Integrand size = 23, antiderivative size = 221 \[ \int \text {csch}^{18}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=-\frac {(a+b)^3 \coth (c+d x)}{d}+\frac {2 (a+b)^2 (4 a+b) \coth ^3(c+d x)}{3 d}-\frac {(a+b) \left (28 a^2+17 a b+b^2\right ) \coth ^5(c+d x)}{5 d}+\frac {4 a \left (14 a^2+15 a b+3 b^2\right ) \coth ^7(c+d x)}{7 d}-\frac {a \left (70 a^2+45 a b+3 b^2\right ) \coth ^9(c+d x)}{9 d}+\frac {2 a^2 (28 a+9 b) \coth ^{11}(c+d x)}{11 d}-\frac {a^2 (28 a+3 b) \coth ^{13}(c+d x)}{13 d}+\frac {8 a^3 \coth ^{15}(c+d x)}{15 d}-\frac {a^3 \coth ^{17}(c+d x)}{17 d} \]
-(a+b)^3*coth(d*x+c)/d+2/3*(a+b)^2*(4*a+b)*coth(d*x+c)^3/d-1/5*(a+b)*(28*a ^2+17*a*b+b^2)*coth(d*x+c)^5/d+4/7*a*(14*a^2+15*a*b+3*b^2)*coth(d*x+c)^7/d -1/9*a*(70*a^2+45*a*b+3*b^2)*coth(d*x+c)^9/d+2/11*a^2*(28*a+9*b)*coth(d*x+ c)^11/d-1/13*a^2*(28*a+3*b)*coth(d*x+c)^13/d+8/15*a^3*coth(d*x+c)^15/d-1/1 7*a^3*coth(d*x+c)^17/d
Leaf count is larger than twice the leaf count of optimal. \(458\) vs. \(2(221)=442\).
Time = 6.01 (sec) , antiderivative size = 458, normalized size of antiderivative = 2.07 \[ \int \text {csch}^{18}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=-\frac {\left (680680 \left (1024 a^3+1152 a^2 b+840 a b^2+231 b^3\right ) \cosh (c+d x)-272272 \left (2048 a^3+5760 a^2 b+4704 a b^2+1353 b^3\right ) \cosh (3 (c+d x))+354844672 a^3 \cosh (5 (c+d x))+1211857920 a^2 b \cosh (5 (c+d x))+1189284096 a b^2 \cosh (5 (c+d x))+372263892 b^3 \cosh (5 (c+d x))-177422336 a^3 \cosh (7 (c+d x))-605928960 a^2 b \cosh (7 (c+d x))-692659968 a b^2 \cosh (7 (c+d x))-242288046 b^3 \cosh (7 (c+d x))+68239360 a^3 \cosh (9 (c+d x))+233049600 a^2 b \cosh (9 (c+d x))+277717440 a b^2 \cosh (9 (c+d x))+108738630 b^3 \cosh (9 (c+d x))-19496960 a^3 \cosh (11 (c+d x))-66585600 a^2 b \cosh (11 (c+d x))-79347840 a b^2 \cosh (11 (c+d x))-33693660 b^3 \cosh (11 (c+d x))+3899392 a^3 \cosh (13 (c+d x))+13317120 a^2 b \cosh (13 (c+d x))+15869568 a b^2 \cosh (13 (c+d x))+6942936 b^3 \cosh (13 (c+d x))-487424 a^3 \cosh (15 (c+d x))-1664640 a^2 b \cosh (15 (c+d x))-1983696 a b^2 \cosh (15 (c+d x))-867867 b^3 \cosh (15 (c+d x))+28672 a^3 \cosh (17 (c+d x))+97920 a^2 b \cosh (17 (c+d x))+116688 a b^2 \cosh (17 (c+d x))+51051 b^3 \cosh (17 (c+d x))\right ) \text {csch}^{17}(c+d x)}{6273146880 d} \]
-1/6273146880*((680680*(1024*a^3 + 1152*a^2*b + 840*a*b^2 + 231*b^3)*Cosh[ c + d*x] - 272272*(2048*a^3 + 5760*a^2*b + 4704*a*b^2 + 1353*b^3)*Cosh[3*( c + d*x)] + 354844672*a^3*Cosh[5*(c + d*x)] + 1211857920*a^2*b*Cosh[5*(c + d*x)] + 1189284096*a*b^2*Cosh[5*(c + d*x)] + 372263892*b^3*Cosh[5*(c + d* x)] - 177422336*a^3*Cosh[7*(c + d*x)] - 605928960*a^2*b*Cosh[7*(c + d*x)] - 692659968*a*b^2*Cosh[7*(c + d*x)] - 242288046*b^3*Cosh[7*(c + d*x)] + 68 239360*a^3*Cosh[9*(c + d*x)] + 233049600*a^2*b*Cosh[9*(c + d*x)] + 2777174 40*a*b^2*Cosh[9*(c + d*x)] + 108738630*b^3*Cosh[9*(c + d*x)] - 19496960*a^ 3*Cosh[11*(c + d*x)] - 66585600*a^2*b*Cosh[11*(c + d*x)] - 79347840*a*b^2* Cosh[11*(c + d*x)] - 33693660*b^3*Cosh[11*(c + d*x)] + 3899392*a^3*Cosh[13 *(c + d*x)] + 13317120*a^2*b*Cosh[13*(c + d*x)] + 15869568*a*b^2*Cosh[13*( c + d*x)] + 6942936*b^3*Cosh[13*(c + d*x)] - 487424*a^3*Cosh[15*(c + d*x)] - 1664640*a^2*b*Cosh[15*(c + d*x)] - 1983696*a*b^2*Cosh[15*(c + d*x)] - 8 67867*b^3*Cosh[15*(c + d*x)] + 28672*a^3*Cosh[17*(c + d*x)] + 97920*a^2*b* Cosh[17*(c + d*x)] + 116688*a*b^2*Cosh[17*(c + d*x)] + 51051*b^3*Cosh[17*( c + d*x)])*Csch[c + d*x]^17)/d
Time = 0.44 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 25, 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.
| \(\displaystyle \int \text {csch}^{18}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\left (a+b \sin (i c+i d x)^4\right )^3}{\sin (i c+i d x)^{18}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\left (b \sin (i c+i d x)^4+a\right )^3}{\sin (i c+i d x)^{18}}dx\) |
| \(\Big \downarrow \) 3696 |
| \(\displaystyle \frac {\int \coth ^{18}(c+d x) \left (1-\tanh ^2(c+d x)\right )^2 \left ((a+b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^3d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1584 |
| \(\displaystyle \frac {\int \left (a^3 \coth ^{18}(c+d x)-8 a^3 \coth ^{16}(c+d x)+a^2 (28 a+3 b) \coth ^{14}(c+d x)-2 a^2 (28 a+9 b) \coth ^{12}(c+d x)+a \left (70 a^2+45 b a+3 b^2\right ) \coth ^{10}(c+d x)-4 a \left (14 a^2+15 b a+3 b^2\right ) \coth ^8(c+d x)+(a+b) \left (28 a^2+17 b a+b^2\right ) \coth ^6(c+d x)-2 (a+b)^2 (4 a+b) \coth ^4(c+d x)+(a+b)^3 \coth ^2(c+d x)\right )d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {1}{17} a^3 \coth ^{17}(c+d x)+\frac {8}{15} a^3 \coth ^{15}(c+d x)-\frac {1}{9} a \left (70 a^2+45 a b+3 b^2\right ) \coth ^9(c+d x)+\frac {4}{7} a \left (14 a^2+15 a b+3 b^2\right ) \coth ^7(c+d x)-\frac {1}{5} (a+b) \left (28 a^2+17 a b+b^2\right ) \coth ^5(c+d x)-\frac {1}{13} a^2 (28 a+3 b) \coth ^{13}(c+d x)+\frac {2}{11} a^2 (28 a+9 b) \coth ^{11}(c+d x)+\frac {2}{3} (a+b)^2 (4 a+b) \coth ^3(c+d x)-(a+b)^3 \coth (c+d x)}{d}\) |
(-((a + b)^3*Coth[c + d*x]) + (2*(a + b)^2*(4*a + b)*Coth[c + d*x]^3)/3 - ((a + b)*(28*a^2 + 17*a*b + b^2)*Coth[c + d*x]^5)/5 + (4*a*(14*a^2 + 15*a* b + 3*b^2)*Coth[c + d*x]^7)/7 - (a*(70*a^2 + 45*a*b + 3*b^2)*Coth[c + d*x] ^9)/9 + (2*a^2*(28*a + 9*b)*Coth[c + d*x]^11)/11 - (a^2*(28*a + 3*b)*Coth[ c + d*x]^13)/13 + (8*a^3*Coth[c + d*x]^15)/15 - (a^3*Coth[c + d*x]^17)/17) /d
3.3.27.3.1 Defintions of rubi rules used
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 = 3.24 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.17
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (-\frac {32768}{109395}-\frac {\operatorname {csch}\left (d x +c \right )^{16}}{17}+\frac {16 \operatorname {csch}\left (d x +c \right )^{14}}{255}-\frac {224 \operatorname {csch}\left (d x +c \right )^{12}}{3315}+\frac {896 \operatorname {csch}\left (d x +c \right )^{10}}{12155}-\frac {1792 \operatorname {csch}\left (d x +c \right )^{8}}{21879}+\frac {2048 \operatorname {csch}\left (d x +c \right )^{6}}{21879}-\frac {4096 \operatorname {csch}\left (d x +c \right )^{4}}{36465}+\frac {16384 \operatorname {csch}\left (d x +c \right )^{2}}{109395}\right ) \coth \left (d x +c \right )+3 a^{2} b \left (-\frac {1024}{3003}-\frac {\operatorname {csch}\left (d x +c \right )^{12}}{13}+\frac {12 \operatorname {csch}\left (d x +c \right )^{10}}{143}-\frac {40 \operatorname {csch}\left (d x +c \right )^{8}}{429}+\frac {320 \operatorname {csch}\left (d x +c \right )^{6}}{3003}-\frac {128 \operatorname {csch}\left (d x +c \right )^{4}}{1001}+\frac {512 \operatorname {csch}\left (d x +c \right )^{2}}{3003}\right ) \coth \left (d x +c \right )+3 a \,b^{2} \left (-\frac {128}{315}-\frac {\operatorname {csch}\left (d x +c \right )^{8}}{9}+\frac {8 \operatorname {csch}\left (d x +c \right )^{6}}{63}-\frac {16 \operatorname {csch}\left (d x +c \right )^{4}}{105}+\frac {64 \operatorname {csch}\left (d x +c \right )^{2}}{315}\right ) \coth \left (d x +c \right )+b^{3} \left (-\frac {8}{15}-\frac {\operatorname {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )}{d}\) | \(258\) |
| default | \(\frac {a^{3} \left (-\frac {32768}{109395}-\frac {\operatorname {csch}\left (d x +c \right )^{16}}{17}+\frac {16 \operatorname {csch}\left (d x +c \right )^{14}}{255}-\frac {224 \operatorname {csch}\left (d x +c \right )^{12}}{3315}+\frac {896 \operatorname {csch}\left (d x +c \right )^{10}}{12155}-\frac {1792 \operatorname {csch}\left (d x +c \right )^{8}}{21879}+\frac {2048 \operatorname {csch}\left (d x +c \right )^{6}}{21879}-\frac {4096 \operatorname {csch}\left (d x +c \right )^{4}}{36465}+\frac {16384 \operatorname {csch}\left (d x +c \right )^{2}}{109395}\right ) \coth \left (d x +c \right )+3 a^{2} b \left (-\frac {1024}{3003}-\frac {\operatorname {csch}\left (d x +c \right )^{12}}{13}+\frac {12 \operatorname {csch}\left (d x +c \right )^{10}}{143}-\frac {40 \operatorname {csch}\left (d x +c \right )^{8}}{429}+\frac {320 \operatorname {csch}\left (d x +c \right )^{6}}{3003}-\frac {128 \operatorname {csch}\left (d x +c \right )^{4}}{1001}+\frac {512 \operatorname {csch}\left (d x +c \right )^{2}}{3003}\right ) \coth \left (d x +c \right )+3 a \,b^{2} \left (-\frac {128}{315}-\frac {\operatorname {csch}\left (d x +c \right )^{8}}{9}+\frac {8 \operatorname {csch}\left (d x +c \right )^{6}}{63}-\frac {16 \operatorname {csch}\left (d x +c \right )^{4}}{105}+\frac {64 \operatorname {csch}\left (d x +c \right )^{2}}{315}\right ) \coth \left (d x +c \right )+b^{3} \left (-\frac {8}{15}-\frac {\operatorname {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )}{d}\) | \(258\) |
| parallelrisch | \(\frac {-45045 \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{17} a^{3}+867867 \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{15} a^{3}-8011080 a^{2} \left (a +\frac {6 b}{17}\right ) \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}+47338200 \left (a +\frac {78 b}{85}\right ) a^{2} \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}+\left (-202502300 a^{3}-318558240 a^{2} b -65345280 a \,b^{2}\right ) \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\left (676936260 a^{3}+1501774560 a^{2} b +756138240 a \,b^{2}\right ) \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (-1895421528 a^{3}-5256210960 a^{2} b -4234374144 a \,b^{2}-627314688 b^{3}\right ) \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (4964199240 a^{3}+15768632880 a^{2} b +16467010560 a \,b^{2}+5227622400 b^{3}\right ) \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-18615747150 a^{3}-63074531520 a^{2} b -74101547520 a \,b^{2}-31365734400 b^{3}\right ) \coth \left (\frac {d x}{2}+\frac {c}{2}\right )-45045 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{16} a^{3}-\frac {289 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{14} a^{3}}{15}+\frac {2312 a^{2} \left (a +\frac {6 b}{17}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{13}-\frac {11560 \left (a +\frac {78 b}{85}\right ) a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{11}+\left (\frac {40460}{9} a^{3}+7072 a^{2} b +\frac {4352}{3} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (-15028 a^{3}-\frac {233376}{7} a^{2} b -\frac {117504}{7} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (\frac {210392}{5} a^{3}+\frac {69632}{5} b^{3}+116688 a^{2} b +\frac {470016}{5} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-\frac {348160}{3} b^{3}-350064 a^{2} b -365568 a \,b^{2}-\frac {330616}{3} a^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+413270 a^{3}+1400256 a^{2} b +1645056 a \,b^{2}+696320 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{100370350080 d}\) | \(481\) |
| risch | \(-\frac {16 \left (510510 b^{3} {\mathrm e}^{28 d x +28 c}-279095817 b^{3} {\mathrm e}^{10 d x +10 c}+68239360 a^{3} {\mathrm e}^{8 d x +8 c}-19496960 a^{3} {\mathrm e}^{6 d x +6 c}+36807771 b^{3} {\mathrm e}^{24 d x +24 c}-129771642 b^{3} {\mathrm e}^{22 d x +22 c}+312227916 b^{3} {\mathrm e}^{20 d x +20 c}-541906365 b^{3} {\mathrm e}^{18 d x +18 c}+699143445 b^{3} {\mathrm e}^{16 d x +16 c}-680611932 b^{3} {\mathrm e}^{14 d x +14 c}+97920 a^{2} b +28672 a^{3}+51051 b^{3}+116688 a \,b^{2}+168030720 a^{2} b \,{\mathrm e}^{20 d x +20 c}+14702688 a \,b^{2} {\mathrm e}^{24 d x +24 c}+354844672 a^{3} {\mathrm e}^{12 d x +12 c}+502035534 b^{3} {\mathrm e}^{12 d x +12 c}-177422336 a^{3} {\mathrm e}^{10 d x +10 c}+115120005 b^{3} {\mathrm e}^{8 d x +8 c}+277717440 \,{\mathrm e}^{8 d x +8 c} a \,b^{2}-66585600 b \,a^{2} {\mathrm e}^{6 d x +6 c}-79347840 a \,b^{2} {\mathrm e}^{6 d x +6 c}-1664640 a^{2} b \,{\mathrm e}^{2 d x +2 c}+13317120 a^{2} b \,{\mathrm e}^{4 d x +4 c}-1983696 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}-6381375 b^{3} {\mathrm e}^{26 d x +26 c}-557613056 a^{3} {\mathrm e}^{14 d x +14 c}-487424 a^{3} {\mathrm e}^{2 d x +2 c}-867867 b^{3} {\mathrm e}^{2 d x +2 c}-34204170 b^{3} {\mathrm e}^{6 d x +6 c}+3899392 a^{3} {\mathrm e}^{4 d x +4 c}+6942936 b^{3} {\mathrm e}^{4 d x +4 c}+697016320 a^{3} {\mathrm e}^{16 d x +16 c}+15869568 \,{\mathrm e}^{4 d x +4 c} b^{2} a +494290368 a \,b^{2} {\mathrm e}^{20 d x +20 c}-1132457040 a \,b^{2} {\mathrm e}^{18 d x +18 c}+1582289280 a^{2} b \,{\mathrm e}^{16 d x +16 c}+1704228240 a \,b^{2} {\mathrm e}^{16 d x +16 c}-1736317440 a^{2} b \,{\mathrm e}^{14 d x +14 c}-127423296 a \,b^{2} {\mathrm e}^{22 d x +22 c}-798145920 a^{2} b \,{\mathrm e}^{18 d x +18 c}-1775057856 a \,b^{2} {\mathrm e}^{14 d x +14 c}+1211857920 a^{2} b \,{\mathrm e}^{12 d x +12 c}+1316707392 a \,b^{2} {\mathrm e}^{12 d x +12 c}-605928960 a^{2} b \,{\mathrm e}^{10 d x +10 c}-707362656 a \,b^{2} {\mathrm e}^{10 d x +10 c}+233049600 a^{2} b \,{\mathrm e}^{8 d x +8 c}\right )}{765765 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{17}}\) | \(680\) |
1/d*(a^3*(-32768/109395-1/17*csch(d*x+c)^16+16/255*csch(d*x+c)^14-224/3315 *csch(d*x+c)^12+896/12155*csch(d*x+c)^10-1792/21879*csch(d*x+c)^8+2048/218 79*csch(d*x+c)^6-4096/36465*csch(d*x+c)^4+16384/109395*csch(d*x+c)^2)*coth (d*x+c)+3*a^2*b*(-1024/3003-1/13*csch(d*x+c)^12+12/143*csch(d*x+c)^10-40/4 29*csch(d*x+c)^8+320/3003*csch(d*x+c)^6-128/1001*csch(d*x+c)^4+512/3003*cs ch(d*x+c)^2)*coth(d*x+c)+3*a*b^2*(-128/315-1/9*csch(d*x+c)^8+8/63*csch(d*x +c)^6-16/105*csch(d*x+c)^4+64/315*csch(d*x+c)^2)*coth(d*x+c)+b^3*(-8/15-1/ 5*csch(d*x+c)^4+4/15*csch(d*x+c)^2)*coth(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 3585 vs. \(2 (205) = 410\).
Time = 0.31 (sec) , antiderivative size = 3585, normalized size of antiderivative = 16.22 \[ \int \text {csch}^{18}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Too large to display} \]
-16/765765*((28672*a^3 + 97920*a^2*b + 116688*a*b^2 + 561561*b^3)*cosh(d*x + c)^14 - 14*(28672*a^3 + 97920*a^2*b + 116688*a*b^2 - 459459*b^3)*cosh(d *x + c)*sinh(d*x + c)^13 + (28672*a^3 + 97920*a^2*b + 116688*a*b^2 + 56156 1*b^3)*sinh(d*x + c)^14 - 34*(14336*a^3 + 48960*a^2*b + 58344*a*b^2 + 2132 13*b^3)*cosh(d*x + c)^12 - (487424*a^3 + 1664640*a^2*b + 1983696*a*b^2 + 7 249242*b^3 - 91*(28672*a^3 + 97920*a^2*b + 116688*a*b^2 + 561561*b^3)*cosh (d*x + c)^2)*sinh(d*x + c)^12 - 4*(91*(28672*a^3 + 97920*a^2*b + 116688*a* b^2 - 459459*b^3)*cosh(d*x + c)^3 - 204*(7168*a^3 + 24480*a^2*b + 29172*a* b^2 - 81081*b^3)*cosh(d*x + c))*sinh(d*x + c)^11 + 17*(229376*a^3 + 783360 *a^2*b + 1798368*a*b^2 + 2573571*b^3)*cosh(d*x + c)^10 + (1001*(28672*a^3 + 97920*a^2*b + 116688*a*b^2 + 561561*b^3)*cosh(d*x + c)^4 + 3899392*a^3 + 13317120*a^2*b + 30572256*a*b^2 + 43750707*b^3 - 2244*(14336*a^3 + 48960* a^2*b + 58344*a*b^2 + 213213*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^10 - 2*(1 001*(28672*a^3 + 97920*a^2*b + 116688*a*b^2 - 459459*b^3)*cosh(d*x + c)^5 - 7480*(7168*a^3 + 24480*a^2*b + 29172*a*b^2 - 81081*b^3)*cosh(d*x + c)^3 + 85*(229376*a^3 + 783360*a^2*b + 68640*a*b^2 - 1756755*b^3)*cosh(d*x + c) )*sinh(d*x + c)^9 - 68*(286720*a^3 + 979200*a^2*b + 3040752*a*b^2 + 241140 9*b^3)*cosh(d*x + c)^8 + (3003*(28672*a^3 + 97920*a^2*b + 116688*a*b^2 + 5 61561*b^3)*cosh(d*x + c)^6 - 16830*(14336*a^3 + 48960*a^2*b + 58344*a*b^2 + 213213*b^3)*cosh(d*x + c)^4 - 19496960*a^3 - 66585600*a^2*b - 2067711...
Timed out. \[ \int \text {csch}^{18}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 3719 vs. \(2 (205) = 410\).
Time = 0.22 (sec) , antiderivative size = 3719, normalized size of antiderivative = 16.83 \[ \int \text {csch}^{18}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Too large to display} \]
-65536/109395*a^3*(17*e^(-2*d*x - 2*c)/(d*(17*e^(-2*d*x - 2*c) - 136*e^(-4 *d*x - 4*c) + 680*e^(-6*d*x - 6*c) - 2380*e^(-8*d*x - 8*c) + 6188*e^(-10*d *x - 10*c) - 12376*e^(-12*d*x - 12*c) + 19448*e^(-14*d*x - 14*c) - 24310*e ^(-16*d*x - 16*c) + 24310*e^(-18*d*x - 18*c) - 19448*e^(-20*d*x - 20*c) + 12376*e^(-22*d*x - 22*c) - 6188*e^(-24*d*x - 24*c) + 2380*e^(-26*d*x - 26* c) - 680*e^(-28*d*x - 28*c) + 136*e^(-30*d*x - 30*c) - 17*e^(-32*d*x - 32* c) + e^(-34*d*x - 34*c) - 1)) - 136*e^(-4*d*x - 4*c)/(d*(17*e^(-2*d*x - 2* c) - 136*e^(-4*d*x - 4*c) + 680*e^(-6*d*x - 6*c) - 2380*e^(-8*d*x - 8*c) + 6188*e^(-10*d*x - 10*c) - 12376*e^(-12*d*x - 12*c) + 19448*e^(-14*d*x - 1 4*c) - 24310*e^(-16*d*x - 16*c) + 24310*e^(-18*d*x - 18*c) - 19448*e^(-20* d*x - 20*c) + 12376*e^(-22*d*x - 22*c) - 6188*e^(-24*d*x - 24*c) + 2380*e^ (-26*d*x - 26*c) - 680*e^(-28*d*x - 28*c) + 136*e^(-30*d*x - 30*c) - 17*e^ (-32*d*x - 32*c) + e^(-34*d*x - 34*c) - 1)) + 680*e^(-6*d*x - 6*c)/(d*(17* e^(-2*d*x - 2*c) - 136*e^(-4*d*x - 4*c) + 680*e^(-6*d*x - 6*c) - 2380*e^(- 8*d*x - 8*c) + 6188*e^(-10*d*x - 10*c) - 12376*e^(-12*d*x - 12*c) + 19448* e^(-14*d*x - 14*c) - 24310*e^(-16*d*x - 16*c) + 24310*e^(-18*d*x - 18*c) - 19448*e^(-20*d*x - 20*c) + 12376*e^(-22*d*x - 22*c) - 6188*e^(-24*d*x - 2 4*c) + 2380*e^(-26*d*x - 26*c) - 680*e^(-28*d*x - 28*c) + 136*e^(-30*d*x - 30*c) - 17*e^(-32*d*x - 32*c) + e^(-34*d*x - 34*c) - 1)) - 2380*e^(-8*d*x - 8*c)/(d*(17*e^(-2*d*x - 2*c) - 136*e^(-4*d*x - 4*c) + 680*e^(-6*d*x ...
Leaf count of result is larger than twice the leaf count of optimal. 679 vs. \(2 (205) = 410\).
Time = 0.54 (sec) , antiderivative size = 679, normalized size of antiderivative = 3.07 \[ \int \text {csch}^{18}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=-\frac {16 \, {\left (510510 \, b^{3} e^{\left (28 \, d x + 28 \, c\right )} - 6381375 \, b^{3} e^{\left (26 \, d x + 26 \, c\right )} + 14702688 \, a b^{2} e^{\left (24 \, d x + 24 \, c\right )} + 36807771 \, b^{3} e^{\left (24 \, d x + 24 \, c\right )} - 127423296 \, a b^{2} e^{\left (22 \, d x + 22 \, c\right )} - 129771642 \, b^{3} e^{\left (22 \, d x + 22 \, c\right )} + 168030720 \, a^{2} b e^{\left (20 \, d x + 20 \, c\right )} + 494290368 \, a b^{2} e^{\left (20 \, d x + 20 \, c\right )} + 312227916 \, b^{3} e^{\left (20 \, d x + 20 \, c\right )} - 798145920 \, a^{2} b e^{\left (18 \, d x + 18 \, c\right )} - 1132457040 \, a b^{2} e^{\left (18 \, d x + 18 \, c\right )} - 541906365 \, b^{3} e^{\left (18 \, d x + 18 \, c\right )} + 697016320 \, a^{3} e^{\left (16 \, d x + 16 \, c\right )} + 1582289280 \, a^{2} b e^{\left (16 \, d x + 16 \, c\right )} + 1704228240 \, a b^{2} e^{\left (16 \, d x + 16 \, c\right )} + 699143445 \, b^{3} e^{\left (16 \, d x + 16 \, c\right )} - 557613056 \, a^{3} e^{\left (14 \, d x + 14 \, c\right )} - 1736317440 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} - 1775057856 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} - 680611932 \, b^{3} e^{\left (14 \, d x + 14 \, c\right )} + 354844672 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 1211857920 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 1316707392 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 502035534 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} - 177422336 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} - 605928960 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} - 707362656 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} - 279095817 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 68239360 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 233049600 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 277717440 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 115120005 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 19496960 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 66585600 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 79347840 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 34204170 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 3899392 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 13317120 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 15869568 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 6942936 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 487424 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 1664640 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 1983696 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 867867 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 28672 \, a^{3} + 97920 \, a^{2} b + 116688 \, a b^{2} + 51051 \, b^{3}\right )}}{765765 \, d {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{17}} \]
-16/765765*(510510*b^3*e^(28*d*x + 28*c) - 6381375*b^3*e^(26*d*x + 26*c) + 14702688*a*b^2*e^(24*d*x + 24*c) + 36807771*b^3*e^(24*d*x + 24*c) - 12742 3296*a*b^2*e^(22*d*x + 22*c) - 129771642*b^3*e^(22*d*x + 22*c) + 168030720 *a^2*b*e^(20*d*x + 20*c) + 494290368*a*b^2*e^(20*d*x + 20*c) + 312227916*b ^3*e^(20*d*x + 20*c) - 798145920*a^2*b*e^(18*d*x + 18*c) - 1132457040*a*b^ 2*e^(18*d*x + 18*c) - 541906365*b^3*e^(18*d*x + 18*c) + 697016320*a^3*e^(1 6*d*x + 16*c) + 1582289280*a^2*b*e^(16*d*x + 16*c) + 1704228240*a*b^2*e^(1 6*d*x + 16*c) + 699143445*b^3*e^(16*d*x + 16*c) - 557613056*a^3*e^(14*d*x + 14*c) - 1736317440*a^2*b*e^(14*d*x + 14*c) - 1775057856*a*b^2*e^(14*d*x + 14*c) - 680611932*b^3*e^(14*d*x + 14*c) + 354844672*a^3*e^(12*d*x + 12*c ) + 1211857920*a^2*b*e^(12*d*x + 12*c) + 1316707392*a*b^2*e^(12*d*x + 12*c ) + 502035534*b^3*e^(12*d*x + 12*c) - 177422336*a^3*e^(10*d*x + 10*c) - 60 5928960*a^2*b*e^(10*d*x + 10*c) - 707362656*a*b^2*e^(10*d*x + 10*c) - 2790 95817*b^3*e^(10*d*x + 10*c) + 68239360*a^3*e^(8*d*x + 8*c) + 233049600*a^2 *b*e^(8*d*x + 8*c) + 277717440*a*b^2*e^(8*d*x + 8*c) + 115120005*b^3*e^(8* d*x + 8*c) - 19496960*a^3*e^(6*d*x + 6*c) - 66585600*a^2*b*e^(6*d*x + 6*c) - 79347840*a*b^2*e^(6*d*x + 6*c) - 34204170*b^3*e^(6*d*x + 6*c) + 3899392 *a^3*e^(4*d*x + 4*c) + 13317120*a^2*b*e^(4*d*x + 4*c) + 15869568*a*b^2*e^( 4*d*x + 4*c) + 6942936*b^3*e^(4*d*x + 4*c) - 487424*a^3*e^(2*d*x + 2*c) - 1664640*a^2*b*e^(2*d*x + 2*c) - 1983696*a*b^2*e^(2*d*x + 2*c) - 867867*...
Time = 1.99 (sec) , antiderivative size = 4490, normalized size of antiderivative = 20.32 \[ \int \text {csch}^{18}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Too large to display} \]
((24*b^3)/(595*d) - (64*exp(10*c + 10*d*x)*(840*a*b^2 + 1152*a^2*b + 1024* a^3 + 231*b^3))/(85*d) + (6864*b^3*exp(20*c + 20*d*x))/(595*d) - (104*b^3* exp(22*c + 22*d*x))/(85*d) + (48*b*exp(8*c + 8*d*x)*(112*a*b + 128*a^2 + 3 3*b^2))/(17*d) + (576*b*exp(12*c + 12*d*x)*(112*a*b + 128*a^2 + 33*b^2))/( 85*d) - (24*b*exp(6*c + 6*d*x)*(448*a*b + 256*a^2 + 165*b^2))/(119*d) - (1 44*b*exp(14*c + 14*d*x)*(448*a*b + 256*a^2 + 165*b^2))/(119*d) - (48*b^2*e xp(2*c + 2*d*x)*(8*a + 11*b))/(595*d) - (528*b^2*exp(18*c + 18*d*x)*(8*a + 11*b))/(119*d) + (16*b^2*exp(4*c + 4*d*x)*(96*a + 55*b))/(119*d) + (264*b ^2*exp(16*c + 16*d*x)*(96*a + 55*b))/(119*d))/(91*exp(4*c + 4*d*x) - 14*ex p(2*c + 2*d*x) - 364*exp(6*c + 6*d*x) + 1001*exp(8*c + 8*d*x) - 2002*exp(1 0*c + 10*d*x) + 3003*exp(12*c + 12*d*x) - 3432*exp(14*c + 14*d*x) + 3003*e xp(16*c + 16*d*x) - 2002*exp(18*c + 18*d*x) + 1001*exp(20*c + 20*d*x) - 36 4*exp(22*c + 22*d*x) + 91*exp(24*c + 24*d*x) - 14*exp(26*c + 26*d*x) + exp (28*c + 28*d*x) + 1) + ((24*b^3)/(595*d) - (4*b^3*exp(2*c + 2*d*x))/(85*d) )/(6*exp(4*c + 4*d*x) - 4*exp(2*c + 2*d*x) - 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1) - ((64*exp(8*c + 8*d*x)*(840*a*b^2 + 1152*a^2*b + 1024*a^3 + 231*b^3))/(221*d) - (1056*b^3*exp(18*c + 18*d*x))/(119*d) + (88*b^3*exp(2 0*c + 20*d*x))/(85*d) + (48*b^2*(8*a + 11*b))/(7735*d) - (192*b*exp(6*c + 6*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(221*d) - (3456*b*exp(10*c + 10*d*x)* (112*a*b + 128*a^2 + 33*b^2))/(1105*d) + (72*b*exp(4*c + 4*d*x)*(448*a*...