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