3.29 \(\int (b \coth ^3(c+d x))^{3/2} \, dx\)
Optimal. Leaf size=134 \[ -\frac{2 b \coth ^2(c+d x) \sqrt{b \coth ^3(c+d x)}}{7 d}-\frac{2 b \sqrt{b \coth ^3(c+d x)}}{3 d}-\frac{b \sqrt{b \coth ^3(c+d x)} \tan ^{-1}\left (\sqrt{\coth (c+d x)}\right )}{d \coth ^{\frac{3}{2}}(c+d x)}+\frac{b \sqrt{b \coth ^3(c+d x)} \tanh ^{-1}\left (\sqrt{\coth (c+d x)}\right )}{d \coth ^{\frac{3}{2}}(c+d x)} \]
[Out]
(-2*b*Sqrt[b*Coth[c + d*x]^3])/(3*d) - (b*ArcTan[Sqrt[Coth[c + d*x]]]*Sqrt[b*Coth[c + d*x]^3])/(d*Coth[c + d*x
]^(3/2)) + (b*ArcTanh[Sqrt[Coth[c + d*x]]]*Sqrt[b*Coth[c + d*x]^3])/(d*Coth[c + d*x]^(3/2)) - (2*b*Coth[c + d*
x]^2*Sqrt[b*Coth[c + d*x]^3])/(7*d)
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Rubi [A] time = 0.0602026, antiderivative size = 134, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used =
{3658, 3473, 3476, 329, 298, 203, 206} \[ -\frac{2 b \coth ^2(c+d x) \sqrt{b \coth ^3(c+d x)}}{7 d}-\frac{2 b \sqrt{b \coth ^3(c+d x)}}{3 d}-\frac{b \sqrt{b \coth ^3(c+d x)} \tan ^{-1}\left (\sqrt{\coth (c+d x)}\right )}{d \coth ^{\frac{3}{2}}(c+d x)}+\frac{b \sqrt{b \coth ^3(c+d x)} \tanh ^{-1}\left (\sqrt{\coth (c+d x)}\right )}{d \coth ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
Int[(b*Coth[c + d*x]^3)^(3/2),x]
[Out]
(-2*b*Sqrt[b*Coth[c + d*x]^3])/(3*d) - (b*ArcTan[Sqrt[Coth[c + d*x]]]*Sqrt[b*Coth[c + d*x]^3])/(d*Coth[c + d*x
]^(3/2)) + (b*ArcTanh[Sqrt[Coth[c + d*x]]]*Sqrt[b*Coth[c + d*x]^3])/(d*Coth[c + d*x]^(3/2)) - (2*b*Coth[c + d*
x]^2*Sqrt[b*Coth[c + d*x]^3])/(7*d)
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
]])
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 3476
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Rule 329
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
Rule 298
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
& !GtQ[a/b, 0]
Rule 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
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 (b \coth ^3(c+d x)\right )^{3/2} \, dx &=\frac{\left (b \sqrt{b \coth ^3(c+d x)}\right ) \int \coth ^{\frac{9}{2}}(c+d x) \, dx}{\coth ^{\frac{3}{2}}(c+d x)}\\ &=-\frac{2 b \coth ^2(c+d x) \sqrt{b \coth ^3(c+d x)}}{7 d}+\frac{\left (b \sqrt{b \coth ^3(c+d x)}\right ) \int \coth ^{\frac{5}{2}}(c+d x) \, dx}{\coth ^{\frac{3}{2}}(c+d x)}\\ &=-\frac{2 b \sqrt{b \coth ^3(c+d x)}}{3 d}-\frac{2 b \coth ^2(c+d x) \sqrt{b \coth ^3(c+d x)}}{7 d}+\frac{\left (b \sqrt{b \coth ^3(c+d x)}\right ) \int \sqrt{\coth (c+d x)} \, dx}{\coth ^{\frac{3}{2}}(c+d x)}\\ &=-\frac{2 b \sqrt{b \coth ^3(c+d x)}}{3 d}-\frac{2 b \coth ^2(c+d x) \sqrt{b \coth ^3(c+d x)}}{7 d}-\frac{\left (b \sqrt{b \coth ^3(c+d x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{-1+x^2} \, dx,x,\coth (c+d x)\right )}{d \coth ^{\frac{3}{2}}(c+d x)}\\ &=-\frac{2 b \sqrt{b \coth ^3(c+d x)}}{3 d}-\frac{2 b \coth ^2(c+d x) \sqrt{b \coth ^3(c+d x)}}{7 d}-\frac{\left (2 b \sqrt{b \coth ^3(c+d x)}\right ) \operatorname{Subst}\left (\int \frac{x^2}{-1+x^4} \, dx,x,\sqrt{\coth (c+d x)}\right )}{d \coth ^{\frac{3}{2}}(c+d x)}\\ &=-\frac{2 b \sqrt{b \coth ^3(c+d x)}}{3 d}-\frac{2 b \coth ^2(c+d x) \sqrt{b \coth ^3(c+d x)}}{7 d}+\frac{\left (b \sqrt{b \coth ^3(c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{\coth (c+d x)}\right )}{d \coth ^{\frac{3}{2}}(c+d x)}-\frac{\left (b \sqrt{b \coth ^3(c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\coth (c+d x)}\right )}{d \coth ^{\frac{3}{2}}(c+d x)}\\ &=-\frac{2 b \sqrt{b \coth ^3(c+d x)}}{3 d}-\frac{b \tan ^{-1}\left (\sqrt{\coth (c+d x)}\right ) \sqrt{b \coth ^3(c+d x)}}{d \coth ^{\frac{3}{2}}(c+d x)}+\frac{b \tanh ^{-1}\left (\sqrt{\coth (c+d x)}\right ) \sqrt{b \coth ^3(c+d x)}}{d \coth ^{\frac{3}{2}}(c+d x)}-\frac{2 b \coth ^2(c+d x) \sqrt{b \coth ^3(c+d x)}}{7 d}\\ \end{align*}
Mathematica [A] time = 0.518629, size = 82, normalized size = 0.61 \[ -\frac{\left (b \coth ^3(c+d x)\right )^{3/2} \left (6 \coth ^{\frac{7}{2}}(c+d x)+14 \coth ^{\frac{3}{2}}(c+d x)+21 \tan ^{-1}\left (\sqrt{\coth (c+d x)}\right )-21 \tanh ^{-1}\left (\sqrt{\coth (c+d x)}\right )\right )}{21 d \coth ^{\frac{9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
Integrate[(b*Coth[c + d*x]^3)^(3/2),x]
[Out]
-((b*Coth[c + d*x]^3)^(3/2)*(21*ArcTan[Sqrt[Coth[c + d*x]]] - 21*ArcTanh[Sqrt[Coth[c + d*x]]] + 14*Coth[c + d*
x]^(3/2) + 6*Coth[c + d*x]^(7/2)))/(21*d*Coth[c + d*x]^(9/2))
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Maple [A] time = 0.04, size = 107, normalized size = 0.8 \begin{align*}{\frac{1}{21\,d \left ({\rm coth} \left (dx+c\right ) \right ) ^{3}{b}^{2}} \left ( b \left ({\rm coth} \left (dx+c\right ) \right ) ^{3} \right ) ^{{\frac{3}{2}}} \left ( 21\,{b}^{7/2}{\it Artanh} \left ({\frac{\sqrt{b{\rm coth} \left (dx+c\right )}}{\sqrt{b}}} \right ) -21\,{b}^{7/2}\arctan \left ({\frac{\sqrt{b{\rm coth} \left (dx+c\right )}}{\sqrt{b}}} \right ) -6\, \left ( b{\rm coth} \left (dx+c\right ) \right ) ^{7/2}-14\,{b}^{2} \left ( b{\rm coth} \left (dx+c\right ) \right ) ^{3/2} \right ) \left ( b{\rm coth} \left (dx+c\right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*coth(d*x+c)^3)^(3/2),x)
[Out]
1/21/d*(b*coth(d*x+c)^3)^(3/2)*(21*b^(7/2)*arctanh((b*coth(d*x+c))^(1/2)/b^(1/2))-21*b^(7/2)*arctan((b*coth(d*
x+c))^(1/2)/b^(1/2))-6*(b*coth(d*x+c))^(7/2)-14*b^2*(b*coth(d*x+c))^(3/2))/coth(d*x+c)^3/(b*coth(d*x+c))^(3/2)
/b^2
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \coth \left (d x + c\right )^{3}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*coth(d*x+c)^3)^(3/2),x, algorithm="maxima")
[Out]
integrate((b*coth(d*x + c)^3)^(3/2), x)
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Fricas [B] time = 3.08659, size = 5846, normalized size = 43.63 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*coth(d*x+c)^3)^(3/2),x, algorithm="fricas")
[Out]
[-1/84*(42*(b*cosh(d*x + c)^6 + 6*b*cosh(d*x + c)*sinh(d*x + c)^5 + b*sinh(d*x + c)^6 - 3*b*cosh(d*x + c)^4 +
3*(5*b*cosh(d*x + c)^2 - b)*sinh(d*x + c)^4 + 4*(5*b*cosh(d*x + c)^3 - 3*b*cosh(d*x + c))*sinh(d*x + c)^3 + 3*
b*cosh(d*x + c)^2 + 3*(5*b*cosh(d*x + c)^4 - 6*b*cosh(d*x + c)^2 + b)*sinh(d*x + c)^2 + 6*(b*cosh(d*x + c)^5 -
2*b*cosh(d*x + c)^3 + b*cosh(d*x + c))*sinh(d*x + c) - b)*sqrt(-b)*arctan((cosh(d*x + c)^2 + 2*cosh(d*x + c)*
sinh(d*x + c) + sinh(d*x + c)^2)*sqrt(-b)*sqrt(b*cosh(d*x + c)/sinh(d*x + c))/(b*cosh(d*x + c)^2 + 2*b*cosh(d*
x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + b)) - 21*(b*cosh(d*x + c)^6 + 6*b*cosh(d*x + c)*sinh(d*x + c)^5 + b
*sinh(d*x + c)^6 - 3*b*cosh(d*x + c)^4 + 3*(5*b*cosh(d*x + c)^2 - b)*sinh(d*x + c)^4 + 4*(5*b*cosh(d*x + c)^3
- 3*b*cosh(d*x + c))*sinh(d*x + c)^3 + 3*b*cosh(d*x + c)^2 + 3*(5*b*cosh(d*x + c)^4 - 6*b*cosh(d*x + c)^2 + b)
*sinh(d*x + c)^2 + 6*(b*cosh(d*x + c)^5 - 2*b*cosh(d*x + c)^3 + b*cosh(d*x + c))*sinh(d*x + c) - b)*sqrt(-b)*l
og(-(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)^3*sinh(d*x + c) + 6*b*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*b*cosh(d*
x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 + 2*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c
)^2 - 1)*sqrt(-b)*sqrt(b*cosh(d*x + c)/sinh(d*x + c)) - 2*b)/(cosh(d*x + c)^4 + 4*cosh(d*x + c)^3*sinh(d*x + c
) + 6*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4)) + 16*(5*b*cosh(d*x
+ c)^6 + 30*b*cosh(d*x + c)*sinh(d*x + c)^5 + 5*b*sinh(d*x + c)^6 + b*cosh(d*x + c)^4 + (75*b*cosh(d*x + c)^2
+ b)*sinh(d*x + c)^4 + 4*(25*b*cosh(d*x + c)^3 + b*cosh(d*x + c))*sinh(d*x + c)^3 + b*cosh(d*x + c)^2 + (75*b
*cosh(d*x + c)^4 + 6*b*cosh(d*x + c)^2 + b)*sinh(d*x + c)^2 + 2*(15*b*cosh(d*x + c)^5 + 2*b*cosh(d*x + c)^3 +
b*cosh(d*x + c))*sinh(d*x + c) + 5*b)*sqrt(b*cosh(d*x + c)/sinh(d*x + c)))/(d*cosh(d*x + c)^6 + 6*d*cosh(d*x +
c)*sinh(d*x + c)^5 + d*sinh(d*x + c)^6 - 3*d*cosh(d*x + c)^4 + 3*(5*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^4 +
4*(5*d*cosh(d*x + c)^3 - 3*d*cosh(d*x + c))*sinh(d*x + c)^3 + 3*d*cosh(d*x + c)^2 + 3*(5*d*cosh(d*x + c)^4 - 6
*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^2 + 6*(d*cosh(d*x + c)^5 - 2*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d
*x + c) - d), -1/84*(42*(b*cosh(d*x + c)^6 + 6*b*cosh(d*x + c)*sinh(d*x + c)^5 + b*sinh(d*x + c)^6 - 3*b*cosh(
d*x + c)^4 + 3*(5*b*cosh(d*x + c)^2 - b)*sinh(d*x + c)^4 + 4*(5*b*cosh(d*x + c)^3 - 3*b*cosh(d*x + c))*sinh(d*
x + c)^3 + 3*b*cosh(d*x + c)^2 + 3*(5*b*cosh(d*x + c)^4 - 6*b*cosh(d*x + c)^2 + b)*sinh(d*x + c)^2 + 6*(b*cosh
(d*x + c)^5 - 2*b*cosh(d*x + c)^3 + b*cosh(d*x + c))*sinh(d*x + c) - b)*sqrt(b)*arctan(sqrt(b)*sqrt(b*cosh(d*x
+ c)/sinh(d*x + c))/(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + b)) - 21*(b*co
sh(d*x + c)^6 + 6*b*cosh(d*x + c)*sinh(d*x + c)^5 + b*sinh(d*x + c)^6 - 3*b*cosh(d*x + c)^4 + 3*(5*b*cosh(d*x
+ c)^2 - b)*sinh(d*x + c)^4 + 4*(5*b*cosh(d*x + c)^3 - 3*b*cosh(d*x + c))*sinh(d*x + c)^3 + 3*b*cosh(d*x + c)^
2 + 3*(5*b*cosh(d*x + c)^4 - 6*b*cosh(d*x + c)^2 + b)*sinh(d*x + c)^2 + 6*(b*cosh(d*x + c)^5 - 2*b*cosh(d*x +
c)^3 + b*cosh(d*x + c))*sinh(d*x + c) - b)*sqrt(b)*log(2*b*cosh(d*x + c)^4 + 8*b*cosh(d*x + c)^3*sinh(d*x + c)
+ 12*b*cosh(d*x + c)^2*sinh(d*x + c)^2 + 8*b*cosh(d*x + c)*sinh(d*x + c)^3 + 2*b*sinh(d*x + c)^4 + 2*(cosh(d*
x + c)^4 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4 + (6*cosh(d*x + c)^2 - 1)*sinh(d*x + c)^2 - cosh(
d*x + c)^2 + 2*(2*cosh(d*x + c)^3 - cosh(d*x + c))*sinh(d*x + c))*sqrt(b)*sqrt(b*cosh(d*x + c)/sinh(d*x + c))
- b) + 16*(5*b*cosh(d*x + c)^6 + 30*b*cosh(d*x + c)*sinh(d*x + c)^5 + 5*b*sinh(d*x + c)^6 + b*cosh(d*x + c)^4
+ (75*b*cosh(d*x + c)^2 + b)*sinh(d*x + c)^4 + 4*(25*b*cosh(d*x + c)^3 + b*cosh(d*x + c))*sinh(d*x + c)^3 + b*
cosh(d*x + c)^2 + (75*b*cosh(d*x + c)^4 + 6*b*cosh(d*x + c)^2 + b)*sinh(d*x + c)^2 + 2*(15*b*cosh(d*x + c)^5 +
2*b*cosh(d*x + c)^3 + b*cosh(d*x + c))*sinh(d*x + c) + 5*b)*sqrt(b*cosh(d*x + c)/sinh(d*x + c)))/(d*cosh(d*x
+ c)^6 + 6*d*cosh(d*x + c)*sinh(d*x + c)^5 + d*sinh(d*x + c)^6 - 3*d*cosh(d*x + c)^4 + 3*(5*d*cosh(d*x + c)^2
- d)*sinh(d*x + c)^4 + 4*(5*d*cosh(d*x + c)^3 - 3*d*cosh(d*x + c))*sinh(d*x + c)^3 + 3*d*cosh(d*x + c)^2 + 3*(
5*d*cosh(d*x + c)^4 - 6*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^2 + 6*(d*cosh(d*x + c)^5 - 2*d*cosh(d*x + c)^3 +
d*cosh(d*x + c))*sinh(d*x + c) - d)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \coth ^{3}{\left (c + d x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*coth(d*x+c)**3)**(3/2),x)
[Out]
Integral((b*coth(c + d*x)**3)**(3/2), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \coth \left (d x + c\right )^{3}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*coth(d*x+c)^3)^(3/2),x, algorithm="giac")
[Out]
integrate((b*coth(d*x + c)^3)^(3/2), x)