### 3.66 $$\int \sinh ^3(c+d x) (a+b \tanh ^3(c+d x))^3 \, dx$$

Optimal. Leaf size=351 $\frac{5 a^2 b \sinh ^3(c+d x)}{2 d}-\frac{15 a^2 b \sinh (c+d x)}{2 d}-\frac{3 a^2 b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}+\frac{15 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{a^3 \cosh ^3(c+d x)}{3 d}-\frac{a^3 \cosh (c+d x)}{d}+\frac{a b^2 \cosh ^3(c+d x)}{d}-\frac{12 a b^2 \cosh (c+d x)}{d}-\frac{3 a b^2 \text{sech}^5(c+d x)}{5 d}+\frac{4 a b^2 \text{sech}^3(c+d x)}{d}-\frac{18 a b^2 \text{sech}(c+d x)}{d}+\frac{385 b^3 \sinh ^3(c+d x)}{128 d}-\frac{1155 b^3 \sinh (c+d x)}{128 d}-\frac{b^3 \sinh ^3(c+d x) \tanh ^8(c+d x)}{8 d}-\frac{11 b^3 \sinh ^3(c+d x) \tanh ^6(c+d x)}{48 d}-\frac{33 b^3 \sinh ^3(c+d x) \tanh ^4(c+d x)}{64 d}-\frac{231 b^3 \sinh ^3(c+d x) \tanh ^2(c+d x)}{128 d}+\frac{1155 b^3 \tan ^{-1}(\sinh (c+d x))}{128 d}$

[Out]

(15*a^2*b*ArcTan[Sinh[c + d*x]])/(2*d) + (1155*b^3*ArcTan[Sinh[c + d*x]])/(128*d) - (a^3*Cosh[c + d*x])/d - (1
2*a*b^2*Cosh[c + d*x])/d + (a^3*Cosh[c + d*x]^3)/(3*d) + (a*b^2*Cosh[c + d*x]^3)/d - (18*a*b^2*Sech[c + d*x])/
d + (4*a*b^2*Sech[c + d*x]^3)/d - (3*a*b^2*Sech[c + d*x]^5)/(5*d) - (15*a^2*b*Sinh[c + d*x])/(2*d) - (1155*b^3
*Sinh[c + d*x])/(128*d) + (5*a^2*b*Sinh[c + d*x]^3)/(2*d) + (385*b^3*Sinh[c + d*x]^3)/(128*d) - (3*a^2*b*Sinh[
c + d*x]^3*Tanh[c + d*x]^2)/(2*d) - (231*b^3*Sinh[c + d*x]^3*Tanh[c + d*x]^2)/(128*d) - (33*b^3*Sinh[c + d*x]^
3*Tanh[c + d*x]^4)/(64*d) - (11*b^3*Sinh[c + d*x]^3*Tanh[c + d*x]^6)/(48*d) - (b^3*Sinh[c + d*x]^3*Tanh[c + d*
x]^8)/(8*d)

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Rubi [A]  time = 0.3499, antiderivative size = 351, normalized size of antiderivative = 1., number of steps used = 20, 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} $\frac{5 a^2 b \sinh ^3(c+d x)}{2 d}-\frac{15 a^2 b \sinh (c+d x)}{2 d}-\frac{3 a^2 b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}+\frac{15 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{a^3 \cosh ^3(c+d x)}{3 d}-\frac{a^3 \cosh (c+d x)}{d}+\frac{a b^2 \cosh ^3(c+d x)}{d}-\frac{12 a b^2 \cosh (c+d x)}{d}-\frac{3 a b^2 \text{sech}^5(c+d x)}{5 d}+\frac{4 a b^2 \text{sech}^3(c+d x)}{d}-\frac{18 a b^2 \text{sech}(c+d x)}{d}+\frac{385 b^3 \sinh ^3(c+d x)}{128 d}-\frac{1155 b^3 \sinh (c+d x)}{128 d}-\frac{b^3 \sinh ^3(c+d x) \tanh ^8(c+d x)}{8 d}-\frac{11 b^3 \sinh ^3(c+d x) \tanh ^6(c+d x)}{48 d}-\frac{33 b^3 \sinh ^3(c+d x) \tanh ^4(c+d x)}{64 d}-\frac{231 b^3 \sinh ^3(c+d x) \tanh ^2(c+d x)}{128 d}+\frac{1155 b^3 \tan ^{-1}(\sinh (c+d x))}{128 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(15*a^2*b*ArcTan[Sinh[c + d*x]])/(2*d) + (1155*b^3*ArcTan[Sinh[c + d*x]])/(128*d) - (a^3*Cosh[c + d*x])/d - (1
2*a*b^2*Cosh[c + d*x])/d + (a^3*Cosh[c + d*x]^3)/(3*d) + (a*b^2*Cosh[c + d*x]^3)/d - (18*a*b^2*Sech[c + d*x])/
d + (4*a*b^2*Sech[c + d*x]^3)/d - (3*a*b^2*Sech[c + d*x]^5)/(5*d) - (15*a^2*b*Sinh[c + d*x])/(2*d) - (1155*b^3
*Sinh[c + d*x])/(128*d) + (5*a^2*b*Sinh[c + d*x]^3)/(2*d) + (385*b^3*Sinh[c + d*x]^3)/(128*d) - (3*a^2*b*Sinh[
c + d*x]^3*Tanh[c + d*x]^2)/(2*d) - (231*b^3*Sinh[c + d*x]^3*Tanh[c + d*x]^2)/(128*d) - (33*b^3*Sinh[c + d*x]^
3*Tanh[c + d*x]^4)/(64*d) - (11*b^3*Sinh[c + d*x]^3*Tanh[c + d*x]^6)/(48*d) - (b^3*Sinh[c + d*x]^3*Tanh[c + d*
x]^8)/(8*d)

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]

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 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 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 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 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 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]

Rubi steps

\begin{align*} \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx &=i \int \left (-i a^3 \sinh ^3(c+d x)-3 i a^2 b \sinh ^3(c+d x) \tanh ^3(c+d x)-3 i a b^2 \sinh ^3(c+d x) \tanh ^6(c+d x)-i b^3 \sinh ^3(c+d x) \tanh ^9(c+d x)\right ) \, dx\\ &=a^3 \int \sinh ^3(c+d x) \, dx+\left (3 a^2 b\right ) \int \sinh ^3(c+d x) \tanh ^3(c+d x) \, dx+\left (3 a b^2\right ) \int \sinh ^3(c+d x) \tanh ^6(c+d x) \, dx+b^3 \int \sinh ^3(c+d x) \tanh ^9(c+d x) \, dx\\ &=-\frac{a^3 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}+\frac{\left (3 a^2 b\right ) \operatorname{Subst}\left (\int \frac{x^6}{\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 )^4}{x^6} \, dx,x,\cosh (c+d x)\right )}{d}+\frac{b^3 \operatorname{Subst}\left (\int \frac{x^{12}}{\left (1+x^2\right )^5} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=-\frac{a^3 \cosh (c+d x)}{d}+\frac{a^3 \cosh ^3(c+d x)}{3 d}-\frac{3 a^2 b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}-\frac{b^3 \sinh ^3(c+d x) \tanh ^8(c+d x)}{8 d}+\frac{\left (15 a^2 b\right ) \operatorname{Subst}\left (\int \frac{x^4}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 d}+\frac{\left (3 a b^2\right ) \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{\left (11 b^3\right ) \operatorname{Subst}\left (\int \frac{x^{10}}{\left (1+x^2\right )^4} \, dx,x,\sinh (c+d x)\right )}{8 d}\\ &=-\frac{a^3 \cosh (c+d x)}{d}-\frac{12 a b^2 \cosh (c+d x)}{d}+\frac{a^3 \cosh ^3(c+d x)}{3 d}+\frac{a b^2 \cosh ^3(c+d x)}{d}-\frac{18 a b^2 \text{sech}(c+d x)}{d}+\frac{4 a b^2 \text{sech}^3(c+d x)}{d}-\frac{3 a b^2 \text{sech}^5(c+d x)}{5 d}-\frac{3 a^2 b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}-\frac{11 b^3 \sinh ^3(c+d x) \tanh ^6(c+d x)}{48 d}-\frac{b^3 \sinh ^3(c+d x) \tanh ^8(c+d x)}{8 d}+\frac{\left (15 a^2 b\right ) \operatorname{Subst}\left (\int \left (-1+x^2+\frac{1}{1+x^2}\right ) \, dx,x,\sinh (c+d x)\right )}{2 d}+\frac{\left (33 b^3\right ) \operatorname{Subst}\left (\int \frac{x^8}{\left (1+x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{16 d}\\ &=-\frac{a^3 \cosh (c+d x)}{d}-\frac{12 a b^2 \cosh (c+d x)}{d}+\frac{a^3 \cosh ^3(c+d x)}{3 d}+\frac{a b^2 \cosh ^3(c+d x)}{d}-\frac{18 a b^2 \text{sech}(c+d x)}{d}+\frac{4 a b^2 \text{sech}^3(c+d x)}{d}-\frac{3 a b^2 \text{sech}^5(c+d x)}{5 d}-\frac{15 a^2 b \sinh (c+d x)}{2 d}+\frac{5 a^2 b \sinh ^3(c+d x)}{2 d}-\frac{3 a^2 b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}-\frac{33 b^3 \sinh ^3(c+d x) \tanh ^4(c+d x)}{64 d}-\frac{11 b^3 \sinh ^3(c+d x) \tanh ^6(c+d x)}{48 d}-\frac{b^3 \sinh ^3(c+d x) \tanh ^8(c+d x)}{8 d}+\frac{\left (15 a^2 b\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 d}+\frac{\left (231 b^3\right ) \operatorname{Subst}\left (\int \frac{x^6}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{64 d}\\ &=\frac{15 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac{a^3 \cosh (c+d x)}{d}-\frac{12 a b^2 \cosh (c+d x)}{d}+\frac{a^3 \cosh ^3(c+d x)}{3 d}+\frac{a b^2 \cosh ^3(c+d x)}{d}-\frac{18 a b^2 \text{sech}(c+d x)}{d}+\frac{4 a b^2 \text{sech}^3(c+d x)}{d}-\frac{3 a b^2 \text{sech}^5(c+d x)}{5 d}-\frac{15 a^2 b \sinh (c+d x)}{2 d}+\frac{5 a^2 b \sinh ^3(c+d x)}{2 d}-\frac{3 a^2 b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}-\frac{231 b^3 \sinh ^3(c+d x) \tanh ^2(c+d x)}{128 d}-\frac{33 b^3 \sinh ^3(c+d x) \tanh ^4(c+d x)}{64 d}-\frac{11 b^3 \sinh ^3(c+d x) \tanh ^6(c+d x)}{48 d}-\frac{b^3 \sinh ^3(c+d x) \tanh ^8(c+d x)}{8 d}+\frac{\left (1155 b^3\right ) \operatorname{Subst}\left (\int \frac{x^4}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{128 d}\\ &=\frac{15 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac{a^3 \cosh (c+d x)}{d}-\frac{12 a b^2 \cosh (c+d x)}{d}+\frac{a^3 \cosh ^3(c+d x)}{3 d}+\frac{a b^2 \cosh ^3(c+d x)}{d}-\frac{18 a b^2 \text{sech}(c+d x)}{d}+\frac{4 a b^2 \text{sech}^3(c+d x)}{d}-\frac{3 a b^2 \text{sech}^5(c+d x)}{5 d}-\frac{15 a^2 b \sinh (c+d x)}{2 d}+\frac{5 a^2 b \sinh ^3(c+d x)}{2 d}-\frac{3 a^2 b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}-\frac{231 b^3 \sinh ^3(c+d x) \tanh ^2(c+d x)}{128 d}-\frac{33 b^3 \sinh ^3(c+d x) \tanh ^4(c+d x)}{64 d}-\frac{11 b^3 \sinh ^3(c+d x) \tanh ^6(c+d x)}{48 d}-\frac{b^3 \sinh ^3(c+d x) \tanh ^8(c+d x)}{8 d}+\frac{\left (1155 b^3\right ) \operatorname{Subst}\left (\int \left (-1+x^2+\frac{1}{1+x^2}\right ) \, dx,x,\sinh (c+d x)\right )}{128 d}\\ &=\frac{15 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac{a^3 \cosh (c+d x)}{d}-\frac{12 a b^2 \cosh (c+d x)}{d}+\frac{a^3 \cosh ^3(c+d x)}{3 d}+\frac{a b^2 \cosh ^3(c+d x)}{d}-\frac{18 a b^2 \text{sech}(c+d x)}{d}+\frac{4 a b^2 \text{sech}^3(c+d x)}{d}-\frac{3 a b^2 \text{sech}^5(c+d x)}{5 d}-\frac{15 a^2 b \sinh (c+d x)}{2 d}-\frac{1155 b^3 \sinh (c+d x)}{128 d}+\frac{5 a^2 b \sinh ^3(c+d x)}{2 d}+\frac{385 b^3 \sinh ^3(c+d x)}{128 d}-\frac{3 a^2 b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}-\frac{231 b^3 \sinh ^3(c+d x) \tanh ^2(c+d x)}{128 d}-\frac{33 b^3 \sinh ^3(c+d x) \tanh ^4(c+d x)}{64 d}-\frac{11 b^3 \sinh ^3(c+d x) \tanh ^6(c+d x)}{48 d}-\frac{b^3 \sinh ^3(c+d x) \tanh ^8(c+d x)}{8 d}+\frac{\left (1155 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{128 d}\\ &=\frac{15 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{1155 b^3 \tan ^{-1}(\sinh (c+d x))}{128 d}-\frac{a^3 \cosh (c+d x)}{d}-\frac{12 a b^2 \cosh (c+d x)}{d}+\frac{a^3 \cosh ^3(c+d x)}{3 d}+\frac{a b^2 \cosh ^3(c+d x)}{d}-\frac{18 a b^2 \text{sech}(c+d x)}{d}+\frac{4 a b^2 \text{sech}^3(c+d x)}{d}-\frac{3 a b^2 \text{sech}^5(c+d x)}{5 d}-\frac{15 a^2 b \sinh (c+d x)}{2 d}-\frac{1155 b^3 \sinh (c+d x)}{128 d}+\frac{5 a^2 b \sinh ^3(c+d x)}{2 d}+\frac{385 b^3 \sinh ^3(c+d x)}{128 d}-\frac{3 a^2 b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}-\frac{231 b^3 \sinh ^3(c+d x) \tanh ^2(c+d x)}{128 d}-\frac{33 b^3 \sinh ^3(c+d x) \tanh ^4(c+d x)}{64 d}-\frac{11 b^3 \sinh ^3(c+d x) \tanh ^6(c+d x)}{48 d}-\frac{b^3 \sinh ^3(c+d x) \tanh ^8(c+d x)}{8 d}\\ \end{align*}

Mathematica [A]  time = 6.54458, size = 291, normalized size = 0.83 $-\frac{3 b \left (9 a^2+7 b^2\right ) \sinh (c+d x)}{4 d}+\frac{b \left (3 a^2+b^2\right ) \sinh (3 (c+d x))}{12 d}-\frac{3 a \left (a^2+15 b^2\right ) \cosh (c+d x)}{4 d}+\frac{a \left (a^2+3 b^2\right ) \cosh (3 (c+d x))}{12 d}+\frac{15 b \left (64 a^2+77 b^2\right ) \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{64 d}-\frac{3 \text{sech}^2(c+d x) \left (64 a^2 b \sinh (c+d x)+255 b^3 \sinh (c+d x)\right )}{128 d}-\frac{3 a b^2 \text{sech}^5(c+d x)}{5 d}+\frac{4 a b^2 \text{sech}^3(c+d x)}{d}-\frac{18 a b^2 \text{sech}(c+d x)}{d}+\frac{b^3 \tanh (c+d x) \text{sech}^7(c+d x)}{8 d}-\frac{41 b^3 \tanh (c+d x) \text{sech}^5(c+d x)}{48 d}+\frac{515 b^3 \tanh (c+d x) \text{sech}^3(c+d x)}{192 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(15*b*(64*a^2 + 77*b^2)*ArcTan[Tanh[(c + d*x)/2]])/(64*d) - (3*a*(a^2 + 15*b^2)*Cosh[c + d*x])/(4*d) + (a*(a^2
+ 3*b^2)*Cosh[3*(c + d*x)])/(12*d) - (18*a*b^2*Sech[c + d*x])/d + (4*a*b^2*Sech[c + d*x]^3)/d - (3*a*b^2*Sech
[c + d*x]^5)/(5*d) - (3*b*(9*a^2 + 7*b^2)*Sinh[c + d*x])/(4*d) - (3*Sech[c + d*x]^2*(64*a^2*b*Sinh[c + d*x] +
255*b^3*Sinh[c + d*x]))/(128*d) + (b*(3*a^2 + b^2)*Sinh[3*(c + d*x)])/(12*d) + (515*b^3*Sech[c + d*x]^3*Tanh[c
+ d*x])/(192*d) - (41*b^3*Sech[c + d*x]^5*Tanh[c + d*x])/(48*d) + (b^3*Sech[c + d*x]^7*Tanh[c + d*x])/(8*d)

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Maple [A]  time = 0.13, size = 554, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

77/16/d*b^3*tanh(d*x+c)*sech(d*x+c)^5+1/3/d*a^3*cosh(d*x+c)*sinh(d*x+c)^2+15/d*a^2*b*arctan(exp(d*x+c))+1/3/d*
b^3*sinh(d*x+c)^11/cosh(d*x+c)^8-11/3/d*b^3*sinh(d*x+c)^9/cosh(d*x+c)^8-33/d*b^3*sinh(d*x+c)^7/cosh(d*x+c)^8-7
7/d*b^3*sinh(d*x+c)^5/cosh(d*x+c)^8-77/d*b^3*sinh(d*x+c)^3/cosh(d*x+c)^8-33/d*b^3*sinh(d*x+c)/cosh(d*x+c)^8+33
/8/d*b^3*tanh(d*x+c)*sech(d*x+c)^7-8/d*a*b^2*sinh(d*x+c)^6/cosh(d*x+c)^5+1155/128*b^3*sech(d*x+c)*tanh(d*x+c)/
d+385/64*b^3*sech(d*x+c)^3*tanh(d*x+c)/d-128/5*a*b^2*cosh(d*x+c)/d-2/3*a^3*cosh(d*x+c)/d-48/d*a*b^2*sinh(d*x+c
)^4/cosh(d*x+c)^5-192/5/d*a*b^2*sinh(d*x+c)^2/cosh(d*x+c)^5+128/5/d*a*b^2*sinh(d*x+c)^2/cosh(d*x+c)^3+128/5/d*
a*b^2*sinh(d*x+c)^2/cosh(d*x+c)+1/d*a^2*b*sinh(d*x+c)^5/cosh(d*x+c)^2-5/d*a^2*b*sinh(d*x+c)^3/cosh(d*x+c)^2-15
/d*a^2*b*sinh(d*x+c)/cosh(d*x+c)^2+1/d*a*b^2*sinh(d*x+c)^8/cosh(d*x+c)^5+15/2*a^2*b*sech(d*x+c)*tanh(d*x+c)/d+
1155/64/d*b^3*arctan(exp(d*x+c))

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Maxima [A]  time = 1.78683, size = 815, normalized size = 2.32 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*b^3*(8*(63*e^(-d*x - c) - e^(-3*d*x - 3*c))/d - 3465*arctan(e^(-d*x - c))/d - (440*e^(-2*d*x - 2*c) + 61
03*e^(-4*d*x - 4*c) + 21019*e^(-6*d*x - 6*c) + 41207*e^(-8*d*x - 8*c) + 40243*e^(-10*d*x - 10*c) + 22589*e^(-1
2*d*x - 12*c) + 505*e^(-14*d*x - 14*c) - 3331*e^(-16*d*x - 16*c) - 1791*e^(-18*d*x - 18*c) - 8)/(d*(e^(-3*d*x
- 3*c) + 8*e^(-5*d*x - 5*c) + 28*e^(-7*d*x - 7*c) + 56*e^(-9*d*x - 9*c) + 70*e^(-11*d*x - 11*c) + 56*e^(-13*d*
x - 13*c) + 28*e^(-15*d*x - 15*c) + 8*e^(-17*d*x - 17*c) + e^(-19*d*x - 19*c)))) - 1/40*a*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)))) + 1/8*a^2*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^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)

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Fricas [B]  time = 3.60726, size = 23883, normalized size = 68.04 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/960*(40*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^22 + 880*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)
*sinh(d*x + c)^21 + 40*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sinh(d*x + c)^22 - 40*(a^3 + 57*a^2*b + 111*a*b^2 + 55*
b^3)*cosh(d*x + c)^20 - 40*(a^3 + 57*a^2*b + 111*a*b^2 + 55*b^3 - 231*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x
+ c)^2)*sinh(d*x + c)^20 + 800*(77*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^3 - (a^3 + 57*a^2*b + 111*a*
b^2 + 55*b^3)*cosh(d*x + c))*sinh(d*x + c)^19 - 5*(424*a^3 + 4440*a^2*b + 15960*a*b^2 + 5599*b^3)*cosh(d*x + c
)^18 + 5*(58520*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 - 424*a^3 - 4440*a^2*b - 15960*a*b^2 - 5599*b^
3 - 1520*(a^3 + 57*a^2*b + 111*a*b^2 + 55*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^18 + 30*(35112*(a^3 + 3*a^2*b +
3*a*b^2 + b^3)*cosh(d*x + c)^5 - 1520*(a^3 + 57*a^2*b + 111*a*b^2 + 55*b^3)*cosh(d*x + c)^3 - 3*(424*a^3 + 444
0*a^2*b + 15960*a*b^2 + 5599*b^3)*cosh(d*x + c))*sinh(d*x + c)^17 - 15*(712*a^3 + 4840*a^2*b + 26584*a*b^2 + 5
665*b^3)*cosh(d*x + c)^16 + 15*(198968*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^6 - 12920*(a^3 + 57*a^2*b
+ 111*a*b^2 + 55*b^3)*cosh(d*x + c)^4 - 712*a^3 - 4840*a^2*b - 26584*a*b^2 - 5665*b^3 - 51*(424*a^3 + 4440*a^
2*b + 15960*a*b^2 + 5599*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^16 + 240*(28424*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*c
osh(d*x + c)^7 - 2584*(a^3 + 57*a^2*b + 111*a*b^2 + 55*b^3)*cosh(d*x + c)^5 - 17*(424*a^3 + 4440*a^2*b + 15960
*a*b^2 + 5599*b^3)*cosh(d*x + c)^3 - (712*a^3 + 4840*a^2*b + 26584*a*b^2 + 5665*b^3)*cosh(d*x + c))*sinh(d*x +
c)^15 - 3*(9040*a^3 + 36400*a^2*b + 344944*a*b^2 + 45265*b^3)*cosh(d*x + c)^14 + 3*(4263600*(a^3 + 3*a^2*b +
3*a*b^2 + b^3)*cosh(d*x + c)^8 - 516800*(a^3 + 57*a^2*b + 111*a*b^2 + 55*b^3)*cosh(d*x + c)^6 - 5100*(424*a^3
+ 4440*a^2*b + 15960*a*b^2 + 5599*b^3)*cosh(d*x + c)^4 - 9040*a^3 - 36400*a^2*b - 344944*a*b^2 - 45265*b^3 - 6
00*(712*a^3 + 4840*a^2*b + 26584*a*b^2 + 5665*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^14 + 2*(9948400*(a^3 + 3*a^2
*b + 3*a*b^2 + b^3)*cosh(d*x + c)^9 - 1550400*(a^3 + 57*a^2*b + 111*a*b^2 + 55*b^3)*cosh(d*x + c)^7 - 21420*(4
24*a^3 + 4440*a^2*b + 15960*a*b^2 + 5599*b^3)*cosh(d*x + c)^5 - 4200*(712*a^3 + 4840*a^2*b + 26584*a*b^2 + 566
5*b^3)*cosh(d*x + c)^3 - 21*(9040*a^3 + 36400*a^2*b + 344944*a*b^2 + 45265*b^3)*cosh(d*x + c))*sinh(d*x + c)^1
3 - 3*(14000*a^3 + 18800*a^2*b + 542672*a*b^2 + 20405*b^3)*cosh(d*x + c)^12 + (25865840*(a^3 + 3*a^2*b + 3*a*b
^2 + b^3)*cosh(d*x + c)^10 - 5038800*(a^3 + 57*a^2*b + 111*a*b^2 + 55*b^3)*cosh(d*x + c)^8 - 92820*(424*a^3 +
4440*a^2*b + 15960*a*b^2 + 5599*b^3)*cosh(d*x + c)^6 - 27300*(712*a^3 + 4840*a^2*b + 26584*a*b^2 + 5665*b^3)*c
osh(d*x + c)^4 - 42000*a^3 - 56400*a^2*b - 1628016*a*b^2 - 61215*b^3 - 273*(9040*a^3 + 36400*a^2*b + 344944*a*
b^2 + 45265*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^12 + 4*(7054320*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^
11 - 1679600*(a^3 + 57*a^2*b + 111*a*b^2 + 55*b^3)*cosh(d*x + c)^9 - 39780*(424*a^3 + 4440*a^2*b + 15960*a*b^2
+ 5599*b^3)*cosh(d*x + c)^7 - 16380*(712*a^3 + 4840*a^2*b + 26584*a*b^2 + 5665*b^3)*cosh(d*x + c)^5 - 273*(90
40*a^3 + 36400*a^2*b + 344944*a*b^2 + 45265*b^3)*cosh(d*x + c)^3 - 9*(14000*a^3 + 18800*a^2*b + 542672*a*b^2 +
20405*b^3)*cosh(d*x + c))*sinh(d*x + c)^11 - 3*(14000*a^3 - 18800*a^2*b + 542672*a*b^2 - 20405*b^3)*cosh(d*x
+ c)^10 + (25865840*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^12 - 7390240*(a^3 + 57*a^2*b + 111*a*b^2 + 5
5*b^3)*cosh(d*x + c)^10 - 218790*(424*a^3 + 4440*a^2*b + 15960*a*b^2 + 5599*b^3)*cosh(d*x + c)^8 - 120120*(712
*a^3 + 4840*a^2*b + 26584*a*b^2 + 5665*b^3)*cosh(d*x + c)^6 - 3003*(9040*a^3 + 36400*a^2*b + 344944*a*b^2 + 45
265*b^3)*cosh(d*x + c)^4 - 42000*a^3 + 56400*a^2*b - 1628016*a*b^2 + 61215*b^3 - 198*(14000*a^3 + 18800*a^2*b
+ 542672*a*b^2 + 20405*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 2*(9948400*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos
h(d*x + c)^13 - 3359200*(a^3 + 57*a^2*b + 111*a*b^2 + 55*b^3)*cosh(d*x + c)^11 - 121550*(424*a^3 + 4440*a^2*b
+ 15960*a*b^2 + 5599*b^3)*cosh(d*x + c)^9 - 85800*(712*a^3 + 4840*a^2*b + 26584*a*b^2 + 5665*b^3)*cosh(d*x + c
)^7 - 3003*(9040*a^3 + 36400*a^2*b + 344944*a*b^2 + 45265*b^3)*cosh(d*x + c)^5 - 330*(14000*a^3 + 18800*a^2*b
+ 542672*a*b^2 + 20405*b^3)*cosh(d*x + c)^3 - 15*(14000*a^3 - 18800*a^2*b + 542672*a*b^2 - 20405*b^3)*cosh(d*x
+ c))*sinh(d*x + c)^9 - 3*(9040*a^3 - 36400*a^2*b + 344944*a*b^2 - 45265*b^3)*cosh(d*x + c)^8 + 3*(4263600*(a
^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^14 - 1679600*(a^3 + 57*a^2*b + 111*a*b^2 + 55*b^3)*cosh(d*x + c)^1
2 - 72930*(424*a^3 + 4440*a^2*b + 15960*a*b^2 + 5599*b^3)*cosh(d*x + c)^10 - 64350*(712*a^3 + 4840*a^2*b + 265
84*a*b^2 + 5665*b^3)*cosh(d*x + c)^8 - 3003*(9040*a^3 + 36400*a^2*b + 344944*a*b^2 + 45265*b^3)*cosh(d*x + c)^
6 - 495*(14000*a^3 + 18800*a^2*b + 542672*a*b^2 + 20405*b^3)*cosh(d*x + c)^4 - 9040*a^3 + 36400*a^2*b - 344944
*a*b^2 + 45265*b^3 - 45*(14000*a^3 - 18800*a^2*b + 542672*a*b^2 - 20405*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8
+ 24*(284240*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^15 - 129200*(a^3 + 57*a^2*b + 111*a*b^2 + 55*b^3)*c
osh(d*x + c)^13 - 6630*(424*a^3 + 4440*a^2*b + 15960*a*b^2 + 5599*b^3)*cosh(d*x + c)^11 - 7150*(712*a^3 + 4840
*a^2*b + 26584*a*b^2 + 5665*b^3)*cosh(d*x + c)^9 - 429*(9040*a^3 + 36400*a^2*b + 344944*a*b^2 + 45265*b^3)*cos
h(d*x + c)^7 - 99*(14000*a^3 + 18800*a^2*b + 542672*a*b^2 + 20405*b^3)*cosh(d*x + c)^5 - 15*(14000*a^3 - 18800
*a^2*b + 542672*a*b^2 - 20405*b^3)*cosh(d*x + c)^3 - (9040*a^3 - 36400*a^2*b + 344944*a*b^2 - 45265*b^3)*cosh(
d*x + c))*sinh(d*x + c)^7 - 15*(712*a^3 - 4840*a^2*b + 26584*a*b^2 - 5665*b^3)*cosh(d*x + c)^6 + 3*(994840*(a^
3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^16 - 516800*(a^3 + 57*a^2*b + 111*a*b^2 + 55*b^3)*cosh(d*x + c)^14
- 30940*(424*a^3 + 4440*a^2*b + 15960*a*b^2 + 5599*b^3)*cosh(d*x + c)^12 - 40040*(712*a^3 + 4840*a^2*b + 26584
*a*b^2 + 5665*b^3)*cosh(d*x + c)^10 - 3003*(9040*a^3 + 36400*a^2*b + 344944*a*b^2 + 45265*b^3)*cosh(d*x + c)^8
- 924*(14000*a^3 + 18800*a^2*b + 542672*a*b^2 + 20405*b^3)*cosh(d*x + c)^6 - 210*(14000*a^3 - 18800*a^2*b + 5
42672*a*b^2 - 20405*b^3)*cosh(d*x + c)^4 - 3560*a^3 + 24200*a^2*b - 132920*a*b^2 + 28325*b^3 - 28*(9040*a^3 -
36400*a^2*b + 344944*a*b^2 - 45265*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 6*(175560*(a^3 + 3*a^2*b + 3*a*b^2
+ b^3)*cosh(d*x + c)^17 - 103360*(a^3 + 57*a^2*b + 111*a*b^2 + 55*b^3)*cosh(d*x + c)^15 - 7140*(424*a^3 + 4440
*a^2*b + 15960*a*b^2 + 5599*b^3)*cosh(d*x + c)^13 - 10920*(712*a^3 + 4840*a^2*b + 26584*a*b^2 + 5665*b^3)*cosh
(d*x + c)^11 - 1001*(9040*a^3 + 36400*a^2*b + 344944*a*b^2 + 45265*b^3)*cosh(d*x + c)^9 - 396*(14000*a^3 + 188
00*a^2*b + 542672*a*b^2 + 20405*b^3)*cosh(d*x + c)^7 - 126*(14000*a^3 - 18800*a^2*b + 542672*a*b^2 - 20405*b^3
)*cosh(d*x + c)^5 - 28*(9040*a^3 - 36400*a^2*b + 344944*a*b^2 - 45265*b^3)*cosh(d*x + c)^3 - 15*(712*a^3 - 484
0*a^2*b + 26584*a*b^2 - 5665*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 - 5*(424*a^3 - 4440*a^2*b + 15960*a*b^2 - 559
9*b^3)*cosh(d*x + c)^4 + (292600*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^18 - 193800*(a^3 + 57*a^2*b + 1
11*a*b^2 + 55*b^3)*cosh(d*x + c)^16 - 15300*(424*a^3 + 4440*a^2*b + 15960*a*b^2 + 5599*b^3)*cosh(d*x + c)^14 -
27300*(712*a^3 + 4840*a^2*b + 26584*a*b^2 + 5665*b^3)*cosh(d*x + c)^12 - 3003*(9040*a^3 + 36400*a^2*b + 34494
4*a*b^2 + 45265*b^3)*cosh(d*x + c)^10 - 1485*(14000*a^3 + 18800*a^2*b + 542672*a*b^2 + 20405*b^3)*cosh(d*x + c
)^8 - 630*(14000*a^3 - 18800*a^2*b + 542672*a*b^2 - 20405*b^3)*cosh(d*x + c)^6 - 210*(9040*a^3 - 36400*a^2*b +
344944*a*b^2 - 45265*b^3)*cosh(d*x + c)^4 - 2120*a^3 + 22200*a^2*b - 79800*a*b^2 + 27995*b^3 - 225*(712*a^3 -
4840*a^2*b + 26584*a*b^2 - 5665*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(15400*(a^3 + 3*a^2*b + 3*a*b^2 + b
^3)*cosh(d*x + c)^19 - 11400*(a^3 + 57*a^2*b + 111*a*b^2 + 55*b^3)*cosh(d*x + c)^17 - 1020*(424*a^3 + 4440*a^2
*b + 15960*a*b^2 + 5599*b^3)*cosh(d*x + c)^15 - 2100*(712*a^3 + 4840*a^2*b + 26584*a*b^2 + 5665*b^3)*cosh(d*x
+ c)^13 - 273*(9040*a^3 + 36400*a^2*b + 344944*a*b^2 + 45265*b^3)*cosh(d*x + c)^11 - 165*(14000*a^3 + 18800*a^
2*b + 542672*a*b^2 + 20405*b^3)*cosh(d*x + c)^9 - 90*(14000*a^3 - 18800*a^2*b + 542672*a*b^2 - 20405*b^3)*cosh
(d*x + c)^7 - 42*(9040*a^3 - 36400*a^2*b + 344944*a*b^2 - 45265*b^3)*cosh(d*x + c)^5 - 75*(712*a^3 - 4840*a^2*
b + 26584*a*b^2 - 5665*b^3)*cosh(d*x + c)^3 - 5*(424*a^3 - 4440*a^2*b + 15960*a*b^2 - 5599*b^3)*cosh(d*x + c))
*sinh(d*x + c)^3 + 40*a^3 - 120*a^2*b + 120*a*b^2 - 40*b^3 - 40*(a^3 - 57*a^2*b + 111*a*b^2 - 55*b^3)*cosh(d*x
+ c)^2 + (9240*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^20 - 7600*(a^3 + 57*a^2*b + 111*a*b^2 + 55*b^3)*
cosh(d*x + c)^18 - 765*(424*a^3 + 4440*a^2*b + 15960*a*b^2 + 5599*b^3)*cosh(d*x + c)^16 - 1800*(712*a^3 + 4840
*a^2*b + 26584*a*b^2 + 5665*b^3)*cosh(d*x + c)^14 - 273*(9040*a^3 + 36400*a^2*b + 344944*a*b^2 + 45265*b^3)*co
sh(d*x + c)^12 - 198*(14000*a^3 + 18800*a^2*b + 542672*a*b^2 + 20405*b^3)*cosh(d*x + c)^10 - 135*(14000*a^3 -
18800*a^2*b + 542672*a*b^2 - 20405*b^3)*cosh(d*x + c)^8 - 84*(9040*a^3 - 36400*a^2*b + 344944*a*b^2 - 45265*b^
3)*cosh(d*x + c)^6 - 225*(712*a^3 - 4840*a^2*b + 26584*a*b^2 - 5665*b^3)*cosh(d*x + c)^4 - 40*a^3 + 2280*a^2*b
- 4440*a*b^2 + 2200*b^3 - 30*(424*a^3 - 4440*a^2*b + 15960*a*b^2 - 5599*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2
+ 225*((64*a^2*b + 77*b^3)*cosh(d*x + c)^19 + 19*(64*a^2*b + 77*b^3)*cosh(d*x + c)*sinh(d*x + c)^18 + (64*a^2
*b + 77*b^3)*sinh(d*x + c)^19 + 8*(64*a^2*b + 77*b^3)*cosh(d*x + c)^17 + (512*a^2*b + 616*b^3 + 171*(64*a^2*b
+ 77*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^17 + 17*(57*(64*a^2*b + 77*b^3)*cosh(d*x + c)^3 + 8*(64*a^2*b + 77*b^
3)*cosh(d*x + c))*sinh(d*x + c)^16 + 28*(64*a^2*b + 77*b^3)*cosh(d*x + c)^15 + 4*(969*(64*a^2*b + 77*b^3)*cosh
(d*x + c)^4 + 448*a^2*b + 539*b^3 + 272*(64*a^2*b + 77*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^15 + 4*(2907*(64*a^
2*b + 77*b^3)*cosh(d*x + c)^5 + 1360*(64*a^2*b + 77*b^3)*cosh(d*x + c)^3 + 105*(64*a^2*b + 77*b^3)*cosh(d*x +
c))*sinh(d*x + c)^14 + 56*(64*a^2*b + 77*b^3)*cosh(d*x + c)^13 + 28*(969*(64*a^2*b + 77*b^3)*cosh(d*x + c)^6 +
680*(64*a^2*b + 77*b^3)*cosh(d*x + c)^4 + 128*a^2*b + 154*b^3 + 105*(64*a^2*b + 77*b^3)*cosh(d*x + c)^2)*sinh
(d*x + c)^13 + 52*(969*(64*a^2*b + 77*b^3)*cosh(d*x + c)^7 + 952*(64*a^2*b + 77*b^3)*cosh(d*x + c)^5 + 245*(64
*a^2*b + 77*b^3)*cosh(d*x + c)^3 + 14*(64*a^2*b + 77*b^3)*cosh(d*x + c))*sinh(d*x + c)^12 + 70*(64*a^2*b + 77*
b^3)*cosh(d*x + c)^11 + 2*(37791*(64*a^2*b + 77*b^3)*cosh(d*x + c)^8 + 49504*(64*a^2*b + 77*b^3)*cosh(d*x + c)
^6 + 19110*(64*a^2*b + 77*b^3)*cosh(d*x + c)^4 + 2240*a^2*b + 2695*b^3 + 2184*(64*a^2*b + 77*b^3)*cosh(d*x + c
)^2)*sinh(d*x + c)^11 + 22*(4199*(64*a^2*b + 77*b^3)*cosh(d*x + c)^9 + 7072*(64*a^2*b + 77*b^3)*cosh(d*x + c)^
7 + 3822*(64*a^2*b + 77*b^3)*cosh(d*x + c)^5 + 728*(64*a^2*b + 77*b^3)*cosh(d*x + c)^3 + 35*(64*a^2*b + 77*b^3
)*cosh(d*x + c))*sinh(d*x + c)^10 + 56*(64*a^2*b + 77*b^3)*cosh(d*x + c)^9 + 2*(46189*(64*a^2*b + 77*b^3)*cosh
(d*x + c)^10 + 97240*(64*a^2*b + 77*b^3)*cosh(d*x + c)^8 + 70070*(64*a^2*b + 77*b^3)*cosh(d*x + c)^6 + 20020*(
64*a^2*b + 77*b^3)*cosh(d*x + c)^4 + 1792*a^2*b + 2156*b^3 + 1925*(64*a^2*b + 77*b^3)*cosh(d*x + c)^2)*sinh(d*
x + c)^9 + 2*(37791*(64*a^2*b + 77*b^3)*cosh(d*x + c)^11 + 97240*(64*a^2*b + 77*b^3)*cosh(d*x + c)^9 + 90090*(
64*a^2*b + 77*b^3)*cosh(d*x + c)^7 + 36036*(64*a^2*b + 77*b^3)*cosh(d*x + c)^5 + 5775*(64*a^2*b + 77*b^3)*cosh
(d*x + c)^3 + 252*(64*a^2*b + 77*b^3)*cosh(d*x + c))*sinh(d*x + c)^8 + 28*(64*a^2*b + 77*b^3)*cosh(d*x + c)^7
+ 4*(12597*(64*a^2*b + 77*b^3)*cosh(d*x + c)^12 + 38896*(64*a^2*b + 77*b^3)*cosh(d*x + c)^10 + 45045*(64*a^2*b
+ 77*b^3)*cosh(d*x + c)^8 + 24024*(64*a^2*b + 77*b^3)*cosh(d*x + c)^6 + 5775*(64*a^2*b + 77*b^3)*cosh(d*x + c
)^4 + 448*a^2*b + 539*b^3 + 504*(64*a^2*b + 77*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 28*(969*(64*a^2*b + 77*
b^3)*cosh(d*x + c)^13 + 3536*(64*a^2*b + 77*b^3)*cosh(d*x + c)^11 + 5005*(64*a^2*b + 77*b^3)*cosh(d*x + c)^9 +
3432*(64*a^2*b + 77*b^3)*cosh(d*x + c)^7 + 1155*(64*a^2*b + 77*b^3)*cosh(d*x + c)^5 + 168*(64*a^2*b + 77*b^3)
*cosh(d*x + c)^3 + 7*(64*a^2*b + 77*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 + 8*(64*a^2*b + 77*b^3)*cosh(d*x + c)^
5 + 4*(2907*(64*a^2*b + 77*b^3)*cosh(d*x + c)^14 + 12376*(64*a^2*b + 77*b^3)*cosh(d*x + c)^12 + 21021*(64*a^2*
b + 77*b^3)*cosh(d*x + c)^10 + 18018*(64*a^2*b + 77*b^3)*cosh(d*x + c)^8 + 8085*(64*a^2*b + 77*b^3)*cosh(d*x +
c)^6 + 1764*(64*a^2*b + 77*b^3)*cosh(d*x + c)^4 + 128*a^2*b + 154*b^3 + 147*(64*a^2*b + 77*b^3)*cosh(d*x + c)
^2)*sinh(d*x + c)^5 + 4*(969*(64*a^2*b + 77*b^3)*cosh(d*x + c)^15 + 4760*(64*a^2*b + 77*b^3)*cosh(d*x + c)^13
+ 9555*(64*a^2*b + 77*b^3)*cosh(d*x + c)^11 + 10010*(64*a^2*b + 77*b^3)*cosh(d*x + c)^9 + 5775*(64*a^2*b + 77*
b^3)*cosh(d*x + c)^7 + 1764*(64*a^2*b + 77*b^3)*cosh(d*x + c)^5 + 245*(64*a^2*b + 77*b^3)*cosh(d*x + c)^3 + 10
*(64*a^2*b + 77*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + (64*a^2*b + 77*b^3)*cosh(d*x + c)^3 + (969*(64*a^2*b + 7
7*b^3)*cosh(d*x + c)^16 + 5440*(64*a^2*b + 77*b^3)*cosh(d*x + c)^14 + 12740*(64*a^2*b + 77*b^3)*cosh(d*x + c)^
12 + 16016*(64*a^2*b + 77*b^3)*cosh(d*x + c)^10 + 11550*(64*a^2*b + 77*b^3)*cosh(d*x + c)^8 + 4704*(64*a^2*b +
77*b^3)*cosh(d*x + c)^6 + 980*(64*a^2*b + 77*b^3)*cosh(d*x + c)^4 + 64*a^2*b + 77*b^3 + 80*(64*a^2*b + 77*b^3
)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + (171*(64*a^2*b + 77*b^3)*cosh(d*x + c)^17 + 1088*(64*a^2*b + 77*b^3)*cosh
(d*x + c)^15 + 2940*(64*a^2*b + 77*b^3)*cosh(d*x + c)^13 + 4368*(64*a^2*b + 77*b^3)*cosh(d*x + c)^11 + 3850*(6
4*a^2*b + 77*b^3)*cosh(d*x + c)^9 + 2016*(64*a^2*b + 77*b^3)*cosh(d*x + c)^7 + 588*(64*a^2*b + 77*b^3)*cosh(d*
x + c)^5 + 80*(64*a^2*b + 77*b^3)*cosh(d*x + c)^3 + 3*(64*a^2*b + 77*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + (19
*(64*a^2*b + 77*b^3)*cosh(d*x + c)^18 + 136*(64*a^2*b + 77*b^3)*cosh(d*x + c)^16 + 420*(64*a^2*b + 77*b^3)*cos
h(d*x + c)^14 + 728*(64*a^2*b + 77*b^3)*cosh(d*x + c)^12 + 770*(64*a^2*b + 77*b^3)*cosh(d*x + c)^10 + 504*(64*
a^2*b + 77*b^3)*cosh(d*x + c)^8 + 196*(64*a^2*b + 77*b^3)*cosh(d*x + c)^6 + 40*(64*a^2*b + 77*b^3)*cosh(d*x +
c)^4 + 3*(64*a^2*b + 77*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) + 2*(440*(a
^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^21 - 400*(a^3 + 57*a^2*b + 111*a*b^2 + 55*b^3)*cosh(d*x + c)^19 -
45*(424*a^3 + 4440*a^2*b + 15960*a*b^2 + 5599*b^3)*cosh(d*x + c)^17 - 120*(712*a^3 + 4840*a^2*b + 26584*a*b^2
+ 5665*b^3)*cosh(d*x + c)^15 - 21*(9040*a^3 + 36400*a^2*b + 344944*a*b^2 + 45265*b^3)*cosh(d*x + c)^13 - 18*(1
4000*a^3 + 18800*a^2*b + 542672*a*b^2 + 20405*b^3)*cosh(d*x + c)^11 - 15*(14000*a^3 - 18800*a^2*b + 542672*a*b
^2 - 20405*b^3)*cosh(d*x + c)^9 - 12*(9040*a^3 - 36400*a^2*b + 344944*a*b^2 - 45265*b^3)*cosh(d*x + c)^7 - 45*
(712*a^3 - 4840*a^2*b + 26584*a*b^2 - 5665*b^3)*cosh(d*x + c)^5 - 10*(424*a^3 - 4440*a^2*b + 15960*a*b^2 - 559
9*b^3)*cosh(d*x + c)^3 - 40*(a^3 - 57*a^2*b + 111*a*b^2 - 55*b^3)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x +
c)^19 + 19*d*cosh(d*x + c)*sinh(d*x + c)^18 + d*sinh(d*x + c)^19 + 8*d*cosh(d*x + c)^17 + (171*d*cosh(d*x + c)
^2 + 8*d)*sinh(d*x + c)^17 + 17*(57*d*cosh(d*x + c)^3 + 8*d*cosh(d*x + c))*sinh(d*x + c)^16 + 28*d*cosh(d*x +
c)^15 + 4*(969*d*cosh(d*x + c)^4 + 272*d*cosh(d*x + c)^2 + 7*d)*sinh(d*x + c)^15 + 4*(2907*d*cosh(d*x + c)^5 +
1360*d*cosh(d*x + c)^3 + 105*d*cosh(d*x + c))*sinh(d*x + c)^14 + 56*d*cosh(d*x + c)^13 + 28*(969*d*cosh(d*x +
c)^6 + 680*d*cosh(d*x + c)^4 + 105*d*cosh(d*x + c)^2 + 2*d)*sinh(d*x + c)^13 + 52*(969*d*cosh(d*x + c)^7 + 95
2*d*cosh(d*x + c)^5 + 245*d*cosh(d*x + c)^3 + 14*d*cosh(d*x + c))*sinh(d*x + c)^12 + 70*d*cosh(d*x + c)^11 + 2
*(37791*d*cosh(d*x + c)^8 + 49504*d*cosh(d*x + c)^6 + 19110*d*cosh(d*x + c)^4 + 2184*d*cosh(d*x + c)^2 + 35*d)
*sinh(d*x + c)^11 + 22*(4199*d*cosh(d*x + c)^9 + 7072*d*cosh(d*x + c)^7 + 3822*d*cosh(d*x + c)^5 + 728*d*cosh(
d*x + c)^3 + 35*d*cosh(d*x + c))*sinh(d*x + c)^10 + 56*d*cosh(d*x + c)^9 + 2*(46189*d*cosh(d*x + c)^10 + 97240
*d*cosh(d*x + c)^8 + 70070*d*cosh(d*x + c)^6 + 20020*d*cosh(d*x + c)^4 + 1925*d*cosh(d*x + c)^2 + 28*d)*sinh(d
*x + c)^9 + 2*(37791*d*cosh(d*x + c)^11 + 97240*d*cosh(d*x + c)^9 + 90090*d*cosh(d*x + c)^7 + 36036*d*cosh(d*x
+ c)^5 + 5775*d*cosh(d*x + c)^3 + 252*d*cosh(d*x + c))*sinh(d*x + c)^8 + 28*d*cosh(d*x + c)^7 + 4*(12597*d*co
sh(d*x + c)^12 + 38896*d*cosh(d*x + c)^10 + 45045*d*cosh(d*x + c)^8 + 24024*d*cosh(d*x + c)^6 + 5775*d*cosh(d*
x + c)^4 + 504*d*cosh(d*x + c)^2 + 7*d)*sinh(d*x + c)^7 + 28*(969*d*cosh(d*x + c)^13 + 3536*d*cosh(d*x + c)^11
+ 5005*d*cosh(d*x + c)^9 + 3432*d*cosh(d*x + c)^7 + 1155*d*cosh(d*x + c)^5 + 168*d*cosh(d*x + c)^3 + 7*d*cosh
(d*x + c))*sinh(d*x + c)^6 + 8*d*cosh(d*x + c)^5 + 4*(2907*d*cosh(d*x + c)^14 + 12376*d*cosh(d*x + c)^12 + 210
21*d*cosh(d*x + c)^10 + 18018*d*cosh(d*x + c)^8 + 8085*d*cosh(d*x + c)^6 + 1764*d*cosh(d*x + c)^4 + 147*d*cosh
(d*x + c)^2 + 2*d)*sinh(d*x + c)^5 + 4*(969*d*cosh(d*x + c)^15 + 4760*d*cosh(d*x + c)^13 + 9555*d*cosh(d*x + c
)^11 + 10010*d*cosh(d*x + c)^9 + 5775*d*cosh(d*x + c)^7 + 1764*d*cosh(d*x + c)^5 + 245*d*cosh(d*x + c)^3 + 10*
d*cosh(d*x + c))*sinh(d*x + c)^4 + d*cosh(d*x + c)^3 + (969*d*cosh(d*x + c)^16 + 5440*d*cosh(d*x + c)^14 + 127
40*d*cosh(d*x + c)^12 + 16016*d*cosh(d*x + c)^10 + 11550*d*cosh(d*x + c)^8 + 4704*d*cosh(d*x + c)^6 + 980*d*co
sh(d*x + c)^4 + 80*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^3 + (171*d*cosh(d*x + c)^17 + 1088*d*cosh(d*x + c)^15
+ 2940*d*cosh(d*x + c)^13 + 4368*d*cosh(d*x + c)^11 + 3850*d*cosh(d*x + c)^9 + 2016*d*cosh(d*x + c)^7 + 588*d*
cosh(d*x + c)^5 + 80*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^2 + (19*d*cosh(d*x + c)^18 + 136*d*c
osh(d*x + c)^16 + 420*d*cosh(d*x + c)^14 + 728*d*cosh(d*x + c)^12 + 770*d*cosh(d*x + c)^10 + 504*d*cosh(d*x +
c)^8 + 196*d*cosh(d*x + c)^6 + 40*d*cosh(d*x + c)^4 + 3*d*cosh(d*x + c)^2)*sinh(d*x + c))

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 3.95178, size = 811, normalized size = 2.31 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/960*(225*(64*a^2*b*e^c + 77*b^3*e^c)*arctan(e^(d*x + c))*e^(-c) - 40*(9*a^3*e^(2*d*x + 2*c) - 81*a^2*b*e^(2*
d*x + 2*c) + 135*a*b^2*e^(2*d*x + 2*c) - 63*b^3*e^(2*d*x + 2*c) - a^3 + 3*a^2*b - 3*a*b^2 + b^3)*e^(-3*d*x - 3
*c) + 40*(a^3*e^(3*d*x + 66*c) + 3*a^2*b*e^(3*d*x + 66*c) + 3*a*b^2*e^(3*d*x + 66*c) + b^3*e^(3*d*x + 66*c) -
9*a^3*e^(d*x + 64*c) - 81*a^2*b*e^(d*x + 64*c) - 135*a*b^2*e^(d*x + 64*c) - 63*b^3*e^(d*x + 64*c))*e^(-63*c) -
(2880*a^2*b*e^(15*d*x + 15*c) + 34560*a*b^2*e^(15*d*x + 15*c) + 11475*b^3*e^(15*d*x + 15*c) + 14400*a^2*b*e^(
13*d*x + 13*c) + 211200*a*b^2*e^(13*d*x + 13*c) + 36775*b^3*e^(13*d*x + 13*c) + 25920*a^2*b*e^(11*d*x + 11*c)
+ 590592*a*b^2*e^(11*d*x + 11*c) + 67715*b^3*e^(11*d*x + 11*c) + 14400*a^2*b*e^(9*d*x + 9*c) + 957696*a*b^2*e^
(9*d*x + 9*c) + 27055*b^3*e^(9*d*x + 9*c) - 14400*a^2*b*e^(7*d*x + 7*c) + 957696*a*b^2*e^(7*d*x + 7*c) - 27055
*b^3*e^(7*d*x + 7*c) - 25920*a^2*b*e^(5*d*x + 5*c) + 590592*a*b^2*e^(5*d*x + 5*c) - 67715*b^3*e^(5*d*x + 5*c)
- 14400*a^2*b*e^(3*d*x + 3*c) + 211200*a*b^2*e^(3*d*x + 3*c) - 36775*b^3*e^(3*d*x + 3*c) - 2880*a^2*b*e^(d*x +
c) + 34560*a*b^2*e^(d*x + c) - 11475*b^3*e^(d*x + c))/(e^(2*d*x + 2*c) + 1)^8)/d