3.1.66 \(\int \sinh ^3(c+d x) (a+b \tanh ^3(c+d x))^3 \, dx\) [66]

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

[Out]

15/2*a^2*b*arctan(sinh(d*x+c))/d+1155/128*b^3*arctan(sinh(d*x+c))/d-a^3*cosh(d*x+c)/d-12*a*b^2*cosh(d*x+c)/d+1
/3*a^3*cosh(d*x+c)^3/d+a*b^2*cosh(d*x+c)^3/d-18*a*b^2*sech(d*x+c)/d+4*a*b^2*sech(d*x+c)^3/d-3/5*a*b^2*sech(d*x
+c)^5/d-15/2*a^2*b*sinh(d*x+c)/d-1155/128*b^3*sinh(d*x+c)/d+5/2*a^2*b*sinh(d*x+c)^3/d+385/128*b^3*sinh(d*x+c)^
3/d-3/2*a^2*b*sinh(d*x+c)^3*tanh(d*x+c)^2/d-231/128*b^3*sinh(d*x+c)^3*tanh(d*x+c)^2/d-33/64*b^3*sinh(d*x+c)^3*
tanh(d*x+c)^4/d-11/48*b^3*sinh(d*x+c)^3*tanh(d*x+c)^6/d-1/8*b^3*sinh(d*x+c)^3*tanh(d*x+c)^8/d

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Rubi [A]
time = 0.25, antiderivative size = 351, normalized size of antiderivative = 1.00, 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 = {3747, 2713, 2672, 294, 308, 209, 2670, 276} \begin {gather*} \frac {a^3 \cosh ^3(c+d x)}{3 d}-\frac {a^3 \cosh (c+d x)}{d}+\frac {15 a^2 b \text {ArcTan}(\sinh (c+d x))}{2 d}+\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 {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 {1155 b^3 \text {ArcTan}(\sinh (c+d x))}{128 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} \end {gather*}

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 209

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

Rule 276

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 294

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 308

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 2670

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 2672

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 2713

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 3747

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 )^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 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}+\frac {\left (3 a^2 b\right ) \text {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 ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^4}{x^6} \, dx,x,\cosh (c+d x)\right )}{d}+\frac {b^3 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 d}+\frac {\left (231 b^3\right ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 6.45, size = 291, normalized size = 0.83 \begin {gather*} \frac {15 b \left (64 a^2+77 b^2\right ) \text {ArcTan}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{64 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 {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 b \left (9 a^2+7 b^2\right ) \sinh (c+d x)}{4 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 {b \left (3 a^2+b^2\right ) \sinh (3 (c+d x))}{12 d}+\frac {515 b^3 \text {sech}^3(c+d x) \tanh (c+d x)}{192 d}-\frac {41 b^3 \text {sech}^5(c+d x) \tanh (c+d x)}{48 d}+\frac {b^3 \text {sech}^7(c+d x) \tanh (c+d x)}{8 d} \end {gather*}

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 [C] Result contains complex when optimal does not.
time = 2.84, size = 675, normalized size = 1.92

method	result	size
risch	\(\frac {{\mathrm e}^{3 d x +3 c} a^{3}}{24 d}+\frac {{\mathrm e}^{3 d x +3 c} a^{2} b}{8 d}+\frac {a \,b^{2} {\mathrm e}^{3 d x +3 c}}{8 d}+\frac {{\mathrm e}^{3 d x +3 c} b^{3}}{24 d}-\frac {3 \,{\mathrm e}^{d x +c} a^{3}}{8 d}-\frac {27 \,{\mathrm e}^{d x +c} a^{2} b}{8 d}-\frac {45 a \,{\mathrm e}^{d x +c} b^{2}}{8 d}-\frac {21 b^{3} {\mathrm e}^{d x +c}}{8 d}-\frac {3 \,{\mathrm e}^{-d x -c} a^{3}}{8 d}+\frac {27 \,{\mathrm e}^{-d x -c} a^{2} b}{8 d}-\frac {45 a \,{\mathrm e}^{-d x -c} b^{2}}{8 d}+\frac {21 \,{\mathrm e}^{-d x -c} b^{3}}{8 d}+\frac {{\mathrm e}^{-3 d x -3 c} a^{3}}{24 d}-\frac {{\mathrm e}^{-3 d x -3 c} a^{2} b}{8 d}+\frac {a \,b^{2} {\mathrm e}^{-3 d x -3 c}}{8 d}-\frac {{\mathrm e}^{-3 d x -3 c} b^{3}}{24 d}-\frac {b \,{\mathrm e}^{d x +c} \left (2880 a^{2} {\mathrm e}^{14 d x +14 c}+34560 a b \,{\mathrm e}^{14 d x +14 c}+11475 b^{2} {\mathrm e}^{14 d x +14 c}+14400 a^{2} {\mathrm e}^{12 d x +12 c}+211200 a b \,{\mathrm e}^{12 d x +12 c}+36775 b^{2} {\mathrm e}^{12 d x +12 c}+25920 a^{2} {\mathrm e}^{10 d x +10 c}+590592 a b \,{\mathrm e}^{10 d x +10 c}+67715 b^{2} {\mathrm e}^{10 d x +10 c}+14400 a^{2} {\mathrm e}^{8 d x +8 c}+957696 a b \,{\mathrm e}^{8 d x +8 c}+27055 b^{2} {\mathrm e}^{8 d x +8 c}-14400 a^{2} {\mathrm e}^{6 d x +6 c}+957696 a b \,{\mathrm e}^{6 d x +6 c}-27055 b^{2} {\mathrm e}^{6 d x +6 c}-25920 a^{2} {\mathrm e}^{4 d x +4 c}+590592 a b \,{\mathrm e}^{4 d x +4 c}-67715 b^{2} {\mathrm e}^{4 d x +4 c}-14400 a^{2} {\mathrm e}^{2 d x +2 c}+211200 a b \,{\mathrm e}^{2 d x +2 c}-36775 b^{2} {\mathrm e}^{2 d x +2 c}-2880 a^{2}+34560 a b -11475 b^{2}\right )}{960 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{8}}-\frac {1155 i b^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{128 d}+\frac {1155 i b^{3} \ln \left ({\mathrm e}^{d x +c}+i\right )}{128 d}-\frac {15 i b \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{2 d}+\frac {15 i b \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{2 d}\)	\(675\)

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,method=_RETURNVERBOSE)

[Out]

1/24/d*exp(3*d*x+3*c)*a^3+1/8/d*exp(3*d*x+3*c)*a^2*b+1/8*a*b^2/d*exp(3*d*x+3*c)+1/24/d*exp(3*d*x+3*c)*b^3-3/8/
d*exp(d*x+c)*a^3-27/8/d*exp(d*x+c)*a^2*b-45/8*a/d*exp(d*x+c)*b^2-21/8*b^3/d*exp(d*x+c)-3/8/d*exp(-d*x-c)*a^3+2
7/8/d*exp(-d*x-c)*a^2*b-45/8*a/d*exp(-d*x-c)*b^2+21/8/d*exp(-d*x-c)*b^3+1/24/d*exp(-3*d*x-3*c)*a^3-1/8/d*exp(-
3*d*x-3*c)*a^2*b+1/8*a*b^2/d*exp(-3*d*x-3*c)-1/24/d*exp(-3*d*x-3*c)*b^3-1/960*b*exp(d*x+c)*(2880*a^2*exp(14*d*
x+14*c)+34560*a*b*exp(14*d*x+14*c)+11475*b^2*exp(14*d*x+14*c)+14400*a^2*exp(12*d*x+12*c)+211200*a*b*exp(12*d*x
+12*c)+36775*b^2*exp(12*d*x+12*c)+25920*a^2*exp(10*d*x+10*c)+590592*a*b*exp(10*d*x+10*c)+67715*b^2*exp(10*d*x+
10*c)+14400*a^2*exp(8*d*x+8*c)+957696*a*b*exp(8*d*x+8*c)+27055*b^2*exp(8*d*x+8*c)-14400*a^2*exp(6*d*x+6*c)+957
696*a*b*exp(6*d*x+6*c)-27055*b^2*exp(6*d*x+6*c)-25920*a^2*exp(4*d*x+4*c)+590592*a*b*exp(4*d*x+4*c)-67715*b^2*e
xp(4*d*x+4*c)-14400*a^2*exp(2*d*x+2*c)+211200*a*b*exp(2*d*x+2*c)-36775*b^2*exp(2*d*x+2*c)-2880*a^2+34560*a*b-1
1475*b^2)/d/(1+exp(2*d*x+2*c))^8-1155/128*I*b^3/d*ln(exp(d*x+c)-I)+1155/128*I*b^3/d*ln(exp(d*x+c)+I)-15/2*I*b/
d*ln(exp(d*x+c)-I)*a^2+15/2*I*b/d*ln(exp(d*x+c)+I)*a^2

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Maxima [A]
time = 0.49, size = 604, normalized size = 1.72 \begin {gather*} \frac {1}{192} \, b^{3} {\left (\frac {8 \, {\left (63 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d} - \frac {3465 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {440 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6103 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21019 \, e^{\left (-6 \, d x - 6 \, c\right )} + 41207 \, e^{\left (-8 \, d x - 8 \, c\right )} + 40243 \, e^{\left (-10 \, d x - 10 \, c\right )} + 22589 \, e^{\left (-12 \, d x - 12 \, c\right )} + 505 \, e^{\left (-14 \, d x - 14 \, c\right )} - 3331 \, e^{\left (-16 \, d x - 16 \, c\right )} - 1791 \, e^{\left (-18 \, d x - 18 \, c\right )} - 8}{d {\left (e^{\left (-3 \, d x - 3 \, c\right )} + 8 \, e^{\left (-5 \, d x - 5 \, c\right )} + 28 \, e^{\left (-7 \, d x - 7 \, c\right )} + 56 \, e^{\left (-9 \, d x - 9 \, c\right )} + 70 \, e^{\left (-11 \, d x - 11 \, c\right )} + 56 \, e^{\left (-13 \, d x - 13 \, c\right )} + 28 \, e^{\left (-15 \, d x - 15 \, c\right )} + 8 \, e^{\left (-17 \, d x - 17 \, c\right )} + e^{\left (-19 \, d x - 19 \, c\right )}\right )}}\right )} - \frac {1}{40} \, a 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}{8} \, a^{2} 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^{3} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 8462 vs. \(2 (325) = 650\).
time = 0.44, size = 8462, normalized size = 24.11 \begin {gather*} \text {Too large to display} \end {gather*}

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)*co...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Giac [A]
time = 0.82, size = 580, normalized size = 1.65 \begin {gather*} \frac {40 \, a^{3} e^{\left (3 \, d x + 3 \, c\right )} + 120 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} + 120 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 40 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 360 \, a^{3} e^{\left (d x + c\right )} - 3240 \, a^{2} b e^{\left (d x + c\right )} - 5400 \, a b^{2} e^{\left (d x + c\right )} - 2520 \, b^{3} e^{\left (d x + c\right )} + 225 \, {\left (64 \, a^{2} b + 77 \, b^{3}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) - 40 \, {\left (9 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 81 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 135 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 63 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - a^{3} + 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} e^{\left (-3 \, d x - 3 \, c\right )} - \frac {2880 \, a^{2} b e^{\left (15 \, d x + 15 \, c\right )} + 34560 \, a b^{2} e^{\left (15 \, d x + 15 \, c\right )} + 11475 \, b^{3} e^{\left (15 \, d x + 15 \, c\right )} + 14400 \, a^{2} b e^{\left (13 \, d x + 13 \, c\right )} + 211200 \, a b^{2} e^{\left (13 \, d x + 13 \, c\right )} + 36775 \, b^{3} e^{\left (13 \, d x + 13 \, c\right )} + 25920 \, a^{2} b e^{\left (11 \, d x + 11 \, c\right )} + 590592 \, a b^{2} e^{\left (11 \, d x + 11 \, c\right )} + 67715 \, b^{3} e^{\left (11 \, d x + 11 \, c\right )} + 14400 \, a^{2} b e^{\left (9 \, d x + 9 \, c\right )} + 957696 \, a b^{2} e^{\left (9 \, d x + 9 \, c\right )} + 27055 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )} - 14400 \, a^{2} b e^{\left (7 \, d x + 7 \, c\right )} + 957696 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} - 27055 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} - 25920 \, a^{2} b e^{\left (5 \, d x + 5 \, c\right )} + 590592 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 67715 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} - 14400 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} + 211200 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 36775 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 2880 \, a^{2} b e^{\left (d x + c\right )} + 34560 \, a b^{2} e^{\left (d x + c\right )} - 11475 \, b^{3} e^{\left (d x + c\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{8}}}{960 \, d} \end {gather*}

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*(40*a^3*e^(3*d*x + 3*c) + 120*a^2*b*e^(3*d*x + 3*c) + 120*a*b^2*e^(3*d*x + 3*c) + 40*b^3*e^(3*d*x + 3*c)
- 360*a^3*e^(d*x + c) - 3240*a^2*b*e^(d*x + c) - 5400*a*b^2*e^(d*x + c) - 2520*b^3*e^(d*x + c) + 225*(64*a^2*
b + 77*b^3)*arctan(e^(d*x + 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) - (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

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Mupad [B]
time = 1.60, size = 757, normalized size = 2.16 \begin {gather*} \frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,{\left (a+b\right )}^3}{24\,d}+\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}\,{\left (a-b\right )}^3}{24\,d}+\frac {15\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (77\,b^3\,\sqrt {d^2}+64\,a^2\,b\,\sqrt {d^2}\right )}{d\,\sqrt {4096\,a^4\,b^2+9856\,a^2\,b^4+5929\,b^6}}\right )\,\sqrt {4096\,a^4\,b^2+9856\,a^2\,b^4+5929\,b^6}}{64\,\sqrt {d^2}}-\frac {3\,{\mathrm {e}}^{-c-d\,x}\,{\left (a-b\right )}^2\,\left (a-7\,b\right )}{8\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (11005\,b^3+6144\,a\,b^2\right )}{120\,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 (3365\,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 {596\,b^3\,{\mathrm {e}}^{c+d\,x}}{3\,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 {3\,{\mathrm {e}}^{c+d\,x}\,{\left (a+b\right )}^2\,\left (a+7\,b\right )}{8\,d}-\frac {3\,{\mathrm {e}}^{c+d\,x}\,\left (64\,a^2\,b+768\,a\,b^2+255\,b^3\right )}{64\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (1625\,b^3+144\,a\,b^2\right )}{15\,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 (576\,a^2\,b+3072\,a\,b^2+4355\,b^3\right )}{96\,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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(exp(3*c + 3*d*x)*(a + b)^3)/(24*d) + (exp(- 3*c - 3*d*x)*(a - b)^3)/(24*d) + (15*atan((exp(d*x)*exp(c)*(77*b^
3*(d^2)^(1/2) + 64*a^2*b*(d^2)^(1/2)))/(d*(5929*b^6 + 9856*a^2*b^4 + 4096*a^4*b^2)^(1/2)))*(5929*b^6 + 9856*a^
2*b^4 + 4096*a^4*b^2)^(1/2))/(64*(d^2)^(1/2)) - (3*exp(- c - d*x)*(a - b)^2*(a - 7*b))/(8*d) - (exp(c + d*x)*(
6144*a*b^2 + 11005*b^3))/(120*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 + 3365*b^3))/(20*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)) + (596*b^3*exp(c + d*x))/(3*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)) - (3*exp(c + d*x)*(a + b)^2*(a + 7*b))/(8*d
) - (3*exp(c + d*x)*(768*a*b^2 + 64*a^2*b + 255*b^3))/(64*d*(exp(2*c + 2*d*x) + 1)) - (2*exp(c + d*x)*(144*a*b
^2 + 1625*b^3))/(15*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)*(3072*a*b^2 + 576*a^2*b + 4355*b^3))/(96*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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