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

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

[Out]

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

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 3.94, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \sin \left (c+d (f+g x)^n\right )\right )^2}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b^{2} \cos \left ({\left (g x + f\right )}^{n} d + c\right )^{2} - 2 \, a b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) - a^{2} - b^{2}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.89, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \sin \left (c +d \left (g x +f \right )^{n}\right )\right )^{2}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, b^{2} \int \frac {\cos \left (2 \, {\left (g x + f\right )}^{n} d + 2 \, c\right )}{x}\,{d x} + 2 \, a b \int \frac {\sin \left ({\left (g x + f\right )}^{n} d + c\right )}{x}\,{d x} + a^{2} \log \relax (x) + \frac {1}{2} \, b^{2} \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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