### 3.156 $$\int \tanh ^4(c+d x) (a+b \tanh ^2(c+d x))^3 \, dx$$

Optimal. Leaf size=114 $-\frac{b \left (3 a^2+3 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac{b^2 (3 a+b) \tanh ^7(c+d x)}{7 d}-\frac{(a+b)^3 \tanh ^3(c+d x)}{3 d}-\frac{(a+b)^3 \tanh (c+d x)}{d}+x (a+b)^3-\frac{b^3 \tanh ^9(c+d x)}{9 d}$

[Out]

(a + b)^3*x - ((a + b)^3*Tanh[c + d*x])/d - ((a + b)^3*Tanh[c + d*x]^3)/(3*d) - (b*(3*a^2 + 3*a*b + b^2)*Tanh[
c + d*x]^5)/(5*d) - (b^2*(3*a + b)*Tanh[c + d*x]^7)/(7*d) - (b^3*Tanh[c + d*x]^9)/(9*d)

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Rubi [A]  time = 0.100523, antiderivative size = 114, 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, 461, 206} $-\frac{b \left (3 a^2+3 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac{b^2 (3 a+b) \tanh ^7(c+d x)}{7 d}-\frac{(a+b)^3 \tanh ^3(c+d x)}{3 d}-\frac{(a+b)^3 \tanh (c+d x)}{d}+x (a+b)^3-\frac{b^3 \tanh ^9(c+d x)}{9 d}$

Antiderivative was successfully veriﬁed.

[In]

Int[Tanh[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

(a + b)^3*x - ((a + b)^3*Tanh[c + d*x])/d - ((a + b)^3*Tanh[c + d*x]^3)/(3*d) - (b*(3*a^2 + 3*a*b + b^2)*Tanh[
c + d*x]^5)/(5*d) - (b^2*(3*a + b)*Tanh[c + d*x]^7)/(7*d) - (b^3*Tanh[c + d*x]^9)/(9*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 461

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[((e*x)^m*(a + b*x^n)^p)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rule 206

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

Rubi steps

\begin{align*} \int \tanh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4 \left (a+b x^2\right )^3}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-(a+b)^3-(a+b)^3 x^2-b \left (3 a^2+3 a b+b^2\right ) x^4-b^2 (3 a+b) x^6-b^3 x^8+\frac{a^3+3 a^2 b+3 a b^2+b^3}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{(a+b)^3 \tanh (c+d x)}{d}-\frac{(a+b)^3 \tanh ^3(c+d x)}{3 d}-\frac{b \left (3 a^2+3 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac{b^2 (3 a+b) \tanh ^7(c+d x)}{7 d}-\frac{b^3 \tanh ^9(c+d x)}{9 d}+\frac{(a+b)^3 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=(a+b)^3 x-\frac{(a+b)^3 \tanh (c+d x)}{d}-\frac{(a+b)^3 \tanh ^3(c+d x)}{3 d}-\frac{b \left (3 a^2+3 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac{b^2 (3 a+b) \tanh ^7(c+d x)}{7 d}-\frac{b^3 \tanh ^9(c+d x)}{9 d}\\ \end{align*}

Mathematica [A]  time = 1.43208, size = 123, normalized size = 1.08 $\frac{\tanh (c+d x) \left (-63 b \left (3 a^2+3 a b+b^2\right ) \tanh ^4(c+d x)-45 b^2 (3 a+b) \tanh ^6(c+d x)-105 (a+b)^3 \tanh ^2(c+d x)+\frac{315 (a+b)^3 \tanh ^{-1}\left (\sqrt{\tanh ^2(c+d x)}\right )}{\sqrt{\tanh ^2(c+d x)}}-315 (a+b)^3-35 b^3 \tanh ^8(c+d x)\right )}{315 d}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Tanh[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

(Tanh[c + d*x]*(-315*(a + b)^3 - 105*(a + b)^3*Tanh[c + d*x]^2 - 63*b*(3*a^2 + 3*a*b + b^2)*Tanh[c + d*x]^4 -
45*b^2*(3*a + b)*Tanh[c + d*x]^6 - 35*b^3*Tanh[c + d*x]^8 + (315*(a + b)^3*ArcTanh[Sqrt[Tanh[c + d*x]^2]])/Sqr
t[Tanh[c + d*x]^2]))/(315*d)

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Maple [B]  time = 0.007, size = 365, normalized size = 3.2 \begin{align*} -{\frac{{a}^{3}\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) }{2\,d}}-{\frac{3\,\ln \left ( \tanh \left ( dx+c \right ) -1 \right ){a}^{2}b}{2\,d}}-{\frac{3\,\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) a{b}^{2}}{2\,d}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ){b}^{3}}{2\,d}}-{\frac{{b}^{3} \left ( \tanh \left ( dx+c \right ) \right ) ^{7}}{7\,d}}-{\frac{{b}^{3} \left ( \tanh \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-{\frac{ \left ( \tanh \left ( dx+c \right ) \right ) ^{3}{a}^{3}}{3\,d}}-{\frac{{b}^{3} \left ( \tanh \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{{b}^{3}\tanh \left ( dx+c \right ) }{d}}-{\frac{{b}^{3} \left ( \tanh \left ( dx+c \right ) \right ) ^{9}}{9\,d}}-{\frac{{a}^{2}b \left ( \tanh \left ( dx+c \right ) \right ) ^{3}}{d}}-{\frac{a{b}^{2} \left ( \tanh \left ( dx+c \right ) \right ) ^{3}}{d}}-{\frac{3\, \left ( \tanh \left ( dx+c \right ) \right ) ^{7}a{b}^{2}}{7\,d}}-{\frac{3\, \left ( \tanh \left ( dx+c \right ) \right ) ^{5}{a}^{2}b}{5\,d}}-{\frac{3\,a{b}^{2} \left ( \tanh \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-3\,{\frac{{a}^{2}b\tanh \left ( dx+c \right ) }{d}}+{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ){a}^{3}}{2\,d}}+{\frac{3\,\ln \left ( \tanh \left ( dx+c \right ) +1 \right ){a}^{2}b}{2\,d}}+{\frac{3\,\ln \left ( \tanh \left ( dx+c \right ) +1 \right ) a{b}^{2}}{2\,d}}+{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ){b}^{3}}{2\,d}}-{\frac{{a}^{3}\tanh \left ( dx+c \right ) }{d}}-3\,{\frac{a{b}^{2}\tanh \left ( dx+c \right ) }{d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(d*x+c)^4*(a+b*tanh(d*x+c)^2)^3,x)

[Out]

-1/2/d*a^3*ln(tanh(d*x+c)-1)-3/2/d*ln(tanh(d*x+c)-1)*a^2*b-3/2/d*ln(tanh(d*x+c)-1)*a*b^2-1/2/d*ln(tanh(d*x+c)-
1)*b^3-1/7*b^3*tanh(d*x+c)^7/d-1/5*b^3*tanh(d*x+c)^5/d-1/3/d*tanh(d*x+c)^3*a^3-1/3*b^3*tanh(d*x+c)^3/d-b^3*tan
h(d*x+c)/d-1/9*b^3*tanh(d*x+c)^9/d-a^2*b*tanh(d*x+c)^3/d-a*b^2*tanh(d*x+c)^3/d-3/7/d*tanh(d*x+c)^7*a*b^2-3/5/d
*tanh(d*x+c)^5*a^2*b-3/5*a*b^2*tanh(d*x+c)^5/d-3*a^2*b*tanh(d*x+c)/d+1/2/d*ln(tanh(d*x+c)+1)*a^3+3/2/d*ln(tanh
(d*x+c)+1)*a^2*b+3/2/d*ln(tanh(d*x+c)+1)*a*b^2+1/2/d*ln(tanh(d*x+c)+1)*b^3-a^3*tanh(d*x+c)/d-3*a*b^2*tanh(d*x+
c)/d

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Maxima [B]  time = 1.15145, size = 787, normalized size = 6.9 \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)^4*(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

1/315*b^3*(315*x + 315*c/d - 2*(3492*e^(-2*d*x - 2*c) + 13968*e^(-4*d*x - 4*c) + 26292*e^(-6*d*x - 6*c) + 3943
8*e^(-8*d*x - 8*c) + 31500*e^(-10*d*x - 10*c) + 21000*e^(-12*d*x - 12*c) + 6300*e^(-14*d*x - 14*c) + 1575*e^(-
16*d*x - 16*c) + 563)/(d*(9*e^(-2*d*x - 2*c) + 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) + 126*e^(-8*d*x - 8*c
) + 126*e^(-10*d*x - 10*c) + 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) + 9*e^(-16*d*x - 16*c) + e^(-18*d*x
- 18*c) + 1))) + 1/35*a*b^2*(105*x + 105*c/d - 8*(203*e^(-2*d*x - 2*c) + 609*e^(-4*d*x - 4*c) + 770*e^(-6*d*x
- 6*c) + 770*e^(-8*d*x - 8*c) + 315*e^(-10*d*x - 10*c) + 105*e^(-12*d*x - 12*c) + 44)/(d*(7*e^(-2*d*x - 2*c)
+ 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*
c) + e^(-14*d*x - 14*c) + 1))) + 1/5*a^2*b*(15*x + 15*c/d - 2*(70*e^(-2*d*x - 2*c) + 140*e^(-4*d*x - 4*c) + 90
*e^(-6*d*x - 6*c) + 45*e^(-8*d*x - 8*c) + 23)/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*
c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1))) + 1/3*a^3*(3*x + 3*c/d - 4*(3*e^(-2*d*x - 2*c) + 3*e^(-4*d
*x - 4*c) + 2)/(d*(3*e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) + 1)))

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Fricas [B]  time = 2.19007, size = 4166, normalized size = 36.54 \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)^4*(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

1/315*((420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^9
+ 9*(420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)*sin
h(d*x + c)^8 - (420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3)*sinh(d*x + c)^9 + 9*(420*a^3 + 1449*a^2*b + 1584*
a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^7 - 9*(280*a^3 + 819*a^2*b + 744*a*b^
2 + 213*b^3 + 4*(420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 21*(4*(420*a^
3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^3 + 3*(420*a^3
+ 1449*a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^6
+ 36*(420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^5 -
9*(14*(420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3)*cosh(d*x + c)^4 + 700*a^3 + 2016*a^2*b + 2136*a*b^2 + 852
*b^3 + 21*(280*a^3 + 819*a^2*b + 744*a*b^2 + 213*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 9*(14*(420*a^3 + 1449
*a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^5 + 35*(420*a^3 + 1449*
a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^3 + 20*(420*a^3 + 1449*a
^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^4 + 84*(42
0*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^3 - 3*(28*(
420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3)*cosh(d*x + c)^6 + 105*(280*a^3 + 819*a^2*b + 744*a*b^2 + 213*b^3)
*cosh(d*x + c)^4 + 2660*a^3 + 8232*a^2*b + 8232*a*b^2 + 1764*b^3 + 120*(175*a^3 + 504*a^2*b + 534*a*b^2 + 213*
b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 9*(4*(420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b
+ 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^7 + 21*(420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b
+ 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^5 + 40*(420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b +
3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^3 + 28*(420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a^3 + 3*a^2*b +
3*a*b^2 + b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^2 + 126*(420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3 + 315*(a
^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c) - 9*((420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3)*cosh(d*x +
c)^8 + 7*(280*a^3 + 819*a^2*b + 744*a*b^2 + 213*b^3)*cosh(d*x + c)^6 + 20*(175*a^3 + 504*a^2*b + 534*a*b^2 +
213*b^3)*cosh(d*x + c)^4 + 420*a^3 + 1386*a^2*b + 1176*a*b^2 + 882*b^3 + 28*(95*a^3 + 294*a^2*b + 294*a*b^2 +
63*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/(d*cosh(d*x + c)^9 + 9*d*cosh(d*x + c)*sinh(d*x + c)^8 + 9*d*cosh(d*x
+ c)^7 + 21*(4*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^6 + 36*d*cosh(d*x + c)^5 + 9*(14*d*cosh(d*
x + c)^5 + 35*d*cosh(d*x + c)^3 + 20*d*cosh(d*x + c))*sinh(d*x + c)^4 + 84*d*cosh(d*x + c)^3 + 9*(4*d*cosh(d*x
+ c)^7 + 21*d*cosh(d*x + c)^5 + 40*d*cosh(d*x + c)^3 + 28*d*cosh(d*x + c))*sinh(d*x + c)^2 + 126*d*cosh(d*x +
c))

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Sympy [A]  time = 2.8266, size = 260, normalized size = 2.28 \begin{align*} \begin{cases} a^{3} x - \frac{a^{3} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac{a^{3} \tanh{\left (c + d x \right )}}{d} + 3 a^{2} b x - \frac{3 a^{2} b \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac{a^{2} b \tanh ^{3}{\left (c + d x \right )}}{d} - \frac{3 a^{2} b \tanh{\left (c + d x \right )}}{d} + 3 a b^{2} x - \frac{3 a b^{2} \tanh ^{7}{\left (c + d x \right )}}{7 d} - \frac{3 a b^{2} \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac{a b^{2} \tanh ^{3}{\left (c + d x \right )}}{d} - \frac{3 a b^{2} \tanh{\left (c + d x \right )}}{d} + b^{3} x - \frac{b^{3} \tanh ^{9}{\left (c + d x \right )}}{9 d} - \frac{b^{3} \tanh ^{7}{\left (c + d x \right )}}{7 d} - \frac{b^{3} \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac{b^{3} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac{b^{3} \tanh{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \tanh ^{2}{\left (c \right )}\right )^{3} \tanh ^{4}{\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)**4*(a+b*tanh(d*x+c)**2)**3,x)

[Out]

Piecewise((a**3*x - a**3*tanh(c + d*x)**3/(3*d) - a**3*tanh(c + d*x)/d + 3*a**2*b*x - 3*a**2*b*tanh(c + d*x)**
5/(5*d) - a**2*b*tanh(c + d*x)**3/d - 3*a**2*b*tanh(c + d*x)/d + 3*a*b**2*x - 3*a*b**2*tanh(c + d*x)**7/(7*d)
- 3*a*b**2*tanh(c + d*x)**5/(5*d) - a*b**2*tanh(c + d*x)**3/d - 3*a*b**2*tanh(c + d*x)/d + b**3*x - b**3*tanh(
c + d*x)**9/(9*d) - b**3*tanh(c + d*x)**7/(7*d) - b**3*tanh(c + d*x)**5/(5*d) - b**3*tanh(c + d*x)**3/(3*d) -
b**3*tanh(c + d*x)/d, Ne(d, 0)), (x*(a + b*tanh(c)**2)**3*tanh(c)**4, True))

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Giac [B]  time = 1.45266, size = 721, normalized size = 6.32 \begin{align*} \frac{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}{\left (d x + c\right )}}{d} + \frac{2 \,{\left (630 \, a^{3} e^{\left (16 \, d x + 16 \, c\right )} + 2835 \, a^{2} b e^{\left (16 \, d x + 16 \, c\right )} + 3780 \, a b^{2} e^{\left (16 \, d x + 16 \, c\right )} + 1575 \, b^{3} e^{\left (16 \, d x + 16 \, c\right )} + 4410 \, a^{3} e^{\left (14 \, d x + 14 \, c\right )} + 17010 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} + 18900 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} + 6300 \, b^{3} e^{\left (14 \, d x + 14 \, c\right )} + 13650 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 48510 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 54180 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 21000 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 24570 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 85050 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 94500 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 31500 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 28350 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 97524 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 105084 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 39438 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 21630 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 73206 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 78876 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 26292 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 10710 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 35154 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 38124 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 13968 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 3150 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 10206 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 10476 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3492 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 420 \, a^{3} + 1449 \, a^{2} b + 1584 \, a b^{2} + 563 \, b^{3}\right )}}{315 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{9}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)^4*(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*(d*x + c)/d + 2/315*(630*a^3*e^(16*d*x + 16*c) + 2835*a^2*b*e^(16*d*x + 16*c)
+ 3780*a*b^2*e^(16*d*x + 16*c) + 1575*b^3*e^(16*d*x + 16*c) + 4410*a^3*e^(14*d*x + 14*c) + 17010*a^2*b*e^(14*d
*x + 14*c) + 18900*a*b^2*e^(14*d*x + 14*c) + 6300*b^3*e^(14*d*x + 14*c) + 13650*a^3*e^(12*d*x + 12*c) + 48510*
a^2*b*e^(12*d*x + 12*c) + 54180*a*b^2*e^(12*d*x + 12*c) + 21000*b^3*e^(12*d*x + 12*c) + 24570*a^3*e^(10*d*x +
10*c) + 85050*a^2*b*e^(10*d*x + 10*c) + 94500*a*b^2*e^(10*d*x + 10*c) + 31500*b^3*e^(10*d*x + 10*c) + 28350*a^
3*e^(8*d*x + 8*c) + 97524*a^2*b*e^(8*d*x + 8*c) + 105084*a*b^2*e^(8*d*x + 8*c) + 39438*b^3*e^(8*d*x + 8*c) + 2
1630*a^3*e^(6*d*x + 6*c) + 73206*a^2*b*e^(6*d*x + 6*c) + 78876*a*b^2*e^(6*d*x + 6*c) + 26292*b^3*e^(6*d*x + 6*
c) + 10710*a^3*e^(4*d*x + 4*c) + 35154*a^2*b*e^(4*d*x + 4*c) + 38124*a*b^2*e^(4*d*x + 4*c) + 13968*b^3*e^(4*d*
x + 4*c) + 3150*a^3*e^(2*d*x + 2*c) + 10206*a^2*b*e^(2*d*x + 2*c) + 10476*a*b^2*e^(2*d*x + 2*c) + 3492*b^3*e^(
2*d*x + 2*c) + 420*a^3 + 1449*a^2*b + 1584*a*b^2 + 563*b^3)/(d*(e^(2*d*x + 2*c) + 1)^9)