Integrand size = 23, antiderivative size = 330 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\frac {15 a^2 b \arctan (\sinh (c+d x))}{2 d}+\frac {1155 b^3 \arctan (\sinh (c+d x))}{128 d}-\frac {a^3 \cosh (c+d x)}{d}-\frac {12 a b^2 \cosh (c+d x)}{d}+\frac {a^3 \cosh ^3(c+d x)}{3 d}+\frac {a b^2 \cosh ^3(c+d x)}{d}-\frac {18 a b^2 \text {sech}(c+d x)}{d}+\frac {4 a b^2 \text {sech}^3(c+d x)}{d}-\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}-\frac {6 a^2 b \sinh (c+d x)}{d}-\frac {5 b^3 \sinh (c+d x)}{d}+\frac {a^2 b \sinh ^3(c+d x)}{d}+\frac {b^3 \sinh ^3(c+d x)}{3 d}-\frac {3 a^2 b \text {sech}(c+d x) \tanh (c+d x)}{2 d}-\frac {765 b^3 \text {sech}(c+d x) \tanh (c+d x)}{128 d}+\frac {515 b^3 \text {sech}^3(c+d x) \tanh (c+d x)}{192 d}-\frac {41 b^3 \text {sech}^5(c+d x) \tanh (c+d x)}{48 d}+\frac {b^3 \text {sech}^7(c+d x) \tanh (c+d x)}{8 d} \] Output:
15/2*a^2*b*arctan(sinh(d*x+c))/d+1155/128*b^3*arctan(sinh(d*x+c))/d-a^3*co sh(d*x+c)/d-12*a*b^2*cosh(d*x+c)/d+1/3*a^3*cosh(d*x+c)^3/d+a*b^2*cosh(d*x+ c)^3/d-18*a*b^2*sech(d*x+c)/d+4*a*b^2*sech(d*x+c)^3/d-3/5*a*b^2*sech(d*x+c )^5/d-6*a^2*b*sinh(d*x+c)/d-5*b^3*sinh(d*x+c)/d+a^2*b*sinh(d*x+c)^3/d+1/3* b^3*sinh(d*x+c)^3/d-3/2*a^2*b*sech(d*x+c)*tanh(d*x+c)/d-765/128*b^3*sech(d *x+c)*tanh(d*x+c)/d+515/192*b^3*sech(d*x+c)^3*tanh(d*x+c)/d-41/48*b^3*sech (d*x+c)^5*tanh(d*x+c)/d+1/8*b^3*sech(d*x+c)^7*tanh(d*x+c)/d
Time = 7.35 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.88 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\frac {15 \left (64 a^2 b+77 b^3\right ) \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{64 d}-\frac {3 a \left (a^2+15 b^2\right ) \cosh (c+d x)}{4 d}+\frac {a \left (a^2+3 b^2\right ) \cosh (3 (c+d x))}{12 d}-\frac {18 a b^2 \text {sech}(c+d x)}{d}+\frac {4 a b^2 \text {sech}^3(c+d x)}{d}-\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}-\frac {3 b \left (9 a^2+7 b^2\right ) \sinh (c+d x)}{4 d}-\frac {3 \text {sech}^2(c+d x) \left (64 a^2 b \sinh (c+d x)+255 b^3 \sinh (c+d x)\right )}{128 d}+\frac {b \left (3 a^2+b^2\right ) \sinh (3 (c+d x))}{12 d}+\frac {515 b^3 \text {sech}^3(c+d x) \tanh (c+d x)}{192 d}-\frac {41 b^3 \text {sech}^5(c+d x) \tanh (c+d x)}{48 d}+\frac {b^3 \text {sech}^7(c+d x) \tanh (c+d x)}{8 d} \] Input:
Integrate[Sinh[c + d*x]^3*(a + b*Tanh[c + d*x]^3)^3,x]
Output:
(15*(64*a^2*b + 77*b^3)*ArcTan[Tanh[(c + d*x)/2]])/(64*d) - (3*a*(a^2 + 15 *b^2)*Cosh[c + d*x])/(4*d) + (a*(a^2 + 3*b^2)*Cosh[3*(c + d*x)])/(12*d) - (18*a*b^2*Sech[c + d*x])/d + (4*a*b^2*Sech[c + d*x]^3)/d - (3*a*b^2*Sech[c + d*x]^5)/(5*d) - (3*b*(9*a^2 + 7*b^2)*Sinh[c + d*x])/(4*d) - (3*Sech[c + d*x]^2*(64*a^2*b*Sinh[c + d*x] + 255*b^3*Sinh[c + d*x]))/(128*d) + (b*(3* a^2 + b^2)*Sinh[3*(c + d*x)])/(12*d) + (515*b^3*Sech[c + d*x]^3*Tanh[c + d *x])/(192*d) - (41*b^3*Sech[c + d*x]^5*Tanh[c + d*x])/(48*d) + (b^3*Sech[c + d*x]^7*Tanh[c + d*x])/(8*d)
Result contains complex when optimal does not.
Time = 0.65 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.19, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 26, 4149, 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.
| \(\displaystyle \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int i \sin (i c+i d x)^3 \left (a+i b \tan (i c+i d x)^3\right )^3dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \sin (i c+i d x)^3 \left (i b \tan (i c+i d x)^3+a\right )^3dx\) |
| \(\Big \downarrow \) 4149 |
| \(\displaystyle i \int \left (-i b^3 \sinh ^3(c+d x) \tanh ^9(c+d x)-3 i a b^2 \sinh ^3(c+d x) \tanh ^6(c+d x)-3 i a^2 b \sinh ^3(c+d x) \tanh ^3(c+d x)-i a^3 \sinh ^3(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle i \left (-\frac {i a^3 \cosh ^3(c+d x)}{3 d}+\frac {i a^3 \cosh (c+d x)}{d}-\frac {15 i a^2 b \arctan (\sinh (c+d x))}{2 d}-\frac {5 i a^2 b \sinh ^3(c+d x)}{2 d}+\frac {15 i a^2 b \sinh (c+d x)}{2 d}+\frac {3 i a^2 b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}-\frac {i a b^2 \cosh ^3(c+d x)}{d}+\frac {12 i a b^2 \cosh (c+d x)}{d}+\frac {3 i a b^2 \text {sech}^5(c+d x)}{5 d}-\frac {4 i a b^2 \text {sech}^3(c+d x)}{d}+\frac {18 i a b^2 \text {sech}(c+d x)}{d}-\frac {1155 i b^3 \arctan (\sinh (c+d x))}{128 d}-\frac {385 i b^3 \sinh ^3(c+d x)}{128 d}+\frac {1155 i b^3 \sinh (c+d x)}{128 d}+\frac {i b^3 \sinh ^3(c+d x) \tanh ^8(c+d x)}{8 d}+\frac {11 i b^3 \sinh ^3(c+d x) \tanh ^6(c+d x)}{48 d}+\frac {33 i b^3 \sinh ^3(c+d x) \tanh ^4(c+d x)}{64 d}+\frac {231 i b^3 \sinh ^3(c+d x) \tanh ^2(c+d x)}{128 d}\right )\) |
Input:
Int[Sinh[c + d*x]^3*(a + b*Tanh[c + d*x]^3)^3,x]
Output:
I*((((-15*I)/2)*a^2*b*ArcTan[Sinh[c + d*x]])/d - (((1155*I)/128)*b^3*ArcTa n[Sinh[c + d*x]])/d + (I*a^3*Cosh[c + d*x])/d + ((12*I)*a*b^2*Cosh[c + d*x ])/d - ((I/3)*a^3*Cosh[c + d*x]^3)/d - (I*a*b^2*Cosh[c + d*x]^3)/d + ((18* I)*a*b^2*Sech[c + d*x])/d - ((4*I)*a*b^2*Sech[c + d*x]^3)/d + (((3*I)/5)*a *b^2*Sech[c + d*x]^5)/d + (((15*I)/2)*a^2*b*Sinh[c + d*x])/d + (((1155*I)/ 128)*b^3*Sinh[c + d*x])/d - (((5*I)/2)*a^2*b*Sinh[c + d*x]^3)/d - (((385*I )/128)*b^3*Sinh[c + d*x]^3)/d + (((3*I)/2)*a^2*b*Sinh[c + d*x]^3*Tanh[c + d*x]^2)/d + (((231*I)/128)*b^3*Sinh[c + d*x]^3*Tanh[c + d*x]^2)/d + (((33* I)/64)*b^3*Sinh[c + d*x]^3*Tanh[c + d*x]^4)/d + (((11*I)/48)*b^3*Sinh[c + d*x]^3*Tanh[c + d*x]^6)/d + ((I/8)*b^3*Sinh[c + d*x]^3*Tanh[c + d*x]^8)/d)
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_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> Int[ExpandTrig[(d*sin[e + f*x])^m*(a + b*(c*tan[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]
Time = 32.04 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.11
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+3 a^{2} b \left (\frac {\sinh \left (d x +c \right )^{5}}{3 \cosh \left (d x +c \right )^{2}}-\frac {5 \sinh \left (d x +c \right )^{3}}{3 \cosh \left (d x +c \right )^{2}}-\frac {5 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}+\frac {5 \,\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+5 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+3 b^{2} a \left (\frac {\sinh \left (d x +c \right )^{8}}{3 \cosh \left (d x +c \right )^{5}}-\frac {8 \sinh \left (d x +c \right )^{6}}{3 \cosh \left (d x +c \right )^{5}}-\frac {16 \sinh \left (d x +c \right )^{4}}{\cosh \left (d x +c \right )^{5}}-\frac {64 \sinh \left (d x +c \right )^{2}}{3 \cosh \left (d x +c \right )^{5}}-\frac {128}{15 \cosh \left (d x +c \right )^{5}}\right )+b^{3} \left (\frac {\sinh \left (d x +c \right )^{11}}{3 \cosh \left (d x +c \right )^{8}}-\frac {11 \sinh \left (d x +c \right )^{9}}{3 \cosh \left (d x +c \right )^{8}}-\frac {33 \sinh \left (d x +c \right )^{7}}{\cosh \left (d x +c \right )^{8}}-\frac {77 \sinh \left (d x +c \right )^{5}}{\cosh \left (d x +c \right )^{8}}-\frac {77 \sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{8}}-\frac {33 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{8}}+33 \left (\frac {\operatorname {sech}\left (d x +c \right )^{7}}{8}+\frac {7 \operatorname {sech}\left (d x +c \right )^{5}}{48}+\frac {35 \operatorname {sech}\left (d x +c \right )^{3}}{192}+\frac {35 \,\operatorname {sech}\left (d x +c \right )}{128}\right ) \tanh \left (d x +c \right )+\frac {1155 \arctan \left ({\mathrm e}^{d x +c}\right )}{64}\right )}{d}\) | \(366\) |
| default | \(\frac {a^{3} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+3 a^{2} b \left (\frac {\sinh \left (d x +c \right )^{5}}{3 \cosh \left (d x +c \right )^{2}}-\frac {5 \sinh \left (d x +c \right )^{3}}{3 \cosh \left (d x +c \right )^{2}}-\frac {5 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}+\frac {5 \,\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+5 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+3 b^{2} a \left (\frac {\sinh \left (d x +c \right )^{8}}{3 \cosh \left (d x +c \right )^{5}}-\frac {8 \sinh \left (d x +c \right )^{6}}{3 \cosh \left (d x +c \right )^{5}}-\frac {16 \sinh \left (d x +c \right )^{4}}{\cosh \left (d x +c \right )^{5}}-\frac {64 \sinh \left (d x +c \right )^{2}}{3 \cosh \left (d x +c \right )^{5}}-\frac {128}{15 \cosh \left (d x +c \right )^{5}}\right )+b^{3} \left (\frac {\sinh \left (d x +c \right )^{11}}{3 \cosh \left (d x +c \right )^{8}}-\frac {11 \sinh \left (d x +c \right )^{9}}{3 \cosh \left (d x +c \right )^{8}}-\frac {33 \sinh \left (d x +c \right )^{7}}{\cosh \left (d x +c \right )^{8}}-\frac {77 \sinh \left (d x +c \right )^{5}}{\cosh \left (d x +c \right )^{8}}-\frac {77 \sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{8}}-\frac {33 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{8}}+33 \left (\frac {\operatorname {sech}\left (d x +c \right )^{7}}{8}+\frac {7 \operatorname {sech}\left (d x +c \right )^{5}}{48}+\frac {35 \operatorname {sech}\left (d x +c \right )^{3}}{192}+\frac {35 \,\operatorname {sech}\left (d x +c \right )}{128}\right ) \tanh \left (d x +c \right )+\frac {1155 \arctan \left ({\mathrm e}^{d x +c}\right )}{64}\right )}{d}\) | \(366\) |
| risch | \(\frac {{\mathrm e}^{3 d x +3 c} a^{3}}{24 d}+\frac {{\mathrm e}^{3 d x +3 c} a^{2} b}{8 d}+\frac {b^{2} a \,{\mathrm e}^{3 d x +3 c}}{8 d}+\frac {b^{3} {\mathrm e}^{3 d x +3 c}}{24 d}-\frac {3 \,{\mathrm e}^{d x +c} a^{3}}{8 d}-\frac {27 \,{\mathrm e}^{d x +c} a^{2} b}{8 d}-\frac {45 \,{\mathrm e}^{d x +c} b^{2} a}{8 d}-\frac {21 \,{\mathrm e}^{d x +c} b^{3}}{8 d}-\frac {3 \,{\mathrm e}^{-d x -c} a^{3}}{8 d}+\frac {27 \,{\mathrm e}^{-d x -c} a^{2} b}{8 d}-\frac {45 b^{2} a \,{\mathrm e}^{-d x -c}}{8 d}+\frac {21 b^{3} {\mathrm e}^{-d x -c}}{8 d}+\frac {{\mathrm e}^{-3 d x -3 c} a^{3}}{24 d}-\frac {{\mathrm e}^{-3 d x -3 c} a^{2} b}{8 d}+\frac {b^{2} a \,{\mathrm e}^{-3 d x -3 c}}{8 d}-\frac {b^{3} {\mathrm e}^{-3 d x -3 c}}{24 d}-\frac {b \,{\mathrm e}^{d x +c} \left (2880 a^{2} {\mathrm e}^{14 d x +14 c}+34560 a b \,{\mathrm e}^{14 d x +14 c}+11475 b^{2} {\mathrm e}^{14 d x +14 c}+14400 \,{\mathrm e}^{12 d x +12 c} a^{2}+211200 \,{\mathrm e}^{12 d x +12 c} a b +36775 \,{\mathrm e}^{12 d x +12 c} b^{2}+25920 \,{\mathrm e}^{10 d x +10 c} a^{2}+590592 \,{\mathrm e}^{10 d x +10 c} a b +67715 \,{\mathrm e}^{10 d x +10 c} b^{2}+14400 \,{\mathrm e}^{8 d x +8 c} a^{2}+957696 \,{\mathrm e}^{8 d x +8 c} a b +27055 \,{\mathrm e}^{8 d x +8 c} b^{2}-14400 \,{\mathrm e}^{6 d x +6 c} a^{2}+957696 \,{\mathrm e}^{6 d x +6 c} a b -27055 \,{\mathrm e}^{6 d x +6 c} b^{2}-25920 \,{\mathrm e}^{4 d x +4 c} a^{2}+590592 \,{\mathrm e}^{4 d x +4 c} a b -67715 \,{\mathrm e}^{4 d x +4 c} b^{2}-14400 \,{\mathrm e}^{2 d x +2 c} a^{2}+211200 \,{\mathrm e}^{2 d x +2 c} b a -36775 b^{2} {\mathrm e}^{2 d x +2 c}-2880 a^{2}+34560 a b -11475 b^{2}\right )}{960 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{8}}-\frac {1155 i b^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{128 d}+\frac {1155 i b^{3} \ln \left ({\mathrm e}^{d x +c}+i\right )}{128 d}-\frac {15 i b \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{2 d}+\frac {15 i b \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{2 d}\) | \(641\) |
Input:
int(sinh(d*x+c)^3*(a+b*tanh(d*x+c)^3)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(a^3*(-2/3+1/3*sinh(d*x+c)^2)*cosh(d*x+c)+3*a^2*b*(1/3*sinh(d*x+c)^5/c osh(d*x+c)^2-5/3*sinh(d*x+c)^3/cosh(d*x+c)^2-5*sinh(d*x+c)/cosh(d*x+c)^2+5 /2*sech(d*x+c)*tanh(d*x+c)+5*arctan(exp(d*x+c)))+3*b^2*a*(1/3*sinh(d*x+c)^ 8/cosh(d*x+c)^5-8/3*sinh(d*x+c)^6/cosh(d*x+c)^5-16*sinh(d*x+c)^4/cosh(d*x+ c)^5-64/3*sinh(d*x+c)^2/cosh(d*x+c)^5-128/15/cosh(d*x+c)^5)+b^3*(1/3*sinh( d*x+c)^11/cosh(d*x+c)^8-11/3*sinh(d*x+c)^9/cosh(d*x+c)^8-33*sinh(d*x+c)^7/ cosh(d*x+c)^8-77*sinh(d*x+c)^5/cosh(d*x+c)^8-77*sinh(d*x+c)^3/cosh(d*x+c)^ 8-33*sinh(d*x+c)/cosh(d*x+c)^8+33*(1/8*sech(d*x+c)^7+7/48*sech(d*x+c)^5+35 /192*sech(d*x+c)^3+35/128*sech(d*x+c))*tanh(d*x+c)+1155/64*arctan(exp(d*x+ c))))
Leaf count of result is larger than twice the leaf count of optimal. 8462 vs. \(2 (310) = 620\).
Time = 0.18 (sec) , antiderivative size = 8462, normalized size of antiderivative = 25.64 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:
integrate(sinh(d*x+c)^3*(a+b*tanh(d*x+c)^3)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\text {Timed out} \] Input:
integrate(sinh(d*x+c)**3*(a+b*tanh(d*x+c)**3)**3,x)
Output:
Timed out
Time = 0.13 (sec) , antiderivative size = 604, normalized size of antiderivative = 1.83 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx =\text {Too large to display} \] Input:
integrate(sinh(d*x+c)^3*(a+b*tanh(d*x+c)^3)^3,x, algorithm="maxima")
Output:
1/192*b^3*(8*(63*e^(-d*x - c) - e^(-3*d*x - 3*c))/d - 3465*arctan(e^(-d*x - c))/d - (440*e^(-2*d*x - 2*c) + 6103*e^(-4*d*x - 4*c) + 21019*e^(-6*d*x - 6*c) + 41207*e^(-8*d*x - 8*c) + 40243*e^(-10*d*x - 10*c) + 22589*e^(-12* d*x - 12*c) + 505*e^(-14*d*x - 14*c) - 3331*e^(-16*d*x - 16*c) - 1791*e^(- 18*d*x - 18*c) - 8)/(d*(e^(-3*d*x - 3*c) + 8*e^(-5*d*x - 5*c) + 28*e^(-7*d *x - 7*c) + 56*e^(-9*d*x - 9*c) + 70*e^(-11*d*x - 11*c) + 56*e^(-13*d*x - 13*c) + 28*e^(-15*d*x - 15*c) + 8*e^(-17*d*x - 17*c) + e^(-19*d*x - 19*c)) )) - 1/40*a*b^2*(5*(45*e^(-d*x - c) - e^(-3*d*x - 3*c))/d + (200*e^(-2*d*x - 2*c) + 2515*e^(-4*d*x - 4*c) + 6680*e^(-6*d*x - 6*c) + 9073*e^(-8*d*x - 8*c) + 5600*e^(-10*d*x - 10*c) + 1665*e^(-12*d*x - 12*c) - 5)/(d*(e^(-3*d *x - 3*c) + 5*e^(-5*d*x - 5*c) + 10*e^(-7*d*x - 7*c) + 10*e^(-9*d*x - 9*c) + 5*e^(-11*d*x - 11*c) + e^(-13*d*x - 13*c)))) + 1/8*a^2*b*((27*e^(-d*x - c) - e^(-3*d*x - 3*c))/d - 120*arctan(e^(-d*x - c))/d - (25*e^(-2*d*x - 2 *c) + 77*e^(-4*d*x - 4*c) + 3*e^(-6*d*x - 6*c) - 1)/(d*(e^(-3*d*x - 3*c) + 2*e^(-5*d*x - 5*c) + e^(-7*d*x - 7*c)))) + 1/24*a^3*(e^(3*d*x + 3*c)/d - 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d + e^(-3*d*x - 3*c)/d)
Time = 0.51 (sec) , antiderivative size = 580, normalized size of antiderivative = 1.76 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx =\text {Too large to display} \] Input:
integrate(sinh(d*x+c)^3*(a+b*tanh(d*x+c)^3)^3,x, algorithm="giac")
Output:
1/960*(40*a^3*e^(3*d*x + 3*c) + 120*a^2*b*e^(3*d*x + 3*c) + 120*a*b^2*e^(3 *d*x + 3*c) + 40*b^3*e^(3*d*x + 3*c) - 360*a^3*e^(d*x + c) - 3240*a^2*b*e^ (d*x + c) - 5400*a*b^2*e^(d*x + c) - 2520*b^3*e^(d*x + c) + 225*(64*a^2*b + 77*b^3)*arctan(e^(d*x + c)) - 40*(9*a^3*e^(2*d*x + 2*c) - 81*a^2*b*e^(2* d*x + 2*c) + 135*a*b^2*e^(2*d*x + 2*c) - 63*b^3*e^(2*d*x + 2*c) - a^3 + 3* a^2*b - 3*a*b^2 + b^3)*e^(-3*d*x - 3*c) - (2880*a^2*b*e^(15*d*x + 15*c) + 34560*a*b^2*e^(15*d*x + 15*c) + 11475*b^3*e^(15*d*x + 15*c) + 14400*a^2*b* e^(13*d*x + 13*c) + 211200*a*b^2*e^(13*d*x + 13*c) + 36775*b^3*e^(13*d*x + 13*c) + 25920*a^2*b*e^(11*d*x + 11*c) + 590592*a*b^2*e^(11*d*x + 11*c) + 67715*b^3*e^(11*d*x + 11*c) + 14400*a^2*b*e^(9*d*x + 9*c) + 957696*a*b^2*e ^(9*d*x + 9*c) + 27055*b^3*e^(9*d*x + 9*c) - 14400*a^2*b*e^(7*d*x + 7*c) + 957696*a*b^2*e^(7*d*x + 7*c) - 27055*b^3*e^(7*d*x + 7*c) - 25920*a^2*b*e^ (5*d*x + 5*c) + 590592*a*b^2*e^(5*d*x + 5*c) - 67715*b^3*e^(5*d*x + 5*c) - 14400*a^2*b*e^(3*d*x + 3*c) + 211200*a*b^2*e^(3*d*x + 3*c) - 36775*b^3*e^ (3*d*x + 3*c) - 2880*a^2*b*e^(d*x + c) + 34560*a*b^2*e^(d*x + c) - 11475*b ^3*e^(d*x + c))/(e^(2*d*x + 2*c) + 1)^8)/d
Time = 2.86 (sec) , antiderivative size = 757, normalized size of antiderivative = 2.29 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx =\text {Too large to display} \] Input:
int(sinh(c + d*x)^3*(a + b*tanh(c + d*x)^3)^3,x)
Output:
(exp(3*c + 3*d*x)*(a + b)^3)/(24*d) + (exp(- 3*c - 3*d*x)*(a - b)^3)/(24*d ) + (15*atan((exp(d*x)*exp(c)*(77*b^3*(d^2)^(1/2) + 64*a^2*b*(d^2)^(1/2))) /(d*(5929*b^6 + 9856*a^2*b^4 + 4096*a^4*b^2)^(1/2)))*(5929*b^6 + 9856*a^2* b^4 + 4096*a^4*b^2)^(1/2))/(64*(d^2)^(1/2)) - (3*exp(- c - d*x)*(a - b)^2* (a - 7*b))/(8*d) - (exp(c + d*x)*(6144*a*b^2 + 11005*b^3))/(120*d*(3*exp(2 *c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) + (exp(c + d*x)* (768*a*b^2 + 3365*b^3))/(20*d*(4*exp(2*c + 2*d*x) + 6*exp(4*c + 4*d*x) + 4 *exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1)) + (596*b^3*exp(c + d*x))/(3*d*( 6*exp(2*c + 2*d*x) + 15*exp(4*c + 4*d*x) + 20*exp(6*c + 6*d*x) + 15*exp(8* c + 8*d*x) + 6*exp(10*c + 10*d*x) + exp(12*c + 12*d*x) + 1)) - (3*exp(c + d*x)*(a + b)^2*(a + 7*b))/(8*d) - (3*exp(c + d*x)*(768*a*b^2 + 64*a^2*b + 255*b^3))/(64*d*(exp(2*c + 2*d*x) + 1)) - (2*exp(c + d*x)*(144*a*b^2 + 162 5*b^3))/(15*d*(5*exp(2*c + 2*d*x) + 10*exp(4*c + 4*d*x) + 10*exp(6*c + 6*d *x) + 5*exp(8*c + 8*d*x) + exp(10*c + 10*d*x) + 1)) - (112*b^3*exp(c + d*x ))/(d*(7*exp(2*c + 2*d*x) + 21*exp(4*c + 4*d*x) + 35*exp(6*c + 6*d*x) + 35 *exp(8*c + 8*d*x) + 21*exp(10*c + 10*d*x) + 7*exp(12*c + 12*d*x) + exp(14* c + 14*d*x) + 1)) + (exp(c + d*x)*(3072*a*b^2 + 576*a^2*b + 4355*b^3))/(96 *d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) + (32*b^3*exp(c + d*x))/(d *(8*exp(2*c + 2*d*x) + 28*exp(4*c + 4*d*x) + 56*exp(6*c + 6*d*x) + 70*exp( 8*c + 8*d*x) + 56*exp(10*c + 10*d*x) + 28*exp(12*c + 12*d*x) + 8*exp(14...
Time = 0.23 (sec) , antiderivative size = 1243, normalized size of antiderivative = 3.77 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx =\text {Too large to display} \] Input:
int(sinh(d*x+c)^3*(a+b*tanh(d*x+c)^3)^3,x)
Output:
(14400*e**(19*c + 19*d*x)*atan(e**(c + d*x))*a**2*b + 17325*e**(19*c + 19* d*x)*atan(e**(c + d*x))*b**3 + 115200*e**(17*c + 17*d*x)*atan(e**(c + d*x) )*a**2*b + 138600*e**(17*c + 17*d*x)*atan(e**(c + d*x))*b**3 + 403200*e**( 15*c + 15*d*x)*atan(e**(c + d*x))*a**2*b + 485100*e**(15*c + 15*d*x)*atan( e**(c + d*x))*b**3 + 806400*e**(13*c + 13*d*x)*atan(e**(c + d*x))*a**2*b + 970200*e**(13*c + 13*d*x)*atan(e**(c + d*x))*b**3 + 1008000*e**(11*c + 11 *d*x)*atan(e**(c + d*x))*a**2*b + 1212750*e**(11*c + 11*d*x)*atan(e**(c + d*x))*b**3 + 806400*e**(9*c + 9*d*x)*atan(e**(c + d*x))*a**2*b + 970200*e* *(9*c + 9*d*x)*atan(e**(c + d*x))*b**3 + 403200*e**(7*c + 7*d*x)*atan(e**( c + d*x))*a**2*b + 485100*e**(7*c + 7*d*x)*atan(e**(c + d*x))*b**3 + 11520 0*e**(5*c + 5*d*x)*atan(e**(c + d*x))*a**2*b + 138600*e**(5*c + 5*d*x)*ata n(e**(c + d*x))*b**3 + 14400*e**(3*c + 3*d*x)*atan(e**(c + d*x))*a**2*b + 17325*e**(3*c + 3*d*x)*atan(e**(c + d*x))*b**3 + 40*e**(22*c + 22*d*x)*a** 3 + 120*e**(22*c + 22*d*x)*a**2*b + 120*e**(22*c + 22*d*x)*a*b**2 + 40*e** (22*c + 22*d*x)*b**3 - 40*e**(20*c + 20*d*x)*a**3 - 2280*e**(20*c + 20*d*x )*a**2*b - 4440*e**(20*c + 20*d*x)*a*b**2 - 2200*e**(20*c + 20*d*x)*b**3 - 2120*e**(18*c + 18*d*x)*a**3 - 22200*e**(18*c + 18*d*x)*a**2*b - 79800*e* *(18*c + 18*d*x)*a*b**2 - 27995*e**(18*c + 18*d*x)*b**3 - 10680*e**(16*c + 16*d*x)*a**3 - 72600*e**(16*c + 16*d*x)*a**2*b - 398760*e**(16*c + 16*d*x )*a*b**2 - 84975*e**(16*c + 16*d*x)*b**3 - 27120*e**(14*c + 14*d*x)*a**...