3.3.71 \(\int \frac {a+b \sin (c+d (f+g x)^n)}{x^2} \, dx\) [271]

Optimal. Leaf size=28 \[ -\frac {a}{x}+b \text {Int}\left (\frac {\sin \left (c+d (f+g x)^n\right )}{x^2},x\right ) \]

[Out]

-a/x+b*Unintegrable(sin(c+d*(g*x+f)^n)/x^2,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 = {} \begin {gather*} \int \frac {a+b \sin \left (c+d (f+g x)^n\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(a + b*Sin[c + d*(f + g*x)^n])/x^2,x]

[Out]

-(a/x) + b*Defer[Int][Sin[c + d*(f + g*x)^n]/x^2, x]

Rubi steps

\begin {align*} \int \frac {a+b \sin \left (c+d (f+g x)^n\right )}{x^2} \, dx &=\int \left (\frac {a}{x^2}+\frac {b \sin \left (c+d (f+g x)^n\right )}{x^2}\right ) \, dx\\ &=-\frac {a}{x}+b \int \frac {\sin \left (c+d (f+g x)^n\right )}{x^2} \, dx\\ \end {align*}

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Mathematica [A]
time = 1.40, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b \sin \left (c+d (f+g x)^n\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(a + b*Sin[c + d*(f + g*x)^n])/x^2,x]

[Out]

Integrate[(a + b*Sin[c + d*(f + g*x)^n])/x^2, x]

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Maple [A]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a +b \sin \left (c +d \left (g x +f \right )^{n}\right )}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sin(c+d*(g*x+f)^n))/x^2,x)

[Out]

int((a+b*sin(c+d*(g*x+f)^n))/x^2,x)

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Maxima [A]
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((a+b*sin(c+d*(g*x+f)^n))/x^2,x, algorithm="maxima")

[Out]

b*integrate(sin((g*x + f)^n*d + c)/x^2, x) - a/x

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(c+d*(g*x+f)^n))/x^2,x, algorithm="fricas")

[Out]

integral((b*sin((g*x + f)^n*d + c) + a)/x^2, x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \sin {\left (c + d \left (f + g x\right )^{n} \right )}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(c+d*(g*x+f)**n))/x**2,x)

[Out]

Integral((a + b*sin(c + d*(f + g*x)**n))/x**2, x)

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(c+d*(g*x+f)^n))/x^2,x, algorithm="giac")

[Out]

integrate((b*sin((g*x + f)^n*d + c) + a)/x^2, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {a+b\,\sin \left (c+d\,{\left (f+g\,x\right )}^n\right )}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sin(c + d*(f + g*x)^n))/x^2,x)

[Out]

int((a + b*sin(c + d*(f + g*x)^n))/x^2, x)

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