Integrand size = 23, antiderivative size = 220 \[ \int \sinh ^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {(a+b)^3 \cosh (c+d x)}{d}-\frac {2 (a+b)^2 (a+4 b) \cosh ^3(c+d x)}{3 d}+\frac {(a+b) \left (a^2+17 a b+28 b^2\right ) \cosh ^5(c+d x)}{5 d}-\frac {4 b \left (3 a^2+15 a b+14 b^2\right ) \cosh ^7(c+d x)}{7 d}+\frac {b \left (3 a^2+45 a b+70 b^2\right ) \cosh ^9(c+d x)}{9 d}-\frac {2 b^2 (9 a+28 b) \cosh ^{11}(c+d x)}{11 d}+\frac {b^2 (3 a+28 b) \cosh ^{13}(c+d x)}{13 d}-\frac {8 b^3 \cosh ^{15}(c+d x)}{15 d}+\frac {b^3 \cosh ^{17}(c+d x)}{17 d} \]
(a+b)^3*cosh(d*x+c)/d-2/3*(a+b)^2*(a+4*b)*cosh(d*x+c)^3/d+1/5*(a+b)*(a^2+1 7*a*b+28*b^2)*cosh(d*x+c)^5/d-4/7*b*(3*a^2+15*a*b+14*b^2)*cosh(d*x+c)^7/d+ 1/9*b*(3*a^2+45*a*b+70*b^2)*cosh(d*x+c)^9/d-2/11*b^2*(9*a+28*b)*cosh(d*x+c )^11/d+1/13*b^2*(3*a+28*b)*cosh(d*x+c)^13/d-8/15*b^3*cosh(d*x+c)^15/d+1/17 *b^3*cosh(d*x+c)^17/d
Time = 11.39 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.31 \[ \int \sinh ^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {1531530 \left (20480 a^3+48384 a^2 b+41184 a b^2+12155 b^3\right ) \cosh (c+d x)-2042040 \left (2560 a^3+8064 a^2 b+7722 a b^2+2431 b^3\right ) \cosh (3 (c+d x))+627314688 a^3 \cosh (5 (c+d x))+4234374144 a^2 b \cosh (5 (c+d x))+5256210960 a b^2 \cosh (5 (c+d x))+1895421528 b^3 \cosh (5 (c+d x))-756138240 a^2 b \cosh (7 (c+d x))-1501774560 a b^2 \cosh (7 (c+d x))-676936260 b^3 \cosh (7 (c+d x))+65345280 a^2 b \cosh (9 (c+d x))+318558240 a b^2 \cosh (9 (c+d x))+202502300 b^3 \cosh (9 (c+d x))-43439760 a b^2 \cosh (11 (c+d x))-47338200 b^3 \cosh (11 (c+d x))+2827440 a b^2 \cosh (13 (c+d x))+8011080 b^3 \cosh (13 (c+d x))-867867 b^3 \cosh (15 (c+d x))+45045 b^3 \cosh (17 (c+d x))}{50185175040 d} \]
(1531530*(20480*a^3 + 48384*a^2*b + 41184*a*b^2 + 12155*b^3)*Cosh[c + d*x] - 2042040*(2560*a^3 + 8064*a^2*b + 7722*a*b^2 + 2431*b^3)*Cosh[3*(c + d*x )] + 627314688*a^3*Cosh[5*(c + d*x)] + 4234374144*a^2*b*Cosh[5*(c + d*x)] + 5256210960*a*b^2*Cosh[5*(c + d*x)] + 1895421528*b^3*Cosh[5*(c + d*x)] - 756138240*a^2*b*Cosh[7*(c + d*x)] - 1501774560*a*b^2*Cosh[7*(c + d*x)] - 6 76936260*b^3*Cosh[7*(c + d*x)] + 65345280*a^2*b*Cosh[9*(c + d*x)] + 318558 240*a*b^2*Cosh[9*(c + d*x)] + 202502300*b^3*Cosh[9*(c + d*x)] - 43439760*a *b^2*Cosh[11*(c + d*x)] - 47338200*b^3*Cosh[11*(c + d*x)] + 2827440*a*b^2* Cosh[13*(c + d*x)] + 8011080*b^3*Cosh[13*(c + d*x)] - 867867*b^3*Cosh[15*( c + d*x)] + 45045*b^3*Cosh[17*(c + d*x)])/(50185175040*d)
Time = 0.45 (sec) , antiderivative size = 197, 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, 26, 3694, 1467, 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.
| \(\displaystyle \int \sinh ^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -i \sin (i c+i d x)^5 \left (a+b \sin (i c+i d x)^4\right )^3dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \sin (i c+i d x)^5 \left (b \sin (i c+i d x)^4+a\right )^3dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle \frac {\int \left (1-\cosh ^2(c+d x)\right )^2 \left (b \cosh ^4(c+d x)-2 b \cosh ^2(c+d x)+a+b\right )^3d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1467 |
| \(\displaystyle \frac {\int \left (b^3 \cosh ^{16}(c+d x)-8 b^3 \cosh ^{14}(c+d x)+b^2 (3 a+28 b) \cosh ^{12}(c+d x)-2 b^2 (9 a+28 b) \cosh ^{10}(c+d x)+b \left (3 a^2+45 b a+70 b^2\right ) \cosh ^8(c+d x)-4 b \left (3 a^2+15 b a+14 b^2\right ) \cosh ^6(c+d x)+(a+b) \left (a^2+17 b a+28 b^2\right ) \cosh ^4(c+d x)-2 (a+b)^2 (a+4 b) \cosh ^2(c+d x)+(a+b)^3\right )d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{9} b \left (3 a^2+45 a b+70 b^2\right ) \cosh ^9(c+d x)-\frac {4}{7} b \left (3 a^2+15 a b+14 b^2\right ) \cosh ^7(c+d x)+\frac {1}{5} (a+b) \left (a^2+17 a b+28 b^2\right ) \cosh ^5(c+d x)+\frac {1}{13} b^2 (3 a+28 b) \cosh ^{13}(c+d x)-\frac {2}{11} b^2 (9 a+28 b) \cosh ^{11}(c+d x)-\frac {2}{3} (a+b)^2 (a+4 b) \cosh ^3(c+d x)+(a+b)^3 \cosh (c+d x)+\frac {1}{17} b^3 \cosh ^{17}(c+d x)-\frac {8}{15} b^3 \cosh ^{15}(c+d x)}{d}\) |
((a + b)^3*Cosh[c + d*x] - (2*(a + b)^2*(a + 4*b)*Cosh[c + d*x]^3)/3 + ((a + b)*(a^2 + 17*a*b + 28*b^2)*Cosh[c + d*x]^5)/5 - (4*b*(3*a^2 + 15*a*b + 14*b^2)*Cosh[c + d*x]^7)/7 + (b*(3*a^2 + 45*a*b + 70*b^2)*Cosh[c + d*x]^9) /9 - (2*b^2*(9*a + 28*b)*Cosh[c + d*x]^11)/11 + (b^2*(3*a + 28*b)*Cosh[c + d*x]^13)/13 - (8*b^3*Cosh[c + d*x]^15)/15 + (b^3*Cosh[c + d*x]^17)/17)/d
3.3.7.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 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[Cos[e + f*x], x]}, Simp[-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]
Time = 37.00 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.11
| method | result | size |
| parallelrisch | \(\frac {\left (-5227622400 a^{3}-16467010560 a^{2} b -15768632880 a \,b^{2}-4964199240 b^{3}\right ) \cosh \left (3 d x +3 c \right )+\left (627314688 a^{3}+4234374144 a^{2} b +5256210960 a \,b^{2}+1895421528 b^{3}\right ) \cosh \left (5 d x +5 c \right )+\left (-756138240 a^{2} b -1501774560 a \,b^{2}-676936260 b^{3}\right ) \cosh \left (7 d x +7 c \right )+65345280 b \left (a^{2}+\frac {39}{8} a b +\frac {595}{192} b^{2}\right ) \cosh \left (9 d x +9 c \right )-43439760 \left (a +\frac {85 b}{78}\right ) b^{2} \cosh \left (11 d x +11 c \right )+2827440 \left (\frac {17 b}{6}+a \right ) b^{2} \cosh \left (13 d x +13 c \right )-867867 b^{3} \cosh \left (15 d x +15 c \right )+45045 b^{3} \cosh \left (17 d x +17 c \right )+\left (31365734400 a^{3}+74101547520 a^{2} b +63074531520 a \,b^{2}+18615747150 b^{3}\right ) \cosh \left (d x +c \right )+26765426688 a^{3}+61178118144 a^{2} b +51338280960 a \,b^{2}+15032385536 b^{3}}{50185175040 d}\) | \(244\) |
| derivativedivides | \(\frac {a^{3} \left (\frac {8}{15}+\frac {\sinh \left (d x +c \right )^{4}}{5}-\frac {4 \sinh \left (d x +c \right )^{2}}{15}\right ) \cosh \left (d x +c \right )+3 a^{2} b \left (\frac {128}{315}+\frac {\sinh \left (d x +c \right )^{8}}{9}-\frac {8 \sinh \left (d x +c \right )^{6}}{63}+\frac {16 \sinh \left (d x +c \right )^{4}}{105}-\frac {64 \sinh \left (d x +c \right )^{2}}{315}\right ) \cosh \left (d x +c \right )+3 a \,b^{2} \left (\frac {1024}{3003}+\frac {\sinh \left (d x +c \right )^{12}}{13}-\frac {12 \sinh \left (d x +c \right )^{10}}{143}+\frac {40 \sinh \left (d x +c \right )^{8}}{429}-\frac {320 \sinh \left (d x +c \right )^{6}}{3003}+\frac {128 \sinh \left (d x +c \right )^{4}}{1001}-\frac {512 \sinh \left (d x +c \right )^{2}}{3003}\right ) \cosh \left (d x +c \right )+b^{3} \left (\frac {32768}{109395}+\frac {\sinh \left (d x +c \right )^{16}}{17}-\frac {16 \sinh \left (d x +c \right )^{14}}{255}+\frac {224 \sinh \left (d x +c \right )^{12}}{3315}-\frac {896 \sinh \left (d x +c \right )^{10}}{12155}+\frac {1792 \sinh \left (d x +c \right )^{8}}{21879}-\frac {2048 \sinh \left (d x +c \right )^{6}}{21879}+\frac {4096 \sinh \left (d x +c \right )^{4}}{36465}-\frac {16384 \sinh \left (d x +c \right )^{2}}{109395}\right ) \cosh \left (d x +c \right )}{d}\) | \(258\) |
| default | \(\frac {a^{3} \left (\frac {8}{15}+\frac {\sinh \left (d x +c \right )^{4}}{5}-\frac {4 \sinh \left (d x +c \right )^{2}}{15}\right ) \cosh \left (d x +c \right )+3 a^{2} b \left (\frac {128}{315}+\frac {\sinh \left (d x +c \right )^{8}}{9}-\frac {8 \sinh \left (d x +c \right )^{6}}{63}+\frac {16 \sinh \left (d x +c \right )^{4}}{105}-\frac {64 \sinh \left (d x +c \right )^{2}}{315}\right ) \cosh \left (d x +c \right )+3 a \,b^{2} \left (\frac {1024}{3003}+\frac {\sinh \left (d x +c \right )^{12}}{13}-\frac {12 \sinh \left (d x +c \right )^{10}}{143}+\frac {40 \sinh \left (d x +c \right )^{8}}{429}-\frac {320 \sinh \left (d x +c \right )^{6}}{3003}+\frac {128 \sinh \left (d x +c \right )^{4}}{1001}-\frac {512 \sinh \left (d x +c \right )^{2}}{3003}\right ) \cosh \left (d x +c \right )+b^{3} \left (\frac {32768}{109395}+\frac {\sinh \left (d x +c \right )^{16}}{17}-\frac {16 \sinh \left (d x +c \right )^{14}}{255}+\frac {224 \sinh \left (d x +c \right )^{12}}{3315}-\frac {896 \sinh \left (d x +c \right )^{10}}{12155}+\frac {1792 \sinh \left (d x +c \right )^{8}}{21879}-\frac {2048 \sinh \left (d x +c \right )^{6}}{21879}+\frac {4096 \sinh \left (d x +c \right )^{4}}{36465}-\frac {16384 \sinh \left (d x +c \right )^{2}}{109395}\right ) \cosh \left (d x +c \right )}{d}\) | \(258\) |
| parts | \(\frac {a^{3} \left (\frac {8}{15}+\frac {\sinh \left (d x +c \right )^{4}}{5}-\frac {4 \sinh \left (d x +c \right )^{2}}{15}\right ) \cosh \left (d x +c \right )}{d}+\frac {b^{3} \left (\frac {32768}{109395}+\frac {\sinh \left (d x +c \right )^{16}}{17}-\frac {16 \sinh \left (d x +c \right )^{14}}{255}+\frac {224 \sinh \left (d x +c \right )^{12}}{3315}-\frac {896 \sinh \left (d x +c \right )^{10}}{12155}+\frac {1792 \sinh \left (d x +c \right )^{8}}{21879}-\frac {2048 \sinh \left (d x +c \right )^{6}}{21879}+\frac {4096 \sinh \left (d x +c \right )^{4}}{36465}-\frac {16384 \sinh \left (d x +c \right )^{2}}{109395}\right ) \cosh \left (d x +c \right )}{d}+\frac {3 a \,b^{2} \left (\frac {1024}{3003}+\frac {\sinh \left (d x +c \right )^{12}}{13}-\frac {12 \sinh \left (d x +c \right )^{10}}{143}+\frac {40 \sinh \left (d x +c \right )^{8}}{429}-\frac {320 \sinh \left (d x +c \right )^{6}}{3003}+\frac {128 \sinh \left (d x +c \right )^{4}}{1001}-\frac {512 \sinh \left (d x +c \right )^{2}}{3003}\right ) \cosh \left (d x +c \right )}{d}+\frac {3 a^{2} b \left (\frac {128}{315}+\frac {\sinh \left (d x +c \right )^{8}}{9}-\frac {8 \sinh \left (d x +c \right )^{6}}{63}+\frac {16 \sinh \left (d x +c \right )^{4}}{105}-\frac {64 \sinh \left (d x +c \right )^{2}}{315}\right ) \cosh \left (d x +c \right )}{d}\) | \(266\) |
| risch | \(-\frac {17 b^{3} {\mathrm e}^{-15 d x -15 c}}{1966080 d}+\frac {b^{3} {\mathrm e}^{-17 d x -17 c}}{2228224 d}+\frac {27 b \,{\mathrm e}^{-5 d x -5 c} a^{2}}{640 d}+\frac {27 b \,{\mathrm e}^{5 d x +5 c} a^{2}}{640 d}-\frac {5 \,{\mathrm e}^{-3 d x -3 c} a^{3}}{96 d}+\frac {{\mathrm e}^{-5 d x -5 c} a^{3}}{160 d}+\frac {17 b^{3} {\mathrm e}^{-13 d x -13 c}}{212992 d}+\frac {17 b^{3} {\mathrm e}^{13 d x +13 c}}{212992 d}+\frac {{\mathrm e}^{5 d x +5 c} a^{3}}{160 d}-\frac {5 \,{\mathrm e}^{3 d x +3 c} a^{3}}{96 d}+\frac {b^{3} {\mathrm e}^{17 d x +17 c}}{2228224 d}-\frac {17 b^{3} {\mathrm e}^{15 d x +15 c}}{1966080 d}-\frac {85 b^{3} {\mathrm e}^{11 d x +11 c}}{180224 d}+\frac {595 b^{3} {\mathrm e}^{9 d x +9 c}}{294912 d}+\frac {595 b^{3} {\mathrm e}^{-9 d x -9 c}}{294912 d}-\frac {85 b^{3} {\mathrm e}^{-11 d x -11 c}}{180224 d}-\frac {221 b^{3} {\mathrm e}^{7 d x +7 c}}{32768 d}+\frac {1547 b^{3} {\mathrm e}^{5 d x +5 c}}{81920 d}-\frac {2431 \,{\mathrm e}^{3 d x +3 c} b^{3}}{49152 d}+\frac {5 \,{\mathrm e}^{d x +c} a^{3}}{16 d}+\frac {12155 \,{\mathrm e}^{d x +c} b^{3}}{65536 d}+\frac {5 \,{\mathrm e}^{-d x -c} a^{3}}{16 d}+\frac {12155 \,{\mathrm e}^{-d x -c} b^{3}}{65536 d}-\frac {2431 \,{\mathrm e}^{-3 d x -3 c} b^{3}}{49152 d}+\frac {1547 b^{3} {\mathrm e}^{-5 d x -5 c}}{81920 d}-\frac {221 b^{3} {\mathrm e}^{-7 d x -7 c}}{32768 d}+\frac {189 \,{\mathrm e}^{-d x -c} a^{2} b}{256 d}+\frac {1287 \,{\mathrm e}^{-d x -c} a \,b^{2}}{2048 d}-\frac {21 \,{\mathrm e}^{-3 d x -3 c} a^{2} b}{128 d}-\frac {1287 \,{\mathrm e}^{-3 d x -3 c} a \,b^{2}}{8192 d}+\frac {429 b^{2} {\mathrm e}^{-5 d x -5 c} a}{8192 d}+\frac {429 b^{2} {\mathrm e}^{5 d x +5 c} a}{8192 d}-\frac {21 \,{\mathrm e}^{3 d x +3 c} a^{2} b}{128 d}-\frac {1287 \,{\mathrm e}^{3 d x +3 c} a \,b^{2}}{8192 d}+\frac {189 \,{\mathrm e}^{d x +c} a^{2} b}{256 d}+\frac {1287 \,{\mathrm e}^{d x +c} a \,b^{2}}{2048 d}-\frac {27 b \,{\mathrm e}^{-7 d x -7 c} a^{2}}{3584 d}+\frac {b \,{\mathrm e}^{-9 d x -9 c} a^{2}}{1536 d}+\frac {13 b^{2} {\mathrm e}^{-9 d x -9 c} a}{4096 d}-\frac {39 b^{2} {\mathrm e}^{-11 d x -11 c} a}{90112 d}+\frac {3 b^{2} {\mathrm e}^{-13 d x -13 c} a}{106496 d}-\frac {429 a \,b^{2} {\mathrm e}^{7 d x +7 c}}{28672 d}-\frac {429 a \,b^{2} {\mathrm e}^{-7 d x -7 c}}{28672 d}+\frac {3 b^{2} {\mathrm e}^{13 d x +13 c} a}{106496 d}-\frac {39 b^{2} {\mathrm e}^{11 d x +11 c} a}{90112 d}+\frac {b \,{\mathrm e}^{9 d x +9 c} a^{2}}{1536 d}+\frac {13 b^{2} {\mathrm e}^{9 d x +9 c} a}{4096 d}-\frac {27 b \,{\mathrm e}^{7 d x +7 c} a^{2}}{3584 d}\) | \(786\) |
1/50185175040*((-5227622400*a^3-16467010560*a^2*b-15768632880*a*b^2-496419 9240*b^3)*cosh(3*d*x+3*c)+(627314688*a^3+4234374144*a^2*b+5256210960*a*b^2 +1895421528*b^3)*cosh(5*d*x+5*c)+(-756138240*a^2*b-1501774560*a*b^2-676936 260*b^3)*cosh(7*d*x+7*c)+65345280*b*(a^2+39/8*a*b+595/192*b^2)*cosh(9*d*x+ 9*c)-43439760*(a+85/78*b)*b^2*cosh(11*d*x+11*c)+2827440*(17/6*b+a)*b^2*cos h(13*d*x+13*c)-867867*b^3*cosh(15*d*x+15*c)+45045*b^3*cosh(17*d*x+17*c)+(3 1365734400*a^3+74101547520*a^2*b+63074531520*a*b^2+18615747150*b^3)*cosh(d *x+c)+26765426688*a^3+61178118144*a^2*b+51338280960*a*b^2+15032385536*b^3) /d
Leaf count of result is larger than twice the leaf count of optimal. 1030 vs. \(2 (204) = 408\).
Time = 0.29 (sec) , antiderivative size = 1030, normalized size of antiderivative = 4.68 \[ \int \sinh ^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Too large to display} \]
1/50185175040*(45045*b^3*cosh(d*x + c)^17 + 765765*b^3*cosh(d*x + c)*sinh( d*x + c)^16 - 867867*b^3*cosh(d*x + c)^15 + 765765*(40*b^3*cosh(d*x + c)^3 - 17*b^3*cosh(d*x + c))*sinh(d*x + c)^14 + 471240*(6*a*b^2 + 17*b^3)*cosh (d*x + c)^13 + 255255*(1092*b^3*cosh(d*x + c)^5 - 1547*b^3*cosh(d*x + c)^3 + 24*(6*a*b^2 + 17*b^3)*cosh(d*x + c))*sinh(d*x + c)^12 - 556920*(78*a*b^ 2 + 85*b^3)*cosh(d*x + c)^11 + 153153*(5720*b^3*cosh(d*x + c)^7 - 17017*b^ 3*cosh(d*x + c)^5 + 880*(6*a*b^2 + 17*b^3)*cosh(d*x + c)^3 - 40*(78*a*b^2 + 85*b^3)*cosh(d*x + c))*sinh(d*x + c)^10 + 340340*(192*a^2*b + 936*a*b^2 + 595*b^3)*cosh(d*x + c)^9 + 765765*(1430*b^3*cosh(d*x + c)^9 - 7293*b^3*c osh(d*x + c)^7 + 792*(6*a*b^2 + 17*b^3)*cosh(d*x + c)^5 - 120*(78*a*b^2 + 85*b^3)*cosh(d*x + c)^3 + 4*(192*a^2*b + 936*a*b^2 + 595*b^3)*cosh(d*x + c ))*sinh(d*x + c)^8 - 437580*(1728*a^2*b + 3432*a*b^2 + 1547*b^3)*cosh(d*x + c)^7 + 255255*(2184*b^3*cosh(d*x + c)^11 - 17017*b^3*cosh(d*x + c)^9 + 3 168*(6*a*b^2 + 17*b^3)*cosh(d*x + c)^7 - 1008*(78*a*b^2 + 85*b^3)*cosh(d*x + c)^5 + 112*(192*a^2*b + 936*a*b^2 + 595*b^3)*cosh(d*x + c)^3 - 12*(1728 *a^2*b + 3432*a*b^2 + 1547*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 + 1225224*( 512*a^3 + 3456*a^2*b + 4290*a*b^2 + 1547*b^3)*cosh(d*x + c)^5 + 765765*(14 0*b^3*cosh(d*x + c)^13 - 1547*b^3*cosh(d*x + c)^11 + 440*(6*a*b^2 + 17*b^3 )*cosh(d*x + c)^9 - 240*(78*a*b^2 + 85*b^3)*cosh(d*x + c)^7 + 56*(192*a^2* b + 936*a*b^2 + 595*b^3)*cosh(d*x + c)^5 - 20*(1728*a^2*b + 3432*a*b^2 ...
Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (204) = 408\).
Time = 10.98 (sec) , antiderivative size = 592, normalized size of antiderivative = 2.69 \[ \int \sinh ^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\begin {cases} \frac {a^{3} \sinh ^{4}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {4 a^{3} \sinh ^{2}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {8 a^{3} \cosh ^{5}{\left (c + d x \right )}}{15 d} + \frac {3 a^{2} b \sinh ^{8}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {8 a^{2} b \sinh ^{6}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {48 a^{2} b \sinh ^{4}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac {192 a^{2} b \sinh ^{2}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{35 d} + \frac {128 a^{2} b \cosh ^{9}{\left (c + d x \right )}}{105 d} + \frac {3 a b^{2} \sinh ^{12}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {12 a b^{2} \sinh ^{10}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {24 a b^{2} \sinh ^{8}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{d} - \frac {192 a b^{2} \sinh ^{6}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{7 d} + \frac {128 a b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{7 d} - \frac {512 a b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{11}{\left (c + d x \right )}}{77 d} + \frac {1024 a b^{2} \cosh ^{13}{\left (c + d x \right )}}{1001 d} + \frac {b^{3} \sinh ^{16}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {16 b^{3} \sinh ^{14}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {224 b^{3} \sinh ^{12}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{15 d} - \frac {128 b^{3} \sinh ^{10}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{5 d} + \frac {256 b^{3} \sinh ^{8}{\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{9 d} - \frac {2048 b^{3} \sinh ^{6}{\left (c + d x \right )} \cosh ^{11}{\left (c + d x \right )}}{99 d} + \frac {4096 b^{3} \sinh ^{4}{\left (c + d x \right )} \cosh ^{13}{\left (c + d x \right )}}{429 d} - \frac {16384 b^{3} \sinh ^{2}{\left (c + d x \right )} \cosh ^{15}{\left (c + d x \right )}}{6435 d} + \frac {32768 b^{3} \cosh ^{17}{\left (c + d x \right )}}{109395 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{4}{\left (c \right )}\right )^{3} \sinh ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((a**3*sinh(c + d*x)**4*cosh(c + d*x)/d - 4*a**3*sinh(c + d*x)**2 *cosh(c + d*x)**3/(3*d) + 8*a**3*cosh(c + d*x)**5/(15*d) + 3*a**2*b*sinh(c + d*x)**8*cosh(c + d*x)/d - 8*a**2*b*sinh(c + d*x)**6*cosh(c + d*x)**3/d + 48*a**2*b*sinh(c + d*x)**4*cosh(c + d*x)**5/(5*d) - 192*a**2*b*sinh(c + d*x)**2*cosh(c + d*x)**7/(35*d) + 128*a**2*b*cosh(c + d*x)**9/(105*d) + 3* a*b**2*sinh(c + d*x)**12*cosh(c + d*x)/d - 12*a*b**2*sinh(c + d*x)**10*cos h(c + d*x)**3/d + 24*a*b**2*sinh(c + d*x)**8*cosh(c + d*x)**5/d - 192*a*b* *2*sinh(c + d*x)**6*cosh(c + d*x)**7/(7*d) + 128*a*b**2*sinh(c + d*x)**4*c osh(c + d*x)**9/(7*d) - 512*a*b**2*sinh(c + d*x)**2*cosh(c + d*x)**11/(77* d) + 1024*a*b**2*cosh(c + d*x)**13/(1001*d) + b**3*sinh(c + d*x)**16*cosh( c + d*x)/d - 16*b**3*sinh(c + d*x)**14*cosh(c + d*x)**3/(3*d) + 224*b**3*s inh(c + d*x)**12*cosh(c + d*x)**5/(15*d) - 128*b**3*sinh(c + d*x)**10*cosh (c + d*x)**7/(5*d) + 256*b**3*sinh(c + d*x)**8*cosh(c + d*x)**9/(9*d) - 20 48*b**3*sinh(c + d*x)**6*cosh(c + d*x)**11/(99*d) + 4096*b**3*sinh(c + d*x )**4*cosh(c + d*x)**13/(429*d) - 16384*b**3*sinh(c + d*x)**2*cosh(c + d*x) **15/(6435*d) + 32768*b**3*cosh(c + d*x)**17/(109395*d), Ne(d, 0)), (x*(a + b*sinh(c)**4)**3*sinh(c)**5, True))
Leaf count of result is larger than twice the leaf count of optimal. 600 vs. \(2 (204) = 408\).
Time = 0.23 (sec) , antiderivative size = 600, normalized size of antiderivative = 2.73 \[ \int \sinh ^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=-\frac {1}{14338621440} \, b^{3} {\left (\frac {{\left (123981 \, e^{\left (-2 \, d x - 2 \, c\right )} - 1144440 \, e^{\left (-4 \, d x - 4 \, c\right )} + 6762600 \, e^{\left (-6 \, d x - 6 \, c\right )} - 28928900 \, e^{\left (-8 \, d x - 8 \, c\right )} + 96705180 \, e^{\left (-10 \, d x - 10 \, c\right )} - 270774504 \, e^{\left (-12 \, d x - 12 \, c\right )} + 709171320 \, e^{\left (-14 \, d x - 14 \, c\right )} - 2659392450 \, e^{\left (-16 \, d x - 16 \, c\right )} - 6435\right )} e^{\left (17 \, d x + 17 \, c\right )}}{d} - \frac {2659392450 \, e^{\left (-d x - c\right )} - 709171320 \, e^{\left (-3 \, d x - 3 \, c\right )} + 270774504 \, e^{\left (-5 \, d x - 5 \, c\right )} - 96705180 \, e^{\left (-7 \, d x - 7 \, c\right )} + 28928900 \, e^{\left (-9 \, d x - 9 \, c\right )} - 6762600 \, e^{\left (-11 \, d x - 11 \, c\right )} + 1144440 \, e^{\left (-13 \, d x - 13 \, c\right )} - 123981 \, e^{\left (-15 \, d x - 15 \, c\right )} + 6435 \, e^{\left (-17 \, d x - 17 \, c\right )}}{d}\right )} - \frac {1}{8200192} \, a b^{2} {\left (\frac {{\left (3549 \, e^{\left (-2 \, d x - 2 \, c\right )} - 26026 \, e^{\left (-4 \, d x - 4 \, c\right )} + 122694 \, e^{\left (-6 \, d x - 6 \, c\right )} - 429429 \, e^{\left (-8 \, d x - 8 \, c\right )} + 1288287 \, e^{\left (-10 \, d x - 10 \, c\right )} - 5153148 \, e^{\left (-12 \, d x - 12 \, c\right )} - 231\right )} e^{\left (13 \, d x + 13 \, c\right )}}{d} - \frac {5153148 \, e^{\left (-d x - c\right )} - 1288287 \, e^{\left (-3 \, d x - 3 \, c\right )} + 429429 \, e^{\left (-5 \, d x - 5 \, c\right )} - 122694 \, e^{\left (-7 \, d x - 7 \, c\right )} + 26026 \, e^{\left (-9 \, d x - 9 \, c\right )} - 3549 \, e^{\left (-11 \, d x - 11 \, c\right )} + 231 \, e^{\left (-13 \, d x - 13 \, c\right )}}{d}\right )} - \frac {1}{53760} \, a^{2} b {\left (\frac {{\left (405 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2268 \, e^{\left (-4 \, d x - 4 \, c\right )} + 8820 \, e^{\left (-6 \, d x - 6 \, c\right )} - 39690 \, e^{\left (-8 \, d x - 8 \, c\right )} - 35\right )} e^{\left (9 \, d x + 9 \, c\right )}}{d} - \frac {39690 \, e^{\left (-d x - c\right )} - 8820 \, e^{\left (-3 \, d x - 3 \, c\right )} + 2268 \, e^{\left (-5 \, d x - 5 \, c\right )} - 405 \, e^{\left (-7 \, d x - 7 \, c\right )} + 35 \, e^{\left (-9 \, d x - 9 \, c\right )}}{d}\right )} + \frac {1}{480} \, a^{3} {\left (\frac {3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} - \frac {25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {150 \, e^{\left (d x + c\right )}}{d} + \frac {150 \, e^{\left (-d x - c\right )}}{d} - \frac {25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} \]
-1/14338621440*b^3*((123981*e^(-2*d*x - 2*c) - 1144440*e^(-4*d*x - 4*c) + 6762600*e^(-6*d*x - 6*c) - 28928900*e^(-8*d*x - 8*c) + 96705180*e^(-10*d*x - 10*c) - 270774504*e^(-12*d*x - 12*c) + 709171320*e^(-14*d*x - 14*c) - 2 659392450*e^(-16*d*x - 16*c) - 6435)*e^(17*d*x + 17*c)/d - (2659392450*e^( -d*x - c) - 709171320*e^(-3*d*x - 3*c) + 270774504*e^(-5*d*x - 5*c) - 9670 5180*e^(-7*d*x - 7*c) + 28928900*e^(-9*d*x - 9*c) - 6762600*e^(-11*d*x - 1 1*c) + 1144440*e^(-13*d*x - 13*c) - 123981*e^(-15*d*x - 15*c) + 6435*e^(-1 7*d*x - 17*c))/d) - 1/8200192*a*b^2*((3549*e^(-2*d*x - 2*c) - 26026*e^(-4* d*x - 4*c) + 122694*e^(-6*d*x - 6*c) - 429429*e^(-8*d*x - 8*c) + 1288287*e ^(-10*d*x - 10*c) - 5153148*e^(-12*d*x - 12*c) - 231)*e^(13*d*x + 13*c)/d - (5153148*e^(-d*x - c) - 1288287*e^(-3*d*x - 3*c) + 429429*e^(-5*d*x - 5* c) - 122694*e^(-7*d*x - 7*c) + 26026*e^(-9*d*x - 9*c) - 3549*e^(-11*d*x - 11*c) + 231*e^(-13*d*x - 13*c))/d) - 1/53760*a^2*b*((405*e^(-2*d*x - 2*c) - 2268*e^(-4*d*x - 4*c) + 8820*e^(-6*d*x - 6*c) - 39690*e^(-8*d*x - 8*c) - 35)*e^(9*d*x + 9*c)/d - (39690*e^(-d*x - c) - 8820*e^(-3*d*x - 3*c) + 226 8*e^(-5*d*x - 5*c) - 405*e^(-7*d*x - 7*c) + 35*e^(-9*d*x - 9*c))/d) + 1/48 0*a^3*(3*e^(5*d*x + 5*c)/d - 25*e^(3*d*x + 3*c)/d + 150*e^(d*x + c)/d + 15 0*e^(-d*x - c)/d - 25*e^(-3*d*x - 3*c)/d + 3*e^(-5*d*x - 5*c)/d)
Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (204) = 408\).
Time = 0.46 (sec) , antiderivative size = 520, normalized size of antiderivative = 2.36 \[ \int \sinh ^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {b^{3} e^{\left (17 \, d x + 17 \, c\right )}}{2228224 \, d} - \frac {17 \, b^{3} e^{\left (15 \, d x + 15 \, c\right )}}{1966080 \, d} - \frac {17 \, b^{3} e^{\left (-15 \, d x - 15 \, c\right )}}{1966080 \, d} + \frac {b^{3} e^{\left (-17 \, d x - 17 \, c\right )}}{2228224 \, d} + \frac {{\left (6 \, a b^{2} + 17 \, b^{3}\right )} e^{\left (13 \, d x + 13 \, c\right )}}{212992 \, d} - \frac {{\left (78 \, a b^{2} + 85 \, b^{3}\right )} e^{\left (11 \, d x + 11 \, c\right )}}{180224 \, d} + \frac {{\left (192 \, a^{2} b + 936 \, a b^{2} + 595 \, b^{3}\right )} e^{\left (9 \, d x + 9 \, c\right )}}{294912 \, d} - \frac {{\left (1728 \, a^{2} b + 3432 \, a b^{2} + 1547 \, b^{3}\right )} e^{\left (7 \, d x + 7 \, c\right )}}{229376 \, d} + \frac {{\left (512 \, a^{3} + 3456 \, a^{2} b + 4290 \, a b^{2} + 1547 \, b^{3}\right )} e^{\left (5 \, d x + 5 \, c\right )}}{81920 \, d} - \frac {{\left (2560 \, a^{3} + 8064 \, a^{2} b + 7722 \, a b^{2} + 2431 \, b^{3}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{49152 \, d} + \frac {{\left (20480 \, a^{3} + 48384 \, a^{2} b + 41184 \, a b^{2} + 12155 \, b^{3}\right )} e^{\left (d x + c\right )}}{65536 \, d} + \frac {{\left (20480 \, a^{3} + 48384 \, a^{2} b + 41184 \, a b^{2} + 12155 \, b^{3}\right )} e^{\left (-d x - c\right )}}{65536 \, d} - \frac {{\left (2560 \, a^{3} + 8064 \, a^{2} b + 7722 \, a b^{2} + 2431 \, b^{3}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{49152 \, d} + \frac {{\left (512 \, a^{3} + 3456 \, a^{2} b + 4290 \, a b^{2} + 1547 \, b^{3}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{81920 \, d} - \frac {{\left (1728 \, a^{2} b + 3432 \, a b^{2} + 1547 \, b^{3}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{229376 \, d} + \frac {{\left (192 \, a^{2} b + 936 \, a b^{2} + 595 \, b^{3}\right )} e^{\left (-9 \, d x - 9 \, c\right )}}{294912 \, d} - \frac {{\left (78 \, a b^{2} + 85 \, b^{3}\right )} e^{\left (-11 \, d x - 11 \, c\right )}}{180224 \, d} + \frac {{\left (6 \, a b^{2} + 17 \, b^{3}\right )} e^{\left (-13 \, d x - 13 \, c\right )}}{212992 \, d} \]
1/2228224*b^3*e^(17*d*x + 17*c)/d - 17/1966080*b^3*e^(15*d*x + 15*c)/d - 1 7/1966080*b^3*e^(-15*d*x - 15*c)/d + 1/2228224*b^3*e^(-17*d*x - 17*c)/d + 1/212992*(6*a*b^2 + 17*b^3)*e^(13*d*x + 13*c)/d - 1/180224*(78*a*b^2 + 85* b^3)*e^(11*d*x + 11*c)/d + 1/294912*(192*a^2*b + 936*a*b^2 + 595*b^3)*e^(9 *d*x + 9*c)/d - 1/229376*(1728*a^2*b + 3432*a*b^2 + 1547*b^3)*e^(7*d*x + 7 *c)/d + 1/81920*(512*a^3 + 3456*a^2*b + 4290*a*b^2 + 1547*b^3)*e^(5*d*x + 5*c)/d - 1/49152*(2560*a^3 + 8064*a^2*b + 7722*a*b^2 + 2431*b^3)*e^(3*d*x + 3*c)/d + 1/65536*(20480*a^3 + 48384*a^2*b + 41184*a*b^2 + 12155*b^3)*e^( d*x + c)/d + 1/65536*(20480*a^3 + 48384*a^2*b + 41184*a*b^2 + 12155*b^3)*e ^(-d*x - c)/d - 1/49152*(2560*a^3 + 8064*a^2*b + 7722*a*b^2 + 2431*b^3)*e^ (-3*d*x - 3*c)/d + 1/81920*(512*a^3 + 3456*a^2*b + 4290*a*b^2 + 1547*b^3)* e^(-5*d*x - 5*c)/d - 1/229376*(1728*a^2*b + 3432*a*b^2 + 1547*b^3)*e^(-7*d *x - 7*c)/d + 1/294912*(192*a^2*b + 936*a*b^2 + 595*b^3)*e^(-9*d*x - 9*c)/ d - 1/180224*(78*a*b^2 + 85*b^3)*e^(-11*d*x - 11*c)/d + 1/212992*(6*a*b^2 + 17*b^3)*e^(-13*d*x - 13*c)/d
Time = 2.48 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.45 \[ \int \sinh ^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {\frac {a^3\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-\frac {2\,a^3\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}+a^3\,\mathrm {cosh}\left (c+d\,x\right )+\frac {a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^9}{3}-\frac {12\,a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}+\frac {18\,a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-4\,a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^3+3\,a^2\,b\,\mathrm {cosh}\left (c+d\,x\right )+\frac {3\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^{13}}{13}-\frac {18\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^{11}}{11}+5\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^9-\frac {60\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}+9\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^5-6\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^3+3\,a\,b^2\,\mathrm {cosh}\left (c+d\,x\right )+\frac {b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^{17}}{17}-\frac {8\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^{15}}{15}+\frac {28\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^{13}}{13}-\frac {56\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^{11}}{11}+\frac {70\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^9}{9}-8\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^7+\frac {28\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-\frac {8\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}+b^3\,\mathrm {cosh}\left (c+d\,x\right )}{d} \]
(a^3*cosh(c + d*x) + b^3*cosh(c + d*x) - (2*a^3*cosh(c + d*x)^3)/3 + (a^3* cosh(c + d*x)^5)/5 - (8*b^3*cosh(c + d*x)^3)/3 + (28*b^3*cosh(c + d*x)^5)/ 5 - 8*b^3*cosh(c + d*x)^7 + (70*b^3*cosh(c + d*x)^9)/9 - (56*b^3*cosh(c + d*x)^11)/11 + (28*b^3*cosh(c + d*x)^13)/13 - (8*b^3*cosh(c + d*x)^15)/15 + (b^3*cosh(c + d*x)^17)/17 - 6*a*b^2*cosh(c + d*x)^3 - 4*a^2*b*cosh(c + d* x)^3 + 9*a*b^2*cosh(c + d*x)^5 + (18*a^2*b*cosh(c + d*x)^5)/5 - (60*a*b^2* cosh(c + d*x)^7)/7 - (12*a^2*b*cosh(c + d*x)^7)/7 + 5*a*b^2*cosh(c + d*x)^ 9 + (a^2*b*cosh(c + d*x)^9)/3 - (18*a*b^2*cosh(c + d*x)^11)/11 + (3*a*b^2* cosh(c + d*x)^13)/13 + 3*a*b^2*cosh(c + d*x) + 3*a^2*b*cosh(c + d*x))/d