Integrand size = 23, antiderivative size = 194 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {3 \left (a^2+8 a b+8 b^2\right ) x}{8 a^4}-\frac {3 \sqrt {b} \sqrt {a+b} (a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 a^4 d}-\frac {(5 a+6 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac {3 b (3 a+4 b) \tanh (c+d x)}{8 a^3 d \left (a+b-b \tanh ^2(c+d x)\right )} \]
3/8*(a^2+8*a*b+8*b^2)*x/a^4-3/2*(a+2*b)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^ (1/2))*b^(1/2)*(a+b)^(1/2)/a^4/d-1/8*(5*a+6*b)*cosh(d*x+c)*sinh(d*x+c)/a^2 /d/(a+b-b*tanh(d*x+c)^2)+1/4*cosh(d*x+c)^3*sinh(d*x+c)/a/d/(a+b-b*tanh(d*x +c)^2)-3/8*b*(3*a+4*b)*tanh(d*x+c)/a^3/d/(a+b-b*tanh(d*x+c)^2)
Result contains complex when optimal does not.
Time = 14.74 (sec) , antiderivative size = 1330, normalized size of antiderivative = 6.86 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=-\frac {(a+2 b+a \cosh (2 c+2 d x))^2 \text {sech}^4(c+d x) \left (16 x+\frac {\left (a^3-6 a^2 b-24 a b^2-16 b^3\right ) \text {arctanh}\left (\frac {\text {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}}\right ) (\cosh (2 c)-\sinh (2 c))}{b (a+b)^{3/2} d \sqrt {b (\cosh (c)-\sinh (c))^4}}+\frac {\left (a^2+8 a b+8 b^2\right ) \text {sech}(2 c) ((a+2 b) \sinh (2 c)-a \sinh (2 d x))}{b (a+b) d (a+2 b+a \cosh (2 (c+d x)))}\right )}{256 a^2 \left (a+b \text {sech}^2(c+d x)\right )^2}+\frac {3 (a+2 b+a \cosh (2 c+2 d x))^2 \text {sech}^4(c+d x) \left (\frac {(a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 b^{3/2} (a+b)^{3/2} d}-\frac {a \sinh (2 (c+d x))}{8 b (a+b) d (a+2 b+a \cosh (2 (c+d x)))}\right )}{128 \left (a+b \text {sech}^2(c+d x)\right )^2}+\frac {(a+2 b+a \cosh (2 c+2 d x))^2 \text {sech}^4(c+d x) \left (\frac {\left (a^5-30 a^4 b-480 a^3 b^2-1600 a^2 b^3-1920 a b^4-768 b^5\right ) \left (-\frac {i \arctan \left (\text {sech}(d x) \left (-\frac {i \cosh (2 c)}{2 \sqrt {a+b} \sqrt {b \cosh (4 c)-b \sinh (4 c)}}+\frac {i \sinh (2 c)}{2 \sqrt {a+b} \sqrt {b \cosh (4 c)-b \sinh (4 c)}}\right ) (-a \sinh (d x)-2 b \sinh (d x)+a \sinh (2 c+d x))\right ) \cosh (2 c)}{8 a^4 b \sqrt {a+b} d \sqrt {b \cosh (4 c)-b \sinh (4 c)}}+\frac {i \arctan \left (\text {sech}(d x) \left (-\frac {i \cosh (2 c)}{2 \sqrt {a+b} \sqrt {b \cosh (4 c)-b \sinh (4 c)}}+\frac {i \sinh (2 c)}{2 \sqrt {a+b} \sqrt {b \cosh (4 c)-b \sinh (4 c)}}\right ) (-a \sinh (d x)-2 b \sinh (d x)+a \sinh (2 c+d x))\right ) \sinh (2 c)}{8 a^4 b \sqrt {a+b} d \sqrt {b \cosh (4 c)-b \sinh (4 c)}}\right )}{a+b}+\frac {\text {sech}(2 c) \left (160 a^4 b d x \cosh (2 c)+1248 a^3 b^2 d x \cosh (2 c)+3392 a^2 b^3 d x \cosh (2 c)+3840 a b^4 d x \cosh (2 c)+1536 b^5 d x \cosh (2 c)+80 a^4 b d x \cosh (2 d x)+464 a^3 b^2 d x \cosh (2 d x)+768 a^2 b^3 d x \cosh (2 d x)+384 a b^4 d x \cosh (2 d x)+80 a^4 b d x \cosh (4 c+2 d x)+464 a^3 b^2 d x \cosh (4 c+2 d x)+768 a^2 b^3 d x \cosh (4 c+2 d x)+384 a b^4 d x \cosh (4 c+2 d x)+a^5 \sinh (2 c)+34 a^4 b \sinh (2 c)+224 a^3 b^2 \sinh (2 c)+576 a^2 b^3 \sinh (2 c)+640 a b^4 \sinh (2 c)+256 b^5 \sinh (2 c)-a^5 \sinh (2 d x)-62 a^4 b \sinh (2 d x)-318 a^3 b^2 \sinh (2 d x)-512 a^2 b^3 \sinh (2 d x)-256 a b^4 \sinh (2 d x)-30 a^4 b \sinh (4 c+2 d x)-158 a^3 b^2 \sinh (4 c+2 d x)-256 a^2 b^3 \sinh (4 c+2 d x)-128 a b^4 \sinh (4 c+2 d x)-12 a^4 b \sinh (2 c+4 d x)-36 a^3 b^2 \sinh (2 c+4 d x)-24 a^2 b^3 \sinh (2 c+4 d x)-12 a^4 b \sinh (6 c+4 d x)-36 a^3 b^2 \sinh (6 c+4 d x)-24 a^2 b^3 \sinh (6 c+4 d x)+2 a^4 b \sinh (4 c+6 d x)+2 a^3 b^2 \sinh (4 c+6 d x)+2 a^4 b \sinh (8 c+6 d x)+2 a^3 b^2 \sinh (8 c+6 d x)\right )}{8 a^4 b (a+b) d (a+2 b+a \cosh (2 c+2 d x))}\right )}{128 \left (a+b \text {sech}^2(c+d x)\right )^2} \]
-1/256*((a + 2*b + a*Cosh[2*c + 2*d*x])^2*Sech[c + d*x]^4*(16*x + ((a^3 - 6*a^2*b - 24*a*b^2 - 16*b^3)*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]))/(b*(a + b)^(3/2)*d*Sqrt[b*(Cosh[c] - Sinh[c])^4]) + ((a^2 + 8*a*b + 8*b^2)*Sech[2*c]*((a + 2*b)*Sinh[2*c] - a *Sinh[2*d*x]))/(b*(a + b)*d*(a + 2*b + a*Cosh[2*(c + d*x)]))))/(a^2*(a + b *Sech[c + d*x]^2)^2) + (3*(a + 2*b + a*Cosh[2*c + 2*d*x])^2*Sech[c + d*x]^ 4*(((a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*b^(3/2)*(a + b)^(3/2)*d) - (a*Sinh[2*(c + d*x)])/(8*b*(a + b)*d*(a + 2*b + a*Cosh[2*( c + d*x)]))))/(128*(a + b*Sech[c + d*x]^2)^2) + ((a + 2*b + a*Cosh[2*c + 2 *d*x])^2*Sech[c + d*x]^4*(((a^5 - 30*a^4*b - 480*a^3*b^2 - 1600*a^2*b^3 - 1920*a*b^4 - 768*b^5)*(((-1/8*I)*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(S qrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*S inh[2*c + d*x])]*Cosh[2*c])/(a^4*b*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh [4*c]]) + ((I/8)*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[ b*Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4 *c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x])] *Sinh[2*c])/(a^4*b*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]])))/(a + b ) + (Sech[2*c]*(160*a^4*b*d*x*Cosh[2*c] + 1248*a^3*b^2*d*x*Cosh[2*c] + ...
Time = 0.46 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.13, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3042, 4620, 372, 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.
| \(\displaystyle \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (i c+i d x)^4}{\left (a+b \sec (i c+i d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 4620 |
| \(\displaystyle \frac {\int \frac {\tanh ^4(c+d x)}{\left (1-\tanh ^2(c+d x)\right )^3 \left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2 \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {\int \frac {(4 a+5 b) \tanh ^2(c+d x)+a+b}{\left (1-\tanh ^2(c+d x)\right )^2 \left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{4 a}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2 \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {\frac {\int -\frac {3 \left (b (5 a+6 b) \tanh ^2(c+d x)+(a+b) (a+2 b)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{2 a}+\frac {(5 a+6 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 a}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2 \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {\frac {(5 a+6 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {3 \int \frac {b (5 a+6 b) \tanh ^2(c+d x)+(a+b) (a+2 b)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{2 a}}{4 a}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2 \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {\frac {(5 a+6 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {3 \left (-\frac {\int -\frac {2 (a+b) \left (b (3 a+4 b) \tanh ^2(c+d x)+(a+b) (a+4 b)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a (a+b)}-\frac {b (3 a+4 b) \tanh (c+d x)}{a \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{2 a}}{4 a}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2 \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {\frac {(5 a+6 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {3 \left (\frac {\int \frac {b (3 a+4 b) \tanh ^2(c+d x)+(a+b) (a+4 b)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{a}-\frac {b (3 a+4 b) \tanh (c+d x)}{a \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{2 a}}{4 a}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2 \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {\frac {(5 a+6 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {3 \left (\frac {\frac {\left (a^2+8 a b+8 b^2\right ) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a}-\frac {4 b (a+b) (a+2 b) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{a}-\frac {b (3 a+4 b) \tanh (c+d x)}{a \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{2 a}}{4 a}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2 \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {\frac {(5 a+6 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {3 \left (\frac {\frac {\left (a^2+8 a b+8 b^2\right ) \text {arctanh}(\tanh (c+d x))}{a}-\frac {4 b (a+b) (a+2 b) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{a}-\frac {b (3 a+4 b) \tanh (c+d x)}{a \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{2 a}}{4 a}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2 \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {\frac {(5 a+6 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {3 \left (\frac {\frac {\left (a^2+8 a b+8 b^2\right ) \text {arctanh}(\tanh (c+d x))}{a}-\frac {4 \sqrt {b} \sqrt {a+b} (a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a}}{a}-\frac {b (3 a+4 b) \tanh (c+d x)}{a \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{2 a}}{4 a}}{d}\) |
(Tanh[c + d*x]/(4*a*(1 - Tanh[c + d*x]^2)^2*(a + b - b*Tanh[c + d*x]^2)) - (((5*a + 6*b)*Tanh[c + d*x])/(2*a*(1 - Tanh[c + d*x]^2)*(a + b - b*Tanh[c + d*x]^2)) - (3*((((a^2 + 8*a*b + 8*b^2)*ArcTanh[Tanh[c + d*x]])/a - (4*S qrt[b]*Sqrt[a + b]*(a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]]) /a)/a - (b*(3*a + 4*b)*Tanh[c + d*x])/(a*(a + b - b*Tanh[c + d*x]^2))))/(2 *a))/(4*a))/d
3.1.33.3.1 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. \(461\) vs. \(2(176)=352\).
Time = 0.23 (sec) , antiderivative size = 462, normalized size of antiderivative = 2.38
\[\frac {\frac {1}{4 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{2 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {a +8 b}{8 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {3 a +8 b}{8 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-3 a^{2}-24 a b -24 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 a^{4}}-\frac {1}{4 a^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {1}{2 a^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {-a -8 b}{8 a^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {3 a +8 b}{8 a^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\left (3 a^{2}+24 a b +24 b^{2}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{4}}+\frac {2 b \left (\frac {\left (-\frac {1}{2} a^{2}-\frac {1}{2} a b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-\frac {a \left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (3 a^{2}+9 a b +6 b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (-\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}-\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{2}\right )}{a^{4}}}{d}\]
1/d*(1/4/a^2/(tanh(1/2*d*x+1/2*c)-1)^4+1/2/a^2/(tanh(1/2*d*x+1/2*c)-1)^3-1 /8*(a+8*b)/a^3/(tanh(1/2*d*x+1/2*c)-1)^2-1/8*(3*a+8*b)/a^3/(tanh(1/2*d*x+1 /2*c)-1)+1/8/a^4*(-3*a^2-24*a*b-24*b^2)*ln(tanh(1/2*d*x+1/2*c)-1)-1/4/a^2/ (1+tanh(1/2*d*x+1/2*c))^4+1/2/a^2/(1+tanh(1/2*d*x+1/2*c))^3-1/8*(-a-8*b)/a ^3/(1+tanh(1/2*d*x+1/2*c))^2-1/8*(3*a+8*b)/a^3/(1+tanh(1/2*d*x+1/2*c))+1/8 /a^4*(3*a^2+24*a*b+24*b^2)*ln(1+tanh(1/2*d*x+1/2*c))+2*b/a^4*(((-1/2*a^2-1 /2*a*b)*tanh(1/2*d*x+1/2*c)^3-1/2*a*(a+b)*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)+1/2*(3*a^2+9*a*b+6*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)))))
Leaf count of result is larger than twice the leaf count of optimal. 2464 vs. \(2 (185) = 370\).
Time = 0.33 (sec) , antiderivative size = 5169, normalized size of antiderivative = 26.64 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Too large to display} \]
\[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int \frac {\sinh ^{4}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 1299 vs. \(2 (185) = 370\).
Time = 0.33 (sec) , antiderivative size = 1299, normalized size of antiderivative = 6.70 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Too large to display} \]
-1/64*(3*a^3*b + 42*a^2*b^2 + 88*a*b^3 + 48*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^5 + a^4*b)*sqrt((a + b)*b)*d) - 1/16*(3*a^2*b + 12*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^4 + a^3*b)*sqrt((a + b)*b)*d) + 1/64* (3*a^3*b + 42*a^2*b^2 + 88*a*b^3 + 48*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^5 + a^4*b)*sqrt((a + b)*b)*d) + 1/16*(3*a^2*b + 12*a*b^2 + 8*b^3)*lo g((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^4 + a^3*b)*sqrt((a + b)*b)*d) + 3/32*(3 *a*b + 2*b^2)*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^3 + a^2*b)*sqrt((a + b) *b)*d) + 1/16*(a^3*b + 8*a^2*b^2 + 8*a*b^3 + (a^3*b + 18*a^2*b^2 + 48*a*b^ 3 + 32*b^4)*e^(2*d*x + 2*c))/((a^6 + a^5*b + (a^6 + a^5*b)*e^(4*d*x + 4*c) + 2*(a^6 + 3*a^5*b + 2*a^4*b^2)*e^(2*d*x + 2*c))*d) - 1/16*(a^3*b + 8*a^2 *b^2 + 8*a*b^3 + (a^3*b + 18*a^2*b^2 + 48*a*b^3 + 32*b^4)*e^(-2*d*x - 2*c) )/((a^6 + a^5*b + 2*(a^6 + 3*a^5*b + 2*a^4*b^2)*e^(-2*d*x - 2*c) + (a^6 + a^5*b)*e^(-4*d*x - 4*c))*d) + 1/4*(a^2*b + 2*a*b^2 + (a^2*b + 8*a*b^2 + 8* b^3)*e^(2*d*x + 2*c))/((a^5 + a^4*b + (a^5 + a^4*b)*e^(4*d*x + 4*c) + 2*(a ^5 + 3*a^4*b + 2*a^3*b^2)*e^(2*d*x + 2*c))*d) - 1/4*(a^2*b + 2*a*b^2 + ...
\[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int { \frac {\sinh \left (d x + c\right )^{4}}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^4\,{\mathrm {sinh}\left (c+d\,x\right )}^4}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^2} \,d x \]