### 3.169 $$\int (a+b \tanh ^2(c+d x))^5 \, dx$$

Optimal. Leaf size=160 $-\frac{b^3 \left (10 a^2+5 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac{b^2 \left (10 a^2 b+10 a^3+5 a b^2+b^3\right ) \tanh ^3(c+d x)}{3 d}-\frac{b \left (10 a^2 b^2+10 a^3 b+5 a^4+5 a b^3+b^4\right ) \tanh (c+d x)}{d}-\frac{b^4 (5 a+b) \tanh ^7(c+d x)}{7 d}+x (a+b)^5-\frac{b^5 \tanh ^9(c+d x)}{9 d}$

[Out]

(a + b)^5*x - (b*(5*a^4 + 10*a^3*b + 10*a^2*b^2 + 5*a*b^3 + b^4)*Tanh[c + d*x])/d - (b^2*(10*a^3 + 10*a^2*b +
5*a*b^2 + b^3)*Tanh[c + d*x]^3)/(3*d) - (b^3*(10*a^2 + 5*a*b + b^2)*Tanh[c + d*x]^5)/(5*d) - (b^4*(5*a + b)*Ta
nh[c + d*x]^7)/(7*d) - (b^5*Tanh[c + d*x]^9)/(9*d)

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Rubi [A]  time = 0.0942217, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.214, Rules used = {3661, 390, 206} $-\frac{b^3 \left (10 a^2+5 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac{b^2 \left (10 a^2 b+10 a^3+5 a b^2+b^3\right ) \tanh ^3(c+d x)}{3 d}-\frac{b \left (10 a^2 b^2+10 a^3 b+5 a^4+5 a b^3+b^4\right ) \tanh (c+d x)}{d}-\frac{b^4 (5 a+b) \tanh ^7(c+d x)}{7 d}+x (a+b)^5-\frac{b^5 \tanh ^9(c+d x)}{9 d}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*Tanh[c + d*x]^2)^5,x]

[Out]

(a + b)^5*x - (b*(5*a^4 + 10*a^3*b + 10*a^2*b^2 + 5*a*b^3 + b^4)*Tanh[c + d*x])/d - (b^2*(10*a^3 + 10*a^2*b +
5*a*b^2 + b^3)*Tanh[c + d*x]^3)/(3*d) - (b^3*(10*a^2 + 5*a*b + b^2)*Tanh[c + d*x]^5)/(5*d) - (b^4*(5*a + b)*Ta
nh[c + d*x]^7)/(7*d) - (b^5*Tanh[c + d*x]^9)/(9*d)

Rule 3661

Int[((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[(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, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

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 \left (a+b \tanh ^2(c+d x)\right )^5 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^5}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-b \left (5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right )-b^2 \left (10 a^3+10 a^2 b+5 a b^2+b^3\right ) x^2-b^3 \left (10 a^2+5 a b+b^2\right ) x^4-b^4 (5 a+b) x^6-b^5 x^8+\frac{(a+b)^5}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{b \left (5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right ) \tanh (c+d x)}{d}-\frac{b^2 \left (10 a^3+10 a^2 b+5 a b^2+b^3\right ) \tanh ^3(c+d x)}{3 d}-\frac{b^3 \left (10 a^2+5 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac{b^4 (5 a+b) \tanh ^7(c+d x)}{7 d}-\frac{b^5 \tanh ^9(c+d x)}{9 d}+\frac{(a+b)^5 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=(a+b)^5 x-\frac{b \left (5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right ) \tanh (c+d x)}{d}-\frac{b^2 \left (10 a^3+10 a^2 b+5 a b^2+b^3\right ) \tanh ^3(c+d x)}{3 d}-\frac{b^3 \left (10 a^2+5 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac{b^4 (5 a+b) \tanh ^7(c+d x)}{7 d}-\frac{b^5 \tanh ^9(c+d x)}{9 d}\\ \end{align*}

Mathematica [A]  time = 2.14821, size = 170, normalized size = 1.06 $\frac{\tanh (c+d x) \left (\frac{315 (a+b)^5 \tanh ^{-1}\left (\sqrt{\tanh ^2(c+d x)}\right )}{\sqrt{\tanh ^2(c+d x)}}-b \left (63 b^2 \left (10 a^2+5 a b+b^2\right ) \tanh ^4(c+d x)+105 b \left (10 a^2 b+10 a^3+5 a b^2+b^3\right ) \tanh ^2(c+d x)+315 \left (10 a^2 b^2+10 a^3 b+5 a^4+5 a b^3+b^4\right )+45 b^3 (5 a+b) \tanh ^6(c+d x)+35 b^4 \tanh ^8(c+d x)\right )\right )}{315 d}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*Tanh[c + d*x]^2)^5,x]

[Out]

(Tanh[c + d*x]*((315*(a + b)^5*ArcTanh[Sqrt[Tanh[c + d*x]^2]])/Sqrt[Tanh[c + d*x]^2] - b*(315*(5*a^4 + 10*a^3*
b + 10*a^2*b^2 + 5*a*b^3 + b^4) + 105*b*(10*a^3 + 10*a^2*b + 5*a*b^2 + b^3)*Tanh[c + d*x]^2 + 63*b^2*(10*a^2 +
5*a*b + b^2)*Tanh[c + d*x]^4 + 45*b^3*(5*a + b)*Tanh[c + d*x]^6 + 35*b^4*Tanh[c + d*x]^8)))/(315*d)

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Maple [B]  time = 0.006, size = 472, normalized size = 3. \begin{align*} -{\frac{{a}^{5}\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) }{2\,d}}-{\frac{ \left ( \tanh \left ( dx+c \right ) \right ) ^{5}a{b}^{4}}{d}}-{\frac{10\, \left ( \tanh \left ( dx+c \right ) \right ) ^{3}{a}^{3}{b}^{2}}{3\,d}}-{\frac{10\, \left ( \tanh \left ( dx+c \right ) \right ) ^{3}{a}^{2}{b}^{3}}{3\,d}}-{\frac{5\, \left ( \tanh \left ( dx+c \right ) \right ) ^{7}a{b}^{4}}{7\,d}}-10\,{\frac{{a}^{2}{b}^{3}\tanh \left ( dx+c \right ) }{d}}-5\,{\frac{{a}^{4}b\tanh \left ( dx+c \right ) }{d}}-2\,{\frac{ \left ( \tanh \left ( dx+c \right ) \right ) ^{5}{a}^{2}{b}^{3}}{d}}-{\frac{5\,\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) a{b}^{4}}{2\,d}}-{\frac{5\, \left ( \tanh \left ( dx+c \right ) \right ) ^{3}a{b}^{4}}{3\,d}}-5\,{\frac{a{b}^{4}\tanh \left ( dx+c \right ) }{d}}-10\,{\frac{{a}^{3}{b}^{2}\tanh \left ( dx+c \right ) }{d}}+5\,{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ){a}^{3}{b}^{2}}{d}}+{\frac{5\,\ln \left ( \tanh \left ( dx+c \right ) +1 \right ){a}^{4}b}{2\,d}}-{\frac{ \left ( \tanh \left ( dx+c \right ) \right ) ^{7}{b}^{5}}{7\,d}}-{\frac{ \left ( \tanh \left ( dx+c \right ) \right ) ^{5}{b}^{5}}{5\,d}}-{\frac{ \left ( \tanh \left ( dx+c \right ) \right ) ^{3}{b}^{5}}{3\,d}}-{\frac{{b}^{5}\tanh \left ( dx+c \right ) }{d}}-{\frac{{b}^{5} \left ( \tanh \left ( dx+c \right ) \right ) ^{9}}{9\,d}}+{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ){a}^{5}}{2\,d}}+{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ){b}^{5}}{2\,d}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ){b}^{5}}{2\,d}}+5\,{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ){a}^{2}{b}^{3}}{d}}-{\frac{5\,\ln \left ( \tanh \left ( dx+c \right ) -1 \right ){a}^{4}b}{2\,d}}-5\,{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ){a}^{3}{b}^{2}}{d}}-5\,{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ){a}^{2}{b}^{3}}{d}}+{\frac{5\,\ln \left ( \tanh \left ( dx+c \right ) +1 \right ) a{b}^{4}}{2\,d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tanh(d*x+c)^2)^5,x)

[Out]

-1/2/d*a^5*ln(tanh(d*x+c)-1)-1/d*tanh(d*x+c)^5*a*b^4-10/3/d*tanh(d*x+c)^3*a^3*b^2-10/3/d*tanh(d*x+c)^3*a^2*b^3
-5/7/d*tanh(d*x+c)^7*a*b^4-10/d*a^2*b^3*tanh(d*x+c)-5/d*a^4*b*tanh(d*x+c)-2/d*tanh(d*x+c)^5*a^2*b^3-5/2/d*ln(t
anh(d*x+c)-1)*a*b^4-5/3/d*tanh(d*x+c)^3*a*b^4-5/d*a*b^4*tanh(d*x+c)-10/d*a^3*b^2*tanh(d*x+c)+5/d*ln(tanh(d*x+c
)+1)*a^3*b^2+5/2/d*ln(tanh(d*x+c)+1)*a^4*b-1/7/d*tanh(d*x+c)^7*b^5-1/5/d*tanh(d*x+c)^5*b^5-1/3/d*tanh(d*x+c)^3
*b^5-1/d*b^5*tanh(d*x+c)-1/9*b^5*tanh(d*x+c)^9/d+1/2/d*ln(tanh(d*x+c)+1)*a^5+1/2/d*ln(tanh(d*x+c)+1)*b^5-1/2/d
*ln(tanh(d*x+c)-1)*b^5+5/d*ln(tanh(d*x+c)+1)*a^2*b^3-5/2/d*ln(tanh(d*x+c)-1)*a^4*b-5/d*ln(tanh(d*x+c)-1)*a^3*b
^2-5/d*ln(tanh(d*x+c)-1)*a^2*b^3+5/2/d*ln(tanh(d*x+c)+1)*a*b^4

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Maxima [B]  time = 1.20509, size = 842, normalized size = 5.26 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tanh(d*x+c)^2)^5,x, algorithm="maxima")

[Out]

1/315*b^5*(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))) + 1/21*a*b^4*(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))) + 2/3*a^2*b^3*(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))) + 10/3*a^3*b^2*(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))) + 5*a^4*b*(x + c/d
- 2/(d*(e^(-2*d*x - 2*c) + 1))) + a^5*x

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Fricas [B]  time = 2.67108, size = 5516, normalized size = 34.48 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tanh(d*x+c)^2)^5,x, algorithm="fricas")

[Out]

1/315*((1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10
*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cosh(d*x + c)^9 + 9*(1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 56
3*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cosh(d*x + c)*sinh(d*x + c)^8 - (15
75*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5)*sinh(d*x + c)^9 + 9*(1575*a^4*b + 4200*a^3*b^2
+ 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cos
h(d*x + c)^7 - 9*(1225*a^4*b + 2800*a^3*b^2 + 2730*a^2*b^3 + 1240*a*b^4 + 213*b^5 + 4*(1575*a^4*b + 4200*a^3*b
^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 21*(4*(1575*a^4*b + 4200*a^3*b^2
+ 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cos
h(d*x + c)^3 + 3*(1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^
3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cosh(d*x + c))*sinh(d*x + c)^6 + 36*(1575*a^4*b + 4200*a^3*b^2 + 4830
*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cosh(d*x
+ c)^5 - 9*(3500*a^4*b + 7000*a^3*b^2 + 6720*a^2*b^3 + 3560*a*b^4 + 852*b^5 + 14*(1575*a^4*b + 4200*a^3*b^2 +
4830*a^2*b^3 + 2640*a*b^4 + 563*b^5)*cosh(d*x + c)^4 + 21*(1225*a^4*b + 2800*a^3*b^2 + 2730*a^2*b^3 + 1240*a*b
^4 + 213*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 9*(14*(1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4
+ 563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cosh(d*x + c)^5 + 35*(1575*a^4*
b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^
4 + b^5)*d*x)*cosh(d*x + c)^3 + 20*(1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5
+ 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cosh(d*x + c))*sinh(d*x + c)^4 + 84*(1575*a^4*b + 4
200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b
^5)*d*x)*cosh(d*x + c)^3 - 3*(28*(1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5)*cosh(d*x +
c)^6 + 14700*a^4*b + 26600*a^3*b^2 + 27440*a^2*b^3 + 13720*a*b^4 + 1764*b^5 + 105*(1225*a^4*b + 2800*a^3*b^2 +
2730*a^2*b^3 + 1240*a*b^4 + 213*b^5)*cosh(d*x + c)^4 + 120*(875*a^4*b + 1750*a^3*b^2 + 1680*a^2*b^3 + 890*a*b
^4 + 213*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 9*(4*(1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 +
563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cosh(d*x + c)^7 + 21*(1575*a^4*b
+ 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4
+ b^5)*d*x)*cosh(d*x + c)^5 + 40*(1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5
+ 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cosh(d*x + c)^3 + 28*(1575*a^4*b + 4200*a^3*b^2 + 48
30*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cosh(d*
x + c))*sinh(d*x + c)^2 + 126*(1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5 + 5*
a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cosh(d*x + c) - 9*((1575*a^4*b + 4200*a^3*b^2 + 4830*a^2
*b^3 + 2640*a*b^4 + 563*b^5)*cosh(d*x + c)^8 + 7*(1225*a^4*b + 2800*a^3*b^2 + 2730*a^2*b^3 + 1240*a*b^4 + 213*
b^5)*cosh(d*x + c)^6 + 2450*a^4*b + 4200*a^3*b^2 + 4620*a^2*b^3 + 1960*a*b^4 + 882*b^5 + 20*(875*a^4*b + 1750*
a^3*b^2 + 1680*a^2*b^3 + 890*a*b^4 + 213*b^5)*cosh(d*x + c)^4 + 28*(525*a^4*b + 950*a^3*b^2 + 980*a^2*b^3 + 49
0*a*b^4 + 63*b^5)*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 = 2.82775, size = 308, normalized size = 1.92 \begin{align*} \begin{cases} a^{5} x + 5 a^{4} b x - \frac{5 a^{4} b \tanh{\left (c + d x \right )}}{d} + 10 a^{3} b^{2} x - \frac{10 a^{3} b^{2} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac{10 a^{3} b^{2} \tanh{\left (c + d x \right )}}{d} + 10 a^{2} b^{3} x - \frac{2 a^{2} b^{3} \tanh ^{5}{\left (c + d x \right )}}{d} - \frac{10 a^{2} b^{3} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac{10 a^{2} b^{3} \tanh{\left (c + d x \right )}}{d} + 5 a b^{4} x - \frac{5 a b^{4} \tanh ^{7}{\left (c + d x \right )}}{7 d} - \frac{a b^{4} \tanh ^{5}{\left (c + d x \right )}}{d} - \frac{5 a b^{4} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac{5 a b^{4} \tanh{\left (c + d x \right )}}{d} + b^{5} x - \frac{b^{5} \tanh ^{9}{\left (c + d x \right )}}{9 d} - \frac{b^{5} \tanh ^{7}{\left (c + d x \right )}}{7 d} - \frac{b^{5} \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac{b^{5} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac{b^{5} \tanh{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \tanh ^{2}{\left (c \right )}\right )^{5} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tanh(d*x+c)**2)**5,x)

[Out]

Piecewise((a**5*x + 5*a**4*b*x - 5*a**4*b*tanh(c + d*x)/d + 10*a**3*b**2*x - 10*a**3*b**2*tanh(c + d*x)**3/(3*
d) - 10*a**3*b**2*tanh(c + d*x)/d + 10*a**2*b**3*x - 2*a**2*b**3*tanh(c + d*x)**5/d - 10*a**2*b**3*tanh(c + d*
x)**3/(3*d) - 10*a**2*b**3*tanh(c + d*x)/d + 5*a*b**4*x - 5*a*b**4*tanh(c + d*x)**7/(7*d) - a*b**4*tanh(c + d*
x)**5/d - 5*a*b**4*tanh(c + d*x)**3/(3*d) - 5*a*b**4*tanh(c + d*x)/d + b**5*x - b**5*tanh(c + d*x)**9/(9*d) -
b**5*tanh(c + d*x)**7/(7*d) - b**5*tanh(c + d*x)**5/(5*d) - b**5*tanh(c + d*x)**3/(3*d) - b**5*tanh(c + d*x)/d
, Ne(d, 0)), (x*(a + b*tanh(c)**2)**5, True))

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Giac [B]  time = 1.22694, size = 973, normalized size = 6.08 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tanh(d*x+c)^2)^5,x, algorithm="giac")

[Out]

(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*(d*x + c)/d + 2/315*(1575*a^4*b*e^(16*d*x + 16*c) +
6300*a^3*b^2*e^(16*d*x + 16*c) + 9450*a^2*b^3*e^(16*d*x + 16*c) + 6300*a*b^4*e^(16*d*x + 16*c) + 1575*b^5*e^(1
6*d*x + 16*c) + 12600*a^4*b*e^(14*d*x + 14*c) + 44100*a^3*b^2*e^(14*d*x + 14*c) + 56700*a^2*b^3*e^(14*d*x + 14
*c) + 31500*a*b^4*e^(14*d*x + 14*c) + 6300*b^5*e^(14*d*x + 14*c) + 44100*a^4*b*e^(12*d*x + 12*c) + 136500*a^3*
b^2*e^(12*d*x + 12*c) + 161700*a^2*b^3*e^(12*d*x + 12*c) + 90300*a*b^4*e^(12*d*x + 12*c) + 21000*b^5*e^(12*d*x
+ 12*c) + 88200*a^4*b*e^(10*d*x + 10*c) + 245700*a^3*b^2*e^(10*d*x + 10*c) + 283500*a^2*b^3*e^(10*d*x + 10*c)
+ 157500*a*b^4*e^(10*d*x + 10*c) + 31500*b^5*e^(10*d*x + 10*c) + 110250*a^4*b*e^(8*d*x + 8*c) + 283500*a^3*b^
2*e^(8*d*x + 8*c) + 325080*a^2*b^3*e^(8*d*x + 8*c) + 175140*a*b^4*e^(8*d*x + 8*c) + 39438*b^5*e^(8*d*x + 8*c)
+ 88200*a^4*b*e^(6*d*x + 6*c) + 216300*a^3*b^2*e^(6*d*x + 6*c) + 244020*a^2*b^3*e^(6*d*x + 6*c) + 131460*a*b^4
*e^(6*d*x + 6*c) + 26292*b^5*e^(6*d*x + 6*c) + 44100*a^4*b*e^(4*d*x + 4*c) + 107100*a^3*b^2*e^(4*d*x + 4*c) +
117180*a^2*b^3*e^(4*d*x + 4*c) + 63540*a*b^4*e^(4*d*x + 4*c) + 13968*b^5*e^(4*d*x + 4*c) + 12600*a^4*b*e^(2*d*
x + 2*c) + 31500*a^3*b^2*e^(2*d*x + 2*c) + 34020*a^2*b^3*e^(2*d*x + 2*c) + 17460*a*b^4*e^(2*d*x + 2*c) + 3492*
b^5*e^(2*d*x + 2*c) + 1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5)/(d*(e^(2*d*x + 2*c) + 1
)^9)