3.160 \(\int x \tan (d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=69 \[ i x^2 \, _2F_1\left (1,-\frac {i}{b d n};1-\frac {i}{b d n};-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )-\frac {i x^2}{2} \]

[Out]

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

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Rubi [F]  time = 0.02, 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 = {} \[ \int x \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x*Tan[d*(a + b*Log[c*x^n])],x]

[Out]

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

Rubi steps

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

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Mathematica [B]  time = 6.01, size = 146, normalized size = 2.12 \[ \frac {x^2 \left (i e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1-\frac {i}{b d n};2-\frac {i}{b d n};-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(b d n-i) \, _2F_1\left (1,-\frac {i}{b d n};1-\frac {i}{b d n};-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{-2-2 i b d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \tan \left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [F]  time = 1.01, size = 0, normalized size = 0.00 \[ \int x \tan \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\mathrm {tan}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*tan(d*(a + b*log(c*x^n))),x)

[Out]

int(x*tan(d*(a + b*log(c*x^n))), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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