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

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 = 0.03, 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 = {} \[ \int \frac {\left (a+b \sin \left (c+d (f+g x)^n\right )\right )^2}{x^2} \, dx \]

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*}

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Mathematica [A]  time = 3.34, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \sin \left (c+d (f+g x)^n\right )\right )^2}{x^2} \, dx \]

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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fricas [A]  time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b^{2} \cos \left ({\left (g x + f\right )}^{n} d + c\right )^{2} - 2 \, a b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) - a^{2} - b^{2}}{x^{2}}, x\right ) \]

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

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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maple [A]  time = 0.78, 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 \[ -\frac {a^{2}}{x} - \frac {b^{2} x \int \frac {\cos \left (2 \, {\left (g x + f\right )}^{n} d + 2 \, c\right )}{x^{2}}\,{d x} - 4 \, a b x \int \frac {\sin \left ({\left (g x + f\right )}^{n} d + c\right )}{x^{2}}\,{d x} + b^{2}}{2 \, x} \]

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

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

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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