3.58 \(\int \sinh ^3(c+d x) (a+b \tanh ^3(c+d x))^2 \, dx\)

Optimal. Leaf size=182 \[ \frac {a^2 \cosh ^3(c+d x)}{3 d}-\frac {a^2 \cosh (c+d x)}{d}+\frac {5 a b \sinh ^3(c+d x)}{3 d}-\frac {5 a b \sinh (c+d x)}{d}-\frac {a b \sinh ^3(c+d x) \tanh ^2(c+d x)}{d}+\frac {5 a b \tan ^{-1}(\sinh (c+d x))}{d}+\frac {b^2 \cosh ^3(c+d x)}{3 d}-\frac {4 b^2 \cosh (c+d x)}{d}-\frac {b^2 \text {sech}^5(c+d x)}{5 d}+\frac {4 b^2 \text {sech}^3(c+d x)}{3 d}-\frac {6 b^2 \text {sech}(c+d x)}{d} \]

[Out]

5*a*b*arctan(sinh(d*x+c))/d-a^2*cosh(d*x+c)/d-4*b^2*cosh(d*x+c)/d+1/3*a^2*cosh(d*x+c)^3/d+1/3*b^2*cosh(d*x+c)^
3/d-6*b^2*sech(d*x+c)/d+4/3*b^2*sech(d*x+c)^3/d-1/5*b^2*sech(d*x+c)^5/d-5*a*b*sinh(d*x+c)/d+5/3*a*b*sinh(d*x+c
)^3/d-a*b*sinh(d*x+c)^3*tanh(d*x+c)^2/d

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Rubi [A]  time = 0.22, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3666, 2633, 2592, 288, 302, 203, 2590, 270} \[ \frac {a^2 \cosh ^3(c+d x)}{3 d}-\frac {a^2 \cosh (c+d x)}{d}+\frac {5 a b \sinh ^3(c+d x)}{3 d}-\frac {5 a b \sinh (c+d x)}{d}-\frac {a b \sinh ^3(c+d x) \tanh ^2(c+d x)}{d}+\frac {5 a b \tan ^{-1}(\sinh (c+d x))}{d}+\frac {b^2 \cosh ^3(c+d x)}{3 d}-\frac {4 b^2 \cosh (c+d x)}{d}-\frac {b^2 \text {sech}^5(c+d x)}{5 d}+\frac {4 b^2 \text {sech}^3(c+d x)}{3 d}-\frac {6 b^2 \text {sech}(c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*a*b*ArcTan[Sinh[c + d*x]])/d - (a^2*Cosh[c + d*x])/d - (4*b^2*Cosh[c + d*x])/d + (a^2*Cosh[c + d*x]^3)/(3*d
) + (b^2*Cosh[c + d*x]^3)/(3*d) - (6*b^2*Sech[c + d*x])/d + (4*b^2*Sech[c + d*x]^3)/(3*d) - (b^2*Sech[c + d*x]
^5)/(5*d) - (5*a*b*Sinh[c + d*x])/d + (5*a*b*Sinh[c + d*x]^3)/(3*d) - (a*b*Sinh[c + d*x]^3*Tanh[c + d*x]^2)/d

Rule 203

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 2590

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[f^(-1), Subst[Int[(1 - x^2
)^((m + n - 1)/2)/x^n, x], x, Cos[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]

Rule 2592

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> With[{ff = FreeFactors[S
in[e + f*x], x]}, Dist[ff/f, Subst[Int[(ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, (a*Sin[e + f*x])/ff
], x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 3666

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> Int[ExpandTrig[(d*sin[e + f*x])^m*(a + b*(c*tan[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n},
x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx &=i \int \left (-i a^2 \sinh ^3(c+d x)-2 i a b \sinh ^3(c+d x) \tanh ^3(c+d x)-i b^2 \sinh ^3(c+d x) \tanh ^6(c+d x)\right ) \, dx\\ &=a^2 \int \sinh ^3(c+d x) \, dx+(2 a b) \int \sinh ^3(c+d x) \tanh ^3(c+d x) \, dx+b^2 \int \sinh ^3(c+d x) \tanh ^6(c+d x) \, dx\\ &=-\frac {a^2 \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}+\frac {(2 a b) \operatorname {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^4}{x^6} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a^2 \cosh (c+d x)}{d}+\frac {a^2 \cosh ^3(c+d x)}{3 d}-\frac {a b \sinh ^3(c+d x) \tanh ^2(c+d x)}{d}+\frac {(5 a b) \operatorname {Subst}\left (\int \frac {x^4}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{d}+\frac {b^2 \operatorname {Subst}\left (\int \left (-4+\frac {1}{x^6}-\frac {4}{x^4}+\frac {6}{x^2}+x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a^2 \cosh (c+d x)}{d}-\frac {4 b^2 \cosh (c+d x)}{d}+\frac {a^2 \cosh ^3(c+d x)}{3 d}+\frac {b^2 \cosh ^3(c+d x)}{3 d}-\frac {6 b^2 \text {sech}(c+d x)}{d}+\frac {4 b^2 \text {sech}^3(c+d x)}{3 d}-\frac {b^2 \text {sech}^5(c+d x)}{5 d}-\frac {a b \sinh ^3(c+d x) \tanh ^2(c+d x)}{d}+\frac {(5 a b) \operatorname {Subst}\left (\int \left (-1+x^2+\frac {1}{1+x^2}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=-\frac {a^2 \cosh (c+d x)}{d}-\frac {4 b^2 \cosh (c+d x)}{d}+\frac {a^2 \cosh ^3(c+d x)}{3 d}+\frac {b^2 \cosh ^3(c+d x)}{3 d}-\frac {6 b^2 \text {sech}(c+d x)}{d}+\frac {4 b^2 \text {sech}^3(c+d x)}{3 d}-\frac {b^2 \text {sech}^5(c+d x)}{5 d}-\frac {5 a b \sinh (c+d x)}{d}+\frac {5 a b \sinh ^3(c+d x)}{3 d}-\frac {a b \sinh ^3(c+d x) \tanh ^2(c+d x)}{d}+\frac {(5 a b) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {5 a b \tan ^{-1}(\sinh (c+d x))}{d}-\frac {a^2 \cosh (c+d x)}{d}-\frac {4 b^2 \cosh (c+d x)}{d}+\frac {a^2 \cosh ^3(c+d x)}{3 d}+\frac {b^2 \cosh ^3(c+d x)}{3 d}-\frac {6 b^2 \text {sech}(c+d x)}{d}+\frac {4 b^2 \text {sech}^3(c+d x)}{3 d}-\frac {b^2 \text {sech}^5(c+d x)}{5 d}-\frac {5 a b \sinh (c+d x)}{d}+\frac {5 a b \sinh ^3(c+d x)}{3 d}-\frac {a b \sinh ^3(c+d x) \tanh ^2(c+d x)}{d}\\ \end {align*}

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Mathematica [A]  time = 0.78, size = 121, normalized size = 0.66 \[ \frac {-45 \left (a^2+5 b^2\right ) \cosh (c+d x)+5 \left (a^2+b^2\right ) \cosh (3 (c+d x))-2 b \left (30 \text {sech}(c+d x) (a \tanh (c+d x)+6 b)-5 a \left (-27 \sinh (c+d x)+\sinh (3 (c+d x))+60 \tan ^{-1}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )\right )+6 b \text {sech}^5(c+d x)-40 b \text {sech}^3(c+d x)\right )}{60 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-45*(a^2 + 5*b^2)*Cosh[c + d*x] + 5*(a^2 + b^2)*Cosh[3*(c + d*x)] - 2*b*(-40*b*Sech[c + d*x]^3 + 6*b*Sech[c +
d*x]^5 - 5*a*(60*ArcTan[Tanh[(c + d*x)/2]] - 27*Sinh[c + d*x] + Sinh[3*(c + d*x)]) + 30*Sech[c + d*x]*(6*b +
a*Tanh[c + d*x])))/(60*d)

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fricas [B]  time = 0.66, size = 3341, normalized size = 18.36 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(5*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^16 + 80*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^15 + 5*(a^2
+ 2*a*b + b^2)*sinh(d*x + c)^16 - 20*(a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)^14 + 20*(30*(a^2 + 2*a*b + b^2)*co
sh(d*x + c)^2 - a^2 - 11*a*b - 10*b^2)*sinh(d*x + c)^14 + 280*(10*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 - (a^2 +
11*a*b + 10*b^2)*cosh(d*x + c))*sinh(d*x + c)^13 - 20*(11*a^2 + 61*a*b + 137*b^2)*cosh(d*x + c)^12 + 20*(455*
(a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 - 91*(a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)^2 - 11*a^2 - 61*a*b - 137*b^2)*
sinh(d*x + c)^12 + 80*(273*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^5 - 91*(a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)^3 -
3*(11*a^2 + 61*a*b + 137*b^2)*cosh(d*x + c))*sinh(d*x + c)^11 - 20*(31*a^2 + 87*a*b + 390*b^2)*cosh(d*x + c)^1
0 + 20*(2002*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 - 1001*(a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)^4 - 66*(11*a^2 +
61*a*b + 137*b^2)*cosh(d*x + c)^2 - 31*a^2 - 87*a*b - 390*b^2)*sinh(d*x + c)^10 + 40*(1430*(a^2 + 2*a*b + b^2
)*cosh(d*x + c)^7 - 1001*(a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)^5 - 110*(11*a^2 + 61*a*b + 137*b^2)*cosh(d*x +
c)^3 - 5*(31*a^2 + 87*a*b + 390*b^2)*cosh(d*x + c))*sinh(d*x + c)^9 - 2*(425*a^2 + 5649*b^2)*cosh(d*x + c)^8 +
2*(32175*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^8 - 30030*(a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)^6 - 4950*(11*a^2 +
61*a*b + 137*b^2)*cosh(d*x + c)^4 - 450*(31*a^2 + 87*a*b + 390*b^2)*cosh(d*x + c)^2 - 425*a^2 - 5649*b^2)*sin
h(d*x + c)^8 + 16*(3575*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^9 - 4290*(a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)^7 - 9
90*(11*a^2 + 61*a*b + 137*b^2)*cosh(d*x + c)^5 - 150*(31*a^2 + 87*a*b + 390*b^2)*cosh(d*x + c)^3 - (425*a^2 +
5649*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 20*(31*a^2 - 87*a*b + 390*b^2)*cosh(d*x + c)^6 + 4*(10010*(a^2 + 2*
a*b + b^2)*cosh(d*x + c)^10 - 15015*(a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)^8 - 4620*(11*a^2 + 61*a*b + 137*b^2)
*cosh(d*x + c)^6 - 1050*(31*a^2 + 87*a*b + 390*b^2)*cosh(d*x + c)^4 - 14*(425*a^2 + 5649*b^2)*cosh(d*x + c)^2
- 155*a^2 + 435*a*b - 1950*b^2)*sinh(d*x + c)^6 + 8*(2730*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^11 - 5005*(a^2 + 1
1*a*b + 10*b^2)*cosh(d*x + c)^9 - 1980*(11*a^2 + 61*a*b + 137*b^2)*cosh(d*x + c)^7 - 630*(31*a^2 + 87*a*b + 39
0*b^2)*cosh(d*x + c)^5 - 14*(425*a^2 + 5649*b^2)*cosh(d*x + c)^3 - 15*(31*a^2 - 87*a*b + 390*b^2)*cosh(d*x + c
))*sinh(d*x + c)^5 - 20*(11*a^2 - 61*a*b + 137*b^2)*cosh(d*x + c)^4 + 20*(455*(a^2 + 2*a*b + b^2)*cosh(d*x + c
)^12 - 1001*(a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)^10 - 495*(11*a^2 + 61*a*b + 137*b^2)*cosh(d*x + c)^8 - 210*(
31*a^2 + 87*a*b + 390*b^2)*cosh(d*x + c)^6 - 7*(425*a^2 + 5649*b^2)*cosh(d*x + c)^4 - 15*(31*a^2 - 87*a*b + 39
0*b^2)*cosh(d*x + c)^2 - 11*a^2 + 61*a*b - 137*b^2)*sinh(d*x + c)^4 + 16*(175*(a^2 + 2*a*b + b^2)*cosh(d*x + c
)^13 - 455*(a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)^11 - 275*(11*a^2 + 61*a*b + 137*b^2)*cosh(d*x + c)^9 - 150*(3
1*a^2 + 87*a*b + 390*b^2)*cosh(d*x + c)^7 - 7*(425*a^2 + 5649*b^2)*cosh(d*x + c)^5 - 25*(31*a^2 - 87*a*b + 390
*b^2)*cosh(d*x + c)^3 - 5*(11*a^2 - 61*a*b + 137*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 20*(a^2 - 11*a*b + 10*b
^2)*cosh(d*x + c)^2 + 4*(150*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^14 - 455*(a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)^
12 - 330*(11*a^2 + 61*a*b + 137*b^2)*cosh(d*x + c)^10 - 225*(31*a^2 + 87*a*b + 390*b^2)*cosh(d*x + c)^8 - 14*(
425*a^2 + 5649*b^2)*cosh(d*x + c)^6 - 75*(31*a^2 - 87*a*b + 390*b^2)*cosh(d*x + c)^4 - 30*(11*a^2 - 61*a*b + 1
37*b^2)*cosh(d*x + c)^2 - 5*a^2 + 55*a*b - 50*b^2)*sinh(d*x + c)^2 + 5*a^2 - 10*a*b + 5*b^2 + 1200*(a*b*cosh(d
*x + c)^13 + 13*a*b*cosh(d*x + c)*sinh(d*x + c)^12 + a*b*sinh(d*x + c)^13 + 5*a*b*cosh(d*x + c)^11 + (78*a*b*c
osh(d*x + c)^2 + 5*a*b)*sinh(d*x + c)^11 + 10*a*b*cosh(d*x + c)^9 + 11*(26*a*b*cosh(d*x + c)^3 + 5*a*b*cosh(d*
x + c))*sinh(d*x + c)^10 + 5*(143*a*b*cosh(d*x + c)^4 + 55*a*b*cosh(d*x + c)^2 + 2*a*b)*sinh(d*x + c)^9 + 10*a
*b*cosh(d*x + c)^7 + 3*(429*a*b*cosh(d*x + c)^5 + 275*a*b*cosh(d*x + c)^3 + 30*a*b*cosh(d*x + c))*sinh(d*x + c
)^8 + 2*(858*a*b*cosh(d*x + c)^6 + 825*a*b*cosh(d*x + c)^4 + 180*a*b*cosh(d*x + c)^2 + 5*a*b)*sinh(d*x + c)^7
+ 5*a*b*cosh(d*x + c)^5 + 2*(858*a*b*cosh(d*x + c)^7 + 1155*a*b*cosh(d*x + c)^5 + 420*a*b*cosh(d*x + c)^3 + 35
*a*b*cosh(d*x + c))*sinh(d*x + c)^6 + (1287*a*b*cosh(d*x + c)^8 + 2310*a*b*cosh(d*x + c)^6 + 1260*a*b*cosh(d*x
+ c)^4 + 210*a*b*cosh(d*x + c)^2 + 5*a*b)*sinh(d*x + c)^5 + a*b*cosh(d*x + c)^3 + 5*(143*a*b*cosh(d*x + c)^9
+ 330*a*b*cosh(d*x + c)^7 + 252*a*b*cosh(d*x + c)^5 + 70*a*b*cosh(d*x + c)^3 + 5*a*b*cosh(d*x + c))*sinh(d*x +
c)^4 + (286*a*b*cosh(d*x + c)^10 + 825*a*b*cosh(d*x + c)^8 + 840*a*b*cosh(d*x + c)^6 + 350*a*b*cosh(d*x + c)^
4 + 50*a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c)^3 + (78*a*b*cosh(d*x + c)^11 + 275*a*b*cosh(d*x + c)^9 + 360*a
*b*cosh(d*x + c)^7 + 210*a*b*cosh(d*x + c)^5 + 50*a*b*cosh(d*x + c)^3 + 3*a*b*cosh(d*x + c))*sinh(d*x + c)^2 +
(13*a*b*cosh(d*x + c)^12 + 55*a*b*cosh(d*x + c)^10 + 90*a*b*cosh(d*x + c)^8 + 70*a*b*cosh(d*x + c)^6 + 25*a*b
*cosh(d*x + c)^4 + 3*a*b*cosh(d*x + c)^2)*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) + 8*(10*(a^2 +
2*a*b + b^2)*cosh(d*x + c)^15 - 35*(a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)^13 - 30*(11*a^2 + 61*a*b + 137*b^2)*c
osh(d*x + c)^11 - 25*(31*a^2 + 87*a*b + 390*b^2)*cosh(d*x + c)^9 - 2*(425*a^2 + 5649*b^2)*cosh(d*x + c)^7 - 15
*(31*a^2 - 87*a*b + 390*b^2)*cosh(d*x + c)^5 - 10*(11*a^2 - 61*a*b + 137*b^2)*cosh(d*x + c)^3 - 5*(a^2 - 11*a*
b + 10*b^2)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^13 + 13*d*cosh(d*x + c)*sinh(d*x + c)^12 + d*sinh(d
*x + c)^13 + 5*d*cosh(d*x + c)^11 + (78*d*cosh(d*x + c)^2 + 5*d)*sinh(d*x + c)^11 + 11*(26*d*cosh(d*x + c)^3 +
5*d*cosh(d*x + c))*sinh(d*x + c)^10 + 10*d*cosh(d*x + c)^9 + 5*(143*d*cosh(d*x + c)^4 + 55*d*cosh(d*x + c)^2
+ 2*d)*sinh(d*x + c)^9 + 3*(429*d*cosh(d*x + c)^5 + 275*d*cosh(d*x + c)^3 + 30*d*cosh(d*x + c))*sinh(d*x + c)^
8 + 10*d*cosh(d*x + c)^7 + 2*(858*d*cosh(d*x + c)^6 + 825*d*cosh(d*x + c)^4 + 180*d*cosh(d*x + c)^2 + 5*d)*sin
h(d*x + c)^7 + 2*(858*d*cosh(d*x + c)^7 + 1155*d*cosh(d*x + c)^5 + 420*d*cosh(d*x + c)^3 + 35*d*cosh(d*x + c))
*sinh(d*x + c)^6 + 5*d*cosh(d*x + c)^5 + (1287*d*cosh(d*x + c)^8 + 2310*d*cosh(d*x + c)^6 + 1260*d*cosh(d*x +
c)^4 + 210*d*cosh(d*x + c)^2 + 5*d)*sinh(d*x + c)^5 + 5*(143*d*cosh(d*x + c)^9 + 330*d*cosh(d*x + c)^7 + 252*d
*cosh(d*x + c)^5 + 70*d*cosh(d*x + c)^3 + 5*d*cosh(d*x + c))*sinh(d*x + c)^4 + d*cosh(d*x + c)^3 + (286*d*cosh
(d*x + c)^10 + 825*d*cosh(d*x + c)^8 + 840*d*cosh(d*x + c)^6 + 350*d*cosh(d*x + c)^4 + 50*d*cosh(d*x + c)^2 +
d)*sinh(d*x + c)^3 + (78*d*cosh(d*x + c)^11 + 275*d*cosh(d*x + c)^9 + 360*d*cosh(d*x + c)^7 + 210*d*cosh(d*x +
c)^5 + 50*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^2 + (13*d*cosh(d*x + c)^12 + 55*d*cosh(d*x + c
)^10 + 90*d*cosh(d*x + c)^8 + 70*d*cosh(d*x + c)^6 + 25*d*cosh(d*x + c)^4 + 3*d*cosh(d*x + c)^2)*sinh(d*x + c)
)

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giac [A]  time = 0.48, size = 300, normalized size = 1.65 \[ \frac {1200 \, a b \arctan \left (e^{\left (d x + c\right )}\right ) - 5 \, {\left (9 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 54 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 45 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - a^{2} + 2 \, a b - b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )} + 5 \, {\left (a^{2} e^{\left (3 \, d x + 48 \, c\right )} + 2 \, a b e^{\left (3 \, d x + 48 \, c\right )} + b^{2} e^{\left (3 \, d x + 48 \, c\right )} - 9 \, a^{2} e^{\left (d x + 46 \, c\right )} - 54 \, a b e^{\left (d x + 46 \, c\right )} - 45 \, b^{2} e^{\left (d x + 46 \, c\right )}\right )} e^{\left (-45 \, c\right )} - \frac {16 \, {\left (15 \, a b e^{\left (9 \, d x + 9 \, c\right )} + 90 \, b^{2} e^{\left (9 \, d x + 9 \, c\right )} + 30 \, a b e^{\left (7 \, d x + 7 \, c\right )} + 280 \, b^{2} e^{\left (7 \, d x + 7 \, c\right )} + 428 \, b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 30 \, a b e^{\left (3 \, d x + 3 \, c\right )} + 280 \, b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 15 \, a b e^{\left (d x + c\right )} + 90 \, b^{2} e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}}{120 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(1200*a*b*arctan(e^(d*x + c)) - 5*(9*a^2*e^(2*d*x + 2*c) - 54*a*b*e^(2*d*x + 2*c) + 45*b^2*e^(2*d*x + 2*
c) - a^2 + 2*a*b - b^2)*e^(-3*d*x - 3*c) + 5*(a^2*e^(3*d*x + 48*c) + 2*a*b*e^(3*d*x + 48*c) + b^2*e^(3*d*x + 4
8*c) - 9*a^2*e^(d*x + 46*c) - 54*a*b*e^(d*x + 46*c) - 45*b^2*e^(d*x + 46*c))*e^(-45*c) - 16*(15*a*b*e^(9*d*x +
9*c) + 90*b^2*e^(9*d*x + 9*c) + 30*a*b*e^(7*d*x + 7*c) + 280*b^2*e^(7*d*x + 7*c) + 428*b^2*e^(5*d*x + 5*c) -
30*a*b*e^(3*d*x + 3*c) + 280*b^2*e^(3*d*x + 3*c) - 15*a*b*e^(d*x + c) + 90*b^2*e^(d*x + c))/(e^(2*d*x + 2*c) +
1)^5)/d

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maple [A]  time = 0.38, size = 250, normalized size = 1.37 \[ -\frac {2 a^{2} \cosh \left (d x +c \right )}{3 d}+\frac {a^{2} \cosh \left (d x +c \right ) \left (\sinh ^{2}\left (d x +c \right )\right )}{3 d}+\frac {2 a b \left (\sinh ^{5}\left (d x +c \right )\right )}{3 d \cosh \left (d x +c \right )^{2}}-\frac {10 a b \left (\sinh ^{3}\left (d x +c \right )\right )}{3 d \cosh \left (d x +c \right )^{2}}-\frac {10 a b \sinh \left (d x +c \right )}{d \cosh \left (d x +c \right )^{2}}+\frac {5 a b \,\mathrm {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{d}+\frac {10 a b \arctan \left ({\mathrm e}^{d x +c}\right )}{d}+\frac {b^{2} \left (\sinh ^{8}\left (d x +c \right )\right )}{3 d \cosh \left (d x +c \right )^{5}}-\frac {8 b^{2} \left (\sinh ^{6}\left (d x +c \right )\right )}{3 d \cosh \left (d x +c \right )^{5}}-\frac {16 b^{2} \left (\sinh ^{4}\left (d x +c \right )\right )}{d \cosh \left (d x +c \right )^{5}}-\frac {64 b^{2} \left (\sinh ^{2}\left (d x +c \right )\right )}{3 d \cosh \left (d x +c \right )^{5}}-\frac {128 b^{2}}{15 d \cosh \left (d x +c \right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*a^2*cosh(d*x+c)/d+1/3/d*a^2*cosh(d*x+c)*sinh(d*x+c)^2+2/3/d*a*b*sinh(d*x+c)^5/cosh(d*x+c)^2-10/3/d*a*b*si
nh(d*x+c)^3/cosh(d*x+c)^2-10/d*a*b*sinh(d*x+c)/cosh(d*x+c)^2+5/d*a*b*sech(d*x+c)*tanh(d*x+c)+10/d*a*b*arctan(e
xp(d*x+c))+1/3/d*b^2*sinh(d*x+c)^8/cosh(d*x+c)^5-8/3/d*b^2*sinh(d*x+c)^6/cosh(d*x+c)^5-16/d*b^2*sinh(d*x+c)^4/
cosh(d*x+c)^5-64/3/d*b^2*sinh(d*x+c)^2/cosh(d*x+c)^5-128/15/d*b^2/cosh(d*x+c)^5

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maxima [B]  time = 0.41, size = 348, normalized size = 1.91 \[ -\frac {1}{120} \, b^{2} {\left (\frac {5 \, {\left (45 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d} + \frac {200 \, e^{\left (-2 \, d x - 2 \, c\right )} + 2515 \, e^{\left (-4 \, d x - 4 \, c\right )} + 6680 \, e^{\left (-6 \, d x - 6 \, c\right )} + 9073 \, e^{\left (-8 \, d x - 8 \, c\right )} + 5600 \, e^{\left (-10 \, d x - 10 \, c\right )} + 1665 \, e^{\left (-12 \, d x - 12 \, c\right )} - 5}{d {\left (e^{\left (-3 \, d x - 3 \, c\right )} + 5 \, e^{\left (-5 \, d x - 5 \, c\right )} + 10 \, e^{\left (-7 \, d x - 7 \, c\right )} + 10 \, e^{\left (-9 \, d x - 9 \, c\right )} + 5 \, e^{\left (-11 \, d x - 11 \, c\right )} + e^{\left (-13 \, d x - 13 \, c\right )}\right )}}\right )} + \frac {1}{12} \, a b {\left (\frac {27 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} - \frac {120 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {25 \, e^{\left (-2 \, d x - 2 \, c\right )} + 77 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3 \, e^{\left (-6 \, d x - 6 \, c\right )} - 1}{d {\left (e^{\left (-3 \, d x - 3 \, c\right )} + 2 \, e^{\left (-5 \, d x - 5 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )}\right )}}\right )} + \frac {1}{24} \, a^{2} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/120*b^2*(5*(45*e^(-d*x - c) - e^(-3*d*x - 3*c))/d + (200*e^(-2*d*x - 2*c) + 2515*e^(-4*d*x - 4*c) + 6680*e^
(-6*d*x - 6*c) + 9073*e^(-8*d*x - 8*c) + 5600*e^(-10*d*x - 10*c) + 1665*e^(-12*d*x - 12*c) - 5)/(d*(e^(-3*d*x
- 3*c) + 5*e^(-5*d*x - 5*c) + 10*e^(-7*d*x - 7*c) + 10*e^(-9*d*x - 9*c) + 5*e^(-11*d*x - 11*c) + e^(-13*d*x -
13*c)))) + 1/12*a*b*((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^2*(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)

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mupad [B]  time = 1.44, size = 397, normalized size = 2.18 \[ \frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,{\left (a+b\right )}^2}{24\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (3\,a^2+18\,a\,b+15\,b^2\right )}{8\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (3\,a^2-18\,a\,b+15\,b^2\right )}{8\,d}+\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}\,{\left (a-b\right )}^2}{24\,d}+\frac {10\,\mathrm {atan}\left (\frac {a\,b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {d^2}}{d\,\sqrt {a^2\,b^2}}\right )\,\sqrt {a^2\,b^2}}{\sqrt {d^2}}-\frac {256\,b^2\,{\mathrm {e}}^{c+d\,x}}{15\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}+\frac {64\,b^2\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {32\,b^2\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (6\,b^2+a\,b\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {4\,{\mathrm {e}}^{c+d\,x}\,\left (8\,b^2+3\,a\,b\right )}{3\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(exp(3*c + 3*d*x)*(a + b)^2)/(24*d) - (exp(c + d*x)*(18*a*b + 3*a^2 + 15*b^2))/(8*d) - (exp(- c - d*x)*(3*a^2
- 18*a*b + 15*b^2))/(8*d) + (exp(- 3*c - 3*d*x)*(a - b)^2)/(24*d) + (10*atan((a*b*exp(d*x)*exp(c)*(d^2)^(1/2))
/(d*(a^2*b^2)^(1/2)))*(a^2*b^2)^(1/2))/(d^2)^(1/2) - (256*b^2*exp(c + d*x))/(15*d*(3*exp(2*c + 2*d*x) + 3*exp(
4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) + (64*b^2*exp(c + d*x))/(5*d*(4*exp(2*c + 2*d*x) + 6*exp(4*c + 4*d*x) +
4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1)) - (32*b^2*exp(c + d*x))/(5*d*(5*exp(2*c + 2*d*x) + 10*exp(4*c + 4*
d*x) + 10*exp(6*c + 6*d*x) + 5*exp(8*c + 8*d*x) + exp(10*c + 10*d*x) + 1)) - (2*exp(c + d*x)*(a*b + 6*b^2))/(d
*(exp(2*c + 2*d*x) + 1)) + (4*exp(c + d*x)*(3*a*b + 8*b^2))/(3*d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right )^{2} \sinh ^{3}{\left (c + d x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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