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

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

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

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Rubi [A]  time = 0.09, antiderivative size = 160, normalized size of antiderivative = 1.00, 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 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])

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 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])

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*}

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Mathematica [A]  time = 2.90, 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]

-1/315*(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))/d

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fricas [B]  time = 0.43, size = 2111, normalized size = 13.19 \[ \text {result too large to display} \]

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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giac [B]  time = 0.31, size = 721, normalized size = 4.51 \[ \frac {315 \, {\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )} {\left (d x + c\right )} - \frac {2 \, {\left (1575 \, a^{4} b e^{\left (16 \, d x + 16 \, c\right )} + 6300 \, a^{3} b^{2} e^{\left (16 \, d x + 16 \, c\right )} + 9450 \, a^{2} b^{3} e^{\left (16 \, d x + 16 \, c\right )} + 6300 \, a b^{4} e^{\left (16 \, d x + 16 \, c\right )} + 1575 \, b^{5} e^{\left (16 \, d x + 16 \, c\right )} - 12600 \, a^{4} b e^{\left (14 \, d x + 14 \, c\right )} - 44100 \, a^{3} b^{2} e^{\left (14 \, d x + 14 \, c\right )} - 56700 \, a^{2} b^{3} e^{\left (14 \, d x + 14 \, c\right )} - 31500 \, a b^{4} e^{\left (14 \, d x + 14 \, c\right )} - 6300 \, b^{5} e^{\left (14 \, d x + 14 \, c\right )} + 44100 \, a^{4} b e^{\left (12 \, d x + 12 \, c\right )} + 136500 \, a^{3} b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 161700 \, a^{2} b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 90300 \, a b^{4} e^{\left (12 \, d x + 12 \, c\right )} + 21000 \, b^{5} e^{\left (12 \, d x + 12 \, c\right )} - 88200 \, a^{4} b e^{\left (10 \, d x + 10 \, c\right )} - 245700 \, a^{3} b^{2} e^{\left (10 \, d x + 10 \, c\right )} - 283500 \, a^{2} b^{3} e^{\left (10 \, d x + 10 \, c\right )} - 157500 \, a b^{4} e^{\left (10 \, d x + 10 \, c\right )} - 31500 \, b^{5} e^{\left (10 \, d x + 10 \, c\right )} + 110250 \, a^{4} b e^{\left (8 \, d x + 8 \, c\right )} + 283500 \, a^{3} b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 325080 \, a^{2} b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 175140 \, a b^{4} e^{\left (8 \, d x + 8 \, c\right )} + 39438 \, b^{5} e^{\left (8 \, d x + 8 \, c\right )} - 88200 \, a^{4} b e^{\left (6 \, d x + 6 \, c\right )} - 216300 \, a^{3} b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 244020 \, a^{2} b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 131460 \, a b^{4} e^{\left (6 \, d x + 6 \, c\right )} - 26292 \, b^{5} e^{\left (6 \, d x + 6 \, c\right )} + 44100 \, a^{4} b e^{\left (4 \, d x + 4 \, c\right )} + 107100 \, a^{3} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 117180 \, a^{2} b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 63540 \, a b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 13968 \, b^{5} e^{\left (4 \, d x + 4 \, c\right )} - 12600 \, a^{4} b e^{\left (2 \, d x + 2 \, c\right )} - 31500 \, a^{3} b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 34020 \, a^{2} b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 17460 \, a b^{4} e^{\left (2 \, d x + 2 \, c\right )} - 3492 \, b^{5} e^{\left (2 \, d x + 2 \, c\right )} + 1575 \, a^{4} b + 4200 \, a^{3} b^{2} + 4830 \, a^{2} b^{3} + 2640 \, a b^{4} + 563 \, b^{5}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{9}}}{315 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/315*(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 + c) - 2*(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^(16*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^(1
2*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)/(e^(2*d*x + 2*c) -
1)^9)/d

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maple [B]  time = 0.02, size = 472, normalized size = 2.95 \[ -\frac {\left (\coth ^{5}\left (d x +c \right )\right ) a \,b^{4}}{d}-\frac {2 \left (\coth ^{5}\left (d x +c \right )\right ) a^{2} b^{3}}{d}-\frac {5 \left (\coth ^{7}\left (d x +c \right )\right ) a \,b^{4}}{7 d}-\frac {10 \left (\coth ^{3}\left (d x +c \right )\right ) a^{3} b^{2}}{3 d}-\frac {10 \left (\coth ^{3}\left (d x +c \right )\right ) a^{2} b^{3}}{3 d}+\frac {5 \ln \left (\coth \left (d x +c \right )+1\right ) a^{3} b^{2}}{d}+\frac {5 \ln \left (\coth \left (d x +c \right )+1\right ) a \,b^{4}}{2 d}-\frac {5 a^{4} b \coth \left (d x +c \right )}{d}-\frac {5 \ln \left (\coth \left (d x +c \right )-1\right ) a \,b^{4}}{2 d}+\frac {5 \ln \left (\coth \left (d x +c \right )+1\right ) a^{4} b}{2 d}-\frac {5 \ln \left (\coth \left (d x +c \right )-1\right ) a^{3} b^{2}}{d}-\frac {5 \ln \left (\coth \left (d x +c \right )-1\right ) a^{2} b^{3}}{d}-\frac {10 a^{2} b^{3} \coth \left (d x +c \right )}{d}-\frac {5 a \,b^{4} \coth \left (d x +c \right )}{d}+\frac {\ln \left (\coth \left (d x +c \right )+1\right ) a^{5}}{2 d}-\frac {\ln \left (\coth \left (d x +c \right )-1\right ) a^{5}}{2 d}-\frac {\ln \left (\coth \left (d x +c \right )-1\right ) b^{5}}{2 d}+\frac {\ln \left (\coth \left (d x +c \right )+1\right ) b^{5}}{2 d}-\frac {b^{5} \coth \left (d x +c \right )}{d}-\frac {5 \left (\coth ^{3}\left (d x +c \right )\right ) a \,b^{4}}{3 d}-\frac {10 a^{3} b^{2} \coth \left (d x +c \right )}{d}-\frac {5 \ln \left (\coth \left (d x +c \right )-1\right ) a^{4} b}{2 d}+\frac {5 \ln \left (\coth \left (d x +c \right )+1\right ) a^{2} b^{3}}{d}-\frac {b^{5} \left (\coth ^{9}\left (d x +c \right )\right )}{9 d}-\frac {\left (\coth ^{5}\left (d x +c \right )\right ) b^{5}}{5 d}-\frac {\left (\coth ^{3}\left (d x +c \right )\right ) b^{5}}{3 d}-\frac {\left (\coth ^{7}\left (d x +c \right )\right ) b^{5}}{7 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.36, size = 624, normalized size = 3.90 \[ \frac {1}{315} \, b^{5} {\left (315 \, x + \frac {315 \, c}{d} - \frac {2 \, {\left (3492 \, e^{\left (-2 \, d x - 2 \, c\right )} - 13968 \, e^{\left (-4 \, d x - 4 \, c\right )} + 26292 \, e^{\left (-6 \, d x - 6 \, c\right )} - 39438 \, e^{\left (-8 \, d x - 8 \, c\right )} + 31500 \, e^{\left (-10 \, d x - 10 \, c\right )} - 21000 \, e^{\left (-12 \, d x - 12 \, c\right )} + 6300 \, e^{\left (-14 \, d x - 14 \, c\right )} - 1575 \, e^{\left (-16 \, d x - 16 \, c\right )} - 563\right )}}{d {\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 36 \, e^{\left (-4 \, d x - 4 \, c\right )} + 84 \, e^{\left (-6 \, d x - 6 \, c\right )} - 126 \, e^{\left (-8 \, d x - 8 \, c\right )} + 126 \, e^{\left (-10 \, d x - 10 \, c\right )} - 84 \, e^{\left (-12 \, d x - 12 \, c\right )} + 36 \, e^{\left (-14 \, d x - 14 \, c\right )} - 9 \, e^{\left (-16 \, d x - 16 \, c\right )} + e^{\left (-18 \, d x - 18 \, c\right )} - 1\right )}}\right )} + \frac {1}{21} \, a b^{4} {\left (105 \, x + \frac {105 \, c}{d} - \frac {8 \, {\left (203 \, e^{\left (-2 \, d x - 2 \, c\right )} - 609 \, e^{\left (-4 \, d x - 4 \, c\right )} + 770 \, e^{\left (-6 \, d x - 6 \, c\right )} - 770 \, e^{\left (-8 \, d x - 8 \, c\right )} + 315 \, e^{\left (-10 \, d x - 10 \, c\right )} - 105 \, e^{\left (-12 \, d x - 12 \, c\right )} - 44\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} - 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} - 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} - 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} - 1\right )}}\right )} + \frac {2}{3} \, a^{2} b^{3} {\left (15 \, x + \frac {15 \, c}{d} - \frac {2 \, {\left (70 \, e^{\left (-2 \, d x - 2 \, c\right )} - 140 \, e^{\left (-4 \, d x - 4 \, c\right )} + 90 \, e^{\left (-6 \, d x - 6 \, c\right )} - 45 \, e^{\left (-8 \, d x - 8 \, c\right )} - 23\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}}\right )} + \frac {10}{3} \, a^{3} b^{2} {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} - 2\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + 5 \, a^{4} b {\left (x + \frac {c}{d} + \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}}\right )} + a^{5} x \]

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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mupad [B]  time = 1.37, size = 158, normalized size = 0.99 \[ x\,{\left (a+b\right )}^5-\frac {{\mathrm {coth}\left (c+d\,x\right )}^3\,\left (10\,a^3\,b^2+10\,a^2\,b^3+5\,a\,b^4+b^5\right )}{3\,d}-\frac {{\mathrm {coth}\left (c+d\,x\right )}^5\,\left (10\,a^2\,b^3+5\,a\,b^4+b^5\right )}{5\,d}-\frac {{\mathrm {coth}\left (c+d\,x\right )}^7\,\left (b^5+5\,a\,b^4\right )}{7\,d}-\frac {b^5\,{\mathrm {coth}\left (c+d\,x\right )}^9}{9\,d}-\frac {b\,\mathrm {coth}\left (c+d\,x\right )\,\left (5\,a^4+10\,a^3\,b+10\,a^2\,b^2+5\,a\,b^3+b^4\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*coth(c + d*x)^2)^5,x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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