Optimal. Leaf size=25 \[ b \text{Unintegrable}\left (\frac{\sin \left (c+d (f+g x)^n\right )}{x},x\right )+a \log (x) \]
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Rubi [A] time = 0.018133, 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 = {} \[ \int \frac{a+b \sin \left (c+d (f+g x)^n\right )}{x} \, dx \]
Verification is Not applicable to the result.
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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*}
Mathematica [A] time = 2.78931, size = 0, normalized size = 0. \[ \int \frac{a+b \sin \left (c+d (f+g x)^n\right )}{x} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.072, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\sin \left ( c+d \left ( gx+f \right ) ^{n} \right ) }{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} b \int \frac{\sin \left ({\left (g x + f\right )}^{n} d + c\right )}{x}\,{d x} + a \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \sin{\left (c + d \left (f + g x\right )^{n} \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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