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

Optimal. Leaf size=269 \[ \frac {a^3 \cosh (c+d x)}{d}+\frac {9 a^2 b \sinh (c+d x)}{2 d}-\frac {9 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac {3 a^2 b \sinh (c+d x) \tanh ^2(c+d x)}{2 d}+\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}-\frac {3 a b^2 \text {sech}^3(c+d x)}{d}+\frac {9 a b^2 \text {sech}(c+d x)}{d}+\frac {315 b^3 \sinh (c+d x)}{128 d}-\frac {315 b^3 \tan ^{-1}(\sinh (c+d x))}{128 d}-\frac {b^3 \sinh (c+d x) \tanh ^8(c+d x)}{8 d}-\frac {3 b^3 \sinh (c+d x) \tanh ^6(c+d x)}{16 d}-\frac {21 b^3 \sinh (c+d x) \tanh ^4(c+d x)}{64 d}-\frac {105 b^3 \sinh (c+d x) \tanh ^2(c+d x)}{128 d} \]

[Out]

-9/2*a^2*b*arctan(sinh(d*x+c))/d-315/128*b^3*arctan(sinh(d*x+c))/d+a^3*cosh(d*x+c)/d+3*a*b^2*cosh(d*x+c)/d+9*a
*b^2*sech(d*x+c)/d-3*a*b^2*sech(d*x+c)^3/d+3/5*a*b^2*sech(d*x+c)^5/d+9/2*a^2*b*sinh(d*x+c)/d+315/128*b^3*sinh(
d*x+c)/d-3/2*a^2*b*sinh(d*x+c)*tanh(d*x+c)^2/d-105/128*b^3*sinh(d*x+c)*tanh(d*x+c)^2/d-21/64*b^3*sinh(d*x+c)*t
anh(d*x+c)^4/d-3/16*b^3*sinh(d*x+c)*tanh(d*x+c)^6/d-1/8*b^3*sinh(d*x+c)*tanh(d*x+c)^8/d

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Rubi [A]  time = 0.25, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3666, 2638, 2592, 288, 321, 203, 2590, 270} \[ \frac {9 a^2 b \sinh (c+d x)}{2 d}-\frac {9 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac {3 a^2 b \sinh (c+d x) \tanh ^2(c+d x)}{2 d}+\frac {a^3 \cosh (c+d x)}{d}+\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}-\frac {3 a b^2 \text {sech}^3(c+d x)}{d}+\frac {9 a b^2 \text {sech}(c+d x)}{d}+\frac {315 b^3 \sinh (c+d x)}{128 d}-\frac {315 b^3 \tan ^{-1}(\sinh (c+d x))}{128 d}-\frac {b^3 \sinh (c+d x) \tanh ^8(c+d x)}{8 d}-\frac {3 b^3 \sinh (c+d x) \tanh ^6(c+d x)}{16 d}-\frac {21 b^3 \sinh (c+d x) \tanh ^4(c+d x)}{64 d}-\frac {105 b^3 \sinh (c+d x) \tanh ^2(c+d x)}{128 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-9*a^2*b*ArcTan[Sinh[c + d*x]])/(2*d) - (315*b^3*ArcTan[Sinh[c + d*x]])/(128*d) + (a^3*Cosh[c + d*x])/d + (3*
a*b^2*Cosh[c + d*x])/d + (9*a*b^2*Sech[c + d*x])/d - (3*a*b^2*Sech[c + d*x]^3)/d + (3*a*b^2*Sech[c + d*x]^5)/(
5*d) + (9*a^2*b*Sinh[c + d*x])/(2*d) + (315*b^3*Sinh[c + d*x])/(128*d) - (3*a^2*b*Sinh[c + d*x]*Tanh[c + d*x]^
2)/(2*d) - (105*b^3*Sinh[c + d*x]*Tanh[c + d*x]^2)/(128*d) - (21*b^3*Sinh[c + d*x]*Tanh[c + d*x]^4)/(64*d) - (
3*b^3*Sinh[c + d*x]*Tanh[c + d*x]^6)/(16*d) - (b^3*Sinh[c + d*x]*Tanh[c + d*x]^8)/(8*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 321

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*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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 (c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx &=-\left (i \int \left (i a^3 \sinh (c+d x)+3 i a^2 b \sinh (c+d x) \tanh ^3(c+d x)+3 i a b^2 \sinh (c+d x) \tanh ^6(c+d x)+i b^3 \sinh (c+d x) \tanh ^9(c+d x)\right ) \, dx\right )\\ &=a^3 \int \sinh (c+d x) \, dx+\left (3 a^2 b\right ) \int \sinh (c+d x) \tanh ^3(c+d x) \, dx+\left (3 a b^2\right ) \int \sinh (c+d x) \tanh ^6(c+d x) \, dx+b^3 \int \sinh (c+d x) \tanh ^9(c+d x) \, dx\\ &=\frac {a^3 \cosh (c+d x)}{d}+\frac {\left (3 a^2 b\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}-\frac {\left (3 a b^2\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^3}{x^6} \, dx,x,\cosh (c+d x)\right )}{d}+\frac {b^3 \operatorname {Subst}\left (\int \frac {x^{10}}{\left (1+x^2\right )^5} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {a^3 \cosh (c+d x)}{d}-\frac {3 a^2 b \sinh (c+d x) \tanh ^2(c+d x)}{2 d}-\frac {b^3 \sinh (c+d x) \tanh ^8(c+d x)}{8 d}+\frac {\left (9 a^2 b\right ) \operatorname {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 d}-\frac {\left (3 a b^2\right ) \operatorname {Subst}\left (\int \left (-1+\frac {1}{x^6}-\frac {3}{x^4}+\frac {3}{x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{d}+\frac {\left (9 b^3\right ) \operatorname {Subst}\left (\int \frac {x^8}{\left (1+x^2\right )^4} \, dx,x,\sinh (c+d x)\right )}{8 d}\\ &=\frac {a^3 \cosh (c+d x)}{d}+\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {9 a b^2 \text {sech}(c+d x)}{d}-\frac {3 a b^2 \text {sech}^3(c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}+\frac {9 a^2 b \sinh (c+d x)}{2 d}-\frac {3 a^2 b \sinh (c+d x) \tanh ^2(c+d x)}{2 d}-\frac {3 b^3 \sinh (c+d x) \tanh ^6(c+d x)}{16 d}-\frac {b^3 \sinh (c+d x) \tanh ^8(c+d x)}{8 d}-\frac {\left (9 a^2 b\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 d}+\frac {\left (21 b^3\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{16 d}\\ &=-\frac {9 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac {a^3 \cosh (c+d x)}{d}+\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {9 a b^2 \text {sech}(c+d x)}{d}-\frac {3 a b^2 \text {sech}^3(c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}+\frac {9 a^2 b \sinh (c+d x)}{2 d}-\frac {3 a^2 b \sinh (c+d x) \tanh ^2(c+d x)}{2 d}-\frac {21 b^3 \sinh (c+d x) \tanh ^4(c+d x)}{64 d}-\frac {3 b^3 \sinh (c+d x) \tanh ^6(c+d x)}{16 d}-\frac {b^3 \sinh (c+d x) \tanh ^8(c+d x)}{8 d}+\frac {\left (105 b^3\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{64 d}\\ &=-\frac {9 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac {a^3 \cosh (c+d x)}{d}+\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {9 a b^2 \text {sech}(c+d x)}{d}-\frac {3 a b^2 \text {sech}^3(c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}+\frac {9 a^2 b \sinh (c+d x)}{2 d}-\frac {3 a^2 b \sinh (c+d x) \tanh ^2(c+d x)}{2 d}-\frac {105 b^3 \sinh (c+d x) \tanh ^2(c+d x)}{128 d}-\frac {21 b^3 \sinh (c+d x) \tanh ^4(c+d x)}{64 d}-\frac {3 b^3 \sinh (c+d x) \tanh ^6(c+d x)}{16 d}-\frac {b^3 \sinh (c+d x) \tanh ^8(c+d x)}{8 d}+\frac {\left (315 b^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{128 d}\\ &=-\frac {9 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac {a^3 \cosh (c+d x)}{d}+\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {9 a b^2 \text {sech}(c+d x)}{d}-\frac {3 a b^2 \text {sech}^3(c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}+\frac {9 a^2 b \sinh (c+d x)}{2 d}+\frac {315 b^3 \sinh (c+d x)}{128 d}-\frac {3 a^2 b \sinh (c+d x) \tanh ^2(c+d x)}{2 d}-\frac {105 b^3 \sinh (c+d x) \tanh ^2(c+d x)}{128 d}-\frac {21 b^3 \sinh (c+d x) \tanh ^4(c+d x)}{64 d}-\frac {3 b^3 \sinh (c+d x) \tanh ^6(c+d x)}{16 d}-\frac {b^3 \sinh (c+d x) \tanh ^8(c+d x)}{8 d}-\frac {\left (315 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{128 d}\\ &=-\frac {9 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac {315 b^3 \tan ^{-1}(\sinh (c+d x))}{128 d}+\frac {a^3 \cosh (c+d x)}{d}+\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {9 a b^2 \text {sech}(c+d x)}{d}-\frac {3 a b^2 \text {sech}^3(c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}+\frac {9 a^2 b \sinh (c+d x)}{2 d}+\frac {315 b^3 \sinh (c+d x)}{128 d}-\frac {3 a^2 b \sinh (c+d x) \tanh ^2(c+d x)}{2 d}-\frac {105 b^3 \sinh (c+d x) \tanh ^2(c+d x)}{128 d}-\frac {21 b^3 \sinh (c+d x) \tanh ^4(c+d x)}{64 d}-\frac {3 b^3 \sinh (c+d x) \tanh ^6(c+d x)}{16 d}-\frac {b^3 \sinh (c+d x) \tanh ^8(c+d x)}{8 d}\\ \end {align*}

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Mathematica [A]  time = 6.33, size = 233, normalized size = 0.87 \[ \frac {\text {sech}^2(c+d x) \left (192 a^2 b \sinh (c+d x)+325 b^3 \sinh (c+d x)\right )}{128 d}+\frac {b \left (3 a^2+b^2\right ) \sinh (c+d x)}{d}+\frac {a \left (a^2+3 b^2\right ) \cosh (c+d x)}{d}-\frac {9 b \left (64 a^2+35 b^2\right ) \tan ^{-1}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{64 d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}-\frac {3 a b^2 \text {sech}^3(c+d x)}{d}+\frac {9 a b^2 \text {sech}(c+d x)}{d}-\frac {b^3 \tanh (c+d x) \text {sech}^7(c+d x)}{8 d}+\frac {11 b^3 \tanh (c+d x) \text {sech}^5(c+d x)}{16 d}-\frac {105 b^3 \tanh (c+d x) \text {sech}^3(c+d x)}{64 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-9*b*(64*a^2 + 35*b^2)*ArcTan[Tanh[(c + d*x)/2]])/(64*d) + (a*(a^2 + 3*b^2)*Cosh[c + d*x])/d + (9*a*b^2*Sech[
c + d*x])/d - (3*a*b^2*Sech[c + d*x]^3)/d + (3*a*b^2*Sech[c + d*x]^5)/(5*d) + (b*(3*a^2 + b^2)*Sinh[c + d*x])/
d + (Sech[c + d*x]^2*(192*a^2*b*Sinh[c + d*x] + 325*b^3*Sinh[c + d*x]))/(128*d) - (105*b^3*Sech[c + d*x]^3*Tan
h[c + d*x])/(64*d) + (11*b^3*Sech[c + d*x]^5*Tanh[c + d*x])/(16*d) - (b^3*Sech[c + d*x]^7*Tanh[c + d*x])/(8*d)

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fricas [B]  time = 0.52, size = 6410, normalized size = 23.83 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/320*(160*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^18 + 2880*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x +
c)*sinh(d*x + c)^17 + 160*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sinh(d*x + c)^18 + 45*(32*a^3 + 96*a^2*b + 224*a*b^2
+ 61*b^3)*cosh(d*x + c)^16 + 45*(32*a^3 + 96*a^2*b + 224*a*b^2 + 61*b^3 + 544*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)
*cosh(d*x + c)^2)*sinh(d*x + c)^16 + 240*(544*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^3 + 3*(32*a^3 + 96
*a^2*b + 224*a*b^2 + 61*b^3)*cosh(d*x + c))*sinh(d*x + c)^15 + 15*(384*a^3 + 960*a^2*b + 3328*a*b^2 + 475*b^3)
*cosh(d*x + c)^14 + 15*(32640*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 + 384*a^3 + 960*a^2*b + 3328*a*b
^2 + 475*b^3 + 360*(32*a^3 + 96*a^2*b + 224*a*b^2 + 61*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^14 + 210*(6528*(a^3
+ 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^5 + 120*(32*a^3 + 96*a^2*b + 224*a*b^2 + 61*b^3)*cosh(d*x + c)^3 + (
384*a^3 + 960*a^2*b + 3328*a*b^2 + 475*b^3)*cosh(d*x + c))*sinh(d*x + c)^13 + 3*(4480*a^3 + 7360*a^2*b + 43008
*a*b^2 + 4515*b^3)*cosh(d*x + c)^12 + 3*(990080*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^6 + 27300*(32*a^
3 + 96*a^2*b + 224*a*b^2 + 61*b^3)*cosh(d*x + c)^4 + 4480*a^3 + 7360*a^2*b + 43008*a*b^2 + 4515*b^3 + 455*(384
*a^3 + 960*a^2*b + 3328*a*b^2 + 475*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^12 + 12*(424320*(a^3 + 3*a^2*b + 3*a*b
^2 + b^3)*cosh(d*x + c)^7 + 16380*(32*a^3 + 96*a^2*b + 224*a*b^2 + 61*b^3)*cosh(d*x + c)^5 + 455*(384*a^3 + 96
0*a^2*b + 3328*a*b^2 + 475*b^3)*cosh(d*x + c)^3 + 3*(4480*a^3 + 7360*a^2*b + 43008*a*b^2 + 4515*b^3)*cosh(d*x
+ c))*sinh(d*x + c)^11 + 3*(6720*a^3 + 3840*a^2*b + 67904*a*b^2 + 1295*b^3)*cosh(d*x + c)^10 + 3*(2333760*(a^3
+ 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^8 + 120120*(32*a^3 + 96*a^2*b + 224*a*b^2 + 61*b^3)*cosh(d*x + c)^6
+ 5005*(384*a^3 + 960*a^2*b + 3328*a*b^2 + 475*b^3)*cosh(d*x + c)^4 + 6720*a^3 + 3840*a^2*b + 67904*a*b^2 + 12
95*b^3 + 66*(4480*a^3 + 7360*a^2*b + 43008*a*b^2 + 4515*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 10*(777920*(a
^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^9 + 51480*(32*a^3 + 96*a^2*b + 224*a*b^2 + 61*b^3)*cosh(d*x + c)^7
+ 3003*(384*a^3 + 960*a^2*b + 3328*a*b^2 + 475*b^3)*cosh(d*x + c)^5 + 66*(4480*a^3 + 7360*a^2*b + 43008*a*b^2
+ 4515*b^3)*cosh(d*x + c)^3 + 3*(6720*a^3 + 3840*a^2*b + 67904*a*b^2 + 1295*b^3)*cosh(d*x + c))*sinh(d*x + c)
^9 + 3*(6720*a^3 - 3840*a^2*b + 67904*a*b^2 - 1295*b^3)*cosh(d*x + c)^8 + 3*(2333760*(a^3 + 3*a^2*b + 3*a*b^2
+ b^3)*cosh(d*x + c)^10 + 193050*(32*a^3 + 96*a^2*b + 224*a*b^2 + 61*b^3)*cosh(d*x + c)^8 + 15015*(384*a^3 + 9
60*a^2*b + 3328*a*b^2 + 475*b^3)*cosh(d*x + c)^6 + 495*(4480*a^3 + 7360*a^2*b + 43008*a*b^2 + 4515*b^3)*cosh(d
*x + c)^4 + 6720*a^3 - 3840*a^2*b + 67904*a*b^2 - 1295*b^3 + 45*(6720*a^3 + 3840*a^2*b + 67904*a*b^2 + 1295*b^
3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 24*(212160*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^11 + 21450*(32*
a^3 + 96*a^2*b + 224*a*b^2 + 61*b^3)*cosh(d*x + c)^9 + 2145*(384*a^3 + 960*a^2*b + 3328*a*b^2 + 475*b^3)*cosh(
d*x + c)^7 + 99*(4480*a^3 + 7360*a^2*b + 43008*a*b^2 + 4515*b^3)*cosh(d*x + c)^5 + 15*(6720*a^3 + 3840*a^2*b +
67904*a*b^2 + 1295*b^3)*cosh(d*x + c)^3 + (6720*a^3 - 3840*a^2*b + 67904*a*b^2 - 1295*b^3)*cosh(d*x + c))*sin
h(d*x + c)^7 + 3*(4480*a^3 - 7360*a^2*b + 43008*a*b^2 - 4515*b^3)*cosh(d*x + c)^6 + 3*(990080*(a^3 + 3*a^2*b +
3*a*b^2 + b^3)*cosh(d*x + c)^12 + 120120*(32*a^3 + 96*a^2*b + 224*a*b^2 + 61*b^3)*cosh(d*x + c)^10 + 15015*(3
84*a^3 + 960*a^2*b + 3328*a*b^2 + 475*b^3)*cosh(d*x + c)^8 + 924*(4480*a^3 + 7360*a^2*b + 43008*a*b^2 + 4515*b
^3)*cosh(d*x + c)^6 + 210*(6720*a^3 + 3840*a^2*b + 67904*a*b^2 + 1295*b^3)*cosh(d*x + c)^4 + 4480*a^3 - 7360*a
^2*b + 43008*a*b^2 - 4515*b^3 + 28*(6720*a^3 - 3840*a^2*b + 67904*a*b^2 - 1295*b^3)*cosh(d*x + c)^2)*sinh(d*x
+ c)^6 + 6*(228480*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^13 + 32760*(32*a^3 + 96*a^2*b + 224*a*b^2 + 6
1*b^3)*cosh(d*x + c)^11 + 5005*(384*a^3 + 960*a^2*b + 3328*a*b^2 + 475*b^3)*cosh(d*x + c)^9 + 396*(4480*a^3 +
7360*a^2*b + 43008*a*b^2 + 4515*b^3)*cosh(d*x + c)^7 + 126*(6720*a^3 + 3840*a^2*b + 67904*a*b^2 + 1295*b^3)*co
sh(d*x + c)^5 + 28*(6720*a^3 - 3840*a^2*b + 67904*a*b^2 - 1295*b^3)*cosh(d*x + c)^3 + 3*(4480*a^3 - 7360*a^2*b
+ 43008*a*b^2 - 4515*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 15*(384*a^3 - 960*a^2*b + 3328*a*b^2 - 475*b^3)*co
sh(d*x + c)^4 + 15*(32640*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^14 + 5460*(32*a^3 + 96*a^2*b + 224*a*b
^2 + 61*b^3)*cosh(d*x + c)^12 + 1001*(384*a^3 + 960*a^2*b + 3328*a*b^2 + 475*b^3)*cosh(d*x + c)^10 + 99*(4480*
a^3 + 7360*a^2*b + 43008*a*b^2 + 4515*b^3)*cosh(d*x + c)^8 + 42*(6720*a^3 + 3840*a^2*b + 67904*a*b^2 + 1295*b^
3)*cosh(d*x + c)^6 + 14*(6720*a^3 - 3840*a^2*b + 67904*a*b^2 - 1295*b^3)*cosh(d*x + c)^4 + 384*a^3 - 960*a^2*b
+ 3328*a*b^2 - 475*b^3 + 3*(4480*a^3 - 7360*a^2*b + 43008*a*b^2 - 4515*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4
+ 12*(10880*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^15 + 2100*(32*a^3 + 96*a^2*b + 224*a*b^2 + 61*b^3)*c
osh(d*x + c)^13 + 455*(384*a^3 + 960*a^2*b + 3328*a*b^2 + 475*b^3)*cosh(d*x + c)^11 + 55*(4480*a^3 + 7360*a^2*
b + 43008*a*b^2 + 4515*b^3)*cosh(d*x + c)^9 + 30*(6720*a^3 + 3840*a^2*b + 67904*a*b^2 + 1295*b^3)*cosh(d*x + c
)^7 + 14*(6720*a^3 - 3840*a^2*b + 67904*a*b^2 - 1295*b^3)*cosh(d*x + c)^5 + 5*(4480*a^3 - 7360*a^2*b + 43008*a
*b^2 - 4515*b^3)*cosh(d*x + c)^3 + 5*(384*a^3 - 960*a^2*b + 3328*a*b^2 - 475*b^3)*cosh(d*x + c))*sinh(d*x + c)
^3 + 160*a^3 - 480*a^2*b + 480*a*b^2 - 160*b^3 + 45*(32*a^3 - 96*a^2*b + 224*a*b^2 - 61*b^3)*cosh(d*x + c)^2 +
3*(8160*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^16 + 1800*(32*a^3 + 96*a^2*b + 224*a*b^2 + 61*b^3)*cosh
(d*x + c)^14 + 455*(384*a^3 + 960*a^2*b + 3328*a*b^2 + 475*b^3)*cosh(d*x + c)^12 + 66*(4480*a^3 + 7360*a^2*b +
43008*a*b^2 + 4515*b^3)*cosh(d*x + c)^10 + 45*(6720*a^3 + 3840*a^2*b + 67904*a*b^2 + 1295*b^3)*cosh(d*x + c)^
8 + 28*(6720*a^3 - 3840*a^2*b + 67904*a*b^2 - 1295*b^3)*cosh(d*x + c)^6 + 15*(4480*a^3 - 7360*a^2*b + 43008*a*
b^2 - 4515*b^3)*cosh(d*x + c)^4 + 480*a^3 - 1440*a^2*b + 3360*a*b^2 - 915*b^3 + 30*(384*a^3 - 960*a^2*b + 3328
*a*b^2 - 475*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 45*((64*a^2*b + 35*b^3)*cosh(d*x + c)^17 + 17*(64*a^2*b +
35*b^3)*cosh(d*x + c)*sinh(d*x + c)^16 + (64*a^2*b + 35*b^3)*sinh(d*x + c)^17 + 8*(64*a^2*b + 35*b^3)*cosh(d*
x + c)^15 + 8*(64*a^2*b + 35*b^3 + 17*(64*a^2*b + 35*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^15 + 40*(17*(64*a^2*b
+ 35*b^3)*cosh(d*x + c)^3 + 3*(64*a^2*b + 35*b^3)*cosh(d*x + c))*sinh(d*x + c)^14 + 28*(64*a^2*b + 35*b^3)*co
sh(d*x + c)^13 + 28*(85*(64*a^2*b + 35*b^3)*cosh(d*x + c)^4 + 64*a^2*b + 35*b^3 + 30*(64*a^2*b + 35*b^3)*cosh(
d*x + c)^2)*sinh(d*x + c)^13 + 364*(17*(64*a^2*b + 35*b^3)*cosh(d*x + c)^5 + 10*(64*a^2*b + 35*b^3)*cosh(d*x +
c)^3 + (64*a^2*b + 35*b^3)*cosh(d*x + c))*sinh(d*x + c)^12 + 56*(64*a^2*b + 35*b^3)*cosh(d*x + c)^11 + 56*(22
1*(64*a^2*b + 35*b^3)*cosh(d*x + c)^6 + 195*(64*a^2*b + 35*b^3)*cosh(d*x + c)^4 + 64*a^2*b + 35*b^3 + 39*(64*a
^2*b + 35*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^11 + 88*(221*(64*a^2*b + 35*b^3)*cosh(d*x + c)^7 + 273*(64*a^2*b
+ 35*b^3)*cosh(d*x + c)^5 + 91*(64*a^2*b + 35*b^3)*cosh(d*x + c)^3 + 7*(64*a^2*b + 35*b^3)*cosh(d*x + c))*sin
h(d*x + c)^10 + 70*(64*a^2*b + 35*b^3)*cosh(d*x + c)^9 + 10*(2431*(64*a^2*b + 35*b^3)*cosh(d*x + c)^8 + 4004*(
64*a^2*b + 35*b^3)*cosh(d*x + c)^6 + 2002*(64*a^2*b + 35*b^3)*cosh(d*x + c)^4 + 448*a^2*b + 245*b^3 + 308*(64*
a^2*b + 35*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^9 + 2*(12155*(64*a^2*b + 35*b^3)*cosh(d*x + c)^9 + 25740*(64*a^
2*b + 35*b^3)*cosh(d*x + c)^7 + 18018*(64*a^2*b + 35*b^3)*cosh(d*x + c)^5 + 4620*(64*a^2*b + 35*b^3)*cosh(d*x
+ c)^3 + 315*(64*a^2*b + 35*b^3)*cosh(d*x + c))*sinh(d*x + c)^8 + 56*(64*a^2*b + 35*b^3)*cosh(d*x + c)^7 + 8*(
2431*(64*a^2*b + 35*b^3)*cosh(d*x + c)^10 + 6435*(64*a^2*b + 35*b^3)*cosh(d*x + c)^8 + 6006*(64*a^2*b + 35*b^3
)*cosh(d*x + c)^6 + 2310*(64*a^2*b + 35*b^3)*cosh(d*x + c)^4 + 448*a^2*b + 245*b^3 + 315*(64*a^2*b + 35*b^3)*c
osh(d*x + c)^2)*sinh(d*x + c)^7 + 56*(221*(64*a^2*b + 35*b^3)*cosh(d*x + c)^11 + 715*(64*a^2*b + 35*b^3)*cosh(
d*x + c)^9 + 858*(64*a^2*b + 35*b^3)*cosh(d*x + c)^7 + 462*(64*a^2*b + 35*b^3)*cosh(d*x + c)^5 + 105*(64*a^2*b
+ 35*b^3)*cosh(d*x + c)^3 + 7*(64*a^2*b + 35*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 + 28*(64*a^2*b + 35*b^3)*cos
h(d*x + c)^5 + 28*(221*(64*a^2*b + 35*b^3)*cosh(d*x + c)^12 + 858*(64*a^2*b + 35*b^3)*cosh(d*x + c)^10 + 1287*
(64*a^2*b + 35*b^3)*cosh(d*x + c)^8 + 924*(64*a^2*b + 35*b^3)*cosh(d*x + c)^6 + 315*(64*a^2*b + 35*b^3)*cosh(d
*x + c)^4 + 64*a^2*b + 35*b^3 + 42*(64*a^2*b + 35*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 140*(17*(64*a^2*b +
35*b^3)*cosh(d*x + c)^13 + 78*(64*a^2*b + 35*b^3)*cosh(d*x + c)^11 + 143*(64*a^2*b + 35*b^3)*cosh(d*x + c)^9 +
132*(64*a^2*b + 35*b^3)*cosh(d*x + c)^7 + 63*(64*a^2*b + 35*b^3)*cosh(d*x + c)^5 + 14*(64*a^2*b + 35*b^3)*cos
h(d*x + c)^3 + (64*a^2*b + 35*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 8*(64*a^2*b + 35*b^3)*cosh(d*x + c)^3 + 8*
(85*(64*a^2*b + 35*b^3)*cosh(d*x + c)^14 + 455*(64*a^2*b + 35*b^3)*cosh(d*x + c)^12 + 1001*(64*a^2*b + 35*b^3)
*cosh(d*x + c)^10 + 1155*(64*a^2*b + 35*b^3)*cosh(d*x + c)^8 + 735*(64*a^2*b + 35*b^3)*cosh(d*x + c)^6 + 245*(
64*a^2*b + 35*b^3)*cosh(d*x + c)^4 + 64*a^2*b + 35*b^3 + 35*(64*a^2*b + 35*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)
^3 + 8*(17*(64*a^2*b + 35*b^3)*cosh(d*x + c)^15 + 105*(64*a^2*b + 35*b^3)*cosh(d*x + c)^13 + 273*(64*a^2*b + 3
5*b^3)*cosh(d*x + c)^11 + 385*(64*a^2*b + 35*b^3)*cosh(d*x + c)^9 + 315*(64*a^2*b + 35*b^3)*cosh(d*x + c)^7 +
147*(64*a^2*b + 35*b^3)*cosh(d*x + c)^5 + 35*(64*a^2*b + 35*b^3)*cosh(d*x + c)^3 + 3*(64*a^2*b + 35*b^3)*cosh(
d*x + c))*sinh(d*x + c)^2 + (64*a^2*b + 35*b^3)*cosh(d*x + c) + (17*(64*a^2*b + 35*b^3)*cosh(d*x + c)^16 + 120
*(64*a^2*b + 35*b^3)*cosh(d*x + c)^14 + 364*(64*a^2*b + 35*b^3)*cosh(d*x + c)^12 + 616*(64*a^2*b + 35*b^3)*cos
h(d*x + c)^10 + 630*(64*a^2*b + 35*b^3)*cosh(d*x + c)^8 + 392*(64*a^2*b + 35*b^3)*cosh(d*x + c)^6 + 140*(64*a^
2*b + 35*b^3)*cosh(d*x + c)^4 + 64*a^2*b + 35*b^3 + 24*(64*a^2*b + 35*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))*arc
tan(cosh(d*x + c) + sinh(d*x + c)) + 6*(480*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^17 + 120*(32*a^3 + 9
6*a^2*b + 224*a*b^2 + 61*b^3)*cosh(d*x + c)^15 + 35*(384*a^3 + 960*a^2*b + 3328*a*b^2 + 475*b^3)*cosh(d*x + c)
^13 + 6*(4480*a^3 + 7360*a^2*b + 43008*a*b^2 + 4515*b^3)*cosh(d*x + c)^11 + 5*(6720*a^3 + 3840*a^2*b + 67904*a
*b^2 + 1295*b^3)*cosh(d*x + c)^9 + 4*(6720*a^3 - 3840*a^2*b + 67904*a*b^2 - 1295*b^3)*cosh(d*x + c)^7 + 3*(448
0*a^3 - 7360*a^2*b + 43008*a*b^2 - 4515*b^3)*cosh(d*x + c)^5 + 10*(384*a^3 - 960*a^2*b + 3328*a*b^2 - 475*b^3)
*cosh(d*x + c)^3 + 15*(32*a^3 - 96*a^2*b + 224*a*b^2 - 61*b^3)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^
17 + 17*d*cosh(d*x + c)*sinh(d*x + c)^16 + d*sinh(d*x + c)^17 + 8*d*cosh(d*x + c)^15 + 8*(17*d*cosh(d*x + c)^2
+ d)*sinh(d*x + c)^15 + 40*(17*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^14 + 28*d*cosh(d*x + c)^1
3 + 28*(85*d*cosh(d*x + c)^4 + 30*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^13 + 364*(17*d*cosh(d*x + c)^5 + 10*d*c
osh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c)^12 + 56*d*cosh(d*x + c)^11 + 56*(221*d*cosh(d*x + c)^6 + 195*d
*cosh(d*x + c)^4 + 39*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^11 + 88*(221*d*cosh(d*x + c)^7 + 273*d*cosh(d*x + c
)^5 + 91*d*cosh(d*x + c)^3 + 7*d*cosh(d*x + c))*sinh(d*x + c)^10 + 70*d*cosh(d*x + c)^9 + 10*(2431*d*cosh(d*x
+ c)^8 + 4004*d*cosh(d*x + c)^6 + 2002*d*cosh(d*x + c)^4 + 308*d*cosh(d*x + c)^2 + 7*d)*sinh(d*x + c)^9 + 2*(1
2155*d*cosh(d*x + c)^9 + 25740*d*cosh(d*x + c)^7 + 18018*d*cosh(d*x + c)^5 + 4620*d*cosh(d*x + c)^3 + 315*d*co
sh(d*x + c))*sinh(d*x + c)^8 + 56*d*cosh(d*x + c)^7 + 8*(2431*d*cosh(d*x + c)^10 + 6435*d*cosh(d*x + c)^8 + 60
06*d*cosh(d*x + c)^6 + 2310*d*cosh(d*x + c)^4 + 315*d*cosh(d*x + c)^2 + 7*d)*sinh(d*x + c)^7 + 56*(221*d*cosh(
d*x + c)^11 + 715*d*cosh(d*x + c)^9 + 858*d*cosh(d*x + c)^7 + 462*d*cosh(d*x + c)^5 + 105*d*cosh(d*x + c)^3 +
7*d*cosh(d*x + c))*sinh(d*x + c)^6 + 28*d*cosh(d*x + c)^5 + 28*(221*d*cosh(d*x + c)^12 + 858*d*cosh(d*x + c)^1
0 + 1287*d*cosh(d*x + c)^8 + 924*d*cosh(d*x + c)^6 + 315*d*cosh(d*x + c)^4 + 42*d*cosh(d*x + c)^2 + d)*sinh(d*
x + c)^5 + 140*(17*d*cosh(d*x + c)^13 + 78*d*cosh(d*x + c)^11 + 143*d*cosh(d*x + c)^9 + 132*d*cosh(d*x + c)^7
+ 63*d*cosh(d*x + c)^5 + 14*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c)^4 + 8*d*cosh(d*x + c)^3 + 8*(85
*d*cosh(d*x + c)^14 + 455*d*cosh(d*x + c)^12 + 1001*d*cosh(d*x + c)^10 + 1155*d*cosh(d*x + c)^8 + 735*d*cosh(d
*x + c)^6 + 245*d*cosh(d*x + c)^4 + 35*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^3 + 8*(17*d*cosh(d*x + c)^15 + 105
*d*cosh(d*x + c)^13 + 273*d*cosh(d*x + c)^11 + 385*d*cosh(d*x + c)^9 + 315*d*cosh(d*x + c)^7 + 147*d*cosh(d*x
+ c)^5 + 35*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^2 + d*cosh(d*x + c) + (17*d*cosh(d*x + c)^16
+ 120*d*cosh(d*x + c)^14 + 364*d*cosh(d*x + c)^12 + 616*d*cosh(d*x + c)^10 + 630*d*cosh(d*x + c)^8 + 392*d*cos
h(d*x + c)^6 + 140*d*cosh(d*x + c)^4 + 24*d*cosh(d*x + c)^2 + d)*sinh(d*x + c))

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giac [A]  time = 0.68, size = 485, normalized size = 1.80 \[ -\frac {45 \, {\left (64 \, a^{2} b e^{c} + 35 \, b^{3} e^{c}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) e^{\left (-c\right )} - 160 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} e^{\left (-d x - c\right )} - 160 \, {\left (a^{3} e^{\left (d x + 20 \, c\right )} + 3 \, a^{2} b e^{\left (d x + 20 \, c\right )} + 3 \, a b^{2} e^{\left (d x + 20 \, c\right )} + b^{3} e^{\left (d x + 20 \, c\right )}\right )} e^{\left (-19 \, c\right )} - \frac {960 \, a^{2} b e^{\left (15 \, d x + 15 \, c\right )} + 5760 \, a b^{2} e^{\left (15 \, d x + 15 \, c\right )} + 1625 \, b^{3} e^{\left (15 \, d x + 15 \, c\right )} + 4800 \, a^{2} b e^{\left (13 \, d x + 13 \, c\right )} + 32640 \, a b^{2} e^{\left (13 \, d x + 13 \, c\right )} + 3925 \, b^{3} e^{\left (13 \, d x + 13 \, c\right )} + 8640 \, a^{2} b e^{\left (11 \, d x + 11 \, c\right )} + 88704 \, a b^{2} e^{\left (11 \, d x + 11 \, c\right )} + 9065 \, b^{3} e^{\left (11 \, d x + 11 \, c\right )} + 4800 \, a^{2} b e^{\left (9 \, d x + 9 \, c\right )} + 143232 \, a b^{2} e^{\left (9 \, d x + 9 \, c\right )} + 1645 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )} - 4800 \, a^{2} b e^{\left (7 \, d x + 7 \, c\right )} + 143232 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} - 1645 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} - 8640 \, a^{2} b e^{\left (5 \, d x + 5 \, c\right )} + 88704 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 9065 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} - 4800 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} + 32640 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 3925 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 960 \, a^{2} b e^{\left (d x + c\right )} + 5760 \, a b^{2} e^{\left (d x + c\right )} - 1625 \, b^{3} e^{\left (d x + c\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{8}}}{320 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/320*(45*(64*a^2*b*e^c + 35*b^3*e^c)*arctan(e^(d*x + c))*e^(-c) - 160*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*e^(-d*
x - c) - 160*(a^3*e^(d*x + 20*c) + 3*a^2*b*e^(d*x + 20*c) + 3*a*b^2*e^(d*x + 20*c) + b^3*e^(d*x + 20*c))*e^(-1
9*c) - (960*a^2*b*e^(15*d*x + 15*c) + 5760*a*b^2*e^(15*d*x + 15*c) + 1625*b^3*e^(15*d*x + 15*c) + 4800*a^2*b*e
^(13*d*x + 13*c) + 32640*a*b^2*e^(13*d*x + 13*c) + 3925*b^3*e^(13*d*x + 13*c) + 8640*a^2*b*e^(11*d*x + 11*c) +
88704*a*b^2*e^(11*d*x + 11*c) + 9065*b^3*e^(11*d*x + 11*c) + 4800*a^2*b*e^(9*d*x + 9*c) + 143232*a*b^2*e^(9*d
*x + 9*c) + 1645*b^3*e^(9*d*x + 9*c) - 4800*a^2*b*e^(7*d*x + 7*c) + 143232*a*b^2*e^(7*d*x + 7*c) - 1645*b^3*e^
(7*d*x + 7*c) - 8640*a^2*b*e^(5*d*x + 5*c) + 88704*a*b^2*e^(5*d*x + 5*c) - 9065*b^3*e^(5*d*x + 5*c) - 4800*a^2
*b*e^(3*d*x + 3*c) + 32640*a*b^2*e^(3*d*x + 3*c) - 3925*b^3*e^(3*d*x + 3*c) - 960*a^2*b*e^(d*x + c) + 5760*a*b
^2*e^(d*x + c) - 1625*b^3*e^(d*x + c))/(e^(2*d*x + 2*c) + 1)^8)/d

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maple [A]  time = 0.43, size = 410, normalized size = 1.52 \[ \frac {a^{3} \cosh \left (d x +c \right )}{d}+\frac {3 a^{2} b \left (\sinh ^{3}\left (d x +c \right )\right )}{d \cosh \left (d x +c \right )^{2}}+\frac {9 a^{2} b \sinh \left (d x +c \right )}{d \cosh \left (d x +c \right )^{2}}-\frac {9 a^{2} b \,\mathrm {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2 d}-\frac {9 a^{2} b \arctan \left ({\mathrm e}^{d x +c}\right )}{d}+\frac {3 a \,b^{2} \left (\sinh ^{6}\left (d x +c \right )\right )}{d \cosh \left (d x +c \right )^{5}}+\frac {18 a \,b^{2} \left (\sinh ^{4}\left (d x +c \right )\right )}{d \cosh \left (d x +c \right )^{5}}+\frac {24 a \,b^{2} \left (\sinh ^{2}\left (d x +c \right )\right )}{d \cosh \left (d x +c \right )^{5}}+\frac {48 a \,b^{2}}{5 d \cosh \left (d x +c \right )^{5}}+\frac {b^{3} \left (\sinh ^{9}\left (d x +c \right )\right )}{d \cosh \left (d x +c \right )^{8}}+\frac {9 b^{3} \left (\sinh ^{7}\left (d x +c \right )\right )}{d \cosh \left (d x +c \right )^{8}}+\frac {21 b^{3} \left (\sinh ^{5}\left (d x +c \right )\right )}{d \cosh \left (d x +c \right )^{8}}+\frac {21 b^{3} \left (\sinh ^{3}\left (d x +c \right )\right )}{d \cosh \left (d x +c \right )^{8}}+\frac {9 b^{3} \sinh \left (d x +c \right )}{d \cosh \left (d x +c \right )^{8}}-\frac {9 b^{3} \tanh \left (d x +c \right ) \mathrm {sech}\left (d x +c \right )^{7}}{8 d}-\frac {21 b^{3} \tanh \left (d x +c \right ) \mathrm {sech}\left (d x +c \right )^{5}}{16 d}-\frac {105 b^{3} \tanh \left (d x +c \right ) \mathrm {sech}\left (d x +c \right )^{3}}{64 d}-\frac {315 b^{3} \mathrm {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{128 d}-\frac {315 b^{3} \arctan \left ({\mathrm e}^{d x +c}\right )}{64 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3*cosh(d*x+c)/d+3/d*a^2*b*sinh(d*x+c)^3/cosh(d*x+c)^2+9/d*a^2*b*sinh(d*x+c)/cosh(d*x+c)^2-9/2/d*a^2*b*sech(d
*x+c)*tanh(d*x+c)-9/d*a^2*b*arctan(exp(d*x+c))+3/d*a*b^2*sinh(d*x+c)^6/cosh(d*x+c)^5+18/d*a*b^2*sinh(d*x+c)^4/
cosh(d*x+c)^5+24/d*a*b^2*sinh(d*x+c)^2/cosh(d*x+c)^5+48/5/d*a*b^2/cosh(d*x+c)^5+1/d*b^3*sinh(d*x+c)^9/cosh(d*x
+c)^8+9/d*b^3*sinh(d*x+c)^7/cosh(d*x+c)^8+21/d*b^3*sinh(d*x+c)^5/cosh(d*x+c)^8+21/d*b^3*sinh(d*x+c)^3/cosh(d*x
+c)^8+9/d*b^3*sinh(d*x+c)/cosh(d*x+c)^8-9/8/d*b^3*tanh(d*x+c)*sech(d*x+c)^7-21/16/d*b^3*tanh(d*x+c)*sech(d*x+c
)^5-105/64/d*b^3*tanh(d*x+c)*sech(d*x+c)^3-315/128/d*b^3*sech(d*x+c)*tanh(d*x+c)-315/64/d*b^3*arctan(exp(d*x+c
))

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maxima [A]  time = 0.42, size = 484, normalized size = 1.80 \[ \frac {1}{64} \, b^{3} {\left (\frac {315 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {32 \, e^{\left (-d x - c\right )}}{d} + \frac {581 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1681 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3605 \, e^{\left (-6 \, d x - 6 \, c\right )} + 2569 \, e^{\left (-8 \, d x - 8 \, c\right )} + 1463 \, e^{\left (-10 \, d x - 10 \, c\right )} - 917 \, e^{\left (-12 \, d x - 12 \, c\right )} - 529 \, e^{\left (-14 \, d x - 14 \, c\right )} - 293 \, e^{\left (-16 \, d x - 16 \, c\right )} + 32}{d {\left (e^{\left (-d x - c\right )} + 8 \, e^{\left (-3 \, d x - 3 \, c\right )} + 28 \, e^{\left (-5 \, d x - 5 \, c\right )} + 56 \, e^{\left (-7 \, d x - 7 \, c\right )} + 70 \, e^{\left (-9 \, d x - 9 \, c\right )} + 56 \, e^{\left (-11 \, d x - 11 \, c\right )} + 28 \, e^{\left (-13 \, d x - 13 \, c\right )} + 8 \, e^{\left (-15 \, d x - 15 \, c\right )} + e^{\left (-17 \, d x - 17 \, c\right )}\right )}}\right )} + \frac {3}{2} \, a^{2} b {\left (\frac {6 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )}}{d} + \frac {4 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + 1}{d {\left (e^{\left (-d x - c\right )} + 2 \, e^{\left (-3 \, d x - 3 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )}\right )}}\right )} + \frac {3}{10} \, a b^{2} {\left (\frac {5 \, e^{\left (-d x - c\right )}}{d} + \frac {85 \, e^{\left (-2 \, d x - 2 \, c\right )} + 210 \, e^{\left (-4 \, d x - 4 \, c\right )} + 314 \, e^{\left (-6 \, d x - 6 \, c\right )} + 185 \, e^{\left (-8 \, d x - 8 \, c\right )} + 65 \, e^{\left (-10 \, d x - 10 \, c\right )} + 5}{d {\left (e^{\left (-d x - c\right )} + 5 \, e^{\left (-3 \, d x - 3 \, c\right )} + 10 \, e^{\left (-5 \, d x - 5 \, c\right )} + 10 \, e^{\left (-7 \, d x - 7 \, c\right )} + 5 \, e^{\left (-9 \, d x - 9 \, c\right )} + e^{\left (-11 \, d x - 11 \, c\right )}\right )}}\right )} + \frac {a^{3} \cosh \left (d x + c\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*b^3*(315*arctan(e^(-d*x - c))/d - 32*e^(-d*x - c)/d + (581*e^(-2*d*x - 2*c) + 1681*e^(-4*d*x - 4*c) + 360
5*e^(-6*d*x - 6*c) + 2569*e^(-8*d*x - 8*c) + 1463*e^(-10*d*x - 10*c) - 917*e^(-12*d*x - 12*c) - 529*e^(-14*d*x
- 14*c) - 293*e^(-16*d*x - 16*c) + 32)/(d*(e^(-d*x - c) + 8*e^(-3*d*x - 3*c) + 28*e^(-5*d*x - 5*c) + 56*e^(-7
*d*x - 7*c) + 70*e^(-9*d*x - 9*c) + 56*e^(-11*d*x - 11*c) + 28*e^(-13*d*x - 13*c) + 8*e^(-15*d*x - 15*c) + e^(
-17*d*x - 17*c)))) + 3/2*a^2*b*(6*arctan(e^(-d*x - c))/d - e^(-d*x - c)/d + (4*e^(-2*d*x - 2*c) - e^(-4*d*x -
4*c) + 1)/(d*(e^(-d*x - c) + 2*e^(-3*d*x - 3*c) + e^(-5*d*x - 5*c)))) + 3/10*a*b^2*(5*e^(-d*x - c)/d + (85*e^(
-2*d*x - 2*c) + 210*e^(-4*d*x - 4*c) + 314*e^(-6*d*x - 6*c) + 185*e^(-8*d*x - 8*c) + 65*e^(-10*d*x - 10*c) + 5
)/(d*(e^(-d*x - c) + 5*e^(-3*d*x - 3*c) + 10*e^(-5*d*x - 5*c) + 10*e^(-7*d*x - 7*c) + 5*e^(-9*d*x - 9*c) + e^(
-11*d*x - 11*c)))) + a^3*cosh(d*x + c)/d

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mupad [B]  time = 1.49, size = 707, normalized size = 2.63 \[ \frac {{\mathrm {e}}^{c+d\,x}\,{\left (a+b\right )}^3}{2\,d}+\frac {{\mathrm {e}}^{-c-d\,x}\,{\left (a-b\right )}^3}{2\,d}-\frac {9\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (35\,b^3\,\sqrt {d^2}+64\,a^2\,b\,\sqrt {d^2}\right )}{d\,\sqrt {4096\,a^4\,b^2+4480\,a^2\,b^4+1225\,b^6}}\right )\,\sqrt {4096\,a^4\,b^2+4480\,a^2\,b^4+1225\,b^6}}{64\,\sqrt {d^2}}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (2455\,b^3+1728\,a\,b^2\right )}{40\,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 {{\mathrm {e}}^{c+d\,x}\,\left (2605\,b^3+768\,a\,b^2\right )}{20\,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 {188\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (6\,{\mathrm {e}}^{2\,c+2\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,d\,x}+20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}+6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (192\,a^2\,b+1152\,a\,b^2+325\,b^3\right )}{64\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (475\,b^3+48\,a\,b^2\right )}{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 {112\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (7\,{\mathrm {e}}^{2\,c+2\,d\,x}+21\,{\mathrm {e}}^{4\,c+4\,d\,x}+35\,{\mathrm {e}}^{6\,c+6\,d\,x}+35\,{\mathrm {e}}^{8\,c+8\,d\,x}+21\,{\mathrm {e}}^{10\,c+10\,d\,x}+7\,{\mathrm {e}}^{12\,c+12\,d\,x}+{\mathrm {e}}^{14\,c+14\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (192\,a^2\,b+768\,a\,b^2+745\,b^3\right )}{32\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {32\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (8\,{\mathrm {e}}^{2\,c+2\,d\,x}+28\,{\mathrm {e}}^{4\,c+4\,d\,x}+56\,{\mathrm {e}}^{6\,c+6\,d\,x}+70\,{\mathrm {e}}^{8\,c+8\,d\,x}+56\,{\mathrm {e}}^{10\,c+10\,d\,x}+28\,{\mathrm {e}}^{12\,c+12\,d\,x}+8\,{\mathrm {e}}^{14\,c+14\,d\,x}+{\mathrm {e}}^{16\,c+16\,d\,x}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(exp(c + d*x)*(a + b)^3)/(2*d) + (exp(- c - d*x)*(a - b)^3)/(2*d) - (9*atan((exp(d*x)*exp(c)*(35*b^3*(d^2)^(1/
2) + 64*a^2*b*(d^2)^(1/2)))/(d*(1225*b^6 + 4480*a^2*b^4 + 4096*a^4*b^2)^(1/2)))*(1225*b^6 + 4480*a^2*b^4 + 409
6*a^4*b^2)^(1/2))/(64*(d^2)^(1/2)) + (exp(c + d*x)*(1728*a*b^2 + 2455*b^3))/(40*d*(3*exp(2*c + 2*d*x) + 3*exp(
4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) - (exp(c + d*x)*(768*a*b^2 + 2605*b^3))/(20*d*(4*exp(2*c + 2*d*x) + 6*ex
p(4*c + 4*d*x) + 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1)) - (188*b^3*exp(c + d*x))/(d*(6*exp(2*c + 2*d*x) +
15*exp(4*c + 4*d*x) + 20*exp(6*c + 6*d*x) + 15*exp(8*c + 8*d*x) + 6*exp(10*c + 10*d*x) + exp(12*c + 12*d*x) +
1)) + (exp(c + d*x)*(1152*a*b^2 + 192*a^2*b + 325*b^3))/(64*d*(exp(2*c + 2*d*x) + 1)) + (2*exp(c + d*x)*(48*a
*b^2 + 475*b^3))/(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) + e
xp(10*c + 10*d*x) + 1)) + (112*b^3*exp(c + d*x))/(d*(7*exp(2*c + 2*d*x) + 21*exp(4*c + 4*d*x) + 35*exp(6*c + 6
*d*x) + 35*exp(8*c + 8*d*x) + 21*exp(10*c + 10*d*x) + 7*exp(12*c + 12*d*x) + exp(14*c + 14*d*x) + 1)) - (exp(c
+ d*x)*(768*a*b^2 + 192*a^2*b + 745*b^3))/(32*d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) - (32*b^3*exp(c
+ d*x))/(d*(8*exp(2*c + 2*d*x) + 28*exp(4*c + 4*d*x) + 56*exp(6*c + 6*d*x) + 70*exp(8*c + 8*d*x) + 56*exp(10*c
+ 10*d*x) + 28*exp(12*c + 12*d*x) + 8*exp(14*c + 14*d*x) + exp(16*c + 16*d*x) + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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