3.56 \(\int (b \coth ^m(c+d x))^{2/3} \, dx\)

Optimal. Leaf size=60 \[ \frac {3 \coth (c+d x) \left (b \coth ^m(c+d x)\right )^{2/3} \, _2F_1\left (1,\frac {1}{6} (2 m+3);\frac {1}{6} (2 m+9);\coth ^2(c+d x)\right )}{d (2 m+3)} \]

[Out]

3*coth(d*x+c)*(b*coth(d*x+c)^m)^(2/3)*hypergeom([1, 1/2+1/3*m],[3/2+1/3*m],coth(d*x+c)^2)/d/(3+2*m)

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Rubi [A]  time = 0.04, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3659, 3476, 364} \[ \frac {3 \coth (c+d x) \left (b \coth ^m(c+d x)\right )^{2/3} \, _2F_1\left (1,\frac {1}{6} (2 m+3);\frac {1}{6} (2 m+9);\coth ^2(c+d x)\right )}{d (2 m+3)} \]

Antiderivative was successfully verified.

[In]

Int[(b*Coth[c + d*x]^m)^(2/3),x]

[Out]

(3*Coth[c + d*x]*(b*Coth[c + d*x]^m)^(2/3)*Hypergeometric2F1[1, (3 + 2*m)/6, (9 + 2*m)/6, Coth[c + d*x]^2])/(d
*(3 + 2*m))

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

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 3659

Int[(u_.)*((b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[(b^IntPart[p]*(b*(c*Tan[e + f*x
])^n)^FracPart[p])/(c*Tan[e + f*x])^(n*FracPart[p]), Int[ActivateTrig[u]*(c*Tan[e + f*x])^(n*p), x], x] /; Fre
eQ[{b, c, 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 ^m(c+d x)\right )^{2/3} \, dx &=\left (\coth ^{-\frac {2 m}{3}}(c+d x) \left (b \coth ^m(c+d x)\right )^{2/3}\right ) \int \coth ^{\frac {2 m}{3}}(c+d x) \, dx\\ &=-\frac {\left (\coth ^{-\frac {2 m}{3}}(c+d x) \left (b \coth ^m(c+d x)\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^{2 m/3}}{-1+x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=\frac {3 \coth (c+d x) \left (b \coth ^m(c+d x)\right )^{2/3} \, _2F_1\left (1,\frac {1}{6} (3+2 m);\frac {1}{6} (9+2 m);\coth ^2(c+d x)\right )}{d (3+2 m)}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 60, normalized size = 1.00 \[ \frac {3 \coth (c+d x) \left (b \coth ^m(c+d x)\right )^{2/3} \, _2F_1\left (1,\frac {1}{6} (2 m+3);\frac {1}{6} (2 m+9);\coth ^2(c+d x)\right )}{d (2 m+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Coth[c + d*x]*(b*Coth[c + d*x]^m)^(2/3)*Hypergeometric2F1[1, (3 + 2*m)/6, (9 + 2*m)/6, Coth[c + d*x]^2])/(d
*(3 + 2*m))

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \coth \left (d x + c\right )^{m}\right )^{\frac {2}{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*coth(d*x + c)^m)^(2/3), x)

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maple [F]  time = 0.42, size = 0, normalized size = 0.00 \[ \int \left (b \left (\coth ^{m}\left (d x +c \right )\right )\right )^{\frac {2}{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*coth(d*x+c)^m)^(2/3),x)

[Out]

int((b*coth(d*x+c)^m)^(2/3),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \coth \left (d x + c\right )^{m}\right )^{\frac {2}{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*coth(d*x + c)^m)^(2/3), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^m\right )}^{2/3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*coth(c + d*x)^m)^(2/3),x)

[Out]

int((b*coth(c + d*x)^m)^(2/3), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \coth ^{m}{\left (c + d x \right )}\right )^{\frac {2}{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*coth(d*x+c)**m)**(2/3),x)

[Out]

Integral((b*coth(c + d*x)**m)**(2/3), x)

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