3.280 \(\int \frac {1}{x (a+b \sin (c+d (f+g x)^n))} \, dx\)

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {1}{x \left (a+b \sin \left (c+d (f+g x)^n\right )\right )},x\right ) \]

[Out]

Unintegrable(1/x/(a+b*sin(c+d*(g*x+f)^n)),x)

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Rubi [A]  time = 0.03, 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 = {} \[ \int \frac {1}{x \left (a+b \sin \left (c+d (f+g x)^n\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x*(a + b*Sin[c + d*(f + g*x)^n])),x]

[Out]

Defer[Int][1/(x*(a + b*Sin[c + d*(f + g*x)^n])), x]

Rubi steps

\begin {align*} \int \frac {1}{x \left (a+b \sin \left (c+d (f+g x)^n\right )\right )} \, dx &=\int \frac {1}{x \left (a+b \sin \left (c+d (f+g x)^n\right )\right )} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.73, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a+b \sin \left (c+d (f+g x)^n\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x*(a + b*Sin[c + d*(f + g*x)^n])),x]

[Out]

Integrate[1/(x*(a + b*Sin[c + d*(f + g*x)^n])), x]

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fricas [A]  time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b x \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+b*sin(c+d*(g*x+f)^n)),x, algorithm="fricas")

[Out]

integral(1/(b*x*sin((g*x + f)^n*d + c) + a*x), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+b*sin(c+d*(g*x+f)^n)),x, algorithm="giac")

[Out]

integrate(1/((b*sin((g*x + f)^n*d + c) + a)*x), x)

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maple [A]  time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a +b \sin \left (c +d \left (g x +f \right )^{n}\right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(a+b*sin(c+d*(g*x+f)^n)),x)

[Out]

int(1/x/(a+b*sin(c+d*(g*x+f)^n)),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+b*sin(c+d*(g*x+f)^n)),x, algorithm="maxima")

[Out]

integrate(1/((b*sin((g*x + f)^n*d + c) + a)*x), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{x\,\left (a+b\,\sin \left (c+d\,{\left (f+g\,x\right )}^n\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*sin(c + d*(f + g*x)^n))),x)

[Out]

int(1/(x*(a + b*sin(c + d*(f + g*x)^n))), x)

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sympy [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(1/x/(a+b*sin(c+d*(g*x+f)**n)),x)

[Out]

Timed out

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