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

Optimal. Leaf size=27 \[ b \text{Unintegrable}\left (\frac{\sin \left (c+d (f+g x)^n\right )}{x^2},x\right )-\frac{a}{x} \]

[Out]

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

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Rubi [A]  time = 0.0184941, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{a+b \sin \left (c+d (f+g x)^n\right )}{x^2} \, dx \]

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*}

Mathematica [A]  time = 2.67523, size = 0, normalized size = 0. \[ \int \frac{a+b \sin \left (c+d (f+g x)^n\right )}{x^2} \, dx \]

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 = 0.118, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\sin \left ( c+d \left ( gx+f \right ) ^{n} \right ) }{{x}^{2}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} b \int \frac{\sin \left ({\left (g x + f\right )}^{n} d + c\right )}{x^{2}}\,{d x} - \frac{a}{x} \end{align*}

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., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a}{x^{2}}, x\right ) \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \sin{\left (c + d \left (f + g x\right )^{n} \right )}}{x^{2}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a}{x^{2}}\,{d x} \end{align*}

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)