3.40 \(\int (b \coth ^4(c+d x))^{3/2} \, dx\)
Optimal. Leaf size=110 \[ -\frac {b \coth (c+d x) \sqrt {b \coth ^4(c+d x)}}{3 d}-\frac {b \coth ^3(c+d x) \sqrt {b \coth ^4(c+d x)}}{5 d}+b x \tanh ^2(c+d x) \sqrt {b \coth ^4(c+d x)}-\frac {b \tanh (c+d x) \sqrt {b \coth ^4(c+d x)}}{d} \]
[Out]
-1/3*b*coth(d*x+c)*(b*coth(d*x+c)^4)^(1/2)/d-1/5*b*coth(d*x+c)^3*(b*coth(d*x+c)^4)^(1/2)/d-b*(b*coth(d*x+c)^4)
^(1/2)*tanh(d*x+c)/d+b*x*(b*coth(d*x+c)^4)^(1/2)*tanh(d*x+c)^2
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Rubi [A] time = 0.05, antiderivative size = 110, normalized size of antiderivative = 1.00,
number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used =
{3658, 3473, 8} \[ -\frac {b \coth ^3(c+d x) \sqrt {b \coth ^4(c+d x)}}{5 d}-\frac {b \coth (c+d x) \sqrt {b \coth ^4(c+d x)}}{3 d}+b x \tanh ^2(c+d x) \sqrt {b \coth ^4(c+d x)}-\frac {b \tanh (c+d x) \sqrt {b \coth ^4(c+d x)}}{d} \]
Antiderivative was successfully verified.
[In]
Int[(b*Coth[c + d*x]^4)^(3/2),x]
[Out]
-(b*Coth[c + d*x]*Sqrt[b*Coth[c + d*x]^4])/(3*d) - (b*Coth[c + d*x]^3*Sqrt[b*Coth[c + d*x]^4])/(5*d) - (b*Sqrt
[b*Coth[c + d*x]^4]*Tanh[c + d*x])/d + b*x*Sqrt[b*Coth[c + d*x]^4]*Tanh[c + d*x]^2
Rule 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
Rule 3473
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Rule 3658
Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Tan[e + f*x]^n)^FracPart[p])/(Tan[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Tan[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])
Rubi steps
\begin {align*} \int \left (b \coth ^4(c+d x)\right )^{3/2} \, dx &=\left (b \sqrt {b \coth ^4(c+d x)} \tanh ^2(c+d x)\right ) \int \coth ^6(c+d x) \, dx\\ &=-\frac {b \coth ^3(c+d x) \sqrt {b \coth ^4(c+d x)}}{5 d}+\left (b \sqrt {b \coth ^4(c+d x)} \tanh ^2(c+d x)\right ) \int \coth ^4(c+d x) \, dx\\ &=-\frac {b \coth (c+d x) \sqrt {b \coth ^4(c+d x)}}{3 d}-\frac {b \coth ^3(c+d x) \sqrt {b \coth ^4(c+d x)}}{5 d}+\left (b \sqrt {b \coth ^4(c+d x)} \tanh ^2(c+d x)\right ) \int \coth ^2(c+d x) \, dx\\ &=-\frac {b \coth (c+d x) \sqrt {b \coth ^4(c+d x)}}{3 d}-\frac {b \coth ^3(c+d x) \sqrt {b \coth ^4(c+d x)}}{5 d}-\frac {b \sqrt {b \coth ^4(c+d x)} \tanh (c+d x)}{d}+\left (b \sqrt {b \coth ^4(c+d x)} \tanh ^2(c+d x)\right ) \int 1 \, dx\\ &=-\frac {b \coth (c+d x) \sqrt {b \coth ^4(c+d x)}}{3 d}-\frac {b \coth ^3(c+d x) \sqrt {b \coth ^4(c+d x)}}{5 d}-\frac {b \sqrt {b \coth ^4(c+d x)} \tanh (c+d x)}{d}+b x \sqrt {b \coth ^4(c+d x)} \tanh ^2(c+d x)\\ \end {align*}
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Mathematica [C] time = 0.07, size = 43, normalized size = 0.39 \[ -\frac {\tanh (c+d x) \left (b \coth ^4(c+d x)\right )^{3/2} \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\tanh ^2(c+d x)\right )}{5 d} \]
Antiderivative was successfully verified.
[In]
Integrate[(b*Coth[c + d*x]^4)^(3/2),x]
[Out]
-1/5*((b*Coth[c + d*x]^4)^(3/2)*Hypergeometric2F1[-5/2, 1, -3/2, Tanh[c + d*x]^2]*Tanh[c + d*x])/d
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fricas [B] time = 0.91, size = 3421, normalized size = 31.10 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*coth(d*x+c)^4)^(3/2),x, algorithm="fricas")
[Out]
1/15*(15*b*d*x*cosh(d*x + c)^10 + 15*(b*d*x*e^(4*d*x + 4*c) - 2*b*d*x*e^(2*d*x + 2*c) + b*d*x)*sinh(d*x + c)^1
0 + 150*(b*d*x*cosh(d*x + c)*e^(4*d*x + 4*c) - 2*b*d*x*cosh(d*x + c)*e^(2*d*x + 2*c) + b*d*x*cosh(d*x + c))*si
nh(d*x + c)^9 - 15*(5*b*d*x + 6*b)*cosh(d*x + c)^8 + 15*(45*b*d*x*cosh(d*x + c)^2 - 5*b*d*x + (45*b*d*x*cosh(d
*x + c)^2 - 5*b*d*x - 6*b)*e^(4*d*x + 4*c) - 2*(45*b*d*x*cosh(d*x + c)^2 - 5*b*d*x - 6*b)*e^(2*d*x + 2*c) - 6*
b)*sinh(d*x + c)^8 + 120*(15*b*d*x*cosh(d*x + c)^3 - (5*b*d*x + 6*b)*cosh(d*x + c) + (15*b*d*x*cosh(d*x + c)^3
- (5*b*d*x + 6*b)*cosh(d*x + c))*e^(4*d*x + 4*c) - 2*(15*b*d*x*cosh(d*x + c)^3 - (5*b*d*x + 6*b)*cosh(d*x + c
))*e^(2*d*x + 2*c))*sinh(d*x + c)^7 + 30*(5*b*d*x + 6*b)*cosh(d*x + c)^6 + 30*(105*b*d*x*cosh(d*x + c)^4 + 5*b
*d*x - 14*(5*b*d*x + 6*b)*cosh(d*x + c)^2 + (105*b*d*x*cosh(d*x + c)^4 + 5*b*d*x - 14*(5*b*d*x + 6*b)*cosh(d*x
+ c)^2 + 6*b)*e^(4*d*x + 4*c) - 2*(105*b*d*x*cosh(d*x + c)^4 + 5*b*d*x - 14*(5*b*d*x + 6*b)*cosh(d*x + c)^2 +
6*b)*e^(2*d*x + 2*c) + 6*b)*sinh(d*x + c)^6 + 60*(63*b*d*x*cosh(d*x + c)^5 - 14*(5*b*d*x + 6*b)*cosh(d*x + c)
^3 + 3*(5*b*d*x + 6*b)*cosh(d*x + c) + (63*b*d*x*cosh(d*x + c)^5 - 14*(5*b*d*x + 6*b)*cosh(d*x + c)^3 + 3*(5*b
*d*x + 6*b)*cosh(d*x + c))*e^(4*d*x + 4*c) - 2*(63*b*d*x*cosh(d*x + c)^5 - 14*(5*b*d*x + 6*b)*cosh(d*x + c)^3
+ 3*(5*b*d*x + 6*b)*cosh(d*x + c))*e^(2*d*x + 2*c))*sinh(d*x + c)^5 - 10*(15*b*d*x + 28*b)*cosh(d*x + c)^4 + 1
0*(315*b*d*x*cosh(d*x + c)^6 - 105*(5*b*d*x + 6*b)*cosh(d*x + c)^4 - 15*b*d*x + 45*(5*b*d*x + 6*b)*cosh(d*x +
c)^2 + (315*b*d*x*cosh(d*x + c)^6 - 105*(5*b*d*x + 6*b)*cosh(d*x + c)^4 - 15*b*d*x + 45*(5*b*d*x + 6*b)*cosh(d
*x + c)^2 - 28*b)*e^(4*d*x + 4*c) - 2*(315*b*d*x*cosh(d*x + c)^6 - 105*(5*b*d*x + 6*b)*cosh(d*x + c)^4 - 15*b*
d*x + 45*(5*b*d*x + 6*b)*cosh(d*x + c)^2 - 28*b)*e^(2*d*x + 2*c) - 28*b)*sinh(d*x + c)^4 + 40*(45*b*d*x*cosh(d
*x + c)^7 - 21*(5*b*d*x + 6*b)*cosh(d*x + c)^5 + 15*(5*b*d*x + 6*b)*cosh(d*x + c)^3 - (15*b*d*x + 28*b)*cosh(d
*x + c) + (45*b*d*x*cosh(d*x + c)^7 - 21*(5*b*d*x + 6*b)*cosh(d*x + c)^5 + 15*(5*b*d*x + 6*b)*cosh(d*x + c)^3
- (15*b*d*x + 28*b)*cosh(d*x + c))*e^(4*d*x + 4*c) - 2*(45*b*d*x*cosh(d*x + c)^7 - 21*(5*b*d*x + 6*b)*cosh(d*x
+ c)^5 + 15*(5*b*d*x + 6*b)*cosh(d*x + c)^3 - (15*b*d*x + 28*b)*cosh(d*x + c))*e^(2*d*x + 2*c))*sinh(d*x + c)
^3 - 15*b*d*x + 5*(15*b*d*x + 28*b)*cosh(d*x + c)^2 + 5*(135*b*d*x*cosh(d*x + c)^8 - 84*(5*b*d*x + 6*b)*cosh(d
*x + c)^6 + 90*(5*b*d*x + 6*b)*cosh(d*x + c)^4 + 15*b*d*x - 12*(15*b*d*x + 28*b)*cosh(d*x + c)^2 + (135*b*d*x*
cosh(d*x + c)^8 - 84*(5*b*d*x + 6*b)*cosh(d*x + c)^6 + 90*(5*b*d*x + 6*b)*cosh(d*x + c)^4 + 15*b*d*x - 12*(15*
b*d*x + 28*b)*cosh(d*x + c)^2 + 28*b)*e^(4*d*x + 4*c) - 2*(135*b*d*x*cosh(d*x + c)^8 - 84*(5*b*d*x + 6*b)*cosh
(d*x + c)^6 + 90*(5*b*d*x + 6*b)*cosh(d*x + c)^4 + 15*b*d*x - 12*(15*b*d*x + 28*b)*cosh(d*x + c)^2 + 28*b)*e^(
2*d*x + 2*c) + 28*b)*sinh(d*x + c)^2 + (15*b*d*x*cosh(d*x + c)^10 - 15*(5*b*d*x + 6*b)*cosh(d*x + c)^8 + 30*(5
*b*d*x + 6*b)*cosh(d*x + c)^6 - 10*(15*b*d*x + 28*b)*cosh(d*x + c)^4 - 15*b*d*x + 5*(15*b*d*x + 28*b)*cosh(d*x
+ c)^2 - 46*b)*e^(4*d*x + 4*c) - 2*(15*b*d*x*cosh(d*x + c)^10 - 15*(5*b*d*x + 6*b)*cosh(d*x + c)^8 + 30*(5*b*
d*x + 6*b)*cosh(d*x + c)^6 - 10*(15*b*d*x + 28*b)*cosh(d*x + c)^4 - 15*b*d*x + 5*(15*b*d*x + 28*b)*cosh(d*x +
c)^2 - 46*b)*e^(2*d*x + 2*c) + 10*(15*b*d*x*cosh(d*x + c)^9 - 12*(5*b*d*x + 6*b)*cosh(d*x + c)^7 + 18*(5*b*d*x
+ 6*b)*cosh(d*x + c)^5 - 4*(15*b*d*x + 28*b)*cosh(d*x + c)^3 + (15*b*d*x + 28*b)*cosh(d*x + c) + (15*b*d*x*co
sh(d*x + c)^9 - 12*(5*b*d*x + 6*b)*cosh(d*x + c)^7 + 18*(5*b*d*x + 6*b)*cosh(d*x + c)^5 - 4*(15*b*d*x + 28*b)*
cosh(d*x + c)^3 + (15*b*d*x + 28*b)*cosh(d*x + c))*e^(4*d*x + 4*c) - 2*(15*b*d*x*cosh(d*x + c)^9 - 12*(5*b*d*x
+ 6*b)*cosh(d*x + c)^7 + 18*(5*b*d*x + 6*b)*cosh(d*x + c)^5 - 4*(15*b*d*x + 28*b)*cosh(d*x + c)^3 + (15*b*d*x
+ 28*b)*cosh(d*x + c))*e^(2*d*x + 2*c))*sinh(d*x + c) - 46*b)*sqrt((b*e^(8*d*x + 8*c) + 4*b*e^(6*d*x + 6*c) +
6*b*e^(4*d*x + 4*c) + 4*b*e^(2*d*x + 2*c) + b)/(e^(8*d*x + 8*c) - 4*e^(6*d*x + 6*c) + 6*e^(4*d*x + 4*c) - 4*e
^(2*d*x + 2*c) + 1))/(d*cosh(d*x + c)^10 + (d*e^(4*d*x + 4*c) + 2*d*e^(2*d*x + 2*c) + d)*sinh(d*x + c)^10 + 10
*(d*cosh(d*x + c)*e^(4*d*x + 4*c) + 2*d*cosh(d*x + c)*e^(2*d*x + 2*c) + d*cosh(d*x + c))*sinh(d*x + c)^9 - 5*d
*cosh(d*x + c)^8 + 5*(9*d*cosh(d*x + c)^2 + (9*d*cosh(d*x + c)^2 - d)*e^(4*d*x + 4*c) + 2*(9*d*cosh(d*x + c)^2
- d)*e^(2*d*x + 2*c) - d)*sinh(d*x + c)^8 + 40*(3*d*cosh(d*x + c)^3 - d*cosh(d*x + c) + (3*d*cosh(d*x + c)^3
- d*cosh(d*x + c))*e^(4*d*x + 4*c) + 2*(3*d*cosh(d*x + c)^3 - d*cosh(d*x + c))*e^(2*d*x + 2*c))*sinh(d*x + c)^
7 + 10*d*cosh(d*x + c)^6 + 10*(21*d*cosh(d*x + c)^4 - 14*d*cosh(d*x + c)^2 + (21*d*cosh(d*x + c)^4 - 14*d*cosh
(d*x + c)^2 + d)*e^(4*d*x + 4*c) + 2*(21*d*cosh(d*x + c)^4 - 14*d*cosh(d*x + c)^2 + d)*e^(2*d*x + 2*c) + d)*si
nh(d*x + c)^6 + 4*(63*d*cosh(d*x + c)^5 - 70*d*cosh(d*x + c)^3 + 15*d*cosh(d*x + c) + (63*d*cosh(d*x + c)^5 -
70*d*cosh(d*x + c)^3 + 15*d*cosh(d*x + c))*e^(4*d*x + 4*c) + 2*(63*d*cosh(d*x + c)^5 - 70*d*cosh(d*x + c)^3 +
15*d*cosh(d*x + c))*e^(2*d*x + 2*c))*sinh(d*x + c)^5 - 10*d*cosh(d*x + c)^4 + 10*(21*d*cosh(d*x + c)^6 - 35*d*
cosh(d*x + c)^4 + 15*d*cosh(d*x + c)^2 + (21*d*cosh(d*x + c)^6 - 35*d*cosh(d*x + c)^4 + 15*d*cosh(d*x + c)^2 -
d)*e^(4*d*x + 4*c) + 2*(21*d*cosh(d*x + c)^6 - 35*d*cosh(d*x + c)^4 + 15*d*cosh(d*x + c)^2 - d)*e^(2*d*x + 2*
c) - d)*sinh(d*x + c)^4 + 40*(3*d*cosh(d*x + c)^7 - 7*d*cosh(d*x + c)^5 + 5*d*cosh(d*x + c)^3 - d*cosh(d*x + c
) + (3*d*cosh(d*x + c)^7 - 7*d*cosh(d*x + c)^5 + 5*d*cosh(d*x + c)^3 - d*cosh(d*x + c))*e^(4*d*x + 4*c) + 2*(3
*d*cosh(d*x + c)^7 - 7*d*cosh(d*x + c)^5 + 5*d*cosh(d*x + c)^3 - d*cosh(d*x + c))*e^(2*d*x + 2*c))*sinh(d*x +
c)^3 + 5*d*cosh(d*x + c)^2 + 5*(9*d*cosh(d*x + c)^8 - 28*d*cosh(d*x + c)^6 + 30*d*cosh(d*x + c)^4 - 12*d*cosh(
d*x + c)^2 + (9*d*cosh(d*x + c)^8 - 28*d*cosh(d*x + c)^6 + 30*d*cosh(d*x + c)^4 - 12*d*cosh(d*x + c)^2 + d)*e^
(4*d*x + 4*c) + 2*(9*d*cosh(d*x + c)^8 - 28*d*cosh(d*x + c)^6 + 30*d*cosh(d*x + c)^4 - 12*d*cosh(d*x + c)^2 +
d)*e^(2*d*x + 2*c) + d)*sinh(d*x + c)^2 + (d*cosh(d*x + c)^10 - 5*d*cosh(d*x + c)^8 + 10*d*cosh(d*x + c)^6 - 1
0*d*cosh(d*x + c)^4 + 5*d*cosh(d*x + c)^2 - d)*e^(4*d*x + 4*c) + 2*(d*cosh(d*x + c)^10 - 5*d*cosh(d*x + c)^8 +
10*d*cosh(d*x + c)^6 - 10*d*cosh(d*x + c)^4 + 5*d*cosh(d*x + c)^2 - d)*e^(2*d*x + 2*c) + 10*(d*cosh(d*x + c)^
9 - 4*d*cosh(d*x + c)^7 + 6*d*cosh(d*x + c)^5 - 4*d*cosh(d*x + c)^3 + d*cosh(d*x + c) + (d*cosh(d*x + c)^9 - 4
*d*cosh(d*x + c)^7 + 6*d*cosh(d*x + c)^5 - 4*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*e^(4*d*x + 4*c) + 2*(d*cosh(
d*x + c)^9 - 4*d*cosh(d*x + c)^7 + 6*d*cosh(d*x + c)^5 - 4*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*e^(2*d*x + 2*c
))*sinh(d*x + c) - d)
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giac [A] time = 0.18, size = 77, normalized size = 0.70 \[ \frac {{\left (15 \, d x + 15 \, c - \frac {2 \, {\left (45 \, e^{\left (8 \, d x + 8 \, c\right )} - 90 \, e^{\left (6 \, d x + 6 \, c\right )} + 140 \, e^{\left (4 \, d x + 4 \, c\right )} - 70 \, e^{\left (2 \, d x + 2 \, c\right )} + 23\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{5}}\right )} b^{\frac {3}{2}}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*coth(d*x+c)^4)^(3/2),x, algorithm="giac")
[Out]
1/15*(15*d*x + 15*c - 2*(45*e^(8*d*x + 8*c) - 90*e^(6*d*x + 6*c) + 140*e^(4*d*x + 4*c) - 70*e^(2*d*x + 2*c) +
23)/(e^(2*d*x + 2*c) - 1)^5)*b^(3/2)/d
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maple [A] time = 0.16, size = 77, normalized size = 0.70 \[ -\frac {\left (b \left (\coth ^{4}\left (d x +c \right )\right )\right )^{\frac {3}{2}} \left (6 \left (\coth ^{5}\left (d x +c \right )\right )+10 \left (\coth ^{3}\left (d x +c \right )\right )+15 \ln \left (\coth \left (d x +c \right )-1\right )-15 \ln \left (\coth \left (d x +c \right )+1\right )+30 \coth \left (d x +c \right )\right )}{30 d \coth \left (d x +c \right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*coth(d*x+c)^4)^(3/2),x)
[Out]
-1/30/d*(b*coth(d*x+c)^4)^(3/2)*(6*coth(d*x+c)^5+10*coth(d*x+c)^3+15*ln(coth(d*x+c)-1)-15*ln(coth(d*x+c)+1)+30
*coth(d*x+c))/coth(d*x+c)^6
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maxima [A] time = 0.42, size = 137, normalized size = 1.25 \[ \frac {{\left (d x + c\right )} b^{\frac {3}{2}}}{d} - \frac {2 \, {\left (70 \, b^{\frac {3}{2}} e^{\left (-2 \, d x - 2 \, c\right )} - 140 \, b^{\frac {3}{2}} e^{\left (-4 \, d x - 4 \, c\right )} + 90 \, b^{\frac {3}{2}} e^{\left (-6 \, d x - 6 \, c\right )} - 45 \, b^{\frac {3}{2}} e^{\left (-8 \, d x - 8 \, c\right )} - 23 \, b^{\frac {3}{2}}\right )}}{15 \, d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*coth(d*x+c)^4)^(3/2),x, algorithm="maxima")
[Out]
(d*x + c)*b^(3/2)/d - 2/15*(70*b^(3/2)*e^(-2*d*x - 2*c) - 140*b^(3/2)*e^(-4*d*x - 4*c) + 90*b^(3/2)*e^(-6*d*x
- 6*c) - 45*b^(3/2)*e^(-8*d*x - 8*c) - 23*b^(3/2))/(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))
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^4\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*coth(c + d*x)^4)^(3/2),x)
[Out]
int((b*coth(c + d*x)^4)^(3/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \coth ^{4}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*coth(d*x+c)**4)**(3/2),x)
[Out]
Integral((b*coth(c + d*x)**4)**(3/2), x)
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