\(\int \cosh ^3(c+d x) (a+b \tanh ^2(c+d x))^3 \, dx\) [98]

Optimal result

Integrand size = 23, antiderivative size = 87 \[ \int \cosh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {b^2 (6 a+5 b) \arctan (\sinh (c+d x))}{2 d}+\frac {(a-2 b) (a+b)^2 \sinh (c+d x)}{d}+\frac {(a+b)^3 \sinh ^3(c+d x)}{3 d}-\frac {b^3 \text {sech}(c+d x) \tanh (c+d x)}{2 d} \]

[Out]

1/2*b^2*(6*a+5*b)*arctan(sinh(d*x+c))/d+(a-2*b)*(a+b)^2*sinh(d*x+c)/d+1/3*(a+b)^3*sinh(d*x+c)^3/d-1/2*b^3*sech
(d*x+c)*tanh(d*x+c)/d

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3757, 398, 393, 209} \[ \int \cosh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {b^2 (6 a+5 b) \arctan (\sinh (c+d x))}{2 d}+\frac {(a+b)^3 \sinh ^3(c+d x)}{3 d}+\frac {(a-2 b) (a+b)^2 \sinh (c+d x)}{d}-\frac {b^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d} \]

[In]

Int[Cosh[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

(b^2*(6*a + 5*b)*ArcTan[Sinh[c + d*x]])/(2*d) + ((a - 2*b)*(a + b)^2*Sinh[c + d*x])/d + ((a + b)^3*Sinh[c + d*
x]^3)/(3*d) - (b^3*Sech[c + d*x]*Tanh[c + d*x])/(2*d)

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 393

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 398

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

Rule 3757

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> With[{ff = F
reeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[ExpandToSum[b*(ff*x)^n + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 -
ff^2*x^2)^((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] &&
IntegerQ[n/2] && IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+(a+b) x^2\right )^3}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left ((a-2 b) (a+b)^2+(a+b)^3 x^2+\frac {b^2 (3 a+2 b)+3 b^2 (a+b) x^2}{\left (1+x^2\right )^2}\right ) \, dx,x,\sinh (c+d x)\right )}{d} \\ & = \frac {(a-2 b) (a+b)^2 \sinh (c+d x)}{d}+\frac {(a+b)^3 \sinh ^3(c+d x)}{3 d}+\frac {\text {Subst}\left (\int \frac {b^2 (3 a+2 b)+3 b^2 (a+b) x^2}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d} \\ & = \frac {(a-2 b) (a+b)^2 \sinh (c+d x)}{d}+\frac {(a+b)^3 \sinh ^3(c+d x)}{3 d}-\frac {b^3 \text {sech}(c+d x) \tanh (c+d x)}{2 d}+\frac {\left (b^2 (6 a+5 b)\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 d} \\ & = \frac {b^2 (6 a+5 b) \arctan (\sinh (c+d x))}{2 d}+\frac {(a-2 b) (a+b)^2 \sinh (c+d x)}{d}+\frac {(a+b)^3 \sinh ^3(c+d x)}{3 d}-\frac {b^3 \text {sech}(c+d x) \tanh (c+d x)}{2 d} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

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

Time = 7.06 (sec) , antiderivative size = 494, normalized size of antiderivative = 5.68 \[ \int \cosh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {\text {csch}^5(c+d x) \left (-256 \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^8(c+d x) \left (a+a \sinh ^2(c+d x)+b \sinh ^2(c+d x)\right )^3-\frac {315 \text {arctanh}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \left (b^3 \sinh ^6(c+d x) \left (2161+1875 \sinh ^2(c+d x)+243 \sinh ^4(c+d x)+\sinh ^6(c+d x)\right )+a^3 \cosh ^6(c+d x) \left (2401+1875 \sinh ^2(c+d x)+243 \sinh ^4(c+d x)+\sinh ^6(c+d x)\right )+3 a^2 b \left (\sinh (c+d x)+\sinh ^3(c+d x)\right )^2 \left (2401+1875 \sinh ^2(c+d x)+243 \sinh ^4(c+d x)+\sinh ^6(c+d x)\right )+3 a b^2 \sinh ^4(c+d x) \left (2401+4180 \sinh ^2(c+d x)+2118 \sinh ^4(c+d x)+244 \sinh ^6(c+d x)+\sinh ^8(c+d x)\right )\right )}{\sqrt {-\sinh ^2(c+d x)}}+21 \left (b^3 \sinh ^6(c+d x) \left (32415+17320 \sinh ^2(c+d x)+753 \sinh ^4(c+d x)\right )+3 a b^2 \sinh ^4(c+d x) \left (36015+50695 \sinh ^2(c+d x)+18073 \sinh ^4(c+d x)+753 \sinh ^6(c+d x)\right )+3 a^2 b \sinh ^2(c+d x) \left (36015+88150 \sinh ^2(c+d x)+69728 \sinh ^4(c+d x)+18826 \sinh ^6(c+d x)+753 \sinh ^8(c+d x)\right )+a^3 \left (36015+124165 \sinh ^2(c+d x)+157878 \sinh ^4(c+d x)+89514 \sinh ^6(c+d x)+19579 \sinh ^8(c+d x)+753 \sinh ^{10}(c+d x)\right )\right )\right )}{30240 d} \]

[In]

Integrate[Cosh[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

(Csch[c + d*x]^5*(-256*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^8
*(a + a*Sinh[c + d*x]^2 + b*Sinh[c + d*x]^2)^3 - (315*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*(b^3*Sinh[c + d*x]^6*(21
61 + 1875*Sinh[c + d*x]^2 + 243*Sinh[c + d*x]^4 + Sinh[c + d*x]^6) + a^3*Cosh[c + d*x]^6*(2401 + 1875*Sinh[c +
d*x]^2 + 243*Sinh[c + d*x]^4 + Sinh[c + d*x]^6) + 3*a^2*b*(Sinh[c + d*x] + Sinh[c + d*x]^3)^2*(2401 + 1875*Si
nh[c + d*x]^2 + 243*Sinh[c + d*x]^4 + Sinh[c + d*x]^6) + 3*a*b^2*Sinh[c + d*x]^4*(2401 + 4180*Sinh[c + d*x]^2
+ 2118*Sinh[c + d*x]^4 + 244*Sinh[c + d*x]^6 + Sinh[c + d*x]^8)))/Sqrt[-Sinh[c + d*x]^2] + 21*(b^3*Sinh[c + d*
x]^6*(32415 + 17320*Sinh[c + d*x]^2 + 753*Sinh[c + d*x]^4) + 3*a*b^2*Sinh[c + d*x]^4*(36015 + 50695*Sinh[c + d
*x]^2 + 18073*Sinh[c + d*x]^4 + 753*Sinh[c + d*x]^6) + 3*a^2*b*Sinh[c + d*x]^2*(36015 + 88150*Sinh[c + d*x]^2
+ 69728*Sinh[c + d*x]^4 + 18826*Sinh[c + d*x]^6 + 753*Sinh[c + d*x]^8) + a^3*(36015 + 124165*Sinh[c + d*x]^2 +
157878*Sinh[c + d*x]^4 + 89514*Sinh[c + d*x]^6 + 19579*Sinh[c + d*x]^8 + 753*Sinh[c + d*x]^10))))/(30240*d)

Maple [A] (verified)

Time = 22.77 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.78

method	result	size
derivativedivides	\(\frac {a^{3} \left (\frac {2}{3}+\frac {\cosh \left (d x +c \right )^{2}}{3}\right ) \sinh \left (d x +c \right )+a^{2} b \sinh \left (d x +c \right )^{3}+3 a \,b^{2} \left (\frac {\sinh \left (d x +c \right )^{3}}{3}-\sinh \left (d x +c \right )+2 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+b^{3} \left (\frac {\sinh \left (d x +c \right )^{5}}{3 \cosh \left (d x +c \right )^{2}}-\frac {5 \sinh \left (d x +c \right )^{3}}{3 \cosh \left (d x +c \right )^{2}}-\frac {5 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}+\frac {5 \,\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+5 \arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\)	\(155\)
default	\(\frac {a^{3} \left (\frac {2}{3}+\frac {\cosh \left (d x +c \right )^{2}}{3}\right ) \sinh \left (d x +c \right )+a^{2} b \sinh \left (d x +c \right )^{3}+3 a \,b^{2} \left (\frac {\sinh \left (d x +c \right )^{3}}{3}-\sinh \left (d x +c \right )+2 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+b^{3} \left (\frac {\sinh \left (d x +c \right )^{5}}{3 \cosh \left (d x +c \right )^{2}}-\frac {5 \sinh \left (d x +c \right )^{3}}{3 \cosh \left (d x +c \right )^{2}}-\frac {5 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}+\frac {5 \,\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+5 \arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\)	\(155\)
risch	\(\frac {{\mathrm e}^{3 d x +3 c} a^{3}}{24 d}+\frac {{\mathrm e}^{3 d x +3 c} a^{2} b}{8 d}+\frac {{\mathrm e}^{3 d x +3 c} a \,b^{2}}{8 d}+\frac {{\mathrm e}^{3 d x +3 c} b^{3}}{24 d}+\frac {3 \,{\mathrm e}^{d x +c} a^{3}}{8 d}-\frac {3 \,{\mathrm e}^{d x +c} a^{2} b}{8 d}-\frac {15 \,{\mathrm e}^{d x +c} a \,b^{2}}{8 d}-\frac {9 b^{3} {\mathrm e}^{d x +c}}{8 d}-\frac {3 \,{\mathrm e}^{-d x -c} a^{3}}{8 d}+\frac {3 \,{\mathrm e}^{-d x -c} a^{2} b}{8 d}+\frac {15 \,{\mathrm e}^{-d x -c} a \,b^{2}}{8 d}+\frac {9 \,{\mathrm e}^{-d x -c} b^{3}}{8 d}-\frac {{\mathrm e}^{-3 d x -3 c} a^{3}}{24 d}-\frac {{\mathrm e}^{-3 d x -3 c} a^{2} b}{8 d}-\frac {{\mathrm e}^{-3 d x -3 c} a \,b^{2}}{8 d}-\frac {{\mathrm e}^{-3 d x -3 c} b^{3}}{24 d}-\frac {b^{3} {\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}+\frac {3 i b^{2} \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{d}+\frac {5 i b^{3} \ln \left ({\mathrm e}^{d x +c}+i\right )}{2 d}-\frac {5 i b^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{2 d}-\frac {3 i b^{2} \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{d}\)	\(386\)

[In]

int(cosh(d*x+c)^3*(a+b*tanh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(a^3*(2/3+1/3*cosh(d*x+c)^2)*sinh(d*x+c)+a^2*b*sinh(d*x+c)^3+3*a*b^2*(1/3*sinh(d*x+c)^3-sinh(d*x+c)+2*arct
an(exp(d*x+c)))+b^3*(1/3*sinh(d*x+c)^5/cosh(d*x+c)^2-5/3*sinh(d*x+c)^3/cosh(d*x+c)^2-5*sinh(d*x+c)/cosh(d*x+c)
^2+5/2*sech(d*x+c)*tanh(d*x+c)+5*arctan(exp(d*x+c))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1840 vs. \(2 (81) = 162\).

Time = 0.28 (sec) , antiderivative size = 1840, normalized size of antiderivative = 21.15 \[ \int \cosh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\text {Too large to display} \]

[In]

integrate(cosh(d*x+c)^3*(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

1/24*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^10 + 10*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)*sinh
(d*x + c)^9 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sinh(d*x + c)^10 + (11*a^3 - 3*a^2*b - 39*a*b^2 - 25*b^3)*cosh(d
*x + c)^8 + (11*a^3 - 3*a^2*b - 39*a*b^2 - 25*b^3 + 45*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d
*x + c)^8 + 8*(15*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^3 + (11*a^3 - 3*a^2*b - 39*a*b^2 - 25*b^3)*cos
h(d*x + c))*sinh(d*x + c)^7 + 2*(5*a^3 - 3*a^2*b - 21*a*b^2 - 25*b^3)*cosh(d*x + c)^6 + 2*(105*(a^3 + 3*a^2*b
+ 3*a*b^2 + b^3)*cosh(d*x + c)^4 + 5*a^3 - 3*a^2*b - 21*a*b^2 - 25*b^3 + 14*(11*a^3 - 3*a^2*b - 39*a*b^2 - 25*
b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^5 + 14*(11*a^3 - 3
*a^2*b - 39*a*b^2 - 25*b^3)*cosh(d*x + c)^3 + 3*(5*a^3 - 3*a^2*b - 21*a*b^2 - 25*b^3)*cosh(d*x + c))*sinh(d*x
+ c)^5 - 2*(5*a^3 - 3*a^2*b - 21*a*b^2 - 25*b^3)*cosh(d*x + c)^4 + 2*(105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh
(d*x + c)^6 + 35*(11*a^3 - 3*a^2*b - 39*a*b^2 - 25*b^3)*cosh(d*x + c)^4 - 5*a^3 + 3*a^2*b + 21*a*b^2 + 25*b^3
+ 15*(5*a^3 - 3*a^2*b - 21*a*b^2 - 25*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(15*(a^3 + 3*a^2*b + 3*a*b^2 +
b^3)*cosh(d*x + c)^7 + 7*(11*a^3 - 3*a^2*b - 39*a*b^2 - 25*b^3)*cosh(d*x + c)^5 + 5*(5*a^3 - 3*a^2*b - 21*a*b
^2 - 25*b^3)*cosh(d*x + c)^3 - (5*a^3 - 3*a^2*b - 21*a*b^2 - 25*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - a^3 - 3*
a^2*b - 3*a*b^2 - b^3 - (11*a^3 - 3*a^2*b - 39*a*b^2 - 25*b^3)*cosh(d*x + c)^2 + (45*(a^3 + 3*a^2*b + 3*a*b^2
+ b^3)*cosh(d*x + c)^8 + 28*(11*a^3 - 3*a^2*b - 39*a*b^2 - 25*b^3)*cosh(d*x + c)^6 + 30*(5*a^3 - 3*a^2*b - 21*
a*b^2 - 25*b^3)*cosh(d*x + c)^4 - 11*a^3 + 3*a^2*b + 39*a*b^2 + 25*b^3 - 12*(5*a^3 - 3*a^2*b - 21*a*b^2 - 25*b
^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 24*((6*a*b^2 + 5*b^3)*cosh(d*x + c)^7 + 7*(6*a*b^2 + 5*b^3)*cosh(d*x +
c)*sinh(d*x + c)^6 + (6*a*b^2 + 5*b^3)*sinh(d*x + c)^7 + 2*(6*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + (12*a*b^2 + 10*
b^3 + 21*(6*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 5*(7*(6*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + 2*(6*a*
b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + (6*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + (35*(6*a*b^2 + 5*b^3)*cosh(d
*x + c)^4 + 6*a*b^2 + 5*b^3 + 20*(6*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + (21*(6*a*b^2 + 5*b^3)*co
sh(d*x + c)^5 + 20*(6*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + 3*(6*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + (7
*(6*a*b^2 + 5*b^3)*cosh(d*x + c)^6 + 10*(6*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + 3*(6*a*b^2 + 5*b^3)*cosh(d*x + c)^
2)*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) + 2*(5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^9
+ 4*(11*a^3 - 3*a^2*b - 39*a*b^2 - 25*b^3)*cosh(d*x + c)^7 + 6*(5*a^3 - 3*a^2*b - 21*a*b^2 - 25*b^3)*cosh(d*x
+ c)^5 - 4*(5*a^3 - 3*a^2*b - 21*a*b^2 - 25*b^3)*cosh(d*x + c)^3 - (11*a^3 - 3*a^2*b - 39*a*b^2 - 25*b^3)*cos
h(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^7 + 7*d*cosh(d*x + c)*sinh(d*x + c)^6 + d*sinh(d*x + c)^7 + 2*d*co
sh(d*x + c)^5 + (21*d*cosh(d*x + c)^2 + 2*d)*sinh(d*x + c)^5 + 5*(7*d*cosh(d*x + c)^3 + 2*d*cosh(d*x + c))*sin
h(d*x + c)^4 + d*cosh(d*x + c)^3 + (35*d*cosh(d*x + c)^4 + 20*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^3 + (21*d*c
osh(d*x + c)^5 + 20*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^2 + (7*d*cosh(d*x + c)^6 + 10*d*cosh(
d*x + c)^4 + 3*d*cosh(d*x + c)^2)*sinh(d*x + c))

Sympy [F]

\[ \int \cosh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3} \cosh ^{3}{\left (c + d x \right )}\, dx \]

[In]

integrate(cosh(d*x+c)**3*(a+b*tanh(d*x+c)**2)**3,x)

[Out]

Integral((a + b*tanh(c + d*x)**2)**3*cosh(c + d*x)**3, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (81) = 162\).

Time = 0.28 (sec) , antiderivative size = 284, normalized size of antiderivative = 3.26 \[ \int \cosh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3}}{8 \, d} - \frac {1}{8} \, a b^{2} {\left (\frac {{\left (15 \, e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )} e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {15 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {48 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d}\right )} + \frac {1}{24} \, b^{3} {\left (\frac {27 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} - \frac {120 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {25 \, e^{\left (-2 \, d x - 2 \, c\right )} + 77 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3 \, e^{\left (-6 \, d x - 6 \, c\right )} - 1}{d {\left (e^{\left (-3 \, d x - 3 \, c\right )} + 2 \, e^{\left (-5 \, d x - 5 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )}\right )}}\right )} + \frac {1}{24} \, a^{3} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} - \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \]

[In]

integrate(cosh(d*x+c)^3*(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

1/8*a^2*b*(e^(d*x + c) - e^(-d*x - c))^3/d - 1/8*a*b^2*((15*e^(-2*d*x - 2*c) - 1)*e^(3*d*x + 3*c)/d - (15*e^(-
d*x - c) - e^(-3*d*x - 3*c))/d + 48*arctan(e^(-d*x - c))/d) + 1/24*b^3*((27*e^(-d*x - c) - e^(-3*d*x - 3*c))/d
- 120*arctan(e^(-d*x - c))/d - (25*e^(-2*d*x - 2*c) + 77*e^(-4*d*x - 4*c) + 3*e^(-6*d*x - 6*c) - 1)/(d*(e^(-3
*d*x - 3*c) + 2*e^(-5*d*x - 5*c) + e^(-7*d*x - 7*c)))) + 1/24*a^3*(e^(3*d*x + 3*c)/d + 9*e^(d*x + c)/d - 9*e^(
-d*x - c)/d - e^(-3*d*x - 3*c)/d)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (81) = 162\).

Time = 0.47 (sec) , antiderivative size = 264, normalized size of antiderivative = 3.03 \[ \int \cosh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 3 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 3 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 12 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 36 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 24 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - \frac {24 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4} + 6 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (6 \, a b^{2} + 5 \, b^{3}\right )}}{24 \, d} \]

[In]

integrate(cosh(d*x+c)^3*(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/24*(a^3*(e^(d*x + c) - e^(-d*x - c))^3 + 3*a^2*b*(e^(d*x + c) - e^(-d*x - c))^3 + 3*a*b^2*(e^(d*x + c) - e^(
-d*x - c))^3 + b^3*(e^(d*x + c) - e^(-d*x - c))^3 + 12*a^3*(e^(d*x + c) - e^(-d*x - c)) - 36*a*b^2*(e^(d*x + c
) - e^(-d*x - c)) - 24*b^3*(e^(d*x + c) - e^(-d*x - c)) - 24*b^3*(e^(d*x + c) - e^(-d*x - c))/((e^(d*x + c) -
e^(-d*x - c))^2 + 4) + 6*(pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(6*a*b^2 + 5*b^3))/d

Mupad [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.67 \[ \int \cosh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (5\,b^3\,\sqrt {d^2}+6\,a\,b^2\,\sqrt {d^2}\right )}{d\,\sqrt {36\,a^2\,b^4+60\,a\,b^5+25\,b^6}}\right )\,\sqrt {36\,a^2\,b^4+60\,a\,b^5+25\,b^6}}{\sqrt {d^2}}-\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}\,{\left (a+b\right )}^3}{24\,d}+\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,{\left (a+b\right )}^3}{24\,d}+\frac {3\,{\mathrm {e}}^{c+d\,x}\,{\left (a+b\right )}^2\,\left (a-3\,b\right )}{8\,d}-\frac {b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {3\,{\mathrm {e}}^{-c-d\,x}\,{\left (a+b\right )}^2\,\left (a-3\,b\right )}{8\,d}+\frac {2\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \]

[In]

int(cosh(c + d*x)^3*(a + b*tanh(c + d*x)^2)^3,x)

[Out]

(atan((exp(d*x)*exp(c)*(5*b^3*(d^2)^(1/2) + 6*a*b^2*(d^2)^(1/2)))/(d*(60*a*b^5 + 25*b^6 + 36*a^2*b^4)^(1/2)))*
(60*a*b^5 + 25*b^6 + 36*a^2*b^4)^(1/2))/(d^2)^(1/2) - (exp(- 3*c - 3*d*x)*(a + b)^3)/(24*d) + (exp(3*c + 3*d*x
)*(a + b)^3)/(24*d) + (3*exp(c + d*x)*(a + b)^2*(a - 3*b))/(8*d) - (b^3*exp(c + d*x))/(d*(exp(2*c + 2*d*x) + 1
)) - (3*exp(- c - d*x)*(a + b)^2*(a - 3*b))/(8*d) + (2*b^3*exp(c + d*x))/(d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*
d*x) + 1))