Integrand size = 23, antiderivative size = 111 \[ \int \text {csch}^7(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx=\frac {a (5 a+16 b) \text {arctanh}(\cosh (c+d x))}{16 d}+\frac {b^2 \cosh (c+d x)}{d}-\frac {a (5 a+16 b) \coth (c+d x) \text {csch}(c+d x)}{16 d}+\frac {5 a^2 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^2 \coth (c+d x) \text {csch}^5(c+d x)}{6 d} \]
1/16*a*(5*a+16*b)*arctanh(cosh(d*x+c))/d+b^2*cosh(d*x+c)/d-1/16*a*(5*a+16* b)*coth(d*x+c)*csch(d*x+c)/d+5/24*a^2*coth(d*x+c)*csch(d*x+c)^3/d-1/6*a^2* coth(d*x+c)*csch(d*x+c)^5/d
Leaf count is larger than twice the leaf count of optimal. \(278\) vs. \(2(111)=222\).
Time = 0.10 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.50 \[ \int \text {csch}^7(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx=\frac {b^2 \cosh (c) \cosh (d x)}{d}-\frac {5 a^2 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a b \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{4 d}+\frac {a^2 \text {csch}^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a^2 \text {csch}^6\left (\frac {1}{2} (c+d x)\right )}{384 d}+\frac {5 a^2 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}+\frac {a b \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )}{d}-\frac {5 a^2 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {a b \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{d}-\frac {5 a^2 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a b \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{4 d}-\frac {a^2 \text {sech}^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a^2 \text {sech}^6\left (\frac {1}{2} (c+d x)\right )}{384 d}+\frac {b^2 \sinh (c) \sinh (d x)}{d} \]
(b^2*Cosh[c]*Cosh[d*x])/d - (5*a^2*Csch[(c + d*x)/2]^2)/(64*d) - (a*b*Csch [(c + d*x)/2]^2)/(4*d) + (a^2*Csch[(c + d*x)/2]^4)/(64*d) - (a^2*Csch[(c + d*x)/2]^6)/(384*d) + (5*a^2*Log[Cosh[(c + d*x)/2]])/(16*d) + (a*b*Log[Cos h[(c + d*x)/2]])/d - (5*a^2*Log[Sinh[(c + d*x)/2]])/(16*d) - (a*b*Log[Sinh [(c + d*x)/2]])/d - (5*a^2*Sech[(c + d*x)/2]^2)/(64*d) - (a*b*Sech[(c + d* x)/2]^2)/(4*d) - (a^2*Sech[(c + d*x)/2]^4)/(64*d) - (a^2*Sech[(c + d*x)/2] ^6)/(384*d) + (b^2*Sinh[c]*Sinh[d*x])/d
Time = 0.45 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.20, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3042, 26, 3694, 1471, 25, 2345, 27, 1471, 25, 299, 219}
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.
| \(\displaystyle \int \text {csch}^7(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \left (a+b \sin (i c+i d x)^4\right )^2}{\sin (i c+i d x)^7}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\left (b \sin (i c+i d x)^4+a\right )^2}{\sin (i c+i d x)^7}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle \frac {\int \frac {\left (b \cosh ^4(c+d x)-2 b \cosh ^2(c+d x)+a+b\right )^2}{\left (1-\cosh ^2(c+d x)\right )^4}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1471 |
| \(\displaystyle \frac {\frac {a^2 \cosh (c+d x)}{6 \left (1-\cosh ^2(c+d x)\right )^3}-\frac {1}{6} \int -\frac {-6 b^2 \cosh ^6(c+d x)+18 b^2 \cosh ^4(c+d x)-6 b (2 a+3 b) \cosh ^2(c+d x)+5 a^2+6 b^2+12 a b}{\left (1-\cosh ^2(c+d x)\right )^3}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{6} \int \frac {-6 b^2 \cosh ^6(c+d x)+18 b^2 \cosh ^4(c+d x)-6 b (2 a+3 b) \cosh ^2(c+d x)+5 a^2+6 b^2+12 a b}{\left (1-\cosh ^2(c+d x)\right )^3}d\cosh (c+d x)+\frac {a^2 \cosh (c+d x)}{6 \left (1-\cosh ^2(c+d x)\right )^3}}{d}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {5 a^2 \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}-\frac {1}{4} \int -\frac {3 \left (8 b^2 \cosh ^4(c+d x)-16 b^2 \cosh ^2(c+d x)+5 a^2+8 b^2+16 a b\right )}{\left (1-\cosh ^2(c+d x)\right )^2}d\cosh (c+d x)\right )+\frac {a^2 \cosh (c+d x)}{6 \left (1-\cosh ^2(c+d x)\right )^3}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{4} \int \frac {8 b^2 \cosh ^4(c+d x)-16 b^2 \cosh ^2(c+d x)+5 a^2+8 b^2+16 a b}{\left (1-\cosh ^2(c+d x)\right )^2}d\cosh (c+d x)+\frac {5 a^2 \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}\right )+\frac {a^2 \cosh (c+d x)}{6 \left (1-\cosh ^2(c+d x)\right )^3}}{d}\) |
| \(\Big \downarrow \) 1471 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{4} \left (\frac {a (5 a+16 b) \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}-\frac {1}{2} \int -\frac {5 a^2+16 b a+16 b^2-16 b^2 \cosh ^2(c+d x)}{1-\cosh ^2(c+d x)}d\cosh (c+d x)\right )+\frac {5 a^2 \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}\right )+\frac {a^2 \cosh (c+d x)}{6 \left (1-\cosh ^2(c+d x)\right )^3}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {5 a^2+16 b a+16 b^2-16 b^2 \cosh ^2(c+d x)}{1-\cosh ^2(c+d x)}d\cosh (c+d x)+\frac {a (5 a+16 b) \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}\right )+\frac {5 a^2 \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}\right )+\frac {a^2 \cosh (c+d x)}{6 \left (1-\cosh ^2(c+d x)\right )^3}}{d}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{4} \left (\frac {1}{2} \left (a (5 a+16 b) \int \frac {1}{1-\cosh ^2(c+d x)}d\cosh (c+d x)+16 b^2 \cosh (c+d x)\right )+\frac {a (5 a+16 b) \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}\right )+\frac {5 a^2 \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}\right )+\frac {a^2 \cosh (c+d x)}{6 \left (1-\cosh ^2(c+d x)\right )^3}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {5 a^2 \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}+\frac {3}{4} \left (\frac {1}{2} \left (a (5 a+16 b) \text {arctanh}(\cosh (c+d x))+16 b^2 \cosh (c+d x)\right )+\frac {a (5 a+16 b) \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}\right )\right )+\frac {a^2 \cosh (c+d x)}{6 \left (1-\cosh ^2(c+d x)\right )^3}}{d}\) |
((a^2*Cosh[c + d*x])/(6*(1 - Cosh[c + d*x]^2)^3) + ((5*a^2*Cosh[c + d*x])/ (4*(1 - Cosh[c + d*x]^2)^2) + (3*((a*(5*a + 16*b)*ArcTanh[Cosh[c + d*x]] + 16*b^2*Cosh[c + d*x])/2 + (a*(5*a + 16*b)*Cosh[c + d*x])/(2*(1 - Cosh[c + d*x]^2))))/4)/6)/d
3.3.6.3.1 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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
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] :> 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 = 0.58 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.83
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (\left (-\frac {\operatorname {csch}\left (d x +c \right )^{5}}{6}+\frac {5 \operatorname {csch}\left (d x +c \right )^{3}}{24}-\frac {5 \,\operatorname {csch}\left (d x +c \right )}{16}\right ) \coth \left (d x +c \right )+\frac {5 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )}{8}\right )+2 a b \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+b^{2} \cosh \left (d x +c \right )}{d}\) | \(92\) |
| default | \(\frac {a^{2} \left (\left (-\frac {\operatorname {csch}\left (d x +c \right )^{5}}{6}+\frac {5 \operatorname {csch}\left (d x +c \right )^{3}}{24}-\frac {5 \,\operatorname {csch}\left (d x +c \right )}{16}\right ) \coth \left (d x +c \right )+\frac {5 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )}{8}\right )+2 a b \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+b^{2} \cosh \left (d x +c \right )}{d}\) | \(92\) |
| parallelrisch | \(\frac {-5120 a \left (a +\frac {16 b}{5}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \left (\cosh \left (5 d x +5 c \right )-\frac {3 \cosh \left (6 d x +6 c \right )}{16}+\frac {66 \cosh \left (d x +c \right )}{5}-\frac {45 \cosh \left (2 d x +2 c \right )}{16}-\frac {17 \cosh \left (3 d x +3 c \right )}{3}+\frac {9 \cosh \left (4 d x +4 c \right )}{8}+\frac {15}{8}\right ) a^{2} \operatorname {sech}\left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \operatorname {csch}\left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+4096 b a \left (\operatorname {sech}\left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-3\right ) \operatorname {csch}\left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+4096 \left (a \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+4 b \left (1+\cosh \left (d x +c \right )\right )\right ) b}{16384 d}\) | \(171\) |
| risch | \(\frac {{\mathrm e}^{d x +c} b^{2}}{2 d}+\frac {{\mathrm e}^{-d x -c} b^{2}}{2 d}-\frac {a \,{\mathrm e}^{d x +c} \left (15 a \,{\mathrm e}^{10 d x +10 c}+48 b \,{\mathrm e}^{10 d x +10 c}-85 \,{\mathrm e}^{8 d x +8 c} a -144 b \,{\mathrm e}^{8 d x +8 c}+198 \,{\mathrm e}^{6 d x +6 c} a +96 b \,{\mathrm e}^{6 d x +6 c}+198 \,{\mathrm e}^{4 d x +4 c} a +96 b \,{\mathrm e}^{4 d x +4 c}-85 a \,{\mathrm e}^{2 d x +2 c}-144 b \,{\mathrm e}^{2 d x +2 c}+15 a +48 b \right )}{24 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{6}}-\frac {5 a^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{16 d}-\frac {a \ln \left ({\mathrm e}^{d x +c}-1\right ) b}{d}+\frac {5 a^{2} \ln \left ({\mathrm e}^{d x +c}+1\right )}{16 d}+\frac {a \ln \left ({\mathrm e}^{d x +c}+1\right ) b}{d}\) | \(250\) |
1/d*(a^2*((-1/6*csch(d*x+c)^5+5/24*csch(d*x+c)^3-5/16*csch(d*x+c))*coth(d* x+c)+5/8*arctanh(exp(d*x+c)))+2*a*b*(-1/2*csch(d*x+c)*coth(d*x+c)+arctanh( exp(d*x+c)))+b^2*cosh(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 4500 vs. \(2 (103) = 206\).
Time = 0.32 (sec) , antiderivative size = 4500, normalized size of antiderivative = 40.54 \[ \int \text {csch}^7(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx=\text {Too large to display} \]
1/48*(24*b^2*cosh(d*x + c)^14 + 336*b^2*cosh(d*x + c)*sinh(d*x + c)^13 + 2 4*b^2*sinh(d*x + c)^14 - 6*(5*a^2 + 16*a*b + 20*b^2)*cosh(d*x + c)^12 + 6* (364*b^2*cosh(d*x + c)^2 - 5*a^2 - 16*a*b - 20*b^2)*sinh(d*x + c)^12 + 24* (364*b^2*cosh(d*x + c)^3 - 3*(5*a^2 + 16*a*b + 20*b^2)*cosh(d*x + c))*sinh (d*x + c)^11 + 2*(85*a^2 + 144*a*b + 108*b^2)*cosh(d*x + c)^10 + 2*(12012* b^2*cosh(d*x + c)^4 - 198*(5*a^2 + 16*a*b + 20*b^2)*cosh(d*x + c)^2 + 85*a ^2 + 144*a*b + 108*b^2)*sinh(d*x + c)^10 + 4*(12012*b^2*cosh(d*x + c)^5 - 330*(5*a^2 + 16*a*b + 20*b^2)*cosh(d*x + c)^3 + 5*(85*a^2 + 144*a*b + 108* b^2)*cosh(d*x + c))*sinh(d*x + c)^9 - 12*(33*a^2 + 16*a*b + 10*b^2)*cosh(d *x + c)^8 + 6*(12012*b^2*cosh(d*x + c)^6 - 495*(5*a^2 + 16*a*b + 20*b^2)*c osh(d*x + c)^4 + 15*(85*a^2 + 144*a*b + 108*b^2)*cosh(d*x + c)^2 - 66*a^2 - 32*a*b - 20*b^2)*sinh(d*x + c)^8 + 48*(1716*b^2*cosh(d*x + c)^7 - 99*(5* a^2 + 16*a*b + 20*b^2)*cosh(d*x + c)^5 + 5*(85*a^2 + 144*a*b + 108*b^2)*co sh(d*x + c)^3 - 2*(33*a^2 + 16*a*b + 10*b^2)*cosh(d*x + c))*sinh(d*x + c)^ 7 - 12*(33*a^2 + 16*a*b + 10*b^2)*cosh(d*x + c)^6 + 12*(6006*b^2*cosh(d*x + c)^8 - 462*(5*a^2 + 16*a*b + 20*b^2)*cosh(d*x + c)^6 + 35*(85*a^2 + 144* a*b + 108*b^2)*cosh(d*x + c)^4 - 28*(33*a^2 + 16*a*b + 10*b^2)*cosh(d*x + c)^2 - 33*a^2 - 16*a*b - 10*b^2)*sinh(d*x + c)^6 + 24*(2002*b^2*cosh(d*x + c)^9 - 198*(5*a^2 + 16*a*b + 20*b^2)*cosh(d*x + c)^7 + 21*(85*a^2 + 144*a *b + 108*b^2)*cosh(d*x + c)^5 - 28*(33*a^2 + 16*a*b + 10*b^2)*cosh(d*x ...
Timed out. \[ \int \text {csch}^7(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (103) = 206\).
Time = 0.22 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.69 \[ \int \text {csch}^7(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx=\frac {1}{2} \, b^{2} {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {1}{48} \, a^{2} {\left (\frac {15 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {15 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (15 \, e^{\left (-d x - c\right )} - 85 \, e^{\left (-3 \, d x - 3 \, c\right )} + 198 \, e^{\left (-5 \, d x - 5 \, c\right )} + 198 \, e^{\left (-7 \, d x - 7 \, c\right )} - 85 \, e^{\left (-9 \, d x - 9 \, c\right )} + 15 \, e^{\left (-11 \, d x - 11 \, c\right )}\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}}\right )} + a b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \]
1/2*b^2*(e^(d*x + c)/d + e^(-d*x - c)/d) + 1/48*a^2*(15*log(e^(-d*x - c) + 1)/d - 15*log(e^(-d*x - c) - 1)/d + 2*(15*e^(-d*x - c) - 85*e^(-3*d*x - 3 *c) + 198*e^(-5*d*x - 5*c) + 198*e^(-7*d*x - 7*c) - 85*e^(-9*d*x - 9*c) + 15*e^(-11*d*x - 11*c))/(d*(6*e^(-2*d*x - 2*c) - 15*e^(-4*d*x - 4*c) + 20*e ^(-6*d*x - 6*c) - 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) - e^(-12*d*x - 12*c) - 1))) + a*b*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d + 2*(e^(-d*x - c) + e^(-3*d*x - 3*c))/(d*(2*e^(-2*d*x - 2*c) - e^(-4*d*x - 4 *c) - 1)))
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (103) = 206\).
Time = 0.38 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.19 \[ \int \text {csch}^7(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx=\frac {48 \, b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 3 \, {\left (5 \, a^{2} + 16 \, a b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - 3 \, {\left (5 \, a^{2} + 16 \, a b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {4 \, {\left (15 \, a^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 48 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 160 \, a^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 384 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 528 \, a^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 768 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4\right )}^{3}}}{96 \, d} \]
1/96*(48*b^2*(e^(d*x + c) + e^(-d*x - c)) + 3*(5*a^2 + 16*a*b)*log(e^(d*x + c) + e^(-d*x - c) + 2) - 3*(5*a^2 + 16*a*b)*log(e^(d*x + c) + e^(-d*x - c) - 2) - 4*(15*a^2*(e^(d*x + c) + e^(-d*x - c))^5 + 48*a*b*(e^(d*x + c) + e^(-d*x - c))^5 - 160*a^2*(e^(d*x + c) + e^(-d*x - c))^3 - 384*a*b*(e^(d* x + c) + e^(-d*x - c))^3 + 528*a^2*(e^(d*x + c) + e^(-d*x - c)) + 768*a*b* (e^(d*x + c) + e^(-d*x - c)))/((e^(d*x + c) + e^(-d*x - c))^2 - 4)^3)/d
Time = 1.56 (sec) , antiderivative size = 535, normalized size of antiderivative = 4.82 \[ \int \text {csch}^7(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx=\frac {b^2\,{\mathrm {e}}^{c+d\,x}}{2\,d}-\frac {\frac {8\,{\mathrm {e}}^{5\,c+5\,d\,x}\,\left (4\,a^2+3\,b\,a\right )}{3\,d}+\frac {4\,a\,b\,{\mathrm {e}}^{c+d\,x}}{3\,d}-\frac {16\,a\,b\,{\mathrm {e}}^{3\,c+3\,d\,x}}{3\,d}-\frac {16\,a\,b\,{\mathrm {e}}^{7\,c+7\,d\,x}}{3\,d}+\frac {4\,a\,b\,{\mathrm {e}}^{9\,c+9\,d\,x}}{3\,d}}{15\,{\mathrm {e}}^{4\,c+4\,d\,x}-6\,{\mathrm {e}}^{2\,c+2\,d\,x}-20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}-6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1}+\frac {b^2\,{\mathrm {e}}^{-c-d\,x}}{2\,d}+\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (5\,a^2\,\sqrt {-d^2}+16\,a\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {25\,a^4+160\,a^3\,b+256\,a^2\,b^2}}\right )\,\sqrt {25\,a^4+160\,a^3\,b+256\,a^2\,b^2}}{8\,\sqrt {-d^2}}-\frac {a^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {22\,a^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {16\,a^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1\right )}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (5\,a^2+16\,b\,a\right )}{8\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (32\,a\,b-5\,a^2\right )}{12\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
(b^2*exp(c + d*x))/(2*d) - ((8*exp(5*c + 5*d*x)*(3*a*b + 4*a^2))/(3*d) + ( 4*a*b*exp(c + d*x))/(3*d) - (16*a*b*exp(3*c + 3*d*x))/(3*d) - (16*a*b*exp( 7*c + 7*d*x))/(3*d) + (4*a*b*exp(9*c + 9*d*x))/(3*d))/(15*exp(4*c + 4*d*x) - 6*exp(2*c + 2*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) + (b^2*exp(- c - d*x))/(2*d) + (a tan((exp(d*x)*exp(c)*(5*a^2*(-d^2)^(1/2) + 16*a*b*(-d^2)^(1/2)))/(d*(160*a ^3*b + 25*a^4 + 256*a^2*b^2)^(1/2)))*(160*a^3*b + 25*a^4 + 256*a^2*b^2)^(1 /2))/(8*(-d^2)^(1/2)) - (a^2*exp(c + d*x))/(3*d*(3*exp(2*c + 2*d*x) - 3*ex p(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1)) - (22*a^2*exp(c + d*x))/(3*d*(6*ex p(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)) - (16*a^2*exp(c + d*x))/(3*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)) - (exp(c + d*x)*(16*a*b + 5*a^2))/(8*d*(exp(2*c + 2*d*x) - 1)) - (exp(c + d*x)*(32*a*b - 5*a^2))/(12*d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1))