\(\int \sinh ^3(c+d x) (a+b \tanh ^2(c+d x))^3 \, dx\) [18]

Optimal result

Integrand size = 23, antiderivative size = 105 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=-\frac {(a+b)^2 (a+4 b) \cosh (c+d x)}{d}+\frac {(a+b)^3 \cosh ^3(c+d x)}{3 d}-\frac {3 b (a+b) (a+2 b) \text {sech}(c+d x)}{d}+\frac {b^2 (3 a+4 b) \text {sech}^3(c+d x)}{3 d}-\frac {b^3 \text {sech}^5(c+d x)}{5 d} \]

[Out]

-(a+b)^2*(a+4*b)*cosh(d*x+c)/d+1/3*(a+b)^3*cosh(d*x+c)^3/d-3*b*(a+b)*(a+2*b)*sech(d*x+c)/d+1/3*b^2*(3*a+4*b)*s
ech(d*x+c)^3/d-1/5*b^3*sech(d*x+c)^5/d

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3745, 459} \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {b^2 (3 a+4 b) \text {sech}^3(c+d x)}{3 d}+\frac {(a+b)^3 \cosh ^3(c+d x)}{3 d}-\frac {(a+b)^2 (a+4 b) \cosh (c+d x)}{d}-\frac {3 b (a+b) (a+2 b) \text {sech}(c+d x)}{d}-\frac {b^3 \text {sech}^5(c+d x)}{5 d} \]

[In]

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

[Out]

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

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rule 3745

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sec[e + f*x], x]}, Dist[1/(f*ff^m), Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*((a - b + b*ff^2*x^2)^p/x^(m
+ 1)), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (-1+x^2\right ) \left (a+b-b x^2\right )^3}{x^4} \, dx,x,\text {sech}(c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (3 (-a-2 b) b (a+b)-\frac {(a+b)^3}{x^4}+\frac {(a+b)^2 (a+4 b)}{x^2}+b^2 (3 a+4 b) x^2-b^3 x^4\right ) \, dx,x,\text {sech}(c+d x)\right )}{d} \\ & = -\frac {(a+b)^2 (a+4 b) \cosh (c+d x)}{d}+\frac {(a+b)^3 \cosh ^3(c+d x)}{3 d}-\frac {3 b (a+b) (a+2 b) \text {sech}(c+d x)}{d}+\frac {b^2 (3 a+4 b) \text {sech}^3(c+d x)}{3 d}-\frac {b^3 \text {sech}^5(c+d x)}{5 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.18 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.87 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {-45 (a+b)^2 (a+5 b) \cosh (c+d x)+5 (a+b)^3 \cosh (3 (c+d x))-180 b (a+b) (a+2 b) \text {sech}(c+d x)+20 b^2 (3 a+4 b) \text {sech}^3(c+d x)-12 b^3 \text {sech}^5(c+d x)}{60 d} \]

[In]

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

[Out]

(-45*(a + b)^2*(a + 5*b)*Cosh[c + d*x] + 5*(a + b)^3*Cosh[3*(c + d*x)] - 180*b*(a + b)*(a + 2*b)*Sech[c + d*x]
+ 20*b^2*(3*a + 4*b)*Sech[c + d*x]^3 - 12*b^3*Sech[c + d*x]^5)/(60*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(238\) vs. \(2(99)=198\).

Time = 5.00 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.28

method	result	size
derivativedivides	\(\frac {a^{3} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+3 a^{2} b \left (\frac {\sinh \left (d x +c \right )^{4}}{3 \cosh \left (d x +c \right )}-\frac {4 \sinh \left (d x +c \right )^{2}}{3 \cosh \left (d x +c \right )}-\frac {8}{3 \cosh \left (d x +c \right )}\right )+3 a \,b^{2} \left (\frac {\sinh \left (d x +c \right )^{6}}{3 \cosh \left (d x +c \right )^{3}}-\frac {2 \sinh \left (d x +c \right )^{4}}{\cosh \left (d x +c \right )^{3}}-\frac {8 \sinh \left (d x +c \right )^{2}}{\cosh \left (d x +c \right )^{3}}-\frac {16}{3 \cosh \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sinh \left (d x +c \right )^{8}}{3 \cosh \left (d x +c \right )^{5}}-\frac {8 \sinh \left (d x +c \right )^{6}}{3 \cosh \left (d x +c \right )^{5}}-\frac {16 \sinh \left (d x +c \right )^{4}}{\cosh \left (d x +c \right )^{5}}-\frac {64 \sinh \left (d x +c \right )^{2}}{3 \cosh \left (d x +c \right )^{5}}-\frac {128}{15 \cosh \left (d x +c \right )^{5}}\right )}{d}\)	\(239\)
default	\(\frac {a^{3} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+3 a^{2} b \left (\frac {\sinh \left (d x +c \right )^{4}}{3 \cosh \left (d x +c \right )}-\frac {4 \sinh \left (d x +c \right )^{2}}{3 \cosh \left (d x +c \right )}-\frac {8}{3 \cosh \left (d x +c \right )}\right )+3 a \,b^{2} \left (\frac {\sinh \left (d x +c \right )^{6}}{3 \cosh \left (d x +c \right )^{3}}-\frac {2 \sinh \left (d x +c \right )^{4}}{\cosh \left (d x +c \right )^{3}}-\frac {8 \sinh \left (d x +c \right )^{2}}{\cosh \left (d x +c \right )^{3}}-\frac {16}{3 \cosh \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sinh \left (d x +c \right )^{8}}{3 \cosh \left (d x +c \right )^{5}}-\frac {8 \sinh \left (d x +c \right )^{6}}{3 \cosh \left (d x +c \right )^{5}}-\frac {16 \sinh \left (d x +c \right )^{4}}{\cosh \left (d x +c \right )^{5}}-\frac {64 \sinh \left (d x +c \right )^{2}}{3 \cosh \left (d x +c \right )^{5}}-\frac {128}{15 \cosh \left (d x +c \right )^{5}}\right )}{d}\)	\(239\)
risch	\(\frac {{\mathrm e}^{3 d x +3 c} a^{3}}{24 d}+\frac {{\mathrm e}^{3 d x +3 c} a^{2} b}{8 d}+\frac {{\mathrm e}^{3 d x +3 c} a \,b^{2}}{8 d}+\frac {{\mathrm e}^{3 d x +3 c} b^{3}}{24 d}-\frac {3 \,{\mathrm e}^{d x +c} a^{3}}{8 d}-\frac {21 \,{\mathrm e}^{d x +c} a^{2} b}{8 d}-\frac {33 \,{\mathrm e}^{d x +c} a \,b^{2}}{8 d}-\frac {15 b^{3} {\mathrm e}^{d x +c}}{8 d}-\frac {3 \,{\mathrm e}^{-d x -c} a^{3}}{8 d}-\frac {21 \,{\mathrm e}^{-d x -c} a^{2} b}{8 d}-\frac {33 \,{\mathrm e}^{-d x -c} a \,b^{2}}{8 d}-\frac {15 \,{\mathrm e}^{-d x -c} b^{3}}{8 d}+\frac {{\mathrm e}^{-3 d x -3 c} a^{3}}{24 d}+\frac {{\mathrm e}^{-3 d x -3 c} a^{2} b}{8 d}+\frac {{\mathrm e}^{-3 d x -3 c} a \,b^{2}}{8 d}+\frac {{\mathrm e}^{-3 d x -3 c} b^{3}}{24 d}-\frac {2 b \,{\mathrm e}^{d x +c} \left (45 a^{2} {\mathrm e}^{8 d x +8 c}+135 a b \,{\mathrm e}^{8 d x +8 c}+90 b^{2} {\mathrm e}^{8 d x +8 c}+180 a^{2} {\mathrm e}^{6 d x +6 c}+480 a b \,{\mathrm e}^{6 d x +6 c}+280 b^{2} {\mathrm e}^{6 d x +6 c}+270 a^{2} {\mathrm e}^{4 d x +4 c}+690 a b \,{\mathrm e}^{4 d x +4 c}+428 \,{\mathrm e}^{4 d x +4 c} b^{2}+180 a^{2} {\mathrm e}^{2 d x +2 c}+480 a b \,{\mathrm e}^{2 d x +2 c}+280 \,{\mathrm e}^{2 d x +2 c} b^{2}+45 a^{2}+135 a b +90 b^{2}\right )}{15 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{5}}\)	\(474\)

[In]

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

[Out]

1/d*(a^3*(-2/3+1/3*sinh(d*x+c)^2)*cosh(d*x+c)+3*a^2*b*(1/3*sinh(d*x+c)^4/cosh(d*x+c)-4/3*sinh(d*x+c)^2/cosh(d*
x+c)-8/3/cosh(d*x+c))+3*a*b^2*(1/3*sinh(d*x+c)^6/cosh(d*x+c)^3-2*sinh(d*x+c)^4/cosh(d*x+c)^3-8*sinh(d*x+c)^2/c
osh(d*x+c)^3-16/3/cosh(d*x+c)^3)+b^3*(1/3*sinh(d*x+c)^8/cosh(d*x+c)^5-8/3*sinh(d*x+c)^6/cosh(d*x+c)^5-16*sinh(
d*x+c)^4/cosh(d*x+c)^5-64/3*sinh(d*x+c)^2/cosh(d*x+c)^5-128/15/cosh(d*x+c)^5))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 540 vs. \(2 (99) = 198\).

Time = 0.25 (sec) , antiderivative size = 540, normalized size of antiderivative = 5.14 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {5 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{8} + 5 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sinh \left (d x + c\right )^{8} - 20 \, {\left (a^{3} + 12 \, a^{2} b + 21 \, a b^{2} + 10 \, b^{3}\right )} \cosh \left (d x + c\right )^{6} - 20 \, {\left (a^{3} + 12 \, a^{2} b + 21 \, a b^{2} + 10 \, b^{3} - 7 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{6} - 20 \, {\left (11 \, a^{3} + 123 \, a^{2} b + 249 \, a b^{2} + 137 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 10 \, {\left (35 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{4} - 22 \, a^{3} - 246 \, a^{2} b - 498 \, a b^{2} - 274 \, b^{3} - 30 \, {\left (a^{3} + 12 \, a^{2} b + 21 \, a b^{2} + 10 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{4} - 425 \, a^{3} - 5235 \, a^{2} b - 10395 \, a b^{2} - 5649 \, b^{3} - 20 \, {\left (31 \, a^{3} + 372 \, a^{2} b + 747 \, a b^{2} + 390 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + 20 \, {\left (7 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{6} - 15 \, {\left (a^{3} + 12 \, a^{2} b + 21 \, a b^{2} + 10 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} - 31 \, a^{3} - 372 \, a^{2} b - 747 \, a b^{2} - 390 \, b^{3} - 6 \, {\left (11 \, a^{3} + 123 \, a^{2} b + 249 \, a b^{2} + 137 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2}}{120 \, {\left (d \cosh \left (d x + c\right )^{5} + 5 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 5 \, d \cosh \left (d x + c\right )^{3} + 5 \, {\left (2 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \, d \cosh \left (d x + c\right )\right )}} \]

[In]

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

[Out]

1/120*(5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^8 + 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sinh(d*x + c)^8 -
20*(a^3 + 12*a^2*b + 21*a*b^2 + 10*b^3)*cosh(d*x + c)^6 - 20*(a^3 + 12*a^2*b + 21*a*b^2 + 10*b^3 - 7*(a^3 + 3
*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 - 20*(11*a^3 + 123*a^2*b + 249*a*b^2 + 137*b^3)*cosh(
d*x + c)^4 + 10*(35*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 - 22*a^3 - 246*a^2*b - 498*a*b^2 - 274*b^3
- 30*(a^3 + 12*a^2*b + 21*a*b^2 + 10*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 - 425*a^3 - 5235*a^2*b - 10395*a*b
^2 - 5649*b^3 - 20*(31*a^3 + 372*a^2*b + 747*a*b^2 + 390*b^3)*cosh(d*x + c)^2 + 20*(7*(a^3 + 3*a^2*b + 3*a*b^2
+ b^3)*cosh(d*x + c)^6 - 15*(a^3 + 12*a^2*b + 21*a*b^2 + 10*b^3)*cosh(d*x + c)^4 - 31*a^3 - 372*a^2*b - 747*a
*b^2 - 390*b^3 - 6*(11*a^3 + 123*a^2*b + 249*a*b^2 + 137*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2)/(d*cosh(d*x +
c)^5 + 5*d*cosh(d*x + c)*sinh(d*x + c)^4 + 5*d*cosh(d*x + c)^3 + 5*(2*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*s
inh(d*x + c)^2 + 10*d*cosh(d*x + c))

Sympy [F]

\[ \int \sinh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3} \sinh ^{3}{\left (c + d x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (99) = 198\).

Time = 0.21 (sec) , antiderivative size = 439, normalized size of antiderivative = 4.18 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=-\frac {1}{120} \, b^{3} {\left (\frac {5 \, {\left (45 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d} + \frac {200 \, e^{\left (-2 \, d x - 2 \, c\right )} + 2515 \, e^{\left (-4 \, d x - 4 \, c\right )} + 6680 \, e^{\left (-6 \, d x - 6 \, c\right )} + 9073 \, e^{\left (-8 \, d x - 8 \, c\right )} + 5600 \, e^{\left (-10 \, d x - 10 \, c\right )} + 1665 \, e^{\left (-12 \, d x - 12 \, c\right )} - 5}{d {\left (e^{\left (-3 \, d x - 3 \, c\right )} + 5 \, e^{\left (-5 \, d x - 5 \, c\right )} + 10 \, e^{\left (-7 \, d x - 7 \, c\right )} + 10 \, e^{\left (-9 \, d x - 9 \, c\right )} + 5 \, e^{\left (-11 \, d x - 11 \, c\right )} + e^{\left (-13 \, d x - 13 \, c\right )}\right )}}\right )} - \frac {1}{8} \, a b^{2} {\left (\frac {33 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {30 \, e^{\left (-2 \, d x - 2 \, c\right )} + 240 \, e^{\left (-4 \, d x - 4 \, c\right )} + 322 \, e^{\left (-6 \, d x - 6 \, c\right )} + 177 \, e^{\left (-8 \, d x - 8 \, c\right )} - 1}{d {\left (e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-5 \, d x - 5 \, c\right )} + 3 \, e^{\left (-7 \, d x - 7 \, c\right )} + e^{\left (-9 \, d x - 9 \, c\right )}\right )}}\right )} - \frac {1}{8} \, a^{2} b {\left (\frac {21 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {20 \, e^{\left (-2 \, d x - 2 \, c\right )} + 69 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1}{d {\left (e^{\left (-3 \, d x - 3 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )}\right )}}\right )} + \frac {1}{24} \, a^{3} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \]

[In]

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

[Out]

-1/120*b^3*(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*b^2*((33*e^(-d*x - c) - e^(-3*d*x - 3*c))/d + (30*e^(-2*d*x - 2*c) + 240*e^(-4*d*x - 4*c) + 3
22*e^(-6*d*x - 6*c) + 177*e^(-8*d*x - 8*c) - 1)/(d*(e^(-3*d*x - 3*c) + 3*e^(-5*d*x - 5*c) + 3*e^(-7*d*x - 7*c)
+ e^(-9*d*x - 9*c)))) - 1/8*a^2*b*((21*e^(-d*x - c) - e^(-3*d*x - 3*c))/d + (20*e^(-2*d*x - 2*c) + 69*e^(-4*d
*x - 4*c) - 1)/(d*(e^(-3*d*x - 3*c) + e^(-5*d*x - 5*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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 330 vs. \(2 (99) = 198\).

Time = 0.50 (sec) , antiderivative size = 330, normalized size of antiderivative = 3.14 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {5 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 15 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 15 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 5 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 60 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 360 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 540 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 240 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - \frac {16 \, {\left (45 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4} + 135 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4} + 90 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4} - 60 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 80 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 48 \, b^{3}\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5}}}{120 \, d} \]

[In]

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

[Out]

1/120*(5*a^3*(e^(d*x + c) + e^(-d*x - c))^3 + 15*a^2*b*(e^(d*x + c) + e^(-d*x - c))^3 + 15*a*b^2*(e^(d*x + c)
+ e^(-d*x - c))^3 + 5*b^3*(e^(d*x + c) + e^(-d*x - c))^3 - 60*a^3*(e^(d*x + c) + e^(-d*x - c)) - 360*a^2*b*(e^
(d*x + c) + e^(-d*x - c)) - 540*a*b^2*(e^(d*x + c) + e^(-d*x - c)) - 240*b^3*(e^(d*x + c) + e^(-d*x - c)) - 16
*(45*a^2*b*(e^(d*x + c) + e^(-d*x - c))^4 + 135*a*b^2*(e^(d*x + c) + e^(-d*x - c))^4 + 90*b^3*(e^(d*x + c) + e
^(-d*x - c))^4 - 60*a*b^2*(e^(d*x + c) + e^(-d*x - c))^2 - 80*b^3*(e^(d*x + c) + e^(-d*x - c))^2 + 48*b^3)/(e^
(d*x + c) + e^(-d*x - c))^5)/d

Mupad [B] (verification not implemented)

Time = 2.03 (sec) , antiderivative size = 361, normalized size of antiderivative = 3.44 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}\,{\left (a+b\right )}^3}{24\,d}+\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,{\left (a+b\right )}^3}{24\,d}+\frac {8\,{\mathrm {e}}^{c+d\,x}\,\left (4\,b^3+3\,a\,b^2\right )}{3\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}+\frac {64\,b^3\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {8\,{\mathrm {e}}^{c+d\,x}\,\left (32\,b^3+15\,a\,b^2\right )}{15\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {32\,b^3\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}-\frac {3\,{\mathrm {e}}^{c+d\,x}\,{\left (a+b\right )}^2\,\left (a+5\,b\right )}{8\,d}-\frac {6\,{\mathrm {e}}^{c+d\,x}\,\left (a^2\,b+3\,a\,b^2+2\,b^3\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {3\,{\mathrm {e}}^{-c-d\,x}\,{\left (a+b\right )}^2\,\left (a+5\,b\right )}{8\,d} \]

[In]

int(sinh(c + d*x)^3*(a + b*tanh(c + d*x)^2)^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) + (8*exp(c + d*x)*(3*a*b^2 + 4*b^3
))/(3*d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) + (64*b^3*exp(c + d*x))/(5*d*(4*exp(2*c + 2*d*x) + 6*exp(
4*c + 4*d*x) + 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1)) - (8*exp(c + d*x)*(15*a*b^2 + 32*b^3))/(15*d*(3*exp
(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) - (32*b^3*exp(c + d*x))/(5*d*(5*exp(2*c + 2*d*x) +
10*exp(4*c + 4*d*x) + 10*exp(6*c + 6*d*x) + 5*exp(8*c + 8*d*x) + exp(10*c + 10*d*x) + 1)) - (3*exp(c + d*x)*(
a + b)^2*(a + 5*b))/(8*d) - (6*exp(c + d*x)*(3*a*b^2 + a^2*b + 2*b^3))/(d*(exp(2*c + 2*d*x) + 1)) - (3*exp(- c
- d*x)*(a + b)^2*(a + 5*b))/(8*d)