\(\int \text {sech}^5(c+d x) (a+b \sinh ^2(c+d x))^3 \, dx\) [276]

Optimal result

Integrand size = 23, antiderivative size = 103 \[ \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {3 (a-b) \left (4 b^2+(a+b)^2\right ) \arctan (\sinh (c+d x))}{8 d}+\frac {b^3 \sinh (c+d x)}{d}+\frac {3 (a-b)^2 (a+3 b) \text {sech}(c+d x) \tanh (c+d x)}{8 d}+\frac {(a-b)^3 \text {sech}^3(c+d x) \tanh (c+d x)}{4 d} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 9.37 (sec) , antiderivative size = 472, normalized size of antiderivative = 4.58 \[ \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=-\frac {\text {csch}^5(c+d x) \left (256 \, _6F_5\left (\frac {3}{2},2,2,2,2,2;1,1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^8(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3+384 \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^8(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \left (7 a+5 b \sinh ^2(c+d x)\right )-21 \left (15 a b^2 \sinh ^4(c+d x) \left (36015+21529 \sinh ^2(c+d x)+1128 \sinh ^4(c+d x)\right )+9 a^2 b \sinh ^2(c+d x) \left (72030+41615 \sinh ^2(c+d x)+2131 \sinh ^4(c+d x)\right )+b^3 \sinh ^6(c+d x) \left (149460+90805 \sinh ^2(c+d x)+4887 \sinh ^4(c+d x)\right )+a^3 \left (252105+140965 \sinh ^2(c+d x)+8226 \sinh ^4(c+d x)\right )\right )+\frac {315 \text {arctanh}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \left (a^3 \left (16807+15000 \sinh ^2(c+d x)+2187 \sinh ^4(c+d x)-62 \sinh ^6(c+d x)\right )+9 a^2 b \sinh ^2(c+d x) \left (4802+4375 \sinh ^2(c+d x)+640 \sinh ^4(c+d x)+3 \sinh ^6(c+d x)\right )+b^3 \sinh ^6(c+d x) \left (9964+9375 \sinh ^2(c+d x)+1458 \sinh ^4(c+d x)+7 \sinh ^6(c+d x)\right )+3 a b^2 \sinh ^4(c+d x) \left (12005+11178 \sinh ^2(c+d x)+1701 \sinh ^4(c+d x)+8 \sinh ^6(c+d x)\right )\right )}{\sqrt {-\sinh ^2(c+d x)}}\right )}{60480 d} \] Input:

Integrate[Sech[c + d*x]^5*(a + b*Sinh[c + d*x]^2)^3,x]
 

Output:

-1/60480*(Csch[c + d*x]^5*(256*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 
 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^8*(a + b*Sinh[c + d*x]^2) 
^3 + 384*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d 
*x]^2]*Sinh[c + d*x]^8*(a + b*Sinh[c + d*x]^2)^2*(7*a + 5*b*Sinh[c + d*x]^ 
2) - 21*(15*a*b^2*Sinh[c + d*x]^4*(36015 + 21529*Sinh[c + d*x]^2 + 1128*Si 
nh[c + d*x]^4) + 9*a^2*b*Sinh[c + d*x]^2*(72030 + 41615*Sinh[c + d*x]^2 + 
2131*Sinh[c + d*x]^4) + b^3*Sinh[c + d*x]^6*(149460 + 90805*Sinh[c + d*x]^ 
2 + 4887*Sinh[c + d*x]^4) + a^3*(252105 + 140965*Sinh[c + d*x]^2 + 8226*Si 
nh[c + d*x]^4)) + (315*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*(a^3*(16807 + 15000 
*Sinh[c + d*x]^2 + 2187*Sinh[c + d*x]^4 - 62*Sinh[c + d*x]^6) + 9*a^2*b*Si 
nh[c + d*x]^2*(4802 + 4375*Sinh[c + d*x]^2 + 640*Sinh[c + d*x]^4 + 3*Sinh[ 
c + d*x]^6) + b^3*Sinh[c + d*x]^6*(9964 + 9375*Sinh[c + d*x]^2 + 1458*Sinh 
[c + d*x]^4 + 7*Sinh[c + d*x]^6) + 3*a*b^2*Sinh[c + d*x]^4*(12005 + 11178* 
Sinh[c + d*x]^2 + 1701*Sinh[c + d*x]^4 + 8*Sinh[c + d*x]^6)))/Sqrt[-Sinh[c 
 + d*x]^2]))/d
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 3669, 300, 2009}

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.

\(\displaystyle \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a-b \sin (i c+i d x)^2\right )^3}{\cos (i c+i d x)^5}dx\)

\(\Big \downarrow \) 3669

\(\displaystyle \frac {\int \frac {\left (b \sinh ^2(c+d x)+a\right )^3}{\left (\sinh ^2(c+d x)+1\right )^3}d\sinh (c+d x)}{d}\)

\(\Big \downarrow \) 300

\(\displaystyle \frac {\int \left (b^3+\frac {3 (a-b) b^2 \sinh ^4(c+d x)+3 b \left (a^2-b^2\right ) \sinh ^2(c+d x)+a^3-b^3}{\left (\sinh ^2(c+d x)+1\right )^3}\right )d\sinh (c+d x)}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {3}{8} \left ((a+b)^2+4 b^2\right ) (a-b) \arctan (\sinh (c+d x))+\frac {(a-b)^3 \sinh (c+d x)}{4 \left (\sinh ^2(c+d x)+1\right )^2}+\frac {3 (a+3 b) (a-b)^2 \sinh (c+d x)}{8 \left (\sinh ^2(c+d x)+1\right )}+b^3 \sinh (c+d x)}{d}\)

Input:

Int[Sech[c + d*x]^5*(a + b*Sinh[c + d*x]^2)^3,x]
 

Output:

((3*(a - b)*(4*b^2 + (a + b)^2)*ArcTan[Sinh[c + d*x]])/8 + b^3*Sinh[c + d* 
x] + ((a - b)^3*Sinh[c + d*x])/(4*(1 + Sinh[c + d*x]^2)^2) + (3*(a - b)^2* 
(a + 3*b)*Sinh[c + d*x])/(8*(1 + Sinh[c + d*x]^2)))/d
 

Defintions of rubi rules used

rule 300

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int 
[PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c 
, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3669

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ 
(p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f   S 
ubst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x] 
/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(272\) vs. \(2(97)=194\).

Time = 0.18 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.65

\[\frac {a^{3} \left (\left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )+\frac {3 \arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )+3 a^{2} b \left (-\frac {\sinh \left (d x +c \right )}{3 \cosh \left (d x +c \right )^{4}}+\frac {\left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )}{3}+\frac {\arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )+3 b^{2} a \left (-\frac {\sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{4}}-\frac {\sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{4}}+\left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )+\frac {3 \arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )+b^{3} \left (\frac {\sinh \left (d x +c \right )^{5}}{\cosh \left (d x +c \right )^{4}}+\frac {5 \sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{4}}+\frac {5 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{4}}-5 \left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )-\frac {15 \arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )}{d}\]

Input:

int(sech(d*x+c)^5*(a+b*sinh(d*x+c)^2)^3,x)
 

Output:

1/d*(a^3*((1/4*sech(d*x+c)^3+3/8*sech(d*x+c))*tanh(d*x+c)+3/4*arctan(exp(d 
*x+c)))+3*a^2*b*(-1/3*sinh(d*x+c)/cosh(d*x+c)^4+1/3*(1/4*sech(d*x+c)^3+3/8 
*sech(d*x+c))*tanh(d*x+c)+1/4*arctan(exp(d*x+c)))+3*b^2*a*(-sinh(d*x+c)^3/ 
cosh(d*x+c)^4-sinh(d*x+c)/cosh(d*x+c)^4+(1/4*sech(d*x+c)^3+3/8*sech(d*x+c) 
)*tanh(d*x+c)+3/4*arctan(exp(d*x+c)))+b^3*(sinh(d*x+c)^5/cosh(d*x+c)^4+5*s 
inh(d*x+c)^3/cosh(d*x+c)^4+5*sinh(d*x+c)/cosh(d*x+c)^4-5*(1/4*sech(d*x+c)^ 
3+3/8*sech(d*x+c))*tanh(d*x+c)-15/4*arctan(exp(d*x+c))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2245 vs. \(2 (97) = 194\).

Time = 0.11 (sec) , antiderivative size = 2245, normalized size of antiderivative = 21.80 \[ \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:

integrate(sech(d*x+c)^5*(a+b*sinh(d*x+c)^2)^3,x, algorithm="fricas")
 

Output:

1/4*(2*b^3*cosh(d*x + c)^10 + 20*b^3*cosh(d*x + c)*sinh(d*x + c)^9 + 2*b^3 
*sinh(d*x + c)^10 + 3*(a^3 + a^2*b - 5*a*b^2 + 5*b^3)*cosh(d*x + c)^8 + 3* 
(30*b^3*cosh(d*x + c)^2 + a^3 + a^2*b - 5*a*b^2 + 5*b^3)*sinh(d*x + c)^8 + 
 24*(10*b^3*cosh(d*x + c)^3 + (a^3 + a^2*b - 5*a*b^2 + 5*b^3)*cosh(d*x + c 
))*sinh(d*x + c)^7 + (11*a^3 - 21*a^2*b + 9*a*b^2 + 5*b^3)*cosh(d*x + c)^6 
 + (420*b^3*cosh(d*x + c)^4 + 11*a^3 - 21*a^2*b + 9*a*b^2 + 5*b^3 + 84*(a^ 
3 + a^2*b - 5*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 6*(84*b^3* 
cosh(d*x + c)^5 + 28*(a^3 + a^2*b - 5*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + (11 
*a^3 - 21*a^2*b + 9*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 - (11*a^ 
3 - 21*a^2*b + 9*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + (420*b^3*cosh(d*x + c)^6 
 + 210*(a^3 + a^2*b - 5*a*b^2 + 5*b^3)*cosh(d*x + c)^4 - 11*a^3 + 21*a^2*b 
 - 9*a*b^2 - 5*b^3 + 15*(11*a^3 - 21*a^2*b + 9*a*b^2 + 5*b^3)*cosh(d*x + c 
)^2)*sinh(d*x + c)^4 + 4*(60*b^3*cosh(d*x + c)^7 + 42*(a^3 + a^2*b - 5*a*b 
^2 + 5*b^3)*cosh(d*x + c)^5 + 5*(11*a^3 - 21*a^2*b + 9*a*b^2 + 5*b^3)*cosh 
(d*x + c)^3 - (11*a^3 - 21*a^2*b + 9*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d* 
x + c)^3 - 2*b^3 - 3*(a^3 + a^2*b - 5*a*b^2 + 5*b^3)*cosh(d*x + c)^2 + 3*( 
30*b^3*cosh(d*x + c)^8 + 28*(a^3 + a^2*b - 5*a*b^2 + 5*b^3)*cosh(d*x + c)^ 
6 + 5*(11*a^3 - 21*a^2*b + 9*a*b^2 + 5*b^3)*cosh(d*x + c)^4 - a^3 - a^2*b 
+ 5*a*b^2 - 5*b^3 - 2*(11*a^3 - 21*a^2*b + 9*a*b^2 + 5*b^3)*cosh(d*x + c)^ 
2)*sinh(d*x + c)^2 + 3*((a^3 + a^2*b + 3*a*b^2 - 5*b^3)*cosh(d*x + c)^9...
 

Sympy [F(-1)]

Timed out. \[ \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\text {Timed out} \] Input:

integrate(sech(d*x+c)**5*(a+b*sinh(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. 489 vs. \(2 (97) = 194\).

Time = 0.14 (sec) , antiderivative size = 489, normalized size of antiderivative = 4.75 \[ \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {1}{4} \, b^{3} {\left (\frac {15 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {2 \, e^{\left (-d x - c\right )}}{d} + \frac {17 \, e^{\left (-2 \, d x - 2 \, c\right )} + 13 \, e^{\left (-4 \, d x - 4 \, c\right )} + 7 \, e^{\left (-6 \, d x - 6 \, c\right )} - 7 \, e^{\left (-8 \, d x - 8 \, c\right )} + 2}{d {\left (e^{\left (-d x - c\right )} + 4 \, e^{\left (-3 \, d x - 3 \, c\right )} + 6 \, e^{\left (-5 \, d x - 5 \, c\right )} + 4 \, e^{\left (-7 \, d x - 7 \, c\right )} + e^{\left (-9 \, d x - 9 \, c\right )}\right )}}\right )} - \frac {3}{4} \, a b^{2} {\left (\frac {3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac {5 \, e^{\left (-d x - c\right )} - 3 \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} - \frac {1}{4} \, a^{3} {\left (\frac {3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {3 \, e^{\left (-d x - c\right )} + 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} - 3 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} - \frac {3}{4} \, a^{2} b {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - 7 \, e^{\left (-3 \, d x - 3 \, c\right )} + 7 \, e^{\left (-5 \, d x - 5 \, c\right )} - e^{\left (-7 \, d x - 7 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} \] Input:

integrate(sech(d*x+c)^5*(a+b*sinh(d*x+c)^2)^3,x, algorithm="maxima")
 

Output:

1/4*b^3*(15*arctan(e^(-d*x - c))/d - 2*e^(-d*x - c)/d + (17*e^(-2*d*x - 2* 
c) + 13*e^(-4*d*x - 4*c) + 7*e^(-6*d*x - 6*c) - 7*e^(-8*d*x - 8*c) + 2)/(d 
*(e^(-d*x - c) + 4*e^(-3*d*x - 3*c) + 6*e^(-5*d*x - 5*c) + 4*e^(-7*d*x - 7 
*c) + e^(-9*d*x - 9*c)))) - 3/4*a*b^2*(3*arctan(e^(-d*x - c))/d + (5*e^(-d 
*x - c) - 3*e^(-3*d*x - 3*c) + 3*e^(-5*d*x - 5*c) - 5*e^(-7*d*x - 7*c))/(d 
*(4*e^(-2*d*x - 2*c) + 6*e^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) + e^(-8*d*x 
 - 8*c) + 1))) - 1/4*a^3*(3*arctan(e^(-d*x - c))/d - (3*e^(-d*x - c) + 11* 
e^(-3*d*x - 3*c) - 11*e^(-5*d*x - 5*c) - 3*e^(-7*d*x - 7*c))/(d*(4*e^(-2*d 
*x - 2*c) + 6*e^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c) + 1 
))) - 3/4*a^2*b*(arctan(e^(-d*x - c))/d - (e^(-d*x - c) - 7*e^(-3*d*x - 3* 
c) + 7*e^(-5*d*x - 5*c) - e^(-7*d*x - 7*c))/(d*(4*e^(-2*d*x - 2*c) + 6*e^( 
-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c) + 1)))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (97) = 194\).

Time = 0.20 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.92 \[ \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {8 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 3 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (a^{3} + a^{2} b + 3 \, a b^{2} - 5 \, b^{3}\right )} + \frac {4 \, {\left (3 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 3 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 15 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 9 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 20 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 12 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 36 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 28 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}^{2}}}{16 \, d} \] Input:

integrate(sech(d*x+c)^5*(a+b*sinh(d*x+c)^2)^3,x, algorithm="giac")
 

Output:

1/16*(8*b^3*(e^(d*x + c) - e^(-d*x - c)) + 3*(pi + 2*arctan(1/2*(e^(2*d*x 
+ 2*c) - 1)*e^(-d*x - c)))*(a^3 + a^2*b + 3*a*b^2 - 5*b^3) + 4*(3*a^3*(e^( 
d*x + c) - e^(-d*x - c))^3 + 3*a^2*b*(e^(d*x + c) - e^(-d*x - c))^3 - 15*a 
*b^2*(e^(d*x + c) - e^(-d*x - c))^3 + 9*b^3*(e^(d*x + c) - e^(-d*x - c))^3 
 + 20*a^3*(e^(d*x + c) - e^(-d*x - c)) - 12*a^2*b*(e^(d*x + c) - e^(-d*x - 
 c)) - 36*a*b^2*(e^(d*x + c) - e^(-d*x - c)) + 28*b^3*(e^(d*x + c) - e^(-d 
*x - c)))/((e^(d*x + c) - e^(-d*x - c))^2 + 4)^2)/d
 

Mupad [B] (verification not implemented)

Time = 1.82 (sec) , antiderivative size = 430, normalized size of antiderivative = 4.17 \[ \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a^3\,\sqrt {d^2}-5\,b^3\,\sqrt {d^2}+3\,a\,b^2\,\sqrt {d^2}+a^2\,b\,\sqrt {d^2}\right )}{d\,\sqrt {a^6+2\,a^5\,b+7\,a^4\,b^2-4\,a^3\,b^3-a^2\,b^4-30\,a\,b^5+25\,b^6}}\right )\,\sqrt {a^6+2\,a^5\,b+7\,a^4\,b^2-4\,a^3\,b^3-a^2\,b^4-30\,a\,b^5+25\,b^6}}{4\,\sqrt {d^2}}+\frac {b^3\,{\mathrm {e}}^{c+d\,x}}{2\,d}-\frac {b^3\,{\mathrm {e}}^{-c-d\,x}}{2\,d}+\frac {3\,{\mathrm {e}}^{c+d\,x}\,\left (a^3+a^2\,b-5\,a\,b^2+3\,b^3\right )}{4\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (a^3-15\,a^2\,b+27\,a\,b^2-13\,b^3\right )}{2\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {6\,{\mathrm {e}}^{c+d\,x}\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}{d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}+\frac {4\,{\mathrm {e}}^{c+d\,x}\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}{d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )} \] Input:

int((a + b*sinh(c + d*x)^2)^3/cosh(c + d*x)^5,x)
 

Output:

(3*atan((exp(d*x)*exp(c)*(a^3*(d^2)^(1/2) - 5*b^3*(d^2)^(1/2) + 3*a*b^2*(d 
^2)^(1/2) + a^2*b*(d^2)^(1/2)))/(d*(2*a^5*b - 30*a*b^5 + a^6 + 25*b^6 - a^ 
2*b^4 - 4*a^3*b^3 + 7*a^4*b^2)^(1/2)))*(2*a^5*b - 30*a*b^5 + a^6 + 25*b^6 
- a^2*b^4 - 4*a^3*b^3 + 7*a^4*b^2)^(1/2))/(4*(d^2)^(1/2)) + (b^3*exp(c + d 
*x))/(2*d) - (b^3*exp(- c - d*x))/(2*d) + (3*exp(c + d*x)*(a^2*b - 5*a*b^2 
 + a^3 + 3*b^3))/(4*d*(exp(2*c + 2*d*x) + 1)) + (exp(c + d*x)*(27*a*b^2 - 
15*a^2*b + a^3 - 13*b^3))/(2*d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1) 
) - (6*exp(c + d*x)*(3*a*b^2 - 3*a^2*b + a^3 - b^3))/(d*(3*exp(2*c + 2*d*x 
) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) + (4*exp(c + d*x)*(3*a*b^2 
 - 3*a^2*b + a^3 - b^3))/(d*(4*exp(2*c + 2*d*x) + 6*exp(4*c + 4*d*x) + 4*e 
xp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 791, normalized size of antiderivative = 7.68 \[ \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {-2 b^{3}+18 e^{5 d x +5 c} \mathit {atan} \left (e^{d x +c}\right ) a^{3}-90 e^{5 d x +5 c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}+12 e^{7 d x +7 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2} b +36 e^{7 d x +7 c} \mathit {atan} \left (e^{d x +c}\right ) a \,b^{2}+12 e^{3 d x +3 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2} b +36 e^{3 d x +3 c} \mathit {atan} \left (e^{d x +c}\right ) a \,b^{2}+18 e^{5 d x +5 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2} b +54 e^{5 d x +5 c} \mathit {atan} \left (e^{d x +c}\right ) a \,b^{2}-15 e^{2 d x +2 c} b^{3}+2 e^{10 d x +10 c} b^{3}+12 e^{7 d x +7 c} \mathit {atan} \left (e^{d x +c}\right ) a^{3}-60 e^{7 d x +7 c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}+12 e^{3 d x +3 c} \mathit {atan} \left (e^{d x +c}\right ) a^{3}-60 e^{3 d x +3 c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}-3 e^{2 d x +2 c} a^{2} b +3 e^{8 d x +8 c} a^{2} b -15 e^{8 d x +8 c} a \,b^{2}-21 e^{6 d x +6 c} a^{2} b +9 e^{6 d x +6 c} a \,b^{2}+21 e^{4 d x +4 c} a^{2} b -9 e^{4 d x +4 c} a \,b^{2}+5 e^{6 d x +6 c} b^{3}-5 e^{4 d x +4 c} b^{3}+3 e^{8 d x +8 c} a^{3}+15 e^{8 d x +8 c} b^{3}+11 e^{6 d x +6 c} a^{3}-11 e^{4 d x +4 c} a^{3}+3 e^{9 d x +9 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2} b +9 e^{9 d x +9 c} \mathit {atan} \left (e^{d x +c}\right ) a \,b^{2}+3 e^{d x +c} \mathit {atan} \left (e^{d x +c}\right ) a^{2} b +9 e^{d x +c} \mathit {atan} \left (e^{d x +c}\right ) a \,b^{2}-3 e^{2 d x +2 c} a^{3}+15 e^{2 d x +2 c} a \,b^{2}+3 e^{9 d x +9 c} \mathit {atan} \left (e^{d x +c}\right ) a^{3}-15 e^{9 d x +9 c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}+3 e^{d x +c} \mathit {atan} \left (e^{d x +c}\right ) a^{3}-15 e^{d x +c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}}{4 e^{d x +c} d \left (e^{8 d x +8 c}+4 e^{6 d x +6 c}+6 e^{4 d x +4 c}+4 e^{2 d x +2 c}+1\right )} \] Input:

int(sech(d*x+c)^5*(a+b*sinh(d*x+c)^2)^3,x)
 

Output:

(3*e**(9*c + 9*d*x)*atan(e**(c + d*x))*a**3 + 3*e**(9*c + 9*d*x)*atan(e**( 
c + d*x))*a**2*b + 9*e**(9*c + 9*d*x)*atan(e**(c + d*x))*a*b**2 - 15*e**(9 
*c + 9*d*x)*atan(e**(c + d*x))*b**3 + 12*e**(7*c + 7*d*x)*atan(e**(c + d*x 
))*a**3 + 12*e**(7*c + 7*d*x)*atan(e**(c + d*x))*a**2*b + 36*e**(7*c + 7*d 
*x)*atan(e**(c + d*x))*a*b**2 - 60*e**(7*c + 7*d*x)*atan(e**(c + d*x))*b** 
3 + 18*e**(5*c + 5*d*x)*atan(e**(c + d*x))*a**3 + 18*e**(5*c + 5*d*x)*atan 
(e**(c + d*x))*a**2*b + 54*e**(5*c + 5*d*x)*atan(e**(c + d*x))*a*b**2 - 90 
*e**(5*c + 5*d*x)*atan(e**(c + d*x))*b**3 + 12*e**(3*c + 3*d*x)*atan(e**(c 
 + d*x))*a**3 + 12*e**(3*c + 3*d*x)*atan(e**(c + d*x))*a**2*b + 36*e**(3*c 
 + 3*d*x)*atan(e**(c + d*x))*a*b**2 - 60*e**(3*c + 3*d*x)*atan(e**(c + d*x 
))*b**3 + 3*e**(c + d*x)*atan(e**(c + d*x))*a**3 + 3*e**(c + d*x)*atan(e** 
(c + d*x))*a**2*b + 9*e**(c + d*x)*atan(e**(c + d*x))*a*b**2 - 15*e**(c + 
d*x)*atan(e**(c + d*x))*b**3 + 2*e**(10*c + 10*d*x)*b**3 + 3*e**(8*c + 8*d 
*x)*a**3 + 3*e**(8*c + 8*d*x)*a**2*b - 15*e**(8*c + 8*d*x)*a*b**2 + 15*e** 
(8*c + 8*d*x)*b**3 + 11*e**(6*c + 6*d*x)*a**3 - 21*e**(6*c + 6*d*x)*a**2*b 
 + 9*e**(6*c + 6*d*x)*a*b**2 + 5*e**(6*c + 6*d*x)*b**3 - 11*e**(4*c + 4*d* 
x)*a**3 + 21*e**(4*c + 4*d*x)*a**2*b - 9*e**(4*c + 4*d*x)*a*b**2 - 5*e**(4 
*c + 4*d*x)*b**3 - 3*e**(2*c + 2*d*x)*a**3 - 3*e**(2*c + 2*d*x)*a**2*b + 1 
5*e**(2*c + 2*d*x)*a*b**2 - 15*e**(2*c + 2*d*x)*b**3 - 2*b**3)/(4*e**(c + 
d*x)*d*(e**(8*c + 8*d*x) + 4*e**(6*c + 6*d*x) + 6*e**(4*c + 4*d*x) + 4*...