*x)] + 18149354496*a^2*b*Cosh[3*(c + d*x)] + 14814072000*a*b^2* Cosh[3*(c + d*x)] + 4260103848*b^3*Cosh[3*(c + d*x)] - 4334174208*a^3*Cosh [5*(c + d*x)] - 14582690304*a^2*b*Cosh[5*(c + d*x)] - 14221509120*a*b^2*Co sh[5*(c + d*x)] - 4440518082*b^3*Cosh[5*(c + d*x)] + 2333786112*a^3*Cosh[7 *(c + d*x)] + 7852217856*a^2*b*Cosh[7*(c + d*x)] + 8803791360*a*b^2*Cosh[7 *(c + d*x)] + 3047642598*b^3*Cosh[7*(c + d*x)] - 1000194048*a^3*Cosh[9*(c + d*x)] - 3365236224*a^2*b*Cosh[9*(c + d*x)] - 3906077760*a*b^2*Cosh[9*(c + d*x)] - 1489040982*b^3*Cosh[9*(c + d*x)] + 333398016*a^3*Cosh[11*(c + d* x)] + 1121745408*a^2*b*Cosh[11*(c + d*x)] + 1302025920*a*b^2*Cosh[11*(c + d*x)] + 527386002*b^3*Cosh[11*(c + d*x)] - 83349504*a^3*Cosh[13*(c + d*x)] - 280436352*a^2*b*Cosh[13*(c + d*x)] - 325506480*a*b^2*Cosh[13*(c + d*x)] - 134271423*b^3*Cosh[13*(c + d*x)] + 14708736*a^3*Cosh[15*(c + d*x)] + 49 488768*a^2*b*Cosh[15*(c + d*x)] + 57442320*a*b^2*Cosh[15*(c + d*x)] + 2369 4957*b^3*Cosh[15*(c + d*x)] - 1634304*a^3*Cosh[17*(c + d*x)] - 5498752*a^2 *b*Cosh[17*(c + d*x)] - 6382480*a*b^2*Cosh[17*(c + d*x)] - 2632773*b^3*Cos h[17*(c + d*x)] + 86016*a^3*Cosh[19*(c + d*x)] + 289408*a^2*b*Cosh[19*(c + d*x)] + 335920*a*b^2*Cosh[19*(c + d*x)] + 138567*b^3*Cosh[19*(c + d*x)])* Csch[c + d*x]^19)/(79459860480*d)
Time = 0.50 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 3696, 1584, 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.
| \(\displaystyle \int \text {csch}^{20}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin (i c+i d x)^4\right )^3}{\sin (i c+i d x)^{20}}dx\) |
| \(\Big \downarrow \) 3696 |
| \(\displaystyle \frac {\int \coth ^{20}(c+d x) \left (1-\tanh ^2(c+d x)\right )^3 \left ((a+b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^3d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1584 |
| \(\displaystyle \frac {\int \left (a^3 \coth ^{20}(c+d x)-9 a^3 \coth ^{18}(c+d x)+3 a^2 (12 a+b) \coth ^{16}(c+d x)-21 a^2 (4 a+b) \coth ^{14}(c+d x)+3 a \left (42 a^2+21 b a+b^2\right ) \coth ^{12}(c+d x)-3 a \left (42 a^2+35 b a+5 b^2\right ) \coth ^{10}(c+d x)+\left (84 a^3+105 b a^2+30 b^2 a+b^3\right ) \coth ^8(c+d x)+3 (a+b) \left (-12 a^2-9 b a-b^2\right ) \coth ^6(c+d x)+3 (a+b)^2 (3 a+b) \coth ^4(c+d x)-(a+b)^3 \coth ^2(c+d x)\right )d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {1}{19} a^3 \coth ^{19}(c+d x)+\frac {9}{17} a^3 \coth ^{17}(c+d x)-\frac {3}{11} a \left (42 a^2+21 a b+b^2\right ) \coth ^{11}(c+d x)+\frac {1}{3} a \left (42 a^2+35 a b+5 b^2\right ) \coth ^9(c+d x)+\frac {3}{5} (a+b) \left (12 a^2+9 a b+b^2\right ) \coth ^5(c+d x)-\frac {1}{5} a^2 (12 a+b) \coth ^{15}(c+d x)+\frac {21}{13} a^2 (4 a+b) \coth ^{13}(c+d x)-\frac {1}{7} \left (84 a^3+105 a^2 b+30 a b^2+b^3\right ) \coth ^7(c+d x)-(a+b)^2 (3 a+b) \coth ^3(c+d x)+(a+b)^3 \coth (c+d x)}{d}\) |
Input:
Int[Csch[c + d*x]^20*(a + b*Sinh[c + d*x]^4)^3,x]
Output:
((a + b)^3*Coth[c + d*x] - (a + b)^2*(3*a + b)*Coth[c + d*x]^3 + (3*(a + b )*(12*a^2 + 9*a*b + b^2)*Coth[c + d*x]^5)/5 - ((84*a^3 + 105*a^2*b + 30*a* b^2 + b^3)*Coth[c + d*x]^7)/7 + (a*(42*a^2 + 35*a*b + 5*b^2)*Coth[c + d*x] ^9)/3 - (3*a*(42*a^2 + 21*a*b + b^2)*Coth[c + d*x]^11)/11 + (21*a^2*(4*a + b)*Coth[c + d*x]^13)/13 - (a^2*(12*a + b)*Coth[c + d*x]^15)/5 + (9*a^3*Co th[c + d*x]^17)/17 - (a^3*Coth[c + d*x]^19)/19)/d
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* (a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
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]
Time = 9.52 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.20
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (\frac {65536}{230945}-\frac {\operatorname {csch}\left (d x +c \right )^{18}}{19}+\frac {18 \operatorname {csch}\left (d x +c \right )^{16}}{323}-\frac {96 \operatorname {csch}\left (d x +c \right )^{14}}{1615}+\frac {1344 \operatorname {csch}\left (d x +c \right )^{12}}{20995}-\frac {16128 \operatorname {csch}\left (d x +c \right )^{10}}{230945}+\frac {3584 \operatorname {csch}\left (d x +c \right )^{8}}{46189}-\frac {4096 \operatorname {csch}\left (d x +c \right )^{6}}{46189}+\frac {24576 \operatorname {csch}\left (d x +c \right )^{4}}{230945}-\frac {32768 \operatorname {csch}\left (d x +c \right )^{2}}{230945}\right ) \coth \left (d x +c \right )+3 a^{2} b \left (\frac {2048}{6435}-\frac {\operatorname {csch}\left (d x +c \right )^{14}}{15}+\frac {14 \operatorname {csch}\left (d x +c \right )^{12}}{195}-\frac {56 \operatorname {csch}\left (d x +c \right )^{10}}{715}+\frac {112 \operatorname {csch}\left (d x +c \right )^{8}}{1287}-\frac {128 \operatorname {csch}\left (d x +c \right )^{6}}{1287}+\frac {256 \operatorname {csch}\left (d x +c \right )^{4}}{2145}-\frac {1024 \operatorname {csch}\left (d x +c \right )^{2}}{6435}\right ) \coth \left (d x +c \right )+3 b^{2} a \left (\frac {256}{693}-\frac {\operatorname {csch}\left (d x +c \right )^{10}}{11}+\frac {10 \operatorname {csch}\left (d x +c \right )^{8}}{99}-\frac {80 \operatorname {csch}\left (d x +c \right )^{6}}{693}+\frac {32 \operatorname {csch}\left (d x +c \right )^{4}}{231}-\frac {128 \operatorname {csch}\left (d x +c \right )^{2}}{693}\right ) \coth \left (d x +c \right )+b^{3} \left (\frac {16}{35}-\frac {\operatorname {csch}\left (d x +c \right )^{6}}{7}+\frac {6 \operatorname {csch}\left (d x +c \right )^{4}}{35}-\frac {8 \operatorname {csch}\left (d x +c \right )^{2}}{35}\right ) \coth \left (d x +c \right )}{d}\) | \(298\) |
| default | \(\frac {a^{3} \left (\frac {65536}{230945}-\frac {\operatorname {csch}\left (d x +c \right )^{18}}{19}+\frac {18 \operatorname {csch}\left (d x +c \right )^{16}}{323}-\frac {96 \operatorname {csch}\left (d x +c \right )^{14}}{1615}+\frac {1344 \operatorname {csch}\left (d x +c \right )^{12}}{20995}-\frac {16128 \operatorname {csch}\left (d x +c \right )^{10}}{230945}+\frac {3584 \operatorname {csch}\left (d x +c \right )^{8}}{46189}-\frac {4096 \operatorname {csch}\left (d x +c \right )^{6}}{46189}+\frac {24576 \operatorname {csch}\left (d x +c \right )^{4}}{230945}-\frac {32768 \operatorname {csch}\left (d x +c \right )^{2}}{230945}\right ) \coth \left (d x +c \right )+3 a^{2} b \left (\frac {2048}{6435}-\frac {\operatorname {csch}\left (d x +c \right )^{14}}{15}+\frac {14 \operatorname {csch}\left (d x +c \right )^{12}}{195}-\frac {56 \operatorname {csch}\left (d x +c \right )^{10}}{715}+\frac {112 \operatorname {csch}\left (d x +c \right )^{8}}{1287}-\frac {128 \operatorname {csch}\left (d x +c \right )^{6}}{1287}+\frac {256 \operatorname {csch}\left (d x +c \right )^{4}}{2145}-\frac {1024 \operatorname {csch}\left (d x +c \right )^{2}}{6435}\right ) \coth \left (d x +c \right )+3 b^{2} a \left (\frac {256}{693}-\frac {\operatorname {csch}\left (d x +c \right )^{10}}{11}+\frac {10 \operatorname {csch}\left (d x +c \right )^{8}}{99}-\frac {80 \operatorname {csch}\left (d x +c \right )^{6}}{693}+\frac {32 \operatorname {csch}\left (d x +c \right )^{4}}{231}-\frac {128 \operatorname {csch}\left (d x +c \right )^{2}}{693}\right ) \coth \left (d x +c \right )+b^{3} \left (\frac {16}{35}-\frac {\operatorname {csch}\left (d x +c \right )^{6}}{7}+\frac {6 \operatorname {csch}\left (d x +c \right )^{4}}{35}-\frac {8 \operatorname {csch}\left (d x +c \right )^{2}}{35}\right ) \coth \left (d x +c \right )}{d}\) | \(298\) |
| parallelrisch | \(-\frac {9 \operatorname {csch}\left (\frac {d x}{2}+\frac {c}{2}\right )^{7} \left (\operatorname {sech}\left (\frac {d x}{2}+\frac {c}{2}\right )^{12} a^{3} \left (-\frac {\cosh \left (11 d x +11 c \right )}{3}+\cosh \left (9 d x +9 c \right )-\frac {7 \cosh \left (7 d x +7 c \right )}{3}+\frac {13 \cosh \left (5 d x +5 c \right )}{3}-\frac {13 \cosh \left (3 d x +3 c \right )}{2}+\frac {143 \cosh \left (d x +c \right )}{18}-\frac {\cosh \left (19 d x +19 c \right )}{11628}+\frac {\cosh \left (17 d x +17 c \right )}{612}-\frac {\cosh \left (15 d x +15 c \right )}{68}+\frac {\cosh \left (13 d x +13 c \right )}{12}\right ) \operatorname {csch}\left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\frac {910 b \,a^{2} \operatorname {sech}\left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \left (-3 \cosh \left (7 d x +7 c \right )+\frac {33 \cosh \left (5 d x +5 c \right )}{5}-11 \cosh \left (3 d x +3 c \right )+\frac {99 \cosh \left (d x +c \right )}{7}-\frac {\cosh \left (15 d x +15 c \right )}{455}+\frac {3 \cosh \left (13 d x +13 c \right )}{91}-\frac {3 \cosh \left (11 d x +11 c \right )}{13}+\cosh \left (9 d x +9 c \right )\right ) \operatorname {csch}\left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{27}+\frac {45760 b^{2} a \operatorname {sech}\left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (\cosh \left (9 d x +9 c \right )-\frac {\cosh \left (11 d x +11 c \right )}{11}+42 \cosh \left (d x +c \right )-30 \cosh \left (3 d x +3 c \right )+15 \cosh \left (5 d x +5 c \right )-5 \cosh \left (7 d x +7 c \right )\right ) \operatorname {csch}\left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{189}+\frac {732160 \left (\cosh \left (d x +c \right )-\frac {3 \cosh \left (3 d x +3 c \right )}{5}+\frac {\cosh \left (5 d x +5 c \right )}{5}-\frac {\cosh \left (7 d x +7 c \right )}{35}\right ) b^{3}}{9}\right ) \operatorname {sech}\left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{374865920 d}\) | \(408\) |
| risch | \(-\frac {32 \left (-532235847 b^{3} {\mathrm e}^{8 d x +8 c}+134271423 b^{3} {\mathrm e}^{6 d x +6 c}-14708736 a^{3} {\mathrm e}^{4 d x +4 c}-23694957 b^{3} {\mathrm e}^{4 d x +4 c}+1634304 a^{3} {\mathrm e}^{2 d x +2 c}+1862340480 a^{2} b \,{\mathrm e}^{22 d x +22 c}+4849845 b^{3} {\mathrm e}^{30 d x +30 c}+7945986048 a^{3} {\mathrm e}^{18 d x +18 c}-1352413920 a \,b^{2} {\mathrm e}^{24 d x +24 c}-8897848960 a^{2} b \,{\mathrm e}^{20 d x +20 c}-335920 b^{2} a +155195040 a \,b^{2} {\mathrm e}^{26 d x +26 c}+17837083264 a^{2} b \,{\mathrm e}^{18 d x +18 c}+5287716720 a \,b^{2} {\mathrm e}^{22 d x +22 c}-86016 a^{3}-6501261312 a^{3} {\mathrm e}^{16 d x +16 c}-61108047 b^{3} {\mathrm e}^{28 d x +28 c}+355978623 b^{3} {\mathrm e}^{26 d x +26 c}+4334174208 a^{3} {\mathrm e}^{14 d x +14 c}-289408 a^{2} b -12256713040 a \,b^{2} {\mathrm e}^{20 d x +20 c}+18774904720 a \,b^{2} {\mathrm e}^{18 d x +18 c}-20011694976 a^{2} b \,{\mathrm e}^{16 d x +16 c}-20101788720 a \,b^{2} {\mathrm e}^{16 d x +16 c}+14582690304 a^{2} b \,{\mathrm e}^{14 d x +14 c}+15573923040 a \,b^{2} {\mathrm e}^{14 d x +14 c}-7852217856 a^{2} b \,{\mathrm e}^{12 d x +12 c}-8958986400 a \,b^{2} {\mathrm e}^{12 d x +12 c}+3365236224 a^{2} b \,{\mathrm e}^{10 d x +10 c}+3906077760 a \,b^{2} {\mathrm e}^{10 d x +10 c}-1121745408 a^{2} b \,{\mathrm e}^{8 d x +8 c}-1302025920 a \,b^{2} {\mathrm e}^{8 d x +8 c}+280436352 a^{2} b \,{\mathrm e}^{6 d x +6 c}+325506480 a \,b^{2} {\mathrm e}^{6 d x +6 c}-49488768 a^{2} b \,{\mathrm e}^{4 d x +4 c}-57442320 a \,b^{2} {\mathrm e}^{4 d x +4 c}+5498752 a^{2} b \,{\mathrm e}^{2 d x +2 c}+6382480 a \,b^{2} {\mathrm e}^{2 d x +2 c}-138567 b^{3}+83349504 a^{3} {\mathrm e}^{6 d x +6 c}+1000194048 a^{3} {\mathrm e}^{10 d x +10 c}+1550149029 b^{3} {\mathrm e}^{10 d x +10 c}-333398016 a^{3} {\mathrm e}^{8 d x +8 c}+5711316039 b^{3} {\mathrm e}^{14 d x +14 c}-2333786112 a^{3} {\mathrm e}^{12 d x +12 c}-3403621221 b^{3} {\mathrm e}^{12 d x +12 c}-5504019807 b^{3} {\mathrm e}^{20 d x +20 c}+7296522519 b^{3} {\mathrm e}^{18 d x +18 c}-7366637421 b^{3} {\mathrm e}^{16 d x +16 c}+3106533573 b^{3} {\mathrm e}^{22 d x +22 c}+2632773 b^{3} {\mathrm e}^{2 d x +2 c}-1270797957 b^{3} {\mathrm e}^{24 d x +24 c}\right )}{4849845 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{19}}\) | \(738\) |
Input:
int(csch(d*x+c)^20*(a+b*sinh(d*x+c)^4)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(a^3*(65536/230945-1/19*csch(d*x+c)^18+18/323*csch(d*x+c)^16-96/1615*c sch(d*x+c)^14+1344/20995*csch(d*x+c)^12-16128/230945*csch(d*x+c)^10+3584/4 6189*csch(d*x+c)^8-4096/46189*csch(d*x+c)^6+24576/230945*csch(d*x+c)^4-327 68/230945*csch(d*x+c)^2)*coth(d*x+c)+3*a^2*b*(2048/6435-1/15*csch(d*x+c)^1 4+14/195*csch(d*x+c)^12-56/715*csch(d*x+c)^10+112/1287*csch(d*x+c)^8-128/1 287*csch(d*x+c)^6+256/2145*csch(d*x+c)^4-1024/6435*csch(d*x+c)^2)*coth(d*x +c)+3*b^2*a*(256/693-1/11*csch(d*x+c)^10+10/99*csch(d*x+c)^8-80/693*csch(d *x+c)^6+32/231*csch(d*x+c)^4-128/693*csch(d*x+c)^2)*coth(d*x+c)+b^3*(16/35 -1/7*csch(d*x+c)^6+6/35*csch(d*x+c)^4-8/35*csch(d*x+c)^2)*coth(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 4259 vs. \(2 (232) = 464\).
Time = 0.11 (sec) , antiderivative size = 4259, normalized size of antiderivative = 17.17 \[ \int \text {csch}^{20}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)^20*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \text {csch}^{20}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Timed out} \] Input:
integrate(csch(d*x+c)**20*(a+b*sinh(d*x+c)**4)**3,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 4883 vs. \(2 (232) = 464\).
Time = 0.07 (sec) , antiderivative size = 4883, normalized size of antiderivative = 19.69 \[ \int \text {csch}^{20}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)^20*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")
Output:
131072/230945*a^3*(19*e^(-2*d*x - 2*c)/(d*(19*e^(-2*d*x - 2*c) - 171*e^(-4 *d*x - 4*c) + 969*e^(-6*d*x - 6*c) - 3876*e^(-8*d*x - 8*c) + 11628*e^(-10* d*x - 10*c) - 27132*e^(-12*d*x - 12*c) + 50388*e^(-14*d*x - 14*c) - 75582* e^(-16*d*x - 16*c) + 92378*e^(-18*d*x - 18*c) - 92378*e^(-20*d*x - 20*c) + 75582*e^(-22*d*x - 22*c) - 50388*e^(-24*d*x - 24*c) + 27132*e^(-26*d*x - 26*c) - 11628*e^(-28*d*x - 28*c) + 3876*e^(-30*d*x - 30*c) - 969*e^(-32*d* x - 32*c) + 171*e^(-34*d*x - 34*c) - 19*e^(-36*d*x - 36*c) + e^(-38*d*x - 38*c) - 1)) - 171*e^(-4*d*x - 4*c)/(d*(19*e^(-2*d*x - 2*c) - 171*e^(-4*d*x - 4*c) + 969*e^(-6*d*x - 6*c) - 3876*e^(-8*d*x - 8*c) + 11628*e^(-10*d*x - 10*c) - 27132*e^(-12*d*x - 12*c) + 50388*e^(-14*d*x - 14*c) - 75582*e^(- 16*d*x - 16*c) + 92378*e^(-18*d*x - 18*c) - 92378*e^(-20*d*x - 20*c) + 755 82*e^(-22*d*x - 22*c) - 50388*e^(-24*d*x - 24*c) + 27132*e^(-26*d*x - 26*c ) - 11628*e^(-28*d*x - 28*c) + 3876*e^(-30*d*x - 30*c) - 969*e^(-32*d*x - 32*c) + 171*e^(-34*d*x - 34*c) - 19*e^(-36*d*x - 36*c) + e^(-38*d*x - 38*c ) - 1)) + 969*e^(-6*d*x - 6*c)/(d*(19*e^(-2*d*x - 2*c) - 171*e^(-4*d*x - 4 *c) + 969*e^(-6*d*x - 6*c) - 3876*e^(-8*d*x - 8*c) + 11628*e^(-10*d*x - 10 *c) - 27132*e^(-12*d*x - 12*c) + 50388*e^(-14*d*x - 14*c) - 75582*e^(-16*d *x - 16*c) + 92378*e^(-18*d*x - 18*c) - 92378*e^(-20*d*x - 20*c) + 75582*e ^(-22*d*x - 22*c) - 50388*e^(-24*d*x - 24*c) + 27132*e^(-26*d*x - 26*c) - 11628*e^(-28*d*x - 28*c) + 3876*e^(-30*d*x - 30*c) - 969*e^(-32*d*x - 3...
Leaf count of result is larger than twice the leaf count of optimal. 737 vs. \(2 (232) = 464\).
Time = 0.36 (sec) , antiderivative size = 737, normalized size of antiderivative = 2.97 \[ \int \text {csch}^{20}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx =\text {Too large to display} \] Input:
integrate(csch(d*x+c)^20*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")
Output:
-32/4849845*(4849845*b^3*e^(30*d*x + 30*c) - 61108047*b^3*e^(28*d*x + 28*c ) + 155195040*a*b^2*e^(26*d*x + 26*c) + 355978623*b^3*e^(26*d*x + 26*c) - 1352413920*a*b^2*e^(24*d*x + 24*c) - 1270797957*b^3*e^(24*d*x + 24*c) + 18 62340480*a^2*b*e^(22*d*x + 22*c) + 5287716720*a*b^2*e^(22*d*x + 22*c) + 31 06533573*b^3*e^(22*d*x + 22*c) - 8897848960*a^2*b*e^(20*d*x + 20*c) - 1225 6713040*a*b^2*e^(20*d*x + 20*c) - 5504019807*b^3*e^(20*d*x + 20*c) + 79459 86048*a^3*e^(18*d*x + 18*c) + 17837083264*a^2*b*e^(18*d*x + 18*c) + 187749 04720*a*b^2*e^(18*d*x + 18*c) + 7296522519*b^3*e^(18*d*x + 18*c) - 6501261 312*a^3*e^(16*d*x + 16*c) - 20011694976*a^2*b*e^(16*d*x + 16*c) - 20101788 720*a*b^2*e^(16*d*x + 16*c) - 7366637421*b^3*e^(16*d*x + 16*c) + 433417420 8*a^3*e^(14*d*x + 14*c) + 14582690304*a^2*b*e^(14*d*x + 14*c) + 1557392304 0*a*b^2*e^(14*d*x + 14*c) + 5711316039*b^3*e^(14*d*x + 14*c) - 2333786112* a^3*e^(12*d*x + 12*c) - 7852217856*a^2*b*e^(12*d*x + 12*c) - 8958986400*a* b^2*e^(12*d*x + 12*c) - 3403621221*b^3*e^(12*d*x + 12*c) + 1000194048*a^3* e^(10*d*x + 10*c) + 3365236224*a^2*b*e^(10*d*x + 10*c) + 3906077760*a*b^2* e^(10*d*x + 10*c) + 1550149029*b^3*e^(10*d*x + 10*c) - 333398016*a^3*e^(8* d*x + 8*c) - 1121745408*a^2*b*e^(8*d*x + 8*c) - 1302025920*a*b^2*e^(8*d*x + 8*c) - 532235847*b^3*e^(8*d*x + 8*c) + 83349504*a^3*e^(6*d*x + 6*c) + 28 0436352*a^2*b*e^(6*d*x + 6*c) + 325506480*a*b^2*e^(6*d*x + 6*c) + 13427142 3*b^3*e^(6*d*x + 6*c) - 14708736*a^3*e^(4*d*x + 4*c) - 49488768*a^2*b*e...
Time = 2.28 (sec) , antiderivative size = 5190, normalized size of antiderivative = 20.93 \[ \int \text {csch}^{20}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:
int((a + b*sinh(c + d*x)^4)^3/sinh(c + d*x)^20,x)
Output:
((512*b*(112*a*b + 128*a^2 + 33*b^2))/(138567*d) + (1792*b^3*exp(8*c + 8*d *x))/(969*d) - (448*b^3*exp(10*c + 10*d*x))/(969*d) - (64*b*exp(2*c + 2*d* x)*(448*a*b + 256*a^2 + 165*b^2))/(12597*d) - (256*b^2*exp(6*c + 6*d*x)*(8 *a + 11*b))/(969*d) + (512*b^2*exp(4*c + 4*d*x)*(96*a + 55*b))/(12597*d))/ (9*exp(2*c + 2*d*x) - 36*exp(4*c + 4*d*x) + 84*exp(6*c + 6*d*x) - 126*exp( 8*c + 8*d*x) + 126*exp(10*c + 10*d*x) - 84*exp(12*c + 12*d*x) + 36*exp(14* c + 14*d*x) - 9*exp(16*c + 16*d*x) + exp(18*c + 18*d*x) - 1) - ((8*b*(448* a*b + 256*a^2 + 165*b^2))/(12597*d) + (128*exp(4*c + 4*d*x)*(840*a*b^2 + 1 152*a^2*b + 1024*a^3 + 231*b^3))/(4199*d) - (2816*b^3*exp(14*c + 14*d*x))/ (323*d) + (440*b^3*exp(16*c + 16*d*x))/(323*d) - (512*b*exp(2*c + 2*d*x)*( 112*a*b + 128*a^2 + 33*b^2))/(12597*d) - (2560*b*exp(6*c + 6*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(4199*d) + (880*b*exp(8*c + 8*d*x)*(448*a*b + 256*a^ 2 + 165*b^2))/(4199*d) + (704*b^2*exp(12*c + 12*d*x)*(8*a + 11*b))/(323*d) - (2816*b^2*exp(10*c + 10*d*x)*(96*a + 55*b))/(4199*d))/(66*exp(4*c + 4*d *x) - 12*exp(2*c + 2*d*x) - 220*exp(6*c + 6*d*x) + 495*exp(8*c + 8*d*x) - 792*exp(10*c + 10*d*x) + 924*exp(12*c + 12*d*x) - 792*exp(14*c + 14*d*x) + 495*exp(16*c + 16*d*x) - 220*exp(18*c + 18*d*x) + 66*exp(20*c + 20*d*x) - 12*exp(22*c + 22*d*x) + exp(24*c + 24*d*x) + 1) - ((512*exp(18*c + 18*d*x )*(840*a*b^2 + 1152*a^2*b + 1024*a^3 + 231*b^3))/(19*d) + (128*b^3*exp(6*c + 6*d*x))/(19*d) - (1536*b^3*exp(8*c + 8*d*x))/(19*d) - (1536*b^3*exp(...
Time = 0.20 (sec) , antiderivative size = 1002, normalized size of antiderivative = 4.04 \[ \int \text {csch}^{20}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx =\text {Too large to display} \] Input:
int(csch(d*x+c)^20*(a+b*sinh(d*x+c)^4)^3,x)
Output:
(32*( - 4849845*e**(30*c + 30*d*x)*b**3 + 61108047*e**(28*c + 28*d*x)*b**3 - 155195040*e**(26*c + 26*d*x)*a*b**2 - 355978623*e**(26*c + 26*d*x)*b**3 + 1352413920*e**(24*c + 24*d*x)*a*b**2 + 1270797957*e**(24*c + 24*d*x)*b* *3 - 1862340480*e**(22*c + 22*d*x)*a**2*b - 5287716720*e**(22*c + 22*d*x)* a*b**2 - 3106533573*e**(22*c + 22*d*x)*b**3 + 8897848960*e**(20*c + 20*d*x )*a**2*b + 12256713040*e**(20*c + 20*d*x)*a*b**2 + 5504019807*e**(20*c + 2 0*d*x)*b**3 - 7945986048*e**(18*c + 18*d*x)*a**3 - 17837083264*e**(18*c + 18*d*x)*a**2*b - 18774904720*e**(18*c + 18*d*x)*a*b**2 - 7296522519*e**(18 *c + 18*d*x)*b**3 + 6501261312*e**(16*c + 16*d*x)*a**3 + 20011694976*e**(1 6*c + 16*d*x)*a**2*b + 20101788720*e**(16*c + 16*d*x)*a*b**2 + 7366637421* e**(16*c + 16*d*x)*b**3 - 4334174208*e**(14*c + 14*d*x)*a**3 - 14582690304 *e**(14*c + 14*d*x)*a**2*b - 15573923040*e**(14*c + 14*d*x)*a*b**2 - 57113 16039*e**(14*c + 14*d*x)*b**3 + 2333786112*e**(12*c + 12*d*x)*a**3 + 78522 17856*e**(12*c + 12*d*x)*a**2*b + 8958986400*e**(12*c + 12*d*x)*a*b**2 + 3 403621221*e**(12*c + 12*d*x)*b**3 - 1000194048*e**(10*c + 10*d*x)*a**3 - 3 365236224*e**(10*c + 10*d*x)*a**2*b - 3906077760*e**(10*c + 10*d*x)*a*b**2 - 1550149029*e**(10*c + 10*d*x)*b**3 + 333398016*e**(8*c + 8*d*x)*a**3 + 1121745408*e**(8*c + 8*d*x)*a**2*b + 1302025920*e**(8*c + 8*d*x)*a*b**2 + 532235847*e**(8*c + 8*d*x)*b**3 - 83349504*e**(6*c + 6*d*x)*a**3 - 2804363 52*e**(6*c + 6*d*x)*a**2*b - 325506480*e**(6*c + 6*d*x)*a*b**2 - 134271...