3.3.76 \(\int \frac {(a+b \sin (c+d (f+g x)^n))^2}{x^2} \, dx\) [276]

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {\left (a+b \sin \left (c+d (f+g x)^n\right )\right )^2}{x^2},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, 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 {\left (a+b \sin \left (c+d (f+g x)^n\right )\right )^2}{x^2} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 2.09, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \sin \left (c+d (f+g x)^n\right )\right )^2}{x^2} \, dx \end {gather*}

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 = 0.14, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \sin \left (c +d \left (g x +f \right )^{n}\right )\right )^{2}}{x^{2}}\, dx\]

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

[Out]

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

[Out]

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

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

[Out]

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

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