\(\int \frac {\sinh ^4(c+d x)}{(a+b \text {sech}^2(c+d x))^2} \, dx\) [33]

Optimal result

Integrand size = 23, antiderivative size = 194 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {3 \left (a^2+8 a b+8 b^2\right ) x}{8 a^4}-\frac {3 \sqrt {b} \sqrt {a+b} (a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 a^4 d}-\frac {(5 a+6 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )}-\frac {3 b (3 a+4 b) \tanh (c+d x)}{8 a^3 d \left (a+b-b \tanh ^2(c+d x)\right )} \] Output:

3/8*(a^2+8*a*b+8*b^2)*x/a^4-3/2*b^(1/2)*(a+b)^(1/2)*(a+2*b)*arctanh(b^(1/2 
)*tanh(d*x+c)/(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)
                                                                                    
                                                                                    
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1080\) vs. \(2(194)=388\).

Time = 11.60 (sec) , antiderivative size = 1080, normalized size of antiderivative = 5.57 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:

Integrate[Sinh[c + d*x]^4/(a + b*Sech[c + d*x]^2)^2,x]
 

Output:

-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)*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]*Sqrt[b*(Cosh[c] - Sinh[c])^4] 
) + (Sech[2*c]*(32*b*(5*a^4 + 39*a^3*b + 106*a^2*b^2 + 120*a*b^3 + 48*b^4) 
*d*x*Cosh[2*c] + 16*a*b*(5*a^3 + 29*a^2*b + 48*a*b^2 + 24*b^3)*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*Sin 
h[2*c] + 640*a*b^4*Sinh[2*c] + 256*b^5*Sinh[2*c] - a^5*Sinh[2*d*x] - 62...
 

Rubi [A] (verified)

Time = 0.49 (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

\(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2 \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {\frac {(5 a+6 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {3 \left (\frac {\frac {\left (a^2+8 a b+8 b^2\right ) \text {arctanh}(\tanh (c+d x))}{a}-\frac {4 \sqrt {b} \sqrt {a+b} (a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a}}{a}-\frac {b (3 a+4 b) \tanh (c+d x)}{a \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{2 a}}{4 a}}{d}\)

Input:

Int[Sinh[c + d*x]^4/(a + b*Sech[c + d*x]^2)^2,x]
 

Output:

(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
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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])
 

rule 221

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]
 

rule 372

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]
 

rule 397

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]
 

rule 402

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4620

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(458\) vs. \(2(176)=352\).

Time = 0.22 (sec) , antiderivative size = 459, normalized size of antiderivative = 2.37

\[\frac {\frac {1}{4 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{2 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {a +8 b}{8 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {3 a +8 b}{8 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-3 a^{2}-24 a b -24 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 a^{4}}-\frac {1}{4 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{2 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-a -8 b}{8 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {3 a +8 b}{8 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (3 a^{2}+24 a b +24 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 a^{4}}+\frac {2 b \left (\frac {\left (-\frac {1}{2} a^{2}-\frac {1}{2} a b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-\frac {\left (a +b \right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (3 a^{2}+9 a b +6 b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{2}\right )}{a^{4}}}{d}\]

Input:

int(sinh(d*x+c)^4/(a+b*sech(d*x+c)^2)^2,x)
 

Output:

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/ 
(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)+2*b/a^4*(((-1/2*a^2-1 
/2*a*b)*tanh(1/2*d*x+1/2*c)^3-1/2*(a+b)*a*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)))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2464 vs. \(2 (185) = 370\).

Time = 0.32 (sec) , antiderivative size = 5169, normalized size of antiderivative = 26.64 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(sinh(d*x+c)^4/(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int \frac {\sinh ^{4}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \] Input:

integrate(sinh(d*x+c)**4/(a+b*sech(d*x+c)**2)**2,x)
 

Output:

Integral(sinh(c + d*x)**4/(a + b*sech(c + d*x)**2)**2, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1299 vs. \(2 (185) = 370\).

Time = 0.20 (sec) , antiderivative size = 1299, normalized size of antiderivative = 6.70 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(sinh(d*x+c)^4/(a+b*sech(d*x+c)^2)^2,x, algorithm="maxima")
 

Output:

-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 + ...
 

Giac [A] (verification not implemented)

Time = 0.88 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.66 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {\frac {24 \, {\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} {\left (d x + c\right )}}{a^{4}} - \frac {{\left (18 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 144 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 144 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 16 \, a b e^{\left (2 \, d x + 2 \, c\right )} + a^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{a^{4}} - \frac {96 \, {\left (a^{2} b + 3 \, a b^{2} + 2 \, b^{3}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{\sqrt {-a b - b^{2}} a^{4}} + \frac {a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 16 \, a b e^{\left (2 \, d x + 2 \, c\right )}}{a^{4}} + \frac {64 \, {\left (a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + a^{2} b + a b^{2}\right )}}{{\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )} a^{4}}}{64 \, d} \] Input:

integrate(sinh(d*x+c)^4/(a+b*sech(d*x+c)^2)^2,x, algorithm="giac")
 

Output:

1/64*(24*(a^2 + 8*a*b + 8*b^2)*(d*x + c)/a^4 - (18*a^2*e^(4*d*x + 4*c) + 1 
44*a*b*e^(4*d*x + 4*c) + 144*b^2*e^(4*d*x + 4*c) - 8*a^2*e^(2*d*x + 2*c) - 
 16*a*b*e^(2*d*x + 2*c) + a^2)*e^(-4*d*x - 4*c)/a^4 - 96*(a^2*b + 3*a*b^2 
+ 2*b^3)*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/(sqrt( 
-a*b - b^2)*a^4) + (a^2*e^(4*d*x + 4*c) - 8*a^2*e^(2*d*x + 2*c) - 16*a*b*e 
^(2*d*x + 2*c))/a^4 + 64*(a^2*b*e^(2*d*x + 2*c) + 3*a*b^2*e^(2*d*x + 2*c) 
+ 2*b^3*e^(2*d*x + 2*c) + a^2*b + a*b^2)/((a*e^(4*d*x + 4*c) + 2*a*e^(2*d* 
x + 2*c) + 4*b*e^(2*d*x + 2*c) + a)*a^4))/d
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^4\,{\mathrm {sinh}\left (c+d\,x\right )}^4}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^2} \,d x \] Input:

int(sinh(c + d*x)^4/(a + b/cosh(c + d*x)^2)^2,x)
 

Output:

int((cosh(c + d*x)^4*sinh(c + d*x)^4)/(b + a*cosh(c + d*x)^2)^2, x)
 

Reduce [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 1434, normalized size of antiderivative = 7.39 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:

int(sinh(d*x+c)^4/(a+b*sech(d*x+c)^2)^2,x)
 

Output:

( - 48*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + 
 b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**2 - 96*e**(8*c + 8*d*x)*sqrt(b)* 
sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sq 
rt(a))*a*b - 48*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sq 
rt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**2 - 96*e**(8*c + 8*d*x)*sq 
rt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x) 
*sqrt(a))*a*b + 48*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a + b)*log(2*sqrt(b)*sqrt 
(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a**2 + 96*e**(8*c + 8*d*x)*sqrt(b) 
*sqrt(a + b)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a*b 
 - 96*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + 
b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**2 - 384*e**(6*c + 6*d*x)*sqrt(b)* 
sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sq 
rt(a))*a*b - 384*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b 
)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*b**2 - 96*e**(6*c + 6*d*x 
)*sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + 
d*x)*sqrt(a))*a**2 - 384*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a + b)*log(sqrt(2*s 
qrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a*b - 384*e**(6*c + 
6*d*x)*sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e** 
(c + d*x)*sqrt(a))*b**2 + 96*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a + b)*log(2*sq 
rt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b)*a**2 + 384*e**(6*c + ...