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

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{\left (a+b \sin \left (c+d (f+g x)^n\right )\right )^2}{x^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0255086, 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 = {} \[ \int \frac{\left (a+b \sin \left (c+d (f+g x)^n\right )\right )^2}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 3.16726, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \sin \left (c+d (f+g x)^n\right )\right )^2}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.293, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\sin \left ( c+d \left ( gx+f \right ) ^{n} \right ) \right ) ^{2}}{{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))^2/x^2,x)

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{2}}{x} - \frac{b^{2} x \int \frac{\cos \left (2 \,{\left (g x + f\right )}^{n} d + 2 \, c\right )}{x^{2}}\,{d x} - 4 \, a b x \int \frac{\sin \left ({\left (g x + f\right )}^{n} d + c\right )}{x^{2}}\,{d x} + b^{2}}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^2/x - 1/2*(b^2*x*integrate(cos(2*(g*x + f)^n*d + 2*c)/x^2, x) - 4*a*b*x*integrate(sin((g*x + f)^n*d + c)/x^
2, x) + b^2)/x

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{b^{2} \cos \left ({\left (g x + f\right )}^{n} d + c\right )^{2} - 2 \, a b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) - a^{2} - b^{2}}{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))^2/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a\right )}^{2}}{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))^2/x^2,x, algorithm="giac")

[Out]

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