3.1 \(\int (a+b \coth ^2(c+d x))^5 \, dx\)

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

[Out]

(a + b)^5*x - (b*(5*a^4 + 10*a^3*b + 10*a^2*b^2 + 5*a*b^3 + b^4)*Coth[c + d*x])/d - (b^2*(10*a^3 + 10*a^2*b +
5*a*b^2 + b^3)*Coth[c + d*x]^3)/(3*d) - (b^3*(10*a^2 + 5*a*b + b^2)*Coth[c + d*x]^5)/(5*d) - (b^4*(5*a + b)*Co
th[c + d*x]^7)/(7*d) - (b^5*Coth[c + d*x]^9)/(9*d)

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Rubi [A]  time = 0.0921927, 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} \[ -\frac{b^3 \left (10 a^2+5 a b+b^2\right ) \coth ^5(c+d x)}{5 d}-\frac{b^2 \left (10 a^2 b+10 a^3+5 a b^2+b^3\right ) \coth ^3(c+d x)}{3 d}-\frac{b \left (10 a^2 b^2+10 a^3 b+5 a^4+5 a b^3+b^4\right ) \coth (c+d x)}{d}-\frac{b^4 (5 a+b) \coth ^7(c+d x)}{7 d}+x (a+b)^5-\frac{b^5 \coth ^9(c+d x)}{9 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Coth[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)*Coth[c + d*x])/d - (b^2*(10*a^3 + 10*a^2*b +
5*a*b^2 + b^3)*Coth[c + d*x]^3)/(3*d) - (b^3*(10*a^2 + 5*a*b + b^2)*Coth[c + d*x]^5)/(5*d) - (b^4*(5*a + b)*Co
th[c + d*x]^7)/(7*d) - (b^5*Coth[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 \coth ^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,\coth (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,\coth (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 ) \coth (c+d x)}{d}-\frac{b^2 \left (10 a^3+10 a^2 b+5 a b^2+b^3\right ) \coth ^3(c+d x)}{3 d}-\frac{b^3 \left (10 a^2+5 a b+b^2\right ) \coth ^5(c+d x)}{5 d}-\frac{b^4 (5 a+b) \coth ^7(c+d x)}{7 d}-\frac{b^5 \coth ^9(c+d x)}{9 d}+\frac{(a+b)^5 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\coth (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 ) \coth (c+d x)}{d}-\frac{b^2 \left (10 a^3+10 a^2 b+5 a b^2+b^3\right ) \coth ^3(c+d x)}{3 d}-\frac{b^3 \left (10 a^2+5 a b+b^2\right ) \coth ^5(c+d x)}{5 d}-\frac{b^4 (5 a+b) \coth ^7(c+d x)}{7 d}-\frac{b^5 \coth ^9(c+d x)}{9 d}\\ \end{align*}

Mathematica [A]  time = 2.78567, size = 231, normalized size = 1.44 \[ -\frac{b^5 \coth ^9(c+d x) \left (\frac{1575 a^4 \tanh ^8(c+d x)}{b^4}+\frac{1050 a^3 \left (3 \tanh ^2(c+d x)+1\right ) \tanh ^6(c+d x)}{b^3}+\frac{210 a^2 \left (15 \tanh ^4(c+d x)+5 \tanh ^2(c+d x)+3\right ) \tanh ^4(c+d x)}{b^2}-\frac{315 (a+b)^5 \tanh ^{-1}\left (\sqrt{\tanh ^2(c+d x)}\right ) \tanh ^{10}(c+d x)}{b^5 \sqrt{\tanh ^2(c+d x)}}+\frac{15 a \left (105 \tanh ^6(c+d x)+35 \tanh ^4(c+d x)+21 \tanh ^2(c+d x)+15\right ) \tanh ^2(c+d x)}{b}+315 \tanh ^8(c+d x)+105 \tanh ^6(c+d x)+63 \tanh ^4(c+d x)+45 \tanh ^2(c+d x)+35\right )}{315 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^5*Coth[c + d*x]^9*(35 + 45*Tanh[c + d*x]^2 + 63*Tanh[c + d*x]^4 + 105*Tanh[c + d*x]^6 + 315*Tanh[c + d*x]^
8 + (1575*a^4*Tanh[c + d*x]^8)/b^4 - (315*(a + b)^5*ArcTanh[Sqrt[Tanh[c + d*x]^2]]*Tanh[c + d*x]^10)/(b^5*Sqrt
[Tanh[c + d*x]^2]) + (1050*a^3*Tanh[c + d*x]^6*(1 + 3*Tanh[c + d*x]^2))/b^3 + (210*a^2*Tanh[c + d*x]^4*(3 + 5*
Tanh[c + d*x]^2 + 15*Tanh[c + d*x]^4))/b^2 + (15*a*Tanh[c + d*x]^2*(15 + 21*Tanh[c + d*x]^2 + 35*Tanh[c + d*x]
^4 + 105*Tanh[c + d*x]^6))/b))/(315*d)

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Maple [B]  time = 0.01, size = 472, normalized size = 3. \begin{align*} -5\,{\frac{{a}^{4}b{\rm coth} \left (dx+c\right )}{d}}-10\,{\frac{{a}^{2}{b}^{3}{\rm coth} \left (dx+c\right )}{d}}-2\,{\frac{ \left ({\rm coth} \left (dx+c\right ) \right ) ^{5}{a}^{2}{b}^{3}}{d}}-{\frac{ \left ({\rm coth} \left (dx+c\right ) \right ) ^{5}a{b}^{4}}{d}}-{\frac{5\, \left ({\rm coth} \left (dx+c\right ) \right ) ^{7}a{b}^{4}}{7\,d}}-{\frac{10\, \left ({\rm coth} \left (dx+c\right ) \right ) ^{3}{a}^{3}{b}^{2}}{3\,d}}-{\frac{10\, \left ({\rm coth} \left (dx+c\right ) \right ) ^{3}{a}^{2}{b}^{3}}{3\,d}}+5\,{\frac{\ln \left ({\rm coth} \left (dx+c\right )+1 \right ){a}^{2}{b}^{3}}{d}}+{\frac{5\,\ln \left ({\rm coth} \left (dx+c\right )+1 \right ) a{b}^{4}}{2\,d}}-{\frac{5\,\ln \left ({\rm coth} \left (dx+c\right )-1 \right ){a}^{4}b}{2\,d}}-5\,{\frac{\ln \left ({\rm coth} \left (dx+c\right )-1 \right ){a}^{3}{b}^{2}}{d}}-5\,{\frac{\ln \left ({\rm coth} \left (dx+c\right )-1 \right ){a}^{2}{b}^{3}}{d}}-{\frac{5\,\ln \left ({\rm coth} \left (dx+c\right )-1 \right ) a{b}^{4}}{2\,d}}-{\frac{5\, \left ({\rm coth} \left (dx+c\right ) \right ) ^{3}a{b}^{4}}{3\,d}}-5\,{\frac{a{b}^{4}{\rm coth} \left (dx+c\right )}{d}}-10\,{\frac{{a}^{3}{b}^{2}{\rm coth} \left (dx+c\right )}{d}}+{\frac{5\,\ln \left ({\rm coth} \left (dx+c\right )+1 \right ){a}^{4}b}{2\,d}}+5\,{\frac{\ln \left ({\rm coth} \left (dx+c\right )+1 \right ){a}^{3}{b}^{2}}{d}}+{\frac{\ln \left ({\rm coth} \left (dx+c\right )+1 \right ){a}^{5}}{2\,d}}+{\frac{\ln \left ({\rm coth} \left (dx+c\right )+1 \right ){b}^{5}}{2\,d}}-{\frac{{b}^{5}{\rm coth} \left (dx+c\right )}{d}}-{\frac{ \left ({\rm coth} \left (dx+c\right ) \right ) ^{5}{b}^{5}}{5\,d}}-{\frac{ \left ({\rm coth} \left (dx+c\right ) \right ) ^{7}{b}^{5}}{7\,d}}-{\frac{ \left ({\rm coth} \left (dx+c\right ) \right ) ^{3}{b}^{5}}{3\,d}}-{\frac{\ln \left ({\rm coth} \left (dx+c\right )-1 \right ){a}^{5}}{2\,d}}-{\frac{{b}^{5} \left ({\rm coth} \left (dx+c\right ) \right ) ^{9}}{9\,d}}-{\frac{\ln \left ({\rm coth} \left (dx+c\right )-1 \right ){b}^{5}}{2\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.14901, 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*coth(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 = 2.8841, size = 5446, normalized size = 34.04 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*coth(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)*cosh(d*x + c)^9 + 9*(1575*a^4*b + 42
00*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5)*cosh(d*x + c)*sinh(d*x + c)^8 - (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)*sinh
(d*x + c)^9 - 9*(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)^7 + 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 - 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)^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)*cosh(d*x + c)^3 - 3*(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))*sinh(d*x + c)^6 + 36*(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)^5 - 9*(6300*a^4*b + 16800*a^3*b^2 + 19320*a^2*b^3 + 10560*a*b^4 + 2252*b^5 + 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)^4 + 1260*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x - 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)^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)*cosh(d*x + c)^5 - 35*(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)^3 + 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))*sinh(d*x + c)^4 -
84*(525*a^4*b + 950*a^3*b^2 + 980*a^2*b^3 + 490*a*b^4 + 63*b^5)*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 + 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)^6 - 44100*a^4*b - 117600*a^3*b^2 - 135240*a^2*b^3 - 73920*a*b^4 - 15764*b^5 - 105*(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)^4 - 8820*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x + 120*(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)^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)*cosh(d*x + c)^7 - 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)^5 + 40*(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)^3 - 28*(525*a^4*b + 9
50*a^3*b^2 + 980*a^2*b^3 + 490*a*b^4 + 63*b^5)*cosh(d*x + c))*sinh(d*x + c)^2 + 126*(175*a^4*b + 300*a^3*b^2 +
330*a^2*b^3 + 140*a*b^4 + 63*b^5)*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 + 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)^8 - 7*(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)^6 + 22050*a^4*b + 58800*a^3*b^2 + 67620*a^2*b^3 + 36960*a*b^4 + 7882*b^5 + 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)^4 + 4410*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x - 28*(1
575*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)^2)*sinh(d*x + c))/(d*sinh(d*x + c)^9 + 9*(4*d*cosh(d*x + c)^2 - d)*sinh(d*
x + c)^7 + 9*(14*d*cosh(d*x + c)^4 - 21*d*cosh(d*x + c)^2 + 4*d)*sinh(d*x + c)^5 + 3*(28*d*cosh(d*x + c)^6 - 1
05*d*cosh(d*x + c)^4 + 120*d*cosh(d*x + c)^2 - 28*d)*sinh(d*x + c)^3 + 9*(d*cosh(d*x + c)^8 - 7*d*cosh(d*x + c
)^6 + 20*d*cosh(d*x + c)^4 - 28*d*cosh(d*x + c)^2 + 14*d)*sinh(d*x + c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.21344, 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*coth(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)