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

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {1}{a+b \sin \left (c+d (f+g x)^n\right )},x\right ) \]

[Out]

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

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

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {1}{a+b \sin \left (c+d (f+g x)^n\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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)

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maple [A]  time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {1}{a +b \sin \left (c +d \left (g x +f \right )^{n}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(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 \[ \int \frac {1}{b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {1}{a+b\,\sin \left (c+d\,{\left (f+g\,x\right )}^n\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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