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

Optimal. Leaf size=26 \[ b \text {Int}\left (\frac {\sin \left (c+d (f+g x)^n\right )}{x},x\right )+a \log (x) \]

[Out]

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

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 3.28, size = 0, normalized size = 0.00 \[ \int \frac {a+b \sin \left (c+d (f+g x)^n\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {a +b \sin \left (c +d \left (g x +f \right )^{n}\right )}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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