Integrand size = 23, antiderivative size = 117 \[ \int \frac {\sinh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {\left (3 a^2+12 a b+8 b^2\right ) x}{8 a^3}-\frac {\sqrt {b} (a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a^3 d}-\frac {(5 a+4 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 a d} \]
1/8*(3*a^2+12*a*b+8*b^2)*x/a^3-1/8*(5*a+4*b)*cosh(d*x+c)*sinh(d*x+c)/a^2/d +1/4*cosh(d*x+c)^3*sinh(d*x+c)/a/d-(a+b)^(3/2)*arctanh(b^(1/2)*tanh(d*x+c) /(a+b)^(1/2))*b^(1/2)/a^3/d
Leaf count is larger than twice the leaf count of optimal. \(294\) vs. \(2(117)=234\).
Time = 3.21 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.51 \[ \int \frac {\sinh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=-\frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^2(c+d x) \left (\sqrt {b} \left (3 a^3+34 a^2 b+64 a b^2+32 b^3\right ) \text {arctanh}\left (\frac {\text {sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right ) (\cosh (2 c)-\sinh (2 c))-\sqrt {b (\cosh (c)-\sinh (c))^4} \left (a^2 (3 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )+\sqrt {b} \sqrt {a+b} \left (-2 a^2 c+12 a^2 d x+48 a b d x+32 b^2 d x-8 a (a+b) \sinh (2 (c+d x))+a^2 \sinh (4 (c+d x))\right )\right )\right )}{64 a^3 \sqrt {b} \sqrt {a+b} d \left (a+b \text {sech}^2(c+d x)\right ) \sqrt {b (\cosh (c)-\sinh (c))^4}} \]
-1/64*((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*(Sqrt[b]*(3*a^3 + 3 4*a^2*b + 64*a*b^2 + 32*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]) - Sqrt[b*(Cosh[c] - Sinh[c])^4]*(a^2 *(3*a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]] + Sqrt[b]*Sqrt[a + b]*(-2*a^2*c + 12*a^2*d*x + 48*a*b*d*x + 32*b^2*d*x - 8*a*(a + b)*Sinh[ 2*(c + d*x)] + a^2*Sinh[4*(c + d*x)]))))/(a^3*Sqrt[b]*Sqrt[a + b]*d*(a + b *Sech[c + d*x]^2)*Sqrt[b*(Cosh[c] - Sinh[c])^4])
Time = 0.38 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.24, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 4620, 372, 402, 25, 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.
| \(\displaystyle \int \frac {\sinh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (i c+i d x)^4}{a+b \sec (i c+i d x)^2}dx\) |
| \(\Big \downarrow \) 4620 |
| \(\displaystyle \frac {\int \frac {\tanh ^4(c+d x)}{\left (1-\tanh ^2(c+d x)\right )^3 \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2}-\frac {\int \frac {(4 a+3 b) \tanh ^2(c+d x)+a+b}{\left (1-\tanh ^2(c+d x)\right )^2 \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{4 a}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2}-\frac {\frac {\int -\frac {b (5 a+4 b) \tanh ^2(c+d x)+(a+b) (3 a+4 b)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a}+\frac {(5 a+4 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right )}}{4 a}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2}-\frac {\frac {(5 a+4 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right )}-\frac {\int \frac {b (5 a+4 b) \tanh ^2(c+d x)+(a+b) (3 a+4 b)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a}}{4 a}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2}-\frac {\frac {(5 a+4 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right )}-\frac {\frac {\left (3 a^2+12 a b+8 b^2\right ) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a}-\frac {8 b (a+b)^2 \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{2 a}}{4 a}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2}-\frac {\frac {(5 a+4 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right )}-\frac {\frac {\left (3 a^2+12 a b+8 b^2\right ) \text {arctanh}(\tanh (c+d x))}{a}-\frac {8 b (a+b)^2 \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{2 a}}{4 a}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2}-\frac {\frac {(5 a+4 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right )}-\frac {\frac {\left (3 a^2+12 a b+8 b^2\right ) \text {arctanh}(\tanh (c+d x))}{a}-\frac {8 \sqrt {b} (a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a}}{2 a}}{4 a}}{d}\) |
(Tanh[c + d*x]/(4*a*(1 - Tanh[c + d*x]^2)^2) - (-1/2*(((3*a^2 + 12*a*b + 8 *b^2)*ArcTanh[Tanh[c + d*x]])/a - (8*Sqrt[b]*(a + b)^(3/2)*ArcTanh[(Sqrt[b ]*Tanh[c + d*x])/Sqrt[a + b]])/a)/a + ((5*a + 4*b)*Tanh[c + d*x])/(2*a*(1 - Tanh[c + d*x]^2)))/(4*a))/d
3.1.25.3.1 Defintions of rubi rules used
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. \(326\) vs. \(2(103)=206\).
Time = 235.07 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.79
| method | result | size |
| risch | \(\frac {3 x}{8 a}+\frac {3 b x}{2 a^{2}}+\frac {x \,b^{2}}{a^{3}}+\frac {{\mathrm e}^{4 d x +4 c}}{64 d a}-\frac {{\mathrm e}^{2 d x +2 c}}{8 a d}-\frac {{\mathrm e}^{2 d x +2 c} b}{8 a^{2} d}+\frac {{\mathrm e}^{-2 d x -2 c}}{8 a d}+\frac {{\mathrm e}^{-2 d x -2 c} b}{8 a^{2} d}-\frac {{\mathrm e}^{-4 d x -4 c}}{64 d a}+\frac {\sqrt {a b +b^{2}}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a +2 \sqrt {a b +b^{2}}+2 b}{a}\right )}{2 d \,a^{2}}+\frac {\sqrt {a b +b^{2}}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a +2 \sqrt {a b +b^{2}}+2 b}{a}\right ) b}{2 d \,a^{3}}-\frac {\sqrt {a b +b^{2}}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-a +2 \sqrt {a b +b^{2}}-2 b}{a}\right )}{2 d \,a^{2}}-\frac {\sqrt {a b +b^{2}}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-a +2 \sqrt {a b +b^{2}}-2 b}{a}\right ) b}{2 d \,a^{3}}\) | \(327\) |
| derivativedivides | \(\frac {\frac {1}{4 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{2 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {a +4 b}{8 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {3 a +4 b}{8 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-3 a^{2}-12 a b -8 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 a^{3}}+\frac {2 b \left (a^{2}+2 a b +b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (-\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}-\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{a^{3}}-\frac {1}{4 a \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {1}{2 a \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {-a -4 b}{8 a^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {3 a +4 b}{8 a^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\left (3 a^{2}+12 a b +8 b^{2}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{3}}}{d}\) | \(357\) |
| default | \(\frac {\frac {1}{4 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{2 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {a +4 b}{8 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {3 a +4 b}{8 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-3 a^{2}-12 a b -8 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 a^{3}}+\frac {2 b \left (a^{2}+2 a b +b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (-\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}-\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{a^{3}}-\frac {1}{4 a \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {1}{2 a \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {-a -4 b}{8 a^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {3 a +4 b}{8 a^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\left (3 a^{2}+12 a b +8 b^{2}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{3}}}{d}\) | \(357\) |
3/8*x/a+3/2*b*x/a^2+x/a^3*b^2+1/64/d/a*exp(4*d*x+4*c)-1/8/a/d*exp(2*d*x+2* c)-1/8/a^2/d*exp(2*d*x+2*c)*b+1/8/a/d*exp(-2*d*x-2*c)+1/8/a^2/d*exp(-2*d*x -2*c)*b-1/64/d/a*exp(-4*d*x-4*c)+1/2*(a*b+b^2)^(1/2)/d/a^2*ln(exp(2*d*x+2* c)+(a+2*(a*b+b^2)^(1/2)+2*b)/a)+1/2*(a*b+b^2)^(1/2)/d/a^3*ln(exp(2*d*x+2*c )+(a+2*(a*b+b^2)^(1/2)+2*b)/a)*b-1/2*(a*b+b^2)^(1/2)/d/a^2*ln(exp(2*d*x+2* c)-(-a+2*(a*b+b^2)^(1/2)-2*b)/a)-1/2*(a*b+b^2)^(1/2)/d/a^3*ln(exp(2*d*x+2* c)-(-a+2*(a*b+b^2)^(1/2)-2*b)/a)*b
Leaf count of result is larger than twice the leaf count of optimal. 720 vs. \(2 (103) = 206\).
Time = 0.29 (sec) , antiderivative size = 1681, normalized size of antiderivative = 14.37 \[ \int \frac {\sinh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\text {Too large to display} \]
[1/64*(a^2*cosh(d*x + c)^8 + 8*a^2*cosh(d*x + c)*sinh(d*x + c)^7 + a^2*sin h(d*x + c)^8 + 8*(3*a^2 + 12*a*b + 8*b^2)*d*x*cosh(d*x + c)^4 - 8*(a^2 + a *b)*cosh(d*x + c)^6 + 4*(7*a^2*cosh(d*x + c)^2 - 2*a^2 - 2*a*b)*sinh(d*x + c)^6 + 8*(7*a^2*cosh(d*x + c)^3 - 6*(a^2 + a*b)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*a^2*cosh(d*x + c)^4 + 4*(3*a^2 + 12*a*b + 8*b^2)*d*x - 60*(a ^2 + a*b)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*a^2*cosh(d*x + c)^5 + 4* (3*a^2 + 12*a*b + 8*b^2)*d*x*cosh(d*x + c) - 20*(a^2 + a*b)*cosh(d*x + c)^ 3)*sinh(d*x + c)^3 + 8*(a^2 + a*b)*cosh(d*x + c)^2 + 4*(7*a^2*cosh(d*x + c )^6 + 12*(3*a^2 + 12*a*b + 8*b^2)*d*x*cosh(d*x + c)^2 - 30*(a^2 + a*b)*cos h(d*x + c)^4 + 2*a^2 + 2*a*b)*sinh(d*x + c)^2 + 32*((a + b)*cosh(d*x + c)^ 4 + 4*(a + b)*cosh(d*x + c)^3*sinh(d*x + c) + 6*(a + b)*cosh(d*x + c)^2*si nh(d*x + c)^2 + 4*(a + b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a + b)*sinh(d*x + c)^4)*sqrt(a*b + b^2)*log((a^2*cosh(d*x + c)^4 + 4*a^2*cosh(d*x + c)*si nh(d*x + c)^3 + a^2*sinh(d*x + c)^4 + 2*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 2* (3*a^2*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 + 8*a*b + 8*b^ 2 + 4*(a^2*cosh(d*x + c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c) + 4*(a*cosh(d*x + c)^2 + 2*a*cosh(d*x + c)*sinh(d*x + c) + a*sinh(d*x + c)^2 + a + 2*b)*sqrt(a*b + b^2))/(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d *x + c)^3 + a*sinh(d*x + c)^4 + 2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh( d*x + c)^2 + a + 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 + (a + 2*b...
\[ \int \frac {\sinh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\int \frac {\sinh ^{4}{\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (103) = 206\).
Time = 0.30 (sec) , antiderivative size = 526, normalized size of antiderivative = 4.50 \[ \int \frac {\sinh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {3 \, b \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{16 \, \sqrt {{\left (a + b\right )} b} a d} + \frac {3 \, {\left (d x + c\right )}}{8 \, a d} - \frac {{\left (8 \, b e^{\left (-2 \, d x - 2 \, c\right )} - a\right )} e^{\left (4 \, d x + 4 \, c\right )}}{64 \, a^{2} d} - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{8 \, a d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, a d} + \frac {b \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (a + 2 \, b\right )} e^{\left (2 \, d x + 2 \, c\right )} + a\right )}{4 \, a^{2} d} - \frac {b \log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{4 \, a^{2} d} - \frac {{\left (a b + 2 \, b^{2}\right )} \log \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{8 \, \sqrt {{\left (a + b\right )} b} a^{2} d} + \frac {{\left (a b + 2 \, b^{2}\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{8 \, \sqrt {{\left (a + b\right )} b} a^{2} d} + \frac {{\left (a b + 2 \, b^{2}\right )} {\left (d x + c\right )}}{2 \, a^{3} d} + \frac {8 \, b e^{\left (-2 \, d x - 2 \, c\right )} - a e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, a^{2} d} + \frac {{\left (a^{2} b + 8 \, a b^{2} + 8 \, b^{3}\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{16 \, \sqrt {{\left (a + b\right )} b} a^{3} d} \]
3/16*b*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)))/(sqrt((a + b)*b)*a*d) + 3/8*(d*x + c)/(a*d) - 1/64*(8*b*e^(-2*d*x - 2*c) - a)*e^(4*d*x + 4*c)/(a^2*d) - 1/8* e^(2*d*x + 2*c)/(a*d) + 1/8*e^(-2*d*x - 2*c)/(a*d) + 1/4*b*log(a*e^(4*d*x + 4*c) + 2*(a + 2*b)*e^(2*d*x + 2*c) + a)/(a^2*d) - 1/4*b*log(2*(a + 2*b)* e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/(a^2*d) - 1/8*(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)))/(sqrt((a + b)*b)*a^2*d) + 1/8*(a*b + 2*b^2)*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)))/(sqrt((a + b)*b)*a^2*d) + 1/2*(a*b + 2*b^2) *(d*x + c)/(a^3*d) + 1/64*(8*b*e^(-2*d*x - 2*c) - a*e^(-4*d*x - 4*c))/(a^2 *d) + 1/16*(a^2*b + 8*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)))/(sqr t((a + b)*b)*a^3*d)
\[ \int \frac {\sinh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\int { \frac {\sinh \left (d x + c\right )^{4}}{b \operatorname {sech}\left (d x + c\right )^{2} + a} \,d x } \]
Time = 3.09 (sec) , antiderivative size = 328, normalized size of antiderivative = 2.80 \[ \int \frac {\sinh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {x\,\left (3\,a^2+12\,a\,b+8\,b^2\right )}{8\,a^3}-\frac {{\mathrm {e}}^{-4\,c-4\,d\,x}}{64\,a\,d}+\frac {{\mathrm {e}}^{4\,c+4\,d\,x}}{64\,a\,d}+\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (a+b\right )}{8\,a^2\,d}-\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+b\right )}{8\,a^2\,d}+\frac {\sqrt {b}\,\ln \left (\frac {4\,b\,{\left (a+b\right )}^3\,\left (2\,a\,b+a^2+a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+8\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+8\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^8}-\frac {8\,b^{3/2}\,{\left (a+b\right )}^{7/2}\,\left (a+2\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}+4\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^8}\right )\,{\left (a+b\right )}^{3/2}}{2\,a^3\,d}-\frac {\sqrt {b}\,\ln \left (\frac {8\,b^{3/2}\,{\left (a+b\right )}^{7/2}\,\left (a+2\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}+4\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^8}+\frac {4\,b\,{\left (a+b\right )}^3\,\left (2\,a\,b+a^2+a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+8\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+8\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^8}\right )\,{\left (a+b\right )}^{3/2}}{2\,a^3\,d} \]
(x*(12*a*b + 3*a^2 + 8*b^2))/(8*a^3) - exp(- 4*c - 4*d*x)/(64*a*d) + exp(4 *c + 4*d*x)/(64*a*d) + (exp(- 2*c - 2*d*x)*(a + b))/(8*a^2*d) - (exp(2*c + 2*d*x)*(a + b))/(8*a^2*d) + (b^(1/2)*log((4*b*(a + b)^3*(2*a*b + a^2 + a^ 2*exp(2*c + 2*d*x) + 8*b^2*exp(2*c + 2*d*x) + 8*a*b*exp(2*c + 2*d*x)))/a^8 - (8*b^(3/2)*(a + b)^(7/2)*(a + 2*a*exp(2*c + 2*d*x) + 4*b*exp(2*c + 2*d* x)))/a^8)*(a + b)^(3/2))/(2*a^3*d) - (b^(1/2)*log((8*b^(3/2)*(a + b)^(7/2) *(a + 2*a*exp(2*c + 2*d*x) + 4*b*exp(2*c + 2*d*x)))/a^8 + (4*b*(a + b)^3*( 2*a*b + a^2 + a^2*exp(2*c + 2*d*x) + 8*b^2*exp(2*c + 2*d*x) + 8*a*b*exp(2* c + 2*d*x)))/a^8)*(a + b)^(3/2))/(2*a^3*d)