\(\int \frac {\sinh ^4(c+d x)}{(a+b \text {sech}^2(c+d x))^3} \, dx\) [41]

Optimal result

Integrand size = 23, antiderivative size = 242 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {3 \left (a^2+12 a b+16 b^2\right ) x}{8 a^5}-\frac {3 \sqrt {b} \left (5 a^2+20 a b+16 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 a^5 \sqrt {a+b} d}-\frac {(5 a+8 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 a d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {b (7 a+12 b) \tanh (c+d x)}{8 a^3 d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {3 b (a+2 b) \tanh (c+d x)}{2 a^4 d \left (a+b-b \tanh ^2(c+d x)\right )} \] Output:

3/8*(a^2+12*a*b+16*b^2)*x/a^5-3/8*b^(1/2)*(5*a^2+20*a*b+16*b^2)*arctanh(b^ 
(1/2)*tanh(d*x+c)/(a+b)^(1/2))/a^5/(a+b)^(1/2)/d-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)
                                                                                    
                                                                                    
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(3080\) vs. \(2(242)=484\).

Time = 20.53 (sec) , antiderivative size = 3080, normalized size of antiderivative = 12.73 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Result too large to show} \] Input:

Integrate[Sinh[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3,x]
 

Output:

(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] - ...
 

Rubi [A] (verified)

Time = 0.57 (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

\(\displaystyle \frac {\frac {\tanh (c+d x)}{4 a \left (1-\tanh ^2(c+d x)\right )^2 \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {(5 a+8 b) \tanh (c+d x)}{2 a \left (1-\tanh ^2(c+d x)\right ) \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {3 \left (\frac {\frac {\left (a^2+12 a b+16 b^2\right ) \text {arctanh}(\tanh (c+d x))}{a}-\frac {\sqrt {b} \left (5 a^2+20 a b+16 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {a+b}}}{a}-\frac {4 b (a+2 b) \tanh (c+d x)}{a \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{a}-\frac {b (7 a+12 b) \tanh (c+d x)}{a \left (a-b \tanh ^2(c+d x)+b\right )^2}}{2 a}}{4 a}}{d}\)

Input:

Int[Sinh[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3,x]
 

Output:

(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
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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. \(537\) vs. \(2(222)=444\).

Time = 0.26 (sec) , antiderivative size = 538, normalized size of antiderivative = 2.22

\[\frac {\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}}-\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}\]

Input:

int(sinh(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x)
 

Output:

1/d*(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-48*b^2)*ln(tanh(1/2*d*x+1/2*c)-1)-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+48* 
b^2)*ln(tanh(1/2*d*x+1/2*c)+1))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6038 vs. \(2 (234) = 468\).

Time = 0.74 (sec) , antiderivative size = 12353, normalized size of antiderivative = 51.05 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(sinh(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Timed out} \] Input:

integrate(sinh(d*x+c)**4/(a+b*sech(d*x+c)**2)**3,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2468 vs. \(2 (234) = 468\).

Time = 0.27 (sec) , antiderivative size = 2468, normalized size of antiderivative = 10.20 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(sinh(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")
 

Output:

-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*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (234) = 468\).

Time = 0.97 (sec) , antiderivative size = 518, normalized size of antiderivative = 2.14 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {\frac {24 \, {\left (a^{2} + 12 \, a b + 16 \, b^{2}\right )} {\left (d x + c\right )}}{a^{5}} - \frac {24 \, {\left (5 \, a^{2} b + 20 \, a b^{2} + 16 \, b^{3}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{\sqrt {-a b - b^{2}} a^{5}} + \frac {a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 24 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )}}{a^{6}} - \frac {6 \, a^{4} e^{\left (12 \, d x + 12 \, c\right )} + 72 \, a^{3} b e^{\left (12 \, d x + 12 \, c\right )} + 96 \, a^{2} b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 16 \, a^{4} e^{\left (10 \, d x + 10 \, c\right )} + 168 \, a^{3} b e^{\left (10 \, d x + 10 \, c\right )} + 384 \, a^{2} b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 256 \, a b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 5 \, a^{4} e^{\left (8 \, d x + 8 \, c\right )} - 64 \, a^{3} b e^{\left (8 \, d x + 8 \, c\right )} - 192 \, a^{2} b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 256 \, a b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 256 \, b^{4} e^{\left (8 \, d x + 8 \, c\right )} - 20 \, a^{4} e^{\left (6 \, d x + 6 \, c\right )} - 360 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} - 1024 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 896 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 20 \, a^{4} e^{\left (4 \, d x + 4 \, c\right )} - 216 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} - 304 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 4 \, a^{4} e^{\left (2 \, d x + 2 \, c\right )} - 16 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} + a^{4}}{{\left (a e^{\left (6 \, d x + 6 \, c\right )} + 2 \, a e^{\left (4 \, d x + 4 \, c\right )} + 4 \, b e^{\left (4 \, d x + 4 \, c\right )} + a e^{\left (2 \, d x + 2 \, c\right )}\right )}^{2} a^{5}}}{64 \, d} \] Input:

integrate(sinh(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")
 

Output:

1/64*(24*(a^2 + 12*a*b + 16*b^2)*(d*x + c)/a^5 - 24*(5*a^2*b + 20*a*b^2 + 
16*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^5) + (a^3*e^(4*d*x + 4*c) - 8*a^3*e^(2*d*x + 2*c) - 24*a^2*b* 
e^(2*d*x + 2*c))/a^6 - (6*a^4*e^(12*d*x + 12*c) + 72*a^3*b*e^(12*d*x + 12* 
c) + 96*a^2*b^2*e^(12*d*x + 12*c) + 16*a^4*e^(10*d*x + 10*c) + 168*a^3*b*e 
^(10*d*x + 10*c) + 384*a^2*b^2*e^(10*d*x + 10*c) + 256*a*b^3*e^(10*d*x + 1 
0*c) + 5*a^4*e^(8*d*x + 8*c) - 64*a^3*b*e^(8*d*x + 8*c) - 192*a^2*b^2*e^(8 
*d*x + 8*c) - 256*a*b^3*e^(8*d*x + 8*c) - 256*b^4*e^(8*d*x + 8*c) - 20*a^4 
*e^(6*d*x + 6*c) - 360*a^3*b*e^(6*d*x + 6*c) - 1024*a^2*b^2*e^(6*d*x + 6*c 
) - 896*a*b^3*e^(6*d*x + 6*c) - 20*a^4*e^(4*d*x + 4*c) - 216*a^3*b*e^(4*d* 
x + 4*c) - 304*a^2*b^2*e^(4*d*x + 4*c) - 4*a^4*e^(2*d*x + 2*c) - 16*a^3*b* 
e^(2*d*x + 2*c) + a^4)/((a*e^(6*d*x + 6*c) + 2*a*e^(4*d*x + 4*c) + 4*b*e^( 
4*d*x + 4*c) + a*e^(2*d*x + 2*c))^2*a^5))/d
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6\,{\mathrm {sinh}\left (c+d\,x\right )}^4}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \] Input:

int(sinh(c + d*x)^4/(a + b/cosh(c + d*x)^2)^3,x)
 

Output:

int((cosh(c + d*x)^6*sinh(c + d*x)^4)/(b + a*cosh(c + d*x)^2)^3, x)
 

Reduce [B] (verification not implemented)

Time = 0.54 (sec) , antiderivative size = 5075, normalized size of antiderivative = 20.97 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:

int(sinh(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x)
 

Output:

( - 60*e**(12*c + 12*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**5 - 360*e**(12*c + 12*d*x)*sqr 
t(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d* 
x)*sqrt(a))*a**4*b - 672*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a + b)*log( - sqr 
t(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a**3*b**2 - 384 
*e**(12*c + 12*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*b**3 - 60*e**(12*c + 12*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**5 - 360*e**(12*c + 12*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**4*b - 672*e**(12*c + 1 
2*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**3*b**2 - 384*e**(12*c + 12*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*b** 
3 + 60*e**(12*c + 12*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**5 + 360*e**(12*c + 12*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**4*b + 6 
72*e**(12*c + 12*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**3*b**2 + 384*e**(12*c + 12*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*b**3 
 - 240*e**(10*c + 10*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqr...