3.2.99 \(\int \frac {1}{x \tanh ^{\frac {5}{2}}(a+b \log (c x^n))} \, dx\) [199]

Optimal. Leaf size=72 \[ \frac {\text {ArcTan}\left (\sqrt {\tanh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\tanh ^{-1}\left (\sqrt {\tanh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}-\frac {2}{3 b n \tanh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[Out]

arctan(tanh(a+b*ln(c*x^n))^(1/2))/b/n+arctanh(tanh(a+b*ln(c*x^n))^(1/2))/b/n-2/3/b/n/tanh(a+b*ln(c*x^n))^(3/2)

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Rubi [A]
time = 0.04, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3555, 3557, 335, 218, 212, 209} \begin {gather*} \frac {\text {ArcTan}\left (\sqrt {\tanh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\tanh ^{-1}\left (\sqrt {\tanh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}-\frac {2}{3 b n \tanh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Tanh[a + b*Log[c*x^n]]^(5/2)),x]

[Out]

ArcTan[Sqrt[Tanh[a + b*Log[c*x^n]]]]/(b*n) + ArcTanh[Sqrt[Tanh[a + b*Log[c*x^n]]]]/(b*n) - 2/(3*b*n*Tanh[a + b
*Log[c*x^n]]^(3/2))

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 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[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 3555

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Tan[c + d*x])^(n + 1)/(b*d*(n + 1)), x] - Dist[
1/b^2, Int[(b*Tan[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[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]

Rubi steps

\begin {align*} \int \frac {1}{x \tanh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\tanh ^{\frac {5}{2}}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {2}{3 b n \tanh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {\tanh (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {2}{3 b n \tanh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {x} \left (-1+x^2\right )} \, dx,x,\tanh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ &=-\frac {2}{3 b n \tanh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {2 \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt {\tanh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}\\ &=-\frac {2}{3 b n \tanh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\tanh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\tanh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}\\ &=\frac {\tan ^{-1}\left (\sqrt {\tanh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\tanh ^{-1}\left (\sqrt {\tanh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}-\frac {2}{3 b n \tanh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.15, size = 46, normalized size = 0.64 \begin {gather*} -\frac {2 \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};\tanh ^2\left (a+b \log \left (c x^n\right )\right )\right )}{3 b n \tanh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Tanh[a + b*Log[c*x^n]]^(5/2)),x]

[Out]

(-2*Hypergeometric2F1[-3/4, 1, 1/4, Tanh[a + b*Log[c*x^n]]^2])/(3*b*n*Tanh[a + b*Log[c*x^n]]^(3/2))

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Maple [A]
time = 4.20, size = 74, normalized size = 1.03

method	result	size
derivativedivides	\(\frac {-\frac {2}{3 \tanh \left (a +b \ln \left (c \,x^{n}\right )\right )^{\frac {3}{2}}}+\frac {\ln \left (\sqrt {\tanh }\left (a +b \ln \left (c \,x^{n}\right )\right )+1\right )}{2}-\frac {\ln \left (\sqrt {\tanh }\left (a +b \ln \left (c \,x^{n}\right )\right )-1\right )}{2}+\arctan \left (\sqrt {\tanh }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{n b}\)	\(74\)
default	\(\frac {-\frac {2}{3 \tanh \left (a +b \ln \left (c \,x^{n}\right )\right )^{\frac {3}{2}}}+\frac {\ln \left (\sqrt {\tanh }\left (a +b \ln \left (c \,x^{n}\right )\right )+1\right )}{2}-\frac {\ln \left (\sqrt {\tanh }\left (a +b \ln \left (c \,x^{n}\right )\right )-1\right )}{2}+\arctan \left (\sqrt {\tanh }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{n b}\)	\(74\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/tanh(a+b*ln(c*x^n))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/n/b*(-2/3/tanh(a+b*ln(c*x^n))^(3/2)+1/2*ln(tanh(a+b*ln(c*x^n))^(1/2)+1)-1/2*ln(tanh(a+b*ln(c*x^n))^(1/2)-1)+
arctan(tanh(a+b*ln(c*x^n))^(1/2)))

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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(1/x/tanh(a+b*log(c*x^n))^(5/2),x, algorithm="maxima")

[Out]

integrate(1/(x*tanh(b*log(c*x^n) + a)^(5/2)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1110 vs. \(2 (64) = 128\).
time = 0.37, size = 1110, normalized size = 15.42 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/tanh(a+b*log(c*x^n))^(5/2),x, algorithm="fricas")

[Out]

-1/6*(4*cosh(b*n*log(x) + b*log(c) + a)^4 + 16*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)
^3 + 4*sinh(b*n*log(x) + b*log(c) + a)^4 + 8*(3*cosh(b*n*log(x) + b*log(c) + a)^2 - 1)*sinh(b*n*log(x) + b*log
(c) + a)^2 - 6*(cosh(b*n*log(x) + b*log(c) + a)^4 + 4*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(
c) + a)^3 + sinh(b*n*log(x) + b*log(c) + a)^4 + 2*(3*cosh(b*n*log(x) + b*log(c) + a)^2 - 1)*sinh(b*n*log(x) +
b*log(c) + a)^2 - 2*cosh(b*n*log(x) + b*log(c) + a)^2 + 4*(cosh(b*n*log(x) + b*log(c) + a)^3 - cosh(b*n*log(x)
+ b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a) + 1)*arctan(-cosh(b*n*log(x) + b*log(c) + a)^2 - 2*cosh(b*n*
log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) - sinh(b*n*log(x) + b*log(c) + a)^2 + (cosh(b*n*log(x)
+ b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*lo
g(c) + a)^2 + 1)*sqrt(sinh(b*n*log(x) + b*log(c) + a)/cosh(b*n*log(x) + b*log(c) + a))) - 8*cosh(b*n*log(x) +
b*log(c) + a)^2 + 3*(cosh(b*n*log(x) + b*log(c) + a)^4 + 4*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b
*log(c) + a)^3 + sinh(b*n*log(x) + b*log(c) + a)^4 + 2*(3*cosh(b*n*log(x) + b*log(c) + a)^2 - 1)*sinh(b*n*log(
x) + b*log(c) + a)^2 - 2*cosh(b*n*log(x) + b*log(c) + a)^2 + 4*(cosh(b*n*log(x) + b*log(c) + a)^3 - cosh(b*n*l
og(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a) + 1)*log(-cosh(b*n*log(x) + b*log(c) + a)^2 - 2*cosh(b*
n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) - sinh(b*n*log(x) + b*log(c) + a)^2 + (cosh(b*n*log(x
) + b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*
log(c) + a)^2 + 1)*sqrt(sinh(b*n*log(x) + b*log(c) + a)/cosh(b*n*log(x) + b*log(c) + a))) + 16*(cosh(b*n*log(x
) + b*log(c) + a)^3 - cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a) + 4*(cosh(b*n*log(x) +
b*log(c) + a)^4 + 4*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^3 + sinh(b*n*log(x) + b*lo
g(c) + a)^4 + 2*(3*cosh(b*n*log(x) + b*log(c) + a)^2 + 1)*sinh(b*n*log(x) + b*log(c) + a)^2 + 2*cosh(b*n*log(x
) + b*log(c) + a)^2 + 4*(cosh(b*n*log(x) + b*log(c) + a)^3 + cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x)
+ b*log(c) + a) + 1)*sqrt(sinh(b*n*log(x) + b*log(c) + a)/cosh(b*n*log(x) + b*log(c) + a)) + 4)/(b*n*cosh(b*n*
log(x) + b*log(c) + a)^4 + 4*b*n*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^3 + b*n*sinh(
b*n*log(x) + b*log(c) + a)^4 - 2*b*n*cosh(b*n*log(x) + b*log(c) + a)^2 + 2*(3*b*n*cosh(b*n*log(x) + b*log(c) +
a)^2 - b*n)*sinh(b*n*log(x) + b*log(c) + a)^2 + b*n + 4*(b*n*cosh(b*n*log(x) + b*log(c) + a)^3 - b*n*cosh(b*n
*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \tanh ^{\frac {5}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/tanh(a+b*ln(c*x**n))**(5/2),x)

[Out]

Integral(1/(x*tanh(a + b*log(c*x**n))**(5/2)), x)

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Giac [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(1/x/tanh(a+b*log(c*x^n))^(5/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 2.48, size = 64, normalized size = 0.89 \begin {gather*} \frac {\mathrm {atan}\left (\sqrt {\mathrm {tanh}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n}+\frac {\mathrm {atanh}\left (\sqrt {\mathrm {tanh}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n}-\frac {2}{3\,b\,n\,{\mathrm {tanh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*tanh(a + b*log(c*x^n))^(5/2)),x)

[Out]

atan(tanh(a + b*log(c*x^n))^(1/2))/(b*n) + atanh(tanh(a + b*log(c*x^n))^(1/2))/(b*n) - 2/(3*b*n*tanh(a + b*log
(c*x^n))^(3/2))

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