Integrand size = 23, antiderivative size = 242 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {3 \left (a^2+12 a b+16 b^2\right ) x}{8 a^5}-\frac {3 \sqrt {b} \left (5 a^2+20 a b+16 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 a^5 \sqrt {a+b} d}-\frac {(5 a+8 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {b (7 a+12 b) \tanh (c+d x)}{8 a^3 d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {3 b (a+2 b) \tanh (c+d x)}{2 a^4 d \left (a+b-b \tanh ^2(c+d x)\right )} \]
3/8*(a^2+12*a*b+16*b^2)*x/a^5-3/8*(5*a^2+20*a*b+16*b^2)*arctanh(b^(1/2)*ta nh(d*x+c)/(a+b)^(1/2))*b^(1/2)/a^5/d/(a+b)^(1/2)-1/8*(5*a+8*b)*cosh(d*x+c) *sinh(d*x+c)/a^2/d/(a+b-b*tanh(d*x+c)^2)^2+1/4*cosh(d*x+c)^3*sinh(d*x+c)/a /d/(a+b-b*tanh(d*x+c)^2)^2-1/8*b*(7*a+12*b)*tanh(d*x+c)/a^3/d/(a+b-b*tanh( d*x+c)^2)^2-3/2*b*(a+2*b)*tanh(d*x+c)/a^4/d/(a+b-b*tanh(d*x+c)^2)
Result contains complex when optimal does not.
Time = 31.09 (sec) , antiderivative size = 3457, normalized size of antiderivative = 14.29 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Result too large to show} \]
(3*(a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*(((3*a^2 + 8*a*b + 8* b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(5/2) - (a*Sqrt [b]*(3*a^2 + 16*a*b + 16*b^2 + 3*a*(a + 2*b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2)))/(16384*b^(5/2)*d* (a + b*Sech[c + d*x]^2)^3) + ((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d *x]^6*((-3*a*(a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(5/2) + (Sqrt[b]*(3*a^3 + 14*a^2*b + 24*a*b^2 + 16*b^3 + a*(3*a^2 + 4*a *b + 4*b^2)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a + b)^2*(a + 2*b + a* Cosh[2*(c + d*x)])^2)))/(16384*b^(5/2)*d*(a + b*Sech[c + d*x]^2)^3) - (3*( a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((-2*(3*a^5 - 10*a^4*b + 80*a^3*b^2 + 480*a^2*b^3 + 640*a*b^4 + 256*b^5)*ArcTanh[(Sech[d*x]*(Cosh[2 *c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b] *Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]))/(Sqrt[a + b]*Sqr t[b*(Cosh[c] - Sinh[c])^4]) + (Sech[2*c]*(256*b^2*(a + b)^2*(3*a^2 + 8*a*b + 8*b^2)*d*x*Cosh[2*c] + 512*a*b^2*(a + b)^2*(a + 2*b)*d*x*Cosh[2*d*x] + 128*a^4*b^2*d*x*Cosh[2*(c + 2*d*x)] + 256*a^3*b^3*d*x*Cosh[2*(c + 2*d*x)] + 128*a^2*b^4*d*x*Cosh[2*(c + 2*d*x)] + 512*a^4*b^2*d*x*Cosh[4*c + 2*d*x] + 2048*a^3*b^3*d*x*Cosh[4*c + 2*d*x] + 2560*a^2*b^4*d*x*Cosh[4*c + 2*d*x] + 1024*a*b^5*d*x*Cosh[4*c + 2*d*x] + 128*a^4*b^2*d*x*Cosh[6*c + 4*d*x] + 2 56*a^3*b^3*d*x*Cosh[6*c + 4*d*x] + 128*a^2*b^4*d*x*Cosh[6*c + 4*d*x] - ...
Time = 0.54 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.11, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3042, 4620, 372, 402, 25, 402, 27, 402, 27, 397, 219, 221}
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.
| \(\displaystyle \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (i c+i d x)^4}{\left (a+b \sec (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 4620 |
| \(\displaystyle \frac {\int \frac {\tanh ^4(c+d x)}{\left (1-\tanh ^2(c+d x)\right )^3 \left (-b \tanh ^2(c+d x)+a+b\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2 \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\int \frac {(4 a+7 b) \tanh ^2(c+d x)+a+b}{\left (1-\tanh ^2(c+d x)\right )^2 \left (-b \tanh ^2(c+d x)+a+b\right )^3}d\tanh (c+d x)}{4 a}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2 \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {\int -\frac {5 b (5 a+8 b) \tanh ^2(c+d x)+(a+b) (3 a+8 b)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^3}d\tanh (c+d x)}{2 a}+\frac {(5 a+8 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{4 a}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2 \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {(5 a+8 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\int \frac {5 b (5 a+8 b) \tanh ^2(c+d x)+(a+b) (3 a+8 b)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^3}d\tanh (c+d x)}{2 a}}{4 a}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2 \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {(5 a+8 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {-\frac {\int -\frac {12 (a+b) \left (b (7 a+12 b) \tanh ^2(c+d x)+(a+b) (a+4 b)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{4 a (a+b)}-\frac {b (7 a+12 b) \tanh (c+d x)}{a \left (a-b \tanh ^2(c+d x)+b\right )^2}}{2 a}}{4 a}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2 \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {(5 a+8 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {3 \int \frac {b (7 a+12 b) \tanh ^2(c+d x)+(a+b) (a+4 b)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{a}-\frac {b (7 a+12 b) \tanh (c+d x)}{a \left (a-b \tanh ^2(c+d x)+b\right )^2}}{2 a}}{4 a}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2 \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {(5 a+8 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {3 \left (-\frac {\int -\frac {2 (a+b) \left (a^2+8 b a+8 b^2+4 b (a+2 b) \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a (a+b)}-\frac {4 b (a+2 b) \tanh (c+d x)}{a \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{a}-\frac {b (7 a+12 b) \tanh (c+d x)}{a \left (a-b \tanh ^2(c+d x)+b\right )^2}}{2 a}}{4 a}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2 \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {(5 a+8 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {3 \left (\frac {\int \frac {a^2+8 b a+8 b^2+4 b (a+2 b) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{a}-\frac {4 b (a+2 b) \tanh (c+d x)}{a \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{a}-\frac {b (7 a+12 b) \tanh (c+d x)}{a \left (a-b \tanh ^2(c+d x)+b\right )^2}}{2 a}}{4 a}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2 \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {(5 a+8 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {3 \left (\frac {\frac {\left (a^2+12 a b+16 b^2\right ) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a}-\frac {b \left (5 a^2+20 a b+16 b^2\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{a}-\frac {4 b (a+2 b) \tanh (c+d x)}{a \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{a}-\frac {b (7 a+12 b) \tanh (c+d x)}{a \left (a-b \tanh ^2(c+d x)+b\right )^2}}{2 a}}{4 a}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2 \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {(5 a+8 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {3 \left (\frac {\frac {\left (a^2+12 a b+16 b^2\right ) \text {arctanh}(\tanh (c+d x))}{a}-\frac {b \left (5 a^2+20 a b+16 b^2\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{a}-\frac {4 b (a+2 b) \tanh (c+d x)}{a \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{a}-\frac {b (7 a+12 b) \tanh (c+d x)}{a \left (a-b \tanh ^2(c+d x)+b\right )^2}}{2 a}}{4 a}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2 \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {(5 a+8 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {3 \left (\frac {\frac {\left (a^2+12 a b+16 b^2\right ) \text {arctanh}(\tanh (c+d x))}{a}-\frac {\sqrt {b} \left (5 a^2+20 a b+16 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {a+b}}}{a}-\frac {4 b (a+2 b) \tanh (c+d x)}{a \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{a}-\frac {b (7 a+12 b) \tanh (c+d x)}{a \left (a-b \tanh ^2(c+d x)+b\right )^2}}{2 a}}{4 a}}{d}\) |
(Tanh[c + d*x]/(4*a*(1 - Tanh[c + d*x]^2)^2*(a + b - b*Tanh[c + d*x]^2)^2) - (((5*a + 8*b)*Tanh[c + d*x])/(2*a*(1 - Tanh[c + d*x]^2)*(a + b - b*Tanh [c + d*x]^2)^2) - (-((b*(7*a + 12*b)*Tanh[c + d*x])/(a*(a + b - b*Tanh[c + d*x]^2)^2)) + (3*((((a^2 + 12*a*b + 16*b^2)*ArcTanh[Tanh[c + d*x]])/a - ( Sqrt[b]*(5*a^2 + 20*a*b + 16*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a*Sqrt[a + b]))/a - (4*b*(a + 2*b)*Tanh[c + d*x])/(a*(a + b - b*Tan h[c + d*x]^2))))/a)/(2*a))/(4*a))/d
3.1.41.3.1 Defintions of rubi rules used
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)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 )^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 )) Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a , b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_ )]^(m_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff^(m + 1)/f Subst[Int[x^m*(ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^p/(1 + f f^2*x^2)^(m/2 + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[n/2]
Leaf count of result is larger than twice the leaf count of optimal. \(540\) vs. \(2(222)=444\).
Time = 0.23 (sec) , antiderivative size = 541, normalized size of antiderivative = 2.24
\[\frac {-\frac {1}{4 a^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {1}{2 a^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {-a -12 b}{8 a^{4} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {3 a +12 b}{8 a^{4} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\left (3 a^{2}+36 a b +48 b^{2}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{5}}+\frac {2 b \left (\frac {\left (-\frac {9}{8} a^{3}-\frac {21}{8} a^{2} b -\frac {3}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-\frac {\left (27 a^{2}+35 a b -12 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8}-\frac {\left (27 a^{2}+35 a b -12 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8}+\left (-\frac {9}{8} a^{3}-\frac {21}{8} a^{2} b -\frac {3}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\left (15 a^{2}+60 a b +48 b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (-\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}-\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{8}\right )}{a^{5}}+\frac {1}{4 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{2 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {a +12 b}{8 a^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {3 a +12 b}{8 a^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-3 a^{2}-36 a b -48 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 a^{5}}}{d}\]
1/d*(-1/4/a^3/(1+tanh(1/2*d*x+1/2*c))^4+1/2/a^3/(1+tanh(1/2*d*x+1/2*c))^3- 1/8*(-a-12*b)/a^4/(1+tanh(1/2*d*x+1/2*c))^2-1/8*(3*a+12*b)/a^4/(1+tanh(1/2 *d*x+1/2*c))+1/8/a^5*(3*a^2+36*a*b+48*b^2)*ln(1+tanh(1/2*d*x+1/2*c))+2*b/a ^5*(((-9/8*a^3-21/8*a^2*b-3/2*a*b^2)*tanh(1/2*d*x+1/2*c)^7-1/8*(27*a^2+35* a*b-12*b^2)*a*tanh(1/2*d*x+1/2*c)^5-1/8*(27*a^2+35*a*b-12*b^2)*a*tanh(1/2* d*x+1/2*c)^3+(-9/8*a^3-21/8*a^2*b-3/2*a*b^2)*tanh(1/2*d*x+1/2*c))/(tanh(1/ 2*d*x+1/2*c)^4*a+tanh(1/2*d*x+1/2*c)^4*b+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh( 1/2*d*x+1/2*c)^2*b+a+b)^2+1/8*(15*a^2+60*a*b+48*b^2)*(-1/4/b^(1/2)/(a+b)^( 1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a +b)^(1/2))+1/4/b^(1/2)/(a+b)^(1/2)*ln(-(a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2 *tanh(1/2*d*x+1/2*c)*b^(1/2)-(a+b)^(1/2))))+1/4/a^3/(tanh(1/2*d*x+1/2*c)-1 )^4+1/2/a^3/(tanh(1/2*d*x+1/2*c)-1)^3-1/8*(a+12*b)/a^4/(tanh(1/2*d*x+1/2*c )-1)^2-1/8*(3*a+12*b)/a^4/(tanh(1/2*d*x+1/2*c)-1)+1/8/a^5*(-3*a^2-36*a*b-4 8*b^2)*ln(tanh(1/2*d*x+1/2*c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 6038 vs. \(2 (234) = 468\).
Time = 0.40 (sec) , antiderivative size = 12353, normalized size of antiderivative = 51.05 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 2468 vs. \(2 (234) = 468\).
Time = 0.41 (sec) , antiderivative size = 2468, normalized size of antiderivative = 10.20 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
-3/256*(5*a^4*b + 100*a^3*b^2 + 320*a^2*b^3 + 352*a*b^4 + 128*b^5)*log((a* e^(2*d*x + 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(2*d*x + 2*c) + a + 2* b + 2*sqrt((a + b)*b)))/((a^7 + 2*a^6*b + a^5*b^2)*sqrt((a + b)*b)*d) - 3/ 64*(5*a^3*b + 30*a^2*b^2 + 40*a*b^3 + 16*b^4)*log((a*e^(2*d*x + 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(2*d*x + 2*c) + a + 2*b + 2*sqrt((a + b)*b) ))/((a^6 + 2*a^5*b + a^4*b^2)*sqrt((a + b)*b)*d) + 3/256*(5*a^4*b + 100*a^ 3*b^2 + 320*a^2*b^3 + 352*a*b^4 + 128*b^5)*log((a*e^(-2*d*x - 2*c) + a + 2 *b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)) )/((a^7 + 2*a^6*b + a^5*b^2)*sqrt((a + b)*b)*d) + 3/64*(5*a^3*b + 30*a^2*b ^2 + 40*a*b^3 + 16*b^4)*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b) *b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^6 + 2*a^5*b + a^4*b^2)*sqrt((a + b)*b)*d) + 3/128*(15*a^2*b + 20*a*b^2 + 8*b^3)*log((a* e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^5 + 2*a^4*b + a^3*b^2)*sqrt((a + b)*b)*d) + 1/64*(9*a^5*b + 110*a^4*b^2 + 216*a^3*b^3 + 112*a^2*b^4 + (9*a^5*b + 228*a ^4*b^2 + 920*a^3*b^3 + 1216*a^2*b^4 + 512*a*b^5)*e^(6*d*x + 6*c) + (27*a^5 *b + 594*a^4*b^2 + 2816*a^3*b^3 + 5696*a^2*b^4 + 5248*a*b^5 + 1792*b^6)*e^ (4*d*x + 4*c) + (27*a^5*b + 476*a^4*b^2 + 1720*a^3*b^3 + 2176*a^2*b^4 + 89 6*a*b^5)*e^(2*d*x + 2*c))/((a^9 + 2*a^8*b + a^7*b^2 + (a^9 + 2*a^8*b + a^7 *b^2)*e^(8*d*x + 8*c) + 4*(a^9 + 4*a^8*b + 5*a^7*b^2 + 2*a^6*b^3)*e^(6*...
\[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int { \frac {\sinh \left (d x + c\right )^{4}}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6\,{\mathrm {sinh}\left (c+d\,x\right )}^4}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \]