\(\int \sinh (c+d x) (a+b \tanh ^3(c+d x))^3 \, dx\) [68]

Optimal result

Integrand size = 21, antiderivative size = 269 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=-\frac {9 a^2 b \arctan (\sinh (c+d x))}{2 d}-\frac {315 b^3 \arctan (\sinh (c+d x))}{128 d}+\frac {a^3 \cosh (c+d x)}{d}+\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {9 a b^2 \text {sech}(c+d x)}{d}-\frac {3 a b^2 \text {sech}^3(c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}+\frac {9 a^2 b \sinh (c+d x)}{2 d}+\frac {315 b^3 \sinh (c+d x)}{128 d}-\frac {3 a^2 b \sinh (c+d x) \tanh ^2(c+d x)}{2 d}-\frac {105 b^3 \sinh (c+d x) \tanh ^2(c+d x)}{128 d}-\frac {21 b^3 \sinh (c+d x) \tanh ^4(c+d x)}{64 d}-\frac {3 b^3 \sinh (c+d x) \tanh ^6(c+d x)}{16 d}-\frac {b^3 \sinh (c+d x) \tanh ^8(c+d x)}{8 d} \]

[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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3747, 2718, 2672, 294, 327, 209, 2670, 276} \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\frac {a^3 \cosh (c+d x)}{d}-\frac {9 a^2 b \arctan (\sinh (c+d x))}{2 d}+\frac {9 a^2 b \sinh (c+d x)}{2 d}-\frac {3 a^2 b \sinh (c+d x) \tanh ^2(c+d x)}{2 d}+\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}-\frac {3 a b^2 \text {sech}^3(c+d x)}{d}+\frac {9 a b^2 \text {sech}(c+d x)}{d}-\frac {315 b^3 \arctan (\sinh (c+d x))}{128 d}+\frac {315 b^3 \sinh (c+d x)}{128 d}-\frac {b^3 \sinh (c+d x) \tanh ^8(c+d x)}{8 d}-\frac {3 b^3 \sinh (c+d x) \tanh ^6(c+d x)}{16 d}-\frac {21 b^3 \sinh (c+d x) \tanh ^4(c+d x)}{64 d}-\frac {105 b^3 \sinh (c+d x) \tanh ^2(c+d x)}{128 d} \]

[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 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 327

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 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 2718

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

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*} \text {integral}& = -\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 ) \text {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 ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{x^6} \, dx,x,\cosh (c+d x)\right )}{d}+\frac {b^3 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 d}+\frac {\left (21 b^3\right ) \text {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 \arctan (\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 ) \text {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 \arctan (\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 ) \text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{128 d} \\ & = -\frac {9 a^2 b \arctan (\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 ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{128 d} \\ & = -\frac {9 a^2 b \arctan (\sinh (c+d x))}{2 d}-\frac {315 b^3 \arctan (\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*}

Mathematica [A] (verified)

Time = 7.69 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.87 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=-\frac {9 \left (64 a^2 b+35 b^3\right ) \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{64 d}+\frac {a \left (a^2+3 b^2\right ) \cosh (c+d x)}{d}+\frac {9 a b^2 \text {sech}(c+d x)}{d}-\frac {3 a b^2 \text {sech}^3(c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}+\frac {b \left (3 a^2+b^2\right ) \sinh (c+d x)}{d}+\frac {\text {sech}^2(c+d x) \left (192 a^2 b \sinh (c+d x)+325 b^3 \sinh (c+d x)\right )}{128 d}-\frac {105 b^3 \text {sech}^3(c+d x) \tanh (c+d x)}{64 d}+\frac {11 b^3 \text {sech}^5(c+d x) \tanh (c+d x)}{16 d}-\frac {b^3 \text {sech}^7(c+d x) \tanh (c+d x)}{8 d} \]

[In]

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

[Out]

(-9*(64*a^2*b + 35*b^3)*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)

Maple [A] (verified)

Time = 5.72 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.10

method	result	size
derivativedivides	\(\frac {a^{3} \cosh \left (d x +c \right )+3 a^{2} b \left (\frac {\sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{2}}+\frac {3 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}-\frac {3 \,\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}-3 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+3 a \,b^{2} \left (\frac {\sinh \left (d x +c \right )^{6}}{\cosh \left (d x +c \right )^{5}}+\frac {6 \sinh \left (d x +c \right )^{4}}{\cosh \left (d x +c \right )^{5}}+\frac {8 \sinh \left (d x +c \right )^{2}}{\cosh \left (d x +c \right )^{5}}+\frac {16}{5 \cosh \left (d x +c \right )^{5}}\right )+b^{3} \left (\frac {\sinh \left (d x +c \right )^{9}}{\cosh \left (d x +c \right )^{8}}+\frac {9 \sinh \left (d x +c \right )^{7}}{\cosh \left (d x +c \right )^{8}}+\frac {21 \sinh \left (d x +c \right )^{5}}{\cosh \left (d x +c \right )^{8}}+\frac {21 \sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{8}}+\frac {9 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{8}}-9 \left (\frac {\operatorname {sech}\left (d x +c \right )^{7}}{8}+\frac {7 \operatorname {sech}\left (d x +c \right )^{5}}{48}+\frac {35 \operatorname {sech}\left (d x +c \right )^{3}}{192}+\frac {35 \,\operatorname {sech}\left (d x +c \right )}{128}\right ) \tanh \left (d x +c \right )-\frac {315 \arctan \left ({\mathrm e}^{d x +c}\right )}{64}\right )}{d}\)	\(297\)
default	\(\frac {a^{3} \cosh \left (d x +c \right )+3 a^{2} b \left (\frac {\sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{2}}+\frac {3 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}-\frac {3 \,\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}-3 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+3 a \,b^{2} \left (\frac {\sinh \left (d x +c \right )^{6}}{\cosh \left (d x +c \right )^{5}}+\frac {6 \sinh \left (d x +c \right )^{4}}{\cosh \left (d x +c \right )^{5}}+\frac {8 \sinh \left (d x +c \right )^{2}}{\cosh \left (d x +c \right )^{5}}+\frac {16}{5 \cosh \left (d x +c \right )^{5}}\right )+b^{3} \left (\frac {\sinh \left (d x +c \right )^{9}}{\cosh \left (d x +c \right )^{8}}+\frac {9 \sinh \left (d x +c \right )^{7}}{\cosh \left (d x +c \right )^{8}}+\frac {21 \sinh \left (d x +c \right )^{5}}{\cosh \left (d x +c \right )^{8}}+\frac {21 \sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{8}}+\frac {9 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{8}}-9 \left (\frac {\operatorname {sech}\left (d x +c \right )^{7}}{8}+\frac {7 \operatorname {sech}\left (d x +c \right )^{5}}{48}+\frac {35 \operatorname {sech}\left (d x +c \right )^{3}}{192}+\frac {35 \,\operatorname {sech}\left (d x +c \right )}{128}\right ) \tanh \left (d x +c \right )-\frac {315 \arctan \left ({\mathrm e}^{d x +c}\right )}{64}\right )}{d}\)	\(297\)
risch	\(\frac {{\mathrm e}^{d x +c} a^{3}}{2 d}+\frac {3 \,{\mathrm e}^{d x +c} a^{2} b}{2 d}+\frac {3 \,{\mathrm e}^{d x +c} a \,b^{2}}{2 d}+\frac {b^{3} {\mathrm e}^{d x +c}}{2 d}+\frac {{\mathrm e}^{-d x -c} a^{3}}{2 d}-\frac {3 \,{\mathrm e}^{-d x -c} a^{2} b}{2 d}+\frac {3 \,{\mathrm e}^{-d x -c} a \,b^{2}}{2 d}-\frac {{\mathrm e}^{-d x -c} b^{3}}{2 d}+\frac {b \,{\mathrm e}^{d x +c} \left (960 a^{2} {\mathrm e}^{14 d x +14 c}+5760 a b \,{\mathrm e}^{14 d x +14 c}+1625 b^{2} {\mathrm e}^{14 d x +14 c}+4800 a^{2} {\mathrm e}^{12 d x +12 c}+32640 a b \,{\mathrm e}^{12 d x +12 c}+3925 b^{2} {\mathrm e}^{12 d x +12 c}+8640 a^{2} {\mathrm e}^{10 d x +10 c}+88704 a b \,{\mathrm e}^{10 d x +10 c}+9065 b^{2} {\mathrm e}^{10 d x +10 c}+4800 a^{2} {\mathrm e}^{8 d x +8 c}+143232 a b \,{\mathrm e}^{8 d x +8 c}+1645 b^{2} {\mathrm e}^{8 d x +8 c}-4800 a^{2} {\mathrm e}^{6 d x +6 c}+143232 a b \,{\mathrm e}^{6 d x +6 c}-1645 b^{2} {\mathrm e}^{6 d x +6 c}-8640 a^{2} {\mathrm e}^{4 d x +4 c}+88704 a b \,{\mathrm e}^{4 d x +4 c}-9065 \,{\mathrm e}^{4 d x +4 c} b^{2}-4800 a^{2} {\mathrm e}^{2 d x +2 c}+32640 a b \,{\mathrm e}^{2 d x +2 c}-3925 \,{\mathrm e}^{2 d x +2 c} b^{2}-960 a^{2}+5760 a b -1625 b^{2}\right )}{320 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{8}}+\frac {9 i b \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{2 d}+\frac {315 i b^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{128 d}-\frac {9 i b \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{2 d}-\frac {315 i b^{3} \ln \left ({\mathrm e}^{d x +c}+i\right )}{128 d}\)	\(509\)

[In]

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

[Out]

1/d*(a^3*cosh(d*x+c)+3*a^2*b*(sinh(d*x+c)^3/cosh(d*x+c)^2+3*sinh(d*x+c)/cosh(d*x+c)^2-3/2*sech(d*x+c)*tanh(d*x
+c)-3*arctan(exp(d*x+c)))+3*a*b^2*(sinh(d*x+c)^6/cosh(d*x+c)^5+6*sinh(d*x+c)^4/cosh(d*x+c)^5+8*sinh(d*x+c)^2/c
osh(d*x+c)^5+16/5/cosh(d*x+c)^5)+b^3*(sinh(d*x+c)^9/cosh(d*x+c)^8+9*sinh(d*x+c)^7/cosh(d*x+c)^8+21*sinh(d*x+c)
^5/cosh(d*x+c)^8+21*sinh(d*x+c)^3/cosh(d*x+c)^8+9*sinh(d*x+c)/cosh(d*x+c)^8-9*(1/8*sech(d*x+c)^7+7/48*sech(d*x
+c)^5+35/192*sech(d*x+c)^3+35/128*sech(d*x+c))*tanh(d*x+c)-315/64*arctan(exp(d*x+c))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6410 vs. \(2 (249) = 498\).

Time = 0.32 (sec) , antiderivative size = 6410, normalized size of antiderivative = 23.83 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[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)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.80 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\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} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.59 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.72 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\frac {160 \, a^{3} e^{\left (d x + c\right )} + 480 \, a^{2} b e^{\left (d x + c\right )} + 480 \, a b^{2} e^{\left (d x + c\right )} + 160 \, b^{3} e^{\left (d x + c\right )} - 45 \, {\left (64 \, a^{2} b + 35 \, b^{3}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) + 160 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} e^{\left (-d x - c\right )} + \frac {960 \, a^{2} b e^{\left (15 \, d x + 15 \, c\right )} + 5760 \, a b^{2} e^{\left (15 \, d x + 15 \, c\right )} + 1625 \, b^{3} e^{\left (15 \, d x + 15 \, c\right )} + 4800 \, a^{2} b e^{\left (13 \, d x + 13 \, c\right )} + 32640 \, a b^{2} e^{\left (13 \, d x + 13 \, c\right )} + 3925 \, b^{3} e^{\left (13 \, d x + 13 \, c\right )} + 8640 \, a^{2} b e^{\left (11 \, d x + 11 \, c\right )} + 88704 \, a b^{2} e^{\left (11 \, d x + 11 \, c\right )} + 9065 \, b^{3} e^{\left (11 \, d x + 11 \, c\right )} + 4800 \, a^{2} b e^{\left (9 \, d x + 9 \, c\right )} + 143232 \, a b^{2} e^{\left (9 \, d x + 9 \, c\right )} + 1645 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )} - 4800 \, a^{2} b e^{\left (7 \, d x + 7 \, c\right )} + 143232 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} - 1645 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} - 8640 \, a^{2} b e^{\left (5 \, d x + 5 \, c\right )} + 88704 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 9065 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} - 4800 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} + 32640 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 3925 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 960 \, a^{2} b e^{\left (d x + c\right )} + 5760 \, a b^{2} e^{\left (d x + c\right )} - 1625 \, b^{3} e^{\left (d x + c\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{8}}}{320 \, d} \]

[In]

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

[Out]

1/320*(160*a^3*e^(d*x + c) + 480*a^2*b*e^(d*x + c) + 480*a*b^2*e^(d*x + c) + 160*b^3*e^(d*x + c) - 45*(64*a^2*
b + 35*b^3)*arctan(e^(d*x + c)) + 160*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*e^(-d*x - 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

Mupad [B] (verification not implemented)

Time = 2.21 (sec) , antiderivative size = 707, normalized size of antiderivative = 2.63 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\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 )} \]

[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))