### 3.145 $$\int \tanh ^3(c+d x) (a+b \tanh ^2(c+d x))^2 \, dx$$

Optimal. Leaf size=76 $-\frac{b (2 a+b) \tanh ^4(c+d x)}{4 d}-\frac{(a+b)^2 \tanh ^2(c+d x)}{2 d}+\frac{(a+b)^2 \log (\cosh (c+d x))}{d}-\frac{b^2 \tanh ^6(c+d x)}{6 d}$

[Out]

((a + b)^2*Log[Cosh[c + d*x]])/d - ((a + b)^2*Tanh[c + d*x]^2)/(2*d) - (b*(2*a + b)*Tanh[c + d*x]^4)/(4*d) - (
b^2*Tanh[c + d*x]^6)/(6*d)

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Rubi [A]  time = 0.10963, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.13, Rules used = {3670, 446, 77} $-\frac{b (2 a+b) \tanh ^4(c+d x)}{4 d}-\frac{(a+b)^2 \tanh ^2(c+d x)}{2 d}+\frac{(a+b)^2 \log (\cosh (c+d x))}{d}-\frac{b^2 \tanh ^6(c+d x)}{6 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((a + b)^2*Log[Cosh[c + d*x]])/d - ((a + b)^2*Tanh[c + d*x]^2)/(2*d) - (b*(2*a + b)*Tanh[c + d*x]^4)/(4*d) - (
b^2*Tanh[c + d*x]^6)/(6*d)

Rule 3670

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff)/f, Subst[Int[(((d*ff*x)/c)^m*(a + b*(ff*x)^n)^p)/(c^
2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \tanh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3 \left (a+b x^2\right )^2}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x (a+b x)^2}{1-x} \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-(a+b)^2-\frac{(a+b)^2}{-1+x}-b (2 a+b) x-b^2 x^2\right ) \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=\frac{(a+b)^2 \log (\cosh (c+d x))}{d}-\frac{(a+b)^2 \tanh ^2(c+d x)}{2 d}-\frac{b (2 a+b) \tanh ^4(c+d x)}{4 d}-\frac{b^2 \tanh ^6(c+d x)}{6 d}\\ \end{align*}

Mathematica [A]  time = 0.407834, size = 66, normalized size = 0.87 $-\frac{3 b (2 a+b) \tanh ^4(c+d x)+6 (a+b)^2 \tanh ^2(c+d x)-12 (a+b)^2 \log (\cosh (c+d x))+2 b^2 \tanh ^6(c+d x)}{12 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(-12*(a + b)^2*Log[Cosh[c + d*x]] + 6*(a + b)^2*Tanh[c + d*x]^2 + 3*b*(2*a + b)*Tanh[c + d*x]^4 + 2*b^2*Tanh[
c + d*x]^6)/(12*d)

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Maple [B]  time = 0.004, size = 196, normalized size = 2.6 \begin{align*} -{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ){a}^{2}}{2\,d}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ) ab}{d}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ){b}^{2}}{2\,d}}-{\frac{{a}^{2}\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) }{2\,d}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) ab}{d}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ){b}^{2}}{2\,d}}-{\frac{ \left ( \tanh \left ( dx+c \right ) \right ) ^{2}{a}^{2}}{2\,d}}-{\frac{{b}^{2} \left ( \tanh \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{ \left ( \tanh \left ( dx+c \right ) \right ) ^{4}{b}^{2}}{4\,d}}-{\frac{{b}^{2} \left ( \tanh \left ( dx+c \right ) \right ) ^{6}}{6\,d}}-{\frac{ab \left ( \tanh \left ( dx+c \right ) \right ) ^{2}}{d}}-{\frac{ \left ( \tanh \left ( dx+c \right ) \right ) ^{4}ab}{2\,d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(d*x+c)^3*(a+b*tanh(d*x+c)^2)^2,x)

[Out]

-1/2/d*ln(tanh(d*x+c)+1)*a^2-1/d*ln(tanh(d*x+c)+1)*a*b-1/2/d*ln(tanh(d*x+c)+1)*b^2-1/2*a^2/d*ln(tanh(d*x+c)-1)
-1/d*ln(tanh(d*x+c)-1)*a*b-1/2/d*ln(tanh(d*x+c)-1)*b^2-1/2/d*tanh(d*x+c)^2*a^2-1/2*b^2*tanh(d*x+c)^2/d-1/4/d*t
anh(d*x+c)^4*b^2-1/6*b^2*tanh(d*x+c)^6/d-a*b*tanh(d*x+c)^2/d-1/2/d*tanh(d*x+c)^4*a*b

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Maxima [B]  time = 1.78954, size = 450, normalized size = 5.92 \begin{align*} \frac{1}{3} \, b^{2}{\left (3 \, x + \frac{3 \, c}{d} + \frac{3 \, \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} + \frac{2 \,{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} + 18 \, e^{\left (-4 \, d x - 4 \, c\right )} + 34 \, e^{\left (-6 \, d x - 6 \, c\right )} + 18 \, e^{\left (-8 \, d x - 8 \, c\right )} + 9 \, e^{\left (-10 \, d x - 10 \, c\right )}\right )}}{d{\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} + 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} + 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} + 1\right )}}\right )} + 2 \, a b{\left (x + \frac{c}{d} + \frac{\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} + \frac{4 \,{\left (e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{d{\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} + a^{2}{\left (x + \frac{c}{d} + \frac{\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} + \frac{2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^2*(3*x + 3*c/d + 3*log(e^(-2*d*x - 2*c) + 1)/d + 2*(9*e^(-2*d*x - 2*c) + 18*e^(-4*d*x - 4*c) + 34*e^(-6*
d*x - 6*c) + 18*e^(-8*d*x - 8*c) + 9*e^(-10*d*x - 10*c))/(d*(6*e^(-2*d*x - 2*c) + 15*e^(-4*d*x - 4*c) + 20*e^(
-6*d*x - 6*c) + 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) + e^(-12*d*x - 12*c) + 1))) + 2*a*b*(x + c/d + log(
e^(-2*d*x - 2*c) + 1)/d + 4*(e^(-2*d*x - 2*c) + e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c))/(d*(4*e^(-2*d*x - 2*c) +
6*e^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c) + 1))) + a^2*(x + c/d + log(e^(-2*d*x - 2*c) + 1)/d
+ 2*e^(-2*d*x - 2*c)/(d*(2*e^(-2*d*x - 2*c) + e^(-4*d*x - 4*c) + 1)))

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Fricas [B]  time = 2.34171, size = 8982, normalized size = 118.18 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^12 + 36*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)*sinh(d*x + c)^11 +
3*(a^2 + 2*a*b + b^2)*d*x*sinh(d*x + c)^12 + 6*(3*(a^2 + 2*a*b + b^2)*d*x - a^2 - 4*a*b - 3*b^2)*cosh(d*x + c
)^10 + 6*(33*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^2 + 3*(a^2 + 2*a*b + b^2)*d*x - a^2 - 4*a*b - 3*b^2)*sinh(d
*x + c)^10 + 60*(11*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^3 + (3*(a^2 + 2*a*b + b^2)*d*x - a^2 - 4*a*b - 3*b^2
)*cosh(d*x + c))*sinh(d*x + c)^9 + 3*(15*(a^2 + 2*a*b + b^2)*d*x - 8*a^2 - 24*a*b - 12*b^2)*cosh(d*x + c)^8 +
3*(495*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^4 + 15*(a^2 + 2*a*b + b^2)*d*x + 90*(3*(a^2 + 2*a*b + b^2)*d*x -
a^2 - 4*a*b - 3*b^2)*cosh(d*x + c)^2 - 8*a^2 - 24*a*b - 12*b^2)*sinh(d*x + c)^8 + 24*(99*(a^2 + 2*a*b + b^2)*d
*x*cosh(d*x + c)^5 + 30*(3*(a^2 + 2*a*b + b^2)*d*x - a^2 - 4*a*b - 3*b^2)*cosh(d*x + c)^3 + (15*(a^2 + 2*a*b +
b^2)*d*x - 8*a^2 - 24*a*b - 12*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 + 4*(15*(a^2 + 2*a*b + b^2)*d*x - 9*a^2 -
24*a*b - 17*b^2)*cosh(d*x + c)^6 + 4*(693*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^6 + 315*(3*(a^2 + 2*a*b + b^2)
*d*x - a^2 - 4*a*b - 3*b^2)*cosh(d*x + c)^4 + 15*(a^2 + 2*a*b + b^2)*d*x + 21*(15*(a^2 + 2*a*b + b^2)*d*x - 8*
a^2 - 24*a*b - 12*b^2)*cosh(d*x + c)^2 - 9*a^2 - 24*a*b - 17*b^2)*sinh(d*x + c)^6 + 24*(99*(a^2 + 2*a*b + b^2)
*d*x*cosh(d*x + c)^7 + 63*(3*(a^2 + 2*a*b + b^2)*d*x - a^2 - 4*a*b - 3*b^2)*cosh(d*x + c)^5 + 7*(15*(a^2 + 2*a
*b + b^2)*d*x - 8*a^2 - 24*a*b - 12*b^2)*cosh(d*x + c)^3 + (15*(a^2 + 2*a*b + b^2)*d*x - 9*a^2 - 24*a*b - 17*b
^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 3*(15*(a^2 + 2*a*b + b^2)*d*x - 8*a^2 - 24*a*b - 12*b^2)*cosh(d*x + c)^4
+ 3*(495*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^8 + 420*(3*(a^2 + 2*a*b + b^2)*d*x - a^2 - 4*a*b - 3*b^2)*cosh(
d*x + c)^6 + 70*(15*(a^2 + 2*a*b + b^2)*d*x - 8*a^2 - 24*a*b - 12*b^2)*cosh(d*x + c)^4 + 15*(a^2 + 2*a*b + b^2
)*d*x + 20*(15*(a^2 + 2*a*b + b^2)*d*x - 9*a^2 - 24*a*b - 17*b^2)*cosh(d*x + c)^2 - 8*a^2 - 24*a*b - 12*b^2)*s
inh(d*x + c)^4 + 4*(165*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^9 + 180*(3*(a^2 + 2*a*b + b^2)*d*x - a^2 - 4*a*b
- 3*b^2)*cosh(d*x + c)^7 + 42*(15*(a^2 + 2*a*b + b^2)*d*x - 8*a^2 - 24*a*b - 12*b^2)*cosh(d*x + c)^5 + 20*(15
*(a^2 + 2*a*b + b^2)*d*x - 9*a^2 - 24*a*b - 17*b^2)*cosh(d*x + c)^3 + 3*(15*(a^2 + 2*a*b + b^2)*d*x - 8*a^2 -
24*a*b - 12*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(a^2 + 2*a*b + b^2)*d*x + 6*(3*(a^2 + 2*a*b + b^2)*d*x - a
^2 - 4*a*b - 3*b^2)*cosh(d*x + c)^2 + 6*(33*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^10 + 45*(3*(a^2 + 2*a*b + b^
2)*d*x - a^2 - 4*a*b - 3*b^2)*cosh(d*x + c)^8 + 14*(15*(a^2 + 2*a*b + b^2)*d*x - 8*a^2 - 24*a*b - 12*b^2)*cosh
(d*x + c)^6 + 10*(15*(a^2 + 2*a*b + b^2)*d*x - 9*a^2 - 24*a*b - 17*b^2)*cosh(d*x + c)^4 + 3*(a^2 + 2*a*b + b^2
)*d*x + 3*(15*(a^2 + 2*a*b + b^2)*d*x - 8*a^2 - 24*a*b - 12*b^2)*cosh(d*x + c)^2 - a^2 - 4*a*b - 3*b^2)*sinh(d
*x + c)^2 - 3*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^12 + 12*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^11 +
(a^2 + 2*a*b + b^2)*sinh(d*x + c)^12 + 6*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^10 + 6*(11*(a^2 + 2*a*b + b^2)*cosh
(d*x + c)^2 + a^2 + 2*a*b + b^2)*sinh(d*x + c)^10 + 20*(11*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + 3*(a^2 + 2*a*
b + b^2)*cosh(d*x + c))*sinh(d*x + c)^9 + 15*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^8 + 15*(33*(a^2 + 2*a*b + b^2)*
cosh(d*x + c)^4 + 18*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b + b^2)*sinh(d*x + c)^8 + 24*(33*(a^2 +
2*a*b + b^2)*cosh(d*x + c)^5 + 30*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + 5*(a^2 + 2*a*b + b^2)*cosh(d*x + c))*s
inh(d*x + c)^7 + 20*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 + 4*(231*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 + 315*(a^
2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 105*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + 5*a^2 + 10*a*b + 5*b^2)*sinh(d*x
+ c)^6 + 24*(33*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^7 + 63*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^5 + 35*(a^2 + 2*a*b
+ b^2)*cosh(d*x + c)^3 + 5*(a^2 + 2*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 15*(a^2 + 2*a*b + b^2)*cosh(d
*x + c)^4 + 15*(33*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^8 + 84*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 + 70*(a^2 + 2*
a*b + b^2)*cosh(d*x + c)^4 + 20*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 + 20*
(11*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^9 + 36*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^7 + 42*(a^2 + 2*a*b + b^2)*cosh
(d*x + c)^5 + 20*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + 3*(a^2 + 2*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c)^3 +
6*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + 6*(11*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^10 + 45*(a^2 + 2*a*b + b^2)*co
sh(d*x + c)^8 + 70*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 + 50*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 15*(a^2 + 2*
a*b + b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b + b^2)*sinh(d*x + c)^2 + a^2 + 2*a*b + b^2 + 12*((a^2 + 2*a*b + b^2)*
cosh(d*x + c)^11 + 5*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^9 + 10*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^7 + 10*(a^2 +
2*a*b + b^2)*cosh(d*x + c)^5 + 5*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + (a^2 + 2*a*b + b^2)*cosh(d*x + c))*sinh
(d*x + c))*log(2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 12*(3*(a^2 + 2*a*b + b^2)*d*x*cosh(d*x + c)^
11 + 5*(3*(a^2 + 2*a*b + b^2)*d*x - a^2 - 4*a*b - 3*b^2)*cosh(d*x + c)^9 + 2*(15*(a^2 + 2*a*b + b^2)*d*x - 8*a
^2 - 24*a*b - 12*b^2)*cosh(d*x + c)^7 + 2*(15*(a^2 + 2*a*b + b^2)*d*x - 9*a^2 - 24*a*b - 17*b^2)*cosh(d*x + c)
^5 + (15*(a^2 + 2*a*b + b^2)*d*x - 8*a^2 - 24*a*b - 12*b^2)*cosh(d*x + c)^3 + (3*(a^2 + 2*a*b + b^2)*d*x - a^2
- 4*a*b - 3*b^2)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^12 + 12*d*cosh(d*x + c)*sinh(d*x + c)^11 + d*
sinh(d*x + c)^12 + 6*d*cosh(d*x + c)^10 + 6*(11*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^10 + 20*(11*d*cosh(d*x +
c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^9 + 15*d*cosh(d*x + c)^8 + 15*(33*d*cosh(d*x + c)^4 + 18*d*cosh(d*x +
c)^2 + d)*sinh(d*x + c)^8 + 24*(33*d*cosh(d*x + c)^5 + 30*d*cosh(d*x + c)^3 + 5*d*cosh(d*x + c))*sinh(d*x + c)
^7 + 20*d*cosh(d*x + c)^6 + 4*(231*d*cosh(d*x + c)^6 + 315*d*cosh(d*x + c)^4 + 105*d*cosh(d*x + c)^2 + 5*d)*si
nh(d*x + c)^6 + 24*(33*d*cosh(d*x + c)^7 + 63*d*cosh(d*x + c)^5 + 35*d*cosh(d*x + c)^3 + 5*d*cosh(d*x + c))*si
nh(d*x + c)^5 + 15*d*cosh(d*x + c)^4 + 15*(33*d*cosh(d*x + c)^8 + 84*d*cosh(d*x + c)^6 + 70*d*cosh(d*x + c)^4
+ 20*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^4 + 20*(11*d*cosh(d*x + c)^9 + 36*d*cosh(d*x + c)^7 + 42*d*cosh(d*x
+ c)^5 + 20*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^3 + 6*d*cosh(d*x + c)^2 + 6*(11*d*cosh(d*x +
c)^10 + 45*d*cosh(d*x + c)^8 + 70*d*cosh(d*x + c)^6 + 50*d*cosh(d*x + c)^4 + 15*d*cosh(d*x + c)^2 + d)*sinh(d*
x + c)^2 + 12*(d*cosh(d*x + c)^11 + 5*d*cosh(d*x + c)^9 + 10*d*cosh(d*x + c)^7 + 10*d*cosh(d*x + c)^5 + 5*d*co
sh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c) + d)

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Sympy [A]  time = 1.21659, size = 170, normalized size = 2.24 \begin{align*} \begin{cases} a^{2} x - \frac{a^{2} \log{\left (\tanh{\left (c + d x \right )} + 1 \right )}}{d} - \frac{a^{2} \tanh ^{2}{\left (c + d x \right )}}{2 d} + 2 a b x - \frac{2 a b \log{\left (\tanh{\left (c + d x \right )} + 1 \right )}}{d} - \frac{a b \tanh ^{4}{\left (c + d x \right )}}{2 d} - \frac{a b \tanh ^{2}{\left (c + d x \right )}}{d} + b^{2} x - \frac{b^{2} \log{\left (\tanh{\left (c + d x \right )} + 1 \right )}}{d} - \frac{b^{2} \tanh ^{6}{\left (c + d x \right )}}{6 d} - \frac{b^{2} \tanh ^{4}{\left (c + d x \right )}}{4 d} - \frac{b^{2} \tanh ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (a + b \tanh ^{2}{\left (c \right )}\right )^{2} \tanh ^{3}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*x - a**2*log(tanh(c + d*x) + 1)/d - a**2*tanh(c + d*x)**2/(2*d) + 2*a*b*x - 2*a*b*log(tanh(c +
d*x) + 1)/d - a*b*tanh(c + d*x)**4/(2*d) - a*b*tanh(c + d*x)**2/d + b**2*x - b**2*log(tanh(c + d*x) + 1)/d -
b**2*tanh(c + d*x)**6/(6*d) - b**2*tanh(c + d*x)**4/(4*d) - b**2*tanh(c + d*x)**2/(2*d), Ne(d, 0)), (x*(a + b*
tanh(c)**2)**2*tanh(c)**3, True))

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Giac [B]  time = 1.30806, size = 262, normalized size = 3.45 \begin{align*} -\frac{{\left (a^{2} + 2 \, a b + b^{2}\right )}{\left (d x + c\right )}}{d} + \frac{{\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{d} + \frac{2 \,{\left (3 \,{\left (a^{2} + 4 \, a b + 3 \, b^{2}\right )} e^{\left (10 \, d x + 10 \, c\right )} + 6 \,{\left (2 \, a^{2} + 6 \, a b + 3 \, b^{2}\right )} e^{\left (8 \, d x + 8 \, c\right )} + 2 \,{\left (9 \, a^{2} + 24 \, a b + 17 \, b^{2}\right )} e^{\left (6 \, d x + 6 \, c\right )} + 6 \,{\left (2 \, a^{2} + 6 \, a b + 3 \, b^{2}\right )} e^{\left (4 \, d x + 4 \, c\right )} + 3 \,{\left (a^{2} + 4 \, a b + 3 \, b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )}}{3 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^2 + 2*a*b + b^2)*(d*x + c)/d + (a^2 + 2*a*b + b^2)*log(e^(2*d*x + 2*c) + 1)/d + 2/3*(3*(a^2 + 4*a*b + 3*b^
2)*e^(10*d*x + 10*c) + 6*(2*a^2 + 6*a*b + 3*b^2)*e^(8*d*x + 8*c) + 2*(9*a^2 + 24*a*b + 17*b^2)*e^(6*d*x + 6*c)
+ 6*(2*a^2 + 6*a*b + 3*b^2)*e^(4*d*x + 4*c) + 3*(a^2 + 4*a*b + 3*b^2)*e^(2*d*x + 2*c))/(d*(e^(2*d*x + 2*c) +
1)^6)