3.2.97 \(\int \coth ^p(d (a+b \log (c x^n))) \, dx\) [197]

Optimal. Leaf size=115 \[ x \left (-1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )^p \left (1+e^{2 a d} \left (c x^n\right )^{2 b d}\right )^{-p} F_1\left (\frac {1}{2 b d n};p,-p;1+\frac {1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right ) \]

[Out]

x*(-1-exp(2*a*d)*(c*x^n)^(2*b*d))^p*AppellF1(1/2/b/d/n,p,-p,1+1/2/b/d/n,exp(2*a*d)*(c*x^n)^(2*b*d),-exp(2*a*d)
*(c*x^n)^(2*b*d))/((1+exp(2*a*d)*(c*x^n)^(2*b*d))^p)

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Rubi [A]
time = 0.09, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {5655, 5657, 525, 524} \begin {gather*} x \left (-e^{2 a d} \left (c x^n\right )^{2 b d}-1\right )^p \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )^{-p} F_1\left (\frac {1}{2 b d n};p,-p;1+\frac {1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[d*(a + b*Log[c*x^n])]^p,x]

[Out]

(x*(-1 - E^(2*a*d)*(c*x^n)^(2*b*d))^p*AppellF1[1/(2*b*d*n), p, -p, 1 + 1/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d),
-(E^(2*a*d)*(c*x^n)^(2*b*d))])/(1 + E^(2*a*d)*(c*x^n)^(2*b*d))^p

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 5655

Int[Coth[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[
x^(1/n - 1)*Coth[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n
, 1])

Rule 5657

Int[Coth[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*((-1 - E^(2*a*d)*x^
(2*b*d))^p/(1 - E^(2*a*d)*x^(2*b*d))^p), x] /; FreeQ[{a, b, d, e, m, p}, x]

Rubi steps

\begin {align*} \int \coth ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int \coth ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(387\) vs. \(2(115)=230\).
time = 2.46, size = 387, normalized size = 3.37 \begin {gather*} \frac {(1+2 b d n) x \left (\frac {1+e^{2 a d} \left (c x^n\right )^{2 b d}}{-1+e^{2 a d} \left (c x^n\right )^{2 b d}}\right )^p F_1\left (\frac {1}{2 b d n};p,-p;1+\frac {1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{2 b d e^{2 a d} n p \left (c x^n\right )^{2 b d} F_1\left (1+\frac {1}{2 b d n};p,1-p;2+\frac {1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )+2 b d e^{2 a d} n p \left (c x^n\right )^{2 b d} F_1\left (1+\frac {1}{2 b d n};1+p,-p;2+\frac {1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )+(1+2 b d n) F_1\left (\frac {1}{2 b d n};p,-p;1+\frac {1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Coth[d*(a + b*Log[c*x^n])]^p,x]

[Out]

((1 + 2*b*d*n)*x*((1 + E^(2*a*d)*(c*x^n)^(2*b*d))/(-1 + E^(2*a*d)*(c*x^n)^(2*b*d)))^p*AppellF1[1/(2*b*d*n), p,
-p, 1 + 1/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d), -(E^(2*a*d)*(c*x^n)^(2*b*d))])/(2*b*d*E^(2*a*d)*n*p*(c*x^n)^(
2*b*d)*AppellF1[1 + 1/(2*b*d*n), p, 1 - p, 2 + 1/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d), -(E^(2*a*d)*(c*x^n)^(2*
b*d))] + 2*b*d*E^(2*a*d)*n*p*(c*x^n)^(2*b*d)*AppellF1[1 + 1/(2*b*d*n), 1 + p, -p, 2 + 1/(2*b*d*n), E^(2*a*d)*(
c*x^n)^(2*b*d), -(E^(2*a*d)*(c*x^n)^(2*b*d))] + (1 + 2*b*d*n)*AppellF1[1/(2*b*d*n), p, -p, 1 + 1/(2*b*d*n), E^
(2*a*d)*(c*x^n)^(2*b*d), -(E^(2*a*d)*(c*x^n)^(2*b*d))])

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Maple [F]
time = 1.18, size = 0, normalized size = 0.00 \[\int \coth ^{p}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*(a+b*ln(c*x^n)))^p,x)

[Out]

int(coth(d*(a+b*ln(c*x^n)))^p,x)

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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(coth(d*(a+b*log(c*x^n)))^p,x, algorithm="maxima")

[Out]

integrate(coth((b*log(c*x^n) + a)*d)^p, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*(a+b*log(c*x^n)))^p,x, algorithm="fricas")

[Out]

integral(coth(b*d*log(c*x^n) + a*d)^p, x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*(a+b*ln(c*x**n)))**p,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*(a+b*log(c*x^n)))^p,x, algorithm="giac")

[Out]

integrate(coth((b*log(c*x^n) + a)*d)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*(a + b*log(c*x^n)))^p,x)

[Out]

int(coth(d*(a + b*log(c*x^n)))^p, x)

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