Integrand size = 23, antiderivative size = 142 \[ \int \text {csch}^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=-\frac {3 a^2 (a+8 b) \text {arctanh}(\cosh (c+d x))}{8 d}-\frac {b^2 (3 a+b) \cosh (c+d x)}{d}+\frac {b^2 (a+b) \cosh ^3(c+d x)}{d}-\frac {3 b^3 \cosh ^5(c+d x)}{5 d}+\frac {b^3 \cosh ^7(c+d x)}{7 d}+\frac {3 a^3 \coth (c+d x) \text {csch}(c+d x)}{8 d}-\frac {a^3 \coth (c+d x) \text {csch}^3(c+d x)}{4 d} \]
-3/8*a^2*(a+8*b)*arctanh(cosh(d*x+c))/d-b^2*(3*a+b)*cosh(d*x+c)/d+b^2*(a+b )*cosh(d*x+c)^3/d-3/5*b^3*cosh(d*x+c)^5/d+1/7*b^3*cosh(d*x+c)^7/d+3/8*a^3* coth(d*x+c)*csch(d*x+c)/d-1/4*a^3*coth(d*x+c)*csch(d*x+c)^3/d
Time = 3.35 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.45 \[ \int \text {csch}^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {-35 b^2 (144 a+35 b) \cosh (c+d x)+35 b^2 (16 a+7 b) \cosh (3 (c+d x))-49 b^3 \cosh (5 (c+d x))+5 b^3 \cosh (7 (c+d x))+210 a^3 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )-35 a^3 \text {csch}^4\left (\frac {1}{2} (c+d x)\right )-840 a^3 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )-6720 a^2 b \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+840 a^3 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+6720 a^2 b \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+210 a^3 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )+35 a^3 \text {sech}^4\left (\frac {1}{2} (c+d x)\right )}{2240 d} \]
(-35*b^2*(144*a + 35*b)*Cosh[c + d*x] + 35*b^2*(16*a + 7*b)*Cosh[3*(c + d* x)] - 49*b^3*Cosh[5*(c + d*x)] + 5*b^3*Cosh[7*(c + d*x)] + 210*a^3*Csch[(c + d*x)/2]^2 - 35*a^3*Csch[(c + d*x)/2]^4 - 840*a^3*Log[Cosh[(c + d*x)/2]] - 6720*a^2*b*Log[Cosh[(c + d*x)/2]] + 840*a^3*Log[Sinh[(c + d*x)/2]] + 67 20*a^2*b*Log[Sinh[(c + d*x)/2]] + 210*a^3*Sech[(c + d*x)/2]^2 + 35*a^3*Sec h[(c + d*x)/2]^4)/(2240*d)
Time = 0.63 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3042, 26, 3694, 1471, 25, 2345, 25, 2341, 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.
| \(\displaystyle \int \text {csch}^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \left (a+b \sin (i c+i d x)^4\right )^3}{\sin (i c+i d x)^5}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\left (b \sin (i c+i d x)^4+a\right )^3}{\sin (i c+i d x)^5}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle -\frac {\int \frac {\left (b \cosh ^4(c+d x)-2 b \cosh ^2(c+d x)+a+b\right )^3}{\left (1-\cosh ^2(c+d x)\right )^3}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1471 |
| \(\displaystyle -\frac {\frac {a^3 \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}-\frac {1}{4} \int -\frac {-4 b^3 \cosh ^{10}(c+d x)+20 b^3 \cosh ^8(c+d x)-4 b^2 (3 a+10 b) \cosh ^6(c+d x)+4 b^2 (9 a+10 b) \cosh ^4(c+d x)-4 b \left (3 a^2+9 b a+5 b^2\right ) \cosh ^2(c+d x)+3 a^3+4 b^3+12 a b^2+12 a^2 b}{\left (1-\cosh ^2(c+d x)\right )^2}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{4} \int \frac {-4 b^3 \cosh ^{10}(c+d x)+20 b^3 \cosh ^8(c+d x)-4 b^2 (3 a+10 b) \cosh ^6(c+d x)+4 b^2 (9 a+10 b) \cosh ^4(c+d x)-4 b \left (3 a^2+9 b a+5 b^2\right ) \cosh ^2(c+d x)+3 a^3+4 b^3+12 a b^2+12 a^2 b}{\left (1-\cosh ^2(c+d x)\right )^2}d\cosh (c+d x)+\frac {a^3 \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {3 a^3 \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}-\frac {1}{2} \int -\frac {8 b^3 \cosh ^8(c+d x)-32 b^3 \cosh ^6(c+d x)+24 b^2 (a+2 b) \cosh ^4(c+d x)-16 b^2 (3 a+2 b) \cosh ^2(c+d x)+3 a^3+8 b^3+24 a b^2+24 a^2 b}{1-\cosh ^2(c+d x)}d\cosh (c+d x)\right )+\frac {a^3 \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \int \frac {8 b^3 \cosh ^8(c+d x)-32 b^3 \cosh ^6(c+d x)+24 b^2 (a+2 b) \cosh ^4(c+d x)-16 b^2 (3 a+2 b) \cosh ^2(c+d x)+3 a^3+8 b^3+24 a b^2+24 a^2 b}{1-\cosh ^2(c+d x)}d\cosh (c+d x)+\frac {3 a^3 \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}\right )+\frac {a^3 \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 2341 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \int \left (-8 b^3 \cosh ^6(c+d x)+24 b^3 \cosh ^4(c+d x)-24 b^2 (a+b) \cosh ^2(c+d x)+8 b^2 (3 a+b)+\frac {3 \left (a^3+8 b a^2\right )}{1-\cosh ^2(c+d x)}\right )d\cosh (c+d x)+\frac {3 a^3 \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}\right )+\frac {a^3 \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {a^3 \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}+\frac {1}{4} \left (\frac {3 a^3 \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}+\frac {1}{2} \left (3 a^2 (a+8 b) \text {arctanh}(\cosh (c+d x))-8 b^2 (a+b) \cosh ^3(c+d x)+8 b^2 (3 a+b) \cosh (c+d x)-\frac {8}{7} b^3 \cosh ^7(c+d x)+\frac {24}{5} b^3 \cosh ^5(c+d x)\right )\right )}{d}\) |
-(((a^3*Cosh[c + d*x])/(4*(1 - Cosh[c + d*x]^2)^2) + ((3*a^3*Cosh[c + d*x] )/(2*(1 - Cosh[c + d*x]^2)) + (3*a^2*(a + 8*b)*ArcTanh[Cosh[c + d*x]] + 8* b^2*(3*a + b)*Cosh[c + d*x] - 8*b^2*(a + b)*Cosh[c + d*x]^3 + (24*b^3*Cosh [c + d*x]^5)/5 - (8*b^3*Cosh[c + d*x]^7)/7)/2)/4)/d)
3.3.12.3.1 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] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 , x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x , 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], 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] && LtQ[q, -1]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq* (a + b*x^2)^p, x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
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 = 2.24 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.88
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (\left (-\frac {\operatorname {csch}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {csch}\left (d x +c \right )}{8}\right ) \coth \left (d x +c \right )-\frac {3 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )}{4}\right )-6 a^{2} b \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )+3 a \,b^{2} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+b^{3} \left (-\frac {16}{35}+\frac {\sinh \left (d x +c \right )^{6}}{7}-\frac {6 \sinh \left (d x +c \right )^{4}}{35}+\frac {8 \sinh \left (d x +c \right )^{2}}{35}\right ) \cosh \left (d x +c \right )}{d}\) | \(125\) |
| default | \(\frac {a^{3} \left (\left (-\frac {\operatorname {csch}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {csch}\left (d x +c \right )}{8}\right ) \coth \left (d x +c \right )-\frac {3 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )}{4}\right )-6 a^{2} b \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )+3 a \,b^{2} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+b^{3} \left (-\frac {16}{35}+\frac {\sinh \left (d x +c \right )^{6}}{7}-\frac {6 \sinh \left (d x +c \right )^{4}}{35}+\frac {8 \sinh \left (d x +c \right )^{2}}{35}\right ) \cosh \left (d x +c \right )}{d}\) | \(125\) |
| parallelrisch | \(\frac {107520 a^{2} \left (a +8 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6160 \operatorname {sech}\left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{3} \left (\cosh \left (d x +c \right )-\frac {7 \cosh \left (2 d x +2 c \right )}{4}-\frac {3 \cosh \left (3 d x +3 c \right )}{11}+\frac {7 \cosh \left (4 d x +4 c \right )}{16}+\frac {21}{16}\right ) \operatorname {csch}\left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-645120 \left (\left (-\frac {a}{9}-\frac {7 b}{144}\right ) \cosh \left (3 d x +3 c \right )+\frac {7 b \cosh \left (5 d x +5 c \right )}{720}-\frac {b \cosh \left (7 d x +7 c \right )}{1008}+\left (a +\frac {35 b}{144}\right ) \cosh \left (d x +c \right )+\frac {8 a}{9}+\frac {64 b}{315}\right ) b^{2}}{286720 d}\) | \(160\) |
| risch | \(\frac {b^{3} {\mathrm e}^{7 d x +7 c}}{896 d}-\frac {7 b^{3} {\mathrm e}^{5 d x +5 c}}{640 d}+\frac {{\mathrm e}^{3 d x +3 c} a \,b^{2}}{8 d}+\frac {7 \,{\mathrm e}^{3 d x +3 c} b^{3}}{128 d}-\frac {9 \,{\mathrm e}^{d x +c} a \,b^{2}}{8 d}-\frac {35 \,{\mathrm e}^{d x +c} b^{3}}{128 d}-\frac {9 \,{\mathrm e}^{-d x -c} a \,b^{2}}{8 d}-\frac {35 \,{\mathrm e}^{-d x -c} b^{3}}{128 d}+\frac {{\mathrm e}^{-3 d x -3 c} a \,b^{2}}{8 d}+\frac {7 \,{\mathrm e}^{-3 d x -3 c} b^{3}}{128 d}-\frac {7 b^{3} {\mathrm e}^{-5 d x -5 c}}{640 d}+\frac {b^{3} {\mathrm e}^{-7 d x -7 c}}{896 d}+\frac {a^{3} {\mathrm e}^{d x +c} \left (3 \,{\mathrm e}^{6 d x +6 c}-11 \,{\mathrm e}^{4 d x +4 c}-11 \,{\mathrm e}^{2 d x +2 c}+3\right )}{4 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{4}}+\frac {3 a^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{8 d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{d x +c}-1\right ) b}{d}-\frac {3 a^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{8 d}-\frac {3 a^{2} \ln \left ({\mathrm e}^{d x +c}+1\right ) b}{d}\) | \(336\) |
1/d*(a^3*((-1/4*csch(d*x+c)^3+3/8*csch(d*x+c))*coth(d*x+c)-3/4*arctanh(exp (d*x+c)))-6*a^2*b*arctanh(exp(d*x+c))+3*a*b^2*(-2/3+1/3*sinh(d*x+c)^2)*cos h(d*x+c)+b^3*(-16/35+1/7*sinh(d*x+c)^6-6/35*sinh(d*x+c)^4+8/35*sinh(d*x+c) ^2)*cosh(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 6441 vs. \(2 (132) = 264\).
Time = 0.32 (sec) , antiderivative size = 6441, normalized size of antiderivative = 45.36 \[ \int \text {csch}^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Too large to display} \]
Timed out. \[ \int \text {csch}^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (132) = 264\).
Time = 0.20 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.39 \[ \int \text {csch}^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=-\frac {1}{4480} \, b^{3} {\left (\frac {{\left (49 \, e^{\left (-2 \, d x - 2 \, c\right )} - 245 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1225 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac {1225 \, e^{\left (-d x - c\right )} - 245 \, e^{\left (-3 \, d x - 3 \, c\right )} + 49 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac {1}{8} \, a b^{2} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} - \frac {1}{8} \, a^{3} {\left (\frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (3 \, e^{\left (-d x - c\right )} - 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} + 3 \, e^{\left (-7 \, d x - 7 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} - 3 \, a^{2} b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d}\right )} \]
-1/4480*b^3*((49*e^(-2*d*x - 2*c) - 245*e^(-4*d*x - 4*c) + 1225*e^(-6*d*x - 6*c) - 5)*e^(7*d*x + 7*c)/d + (1225*e^(-d*x - c) - 245*e^(-3*d*x - 3*c) + 49*e^(-5*d*x - 5*c) - 5*e^(-7*d*x - 7*c))/d) + 1/8*a*b^2*(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) - 1/8*a^3*( 3*log(e^(-d*x - c) + 1)/d - 3*log(e^(-d*x - c) - 1)/d + 2*(3*e^(-d*x - c) - 11*e^(-3*d*x - 3*c) - 11*e^(-5*d*x - 5*c) + 3*e^(-7*d*x - 7*c))/(d*(4*e^ (-2*d*x - 2*c) - 6*e^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) - e^(-8*d*x - 8*c ) - 1))) - 3*a^2*b*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d)
Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (132) = 264\).
Time = 0.51 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.91 \[ \int \text {csch}^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {5 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} - 84 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 560 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 560 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 6720 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 2240 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 840 \, {\left (a^{3} + 8 \, a^{2} b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) + 840 \, {\left (a^{3} + 8 \, a^{2} b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) + \frac {1120 \, {\left (3 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 20 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4\right )}^{2}}}{4480 \, d} \]
1/4480*(5*b^3*(e^(d*x + c) + e^(-d*x - c))^7 - 84*b^3*(e^(d*x + c) + e^(-d *x - c))^5 + 560*a*b^2*(e^(d*x + c) + e^(-d*x - c))^3 + 560*b^3*(e^(d*x + c) + e^(-d*x - c))^3 - 6720*a*b^2*(e^(d*x + c) + e^(-d*x - c)) - 2240*b^3* (e^(d*x + c) + e^(-d*x - c)) - 840*(a^3 + 8*a^2*b)*log(e^(d*x + c) + e^(-d *x - c) + 2) + 840*(a^3 + 8*a^2*b)*log(e^(d*x + c) + e^(-d*x - c) - 2) + 1 120*(3*a^3*(e^(d*x + c) + e^(-d*x - c))^3 - 20*a^3*(e^(d*x + c) + e^(-d*x - c)))/((e^(d*x + c) + e^(-d*x - c))^2 - 4)^2)/d
Time = 1.83 (sec) , antiderivative size = 421, normalized size of antiderivative = 2.96 \[ \int \text {csch}^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {b^3\,{\mathrm {e}}^{-7\,c-7\,d\,x}}{896\,d}-\frac {7\,b^3\,{\mathrm {e}}^{-5\,c-5\,d\,x}}{640\,d}-\frac {7\,b^3\,{\mathrm {e}}^{5\,c+5\,d\,x}}{640\,d}-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a^3\,\sqrt {-d^2}+8\,a^2\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {a^6+16\,a^5\,b+64\,a^4\,b^2}}\right )\,\sqrt {a^6+16\,a^5\,b+64\,a^4\,b^2}}{4\,\sqrt {-d^2}}+\frac {b^3\,{\mathrm {e}}^{7\,c+7\,d\,x}}{896\,d}-\frac {6\,a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {b^2\,{\mathrm {e}}^{c+d\,x}\,\left (144\,a+35\,b\right )}{128\,d}-\frac {4\,a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}+\frac {b^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}\,\left (16\,a+7\,b\right )}{128\,d}+\frac {b^2\,{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (16\,a+7\,b\right )}{128\,d}-\frac {b^2\,{\mathrm {e}}^{-c-d\,x}\,\left (144\,a+35\,b\right )}{128\,d}+\frac {3\,a^3\,{\mathrm {e}}^{c+d\,x}}{4\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {a^3\,{\mathrm {e}}^{c+d\,x}}{2\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
(b^3*exp(- 7*c - 7*d*x))/(896*d) - (7*b^3*exp(- 5*c - 5*d*x))/(640*d) - (7 *b^3*exp(5*c + 5*d*x))/(640*d) - (3*atan((exp(d*x)*exp(c)*(a^3*(-d^2)^(1/2 ) + 8*a^2*b*(-d^2)^(1/2)))/(d*(16*a^5*b + a^6 + 64*a^4*b^2)^(1/2)))*(16*a^ 5*b + a^6 + 64*a^4*b^2)^(1/2))/(4*(-d^2)^(1/2)) + (b^3*exp(7*c + 7*d*x))/( 896*d) - (6*a^3*exp(c + d*x))/(d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1)) - (b^2*exp(c + d*x)*(144*a + 35*b))/(128*d) - (4* a^3*exp(c + d*x))/(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)) + (b^2*exp(- 3*c - 3*d*x)*(16*a + 7*b))/ (128*d) + (b^2*exp(3*c + 3*d*x)*(16*a + 7*b))/(128*d) - (b^2*exp(- c - d*x )*(144*a + 35*b))/(128*d) + (3*a^3*exp(c + d*x))/(4*d*(exp(2*c + 2*d*x) - 1)) - (a^3*exp(c + d*x))/(2*d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1))