3.1.29 \(\int (b \coth ^3(c+d x))^{3/2} \, dx\) [29]

Optimal. Leaf size=134 \[ -\frac {2 b \sqrt {b \coth ^3(c+d x)}}{3 d}-\frac {b \text {ArcTan}\left (\sqrt {\coth (c+d x)}\right ) \sqrt {b \coth ^3(c+d x)}}{d \coth ^{\frac {3}{2}}(c+d x)}+\frac {b \tanh ^{-1}\left (\sqrt {\coth (c+d x)}\right ) \sqrt {b \coth ^3(c+d x)}}{d \coth ^{\frac {3}{2}}(c+d x)}-\frac {2 b \coth ^2(c+d x) \sqrt {b \coth ^3(c+d x)}}{7 d} \]

[Out]

-2/3*b*(b*coth(d*x+c)^3)^(1/2)/d-b*arctan(coth(d*x+c)^(1/2))*(b*coth(d*x+c)^3)^(1/2)/d/coth(d*x+c)^(3/2)+b*arc
tanh(coth(d*x+c)^(1/2))*(b*coth(d*x+c)^3)^(1/2)/d/coth(d*x+c)^(3/2)-2/7*b*coth(d*x+c)^2*(b*coth(d*x+c)^3)^(1/2
)/d

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Rubi [A]
time = 0.05, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3739, 3554, 3557, 335, 304, 209, 212} \begin {gather*} -\frac {b \text {ArcTan}\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 \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 {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)} \end {gather*}

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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 304

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 335

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 3554

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 3557

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 3739

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 ^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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.35, size = 82, normalized size = 0.61 \begin {gather*} -\frac {\left (b \coth ^3(c+d x)\right )^{3/2} \left (21 \text {ArcTan}\left (\sqrt {\coth (c+d x)}\right )-21 \tanh ^{-1}\left (\sqrt {\coth (c+d x)}\right )+14 \coth ^{\frac {3}{2}}(c+d x)+6 \coth ^{\frac {7}{2}}(c+d x)\right )}{21 d \coth ^{\frac {9}{2}}(c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*Coth[c + d*x]^3)^(3/2),x]

[Out]

-1/21*((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)))/(d*Coth[c + d*x]^(9/2))

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Maple [A]
time = 2.29, size = 107, normalized size = 0.80

method	result	size
derivativedivides	\(-\frac {\left (b \left (\coth ^{3}\left (d x +c \right )\right )\right )^{\frac {3}{2}} \left (-21 b^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )+21 b^{\frac {7}{2}} \arctan \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )+6 \left (b \coth \left (d x +c \right )\right )^{\frac {7}{2}}+14 b^{2} \left (b \coth \left (d x +c \right )\right )^{\frac {3}{2}}\right )}{21 d \coth \left (d x +c \right )^{3} \left (b \coth \left (d x +c \right )\right )^{\frac {3}{2}} b^{2}}\)	\(107\)
default	\(-\frac {\left (b \left (\coth ^{3}\left (d x +c \right )\right )\right )^{\frac {3}{2}} \left (-21 b^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )+21 b^{\frac {7}{2}} \arctan \left (\frac {\sqrt {b \coth \left (d x +c \right )}}{\sqrt {b}}\right )+6 \left (b \coth \left (d x +c \right )\right )^{\frac {7}{2}}+14 b^{2} \left (b \coth \left (d x +c \right )\right )^{\frac {3}{2}}\right )}{21 d \coth \left (d x +c \right )^{3} \left (b \coth \left (d x +c \right )\right )^{\frac {3}{2}} b^{2}}\)	\(107\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*coth(d*x+c)^3)^(3/2),x,method=_RETURNVERBOSE)

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 1051 vs. \(2 (114) = 228\).
time = 0.42, size = 2152, normalized size = 16.06 \begin {gather*} \text {Too large to display} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b \coth ^{3}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 788 vs. \(2 (114) = 228\).
time = 0.62, size = 788, normalized size = 5.88 \begin {gather*} \frac {{\left (42 \, \sqrt {b} \arctan \left (-\frac {\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b}}{\sqrt {b}}\right ) \mathrm {sgn}\left (e^{\left (6 \, d x + 6 \, c\right )} - 3 \, e^{\left (4 \, d x + 4 \, c\right )} + 3 \, e^{\left (2 \, d x + 2 \, c\right )} - 1\right ) \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right ) - 21 \, \sqrt {b} \log \left ({\left | -\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} + \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b} \right |}\right ) \mathrm {sgn}\left (e^{\left (6 \, d x + 6 \, c\right )} - 3 \, e^{\left (4 \, d x + 4 \, c\right )} + 3 \, e^{\left (2 \, d x + 2 \, c\right )} - 1\right ) \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right ) + \frac {16 \, {\left (21 \, {\left (\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b}\right )}^{6} b \mathrm {sgn}\left (e^{\left (6 \, d x + 6 \, c\right )} - 3 \, e^{\left (4 \, d x + 4 \, c\right )} + 3 \, e^{\left (2 \, d x + 2 \, c\right )} - 1\right ) \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right ) - 42 \, {\left (\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b}\right )}^{5} b^{\frac {3}{2}} \mathrm {sgn}\left (e^{\left (6 \, d x + 6 \, c\right )} - 3 \, e^{\left (4 \, d x + 4 \, c\right )} + 3 \, e^{\left (2 \, d x + 2 \, c\right )} - 1\right ) \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right ) + 119 \, {\left (\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b}\right )}^{4} b^{2} \mathrm {sgn}\left (e^{\left (6 \, d x + 6 \, c\right )} - 3 \, e^{\left (4 \, d x + 4 \, c\right )} + 3 \, e^{\left (2 \, d x + 2 \, c\right )} - 1\right ) \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right ) - 56 \, {\left (\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b}\right )}^{3} b^{\frac {5}{2}} \mathrm {sgn}\left (e^{\left (6 \, d x + 6 \, c\right )} - 3 \, e^{\left (4 \, d x + 4 \, c\right )} + 3 \, e^{\left (2 \, d x + 2 \, c\right )} - 1\right ) \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right ) + 63 \, {\left (\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b}\right )}^{2} b^{3} \mathrm {sgn}\left (e^{\left (6 \, d x + 6 \, c\right )} - 3 \, e^{\left (4 \, d x + 4 \, c\right )} + 3 \, e^{\left (2 \, d x + 2 \, c\right )} - 1\right ) \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right ) - 14 \, {\left (\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b}\right )} b^{\frac {7}{2}} \mathrm {sgn}\left (e^{\left (6 \, d x + 6 \, c\right )} - 3 \, e^{\left (4 \, d x + 4 \, c\right )} + 3 \, e^{\left (2 \, d x + 2 \, c\right )} - 1\right ) \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right ) + 5 \, b^{4} \mathrm {sgn}\left (e^{\left (6 \, d x + 6 \, c\right )} - 3 \, e^{\left (4 \, d x + 4 \, c\right )} + 3 \, e^{\left (2 \, d x + 2 \, c\right )} - 1\right ) \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )\right )}}{{\left (\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b} - \sqrt {b}\right )}^{7}}\right )} b}{42 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*coth(d*x+c)^3)^(3/2),x, algorithm="giac")

[Out]

1/42*(42*sqrt(b)*arctan(-(sqrt(b)*e^(2*d*x + 2*c) - sqrt(b*e^(4*d*x + 4*c) - b))/sqrt(b))*sgn(e^(6*d*x + 6*c)
- 3*e^(4*d*x + 4*c) + 3*e^(2*d*x + 2*c) - 1)*sgn(e^(4*d*x + 4*c) - 1) - 21*sqrt(b)*log(abs(-sqrt(b)*e^(2*d*x +
2*c) + sqrt(b*e^(4*d*x + 4*c) - b)))*sgn(e^(6*d*x + 6*c) - 3*e^(4*d*x + 4*c) + 3*e^(2*d*x + 2*c) - 1)*sgn(e^(
4*d*x + 4*c) - 1) + 16*(21*(sqrt(b)*e^(2*d*x + 2*c) - sqrt(b*e^(4*d*x + 4*c) - b))^6*b*sgn(e^(6*d*x + 6*c) - 3
*e^(4*d*x + 4*c) + 3*e^(2*d*x + 2*c) - 1)*sgn(e^(4*d*x + 4*c) - 1) - 42*(sqrt(b)*e^(2*d*x + 2*c) - sqrt(b*e^(4
*d*x + 4*c) - b))^5*b^(3/2)*sgn(e^(6*d*x + 6*c) - 3*e^(4*d*x + 4*c) + 3*e^(2*d*x + 2*c) - 1)*sgn(e^(4*d*x + 4*
c) - 1) + 119*(sqrt(b)*e^(2*d*x + 2*c) - sqrt(b*e^(4*d*x + 4*c) - b))^4*b^2*sgn(e^(6*d*x + 6*c) - 3*e^(4*d*x +
4*c) + 3*e^(2*d*x + 2*c) - 1)*sgn(e^(4*d*x + 4*c) - 1) - 56*(sqrt(b)*e^(2*d*x + 2*c) - sqrt(b*e^(4*d*x + 4*c)
- b))^3*b^(5/2)*sgn(e^(6*d*x + 6*c) - 3*e^(4*d*x + 4*c) + 3*e^(2*d*x + 2*c) - 1)*sgn(e^(4*d*x + 4*c) - 1) + 6
3*(sqrt(b)*e^(2*d*x + 2*c) - sqrt(b*e^(4*d*x + 4*c) - b))^2*b^3*sgn(e^(6*d*x + 6*c) - 3*e^(4*d*x + 4*c) + 3*e^
(2*d*x + 2*c) - 1)*sgn(e^(4*d*x + 4*c) - 1) - 14*(sqrt(b)*e^(2*d*x + 2*c) - sqrt(b*e^(4*d*x + 4*c) - b))*b^(7/
2)*sgn(e^(6*d*x + 6*c) - 3*e^(4*d*x + 4*c) + 3*e^(2*d*x + 2*c) - 1)*sgn(e^(4*d*x + 4*c) - 1) + 5*b^4*sgn(e^(6*
d*x + 6*c) - 3*e^(4*d*x + 4*c) + 3*e^(2*d*x + 2*c) - 1)*sgn(e^(4*d*x + 4*c) - 1))/(sqrt(b)*e^(2*d*x + 2*c) - s
qrt(b*e^(4*d*x + 4*c) - b) - sqrt(b))^7)*b/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^3\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*coth(c + d*x)^3)^(3/2),x)

[Out]

int((b*coth(c + d*x)^3)^(3/2), x)

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