3.175 \(\int \tanh (d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=53 \[ x-2 x \, _2F_1\left (1,\frac {1}{2 b d n};1+\frac {1}{2 b d n};-e^{2 a d} \left (c x^n\right )^{2 b d}\right ) \]

[Out]

x-2*x*hypergeom([1, 1/2/b/d/n],[1+1/2/b/d/n],-exp(2*a*d)*(c*x^n)^(2*b*d))

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Rubi [F]  time = 0.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Tanh[d*(a + b*Log[c*x^n])],x]

[Out]

Defer[Int][Tanh[d*(a + b*Log[c*x^n])], x]

Rubi steps

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

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Mathematica [B]  time = 8.73, size = 126, normalized size = 2.38 \[ \frac {x e^{2 d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1+\frac {1}{2 b d n};2+\frac {1}{2 b d n};-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )}{2 b d n+1}-x \, _2F_1\left (1,\frac {1}{2 b d n};1+\frac {1}{2 b d n};-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^(2*d*(a + b*Log[c*x^n]))*x*Hypergeometric2F1[1, 1 + 1/(2*b*d*n), 2 + 1/(2*b*d*n), -E^(2*d*(a + b*Log[c*x^n]
))])/(1 + 2*b*d*n) - x*Hypergeometric2F1[1, 1/(2*b*d*n), 1 + 1/(2*b*d*n), -E^(2*d*(a + b*Log[c*x^n]))]

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fricas [F]  time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\tanh \left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.96, size = 0, normalized size = 0.00 \[ \int \tanh \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ x - 2 \, \int \frac {1}{c^{2 \, b d} e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2*integrate(1/(c^(2*b*d)*e^(2*b*d*log(x^n) + 2*a*d) + 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \mathrm {tanh}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(d*(a + b*log(c*x^n))),x)

[Out]

int(tanh(d*(a + b*log(c*x^n))), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \tanh {\left (d \left (a + b \log {\left (c x^{n} \right )}\right ) \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(tanh(d*(a + b*log(c*x**n))), x)

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