Optimal. Leaf size=79 \[ \frac {b^2 (1-n) \text {Int}\left ((b x)^{-n} \sin ^{n-2}(a x),x\right )}{a^2 c^2}+\frac {b (b x)^{1-n} \sin ^{n-1}(a x)}{a^2 \left (a c^2 x \cos (a x)-c^2 \sin (a x)\right )} \]
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Rubi [A] time = 0.16, 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 = {} \[ \int \frac {(b x)^{2-n} \sin ^n(a x)}{(a c x \cos (a x)-c \sin (a x))^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(b x)^{2-n} \sin ^n(a x)}{(a c x \cos (a x)-c \sin (a x))^2} \, dx &=\frac {b (b x)^{1-n} \sin ^{-1+n}(a x)}{a^2 \left (a c^2 x \cos (a x)-c^2 \sin (a x)\right )}+\frac {\left (b^2 (1-n)\right ) \int (b x)^{-n} \sin ^{-2+n}(a x) \, dx}{a^2 c^2}\\ \end {align*}
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Mathematica [A] time = 5.46, size = 0, normalized size = 0.00 \[ \int \frac {(b x)^{2-n} \sin ^n(a x)}{(a c x \cos (a x)-c \sin (a x))^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\left (b x\right )^{-n + 2} \sin \left (a x\right )^{n}}{2 \, a c^{2} x \cos \left (a x\right ) \sin \left (a x\right ) - {\left (a^{2} c^{2} x^{2} - c^{2}\right )} \cos \left (a x\right )^{2} - c^{2}}, x\right ) \]
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 \[ \int \frac {\left (b x\right )^{-n + 2} \sin \left (a x\right )^{n}}{{\left (a c x \cos \left (a x\right ) - c \sin \left (a x\right )\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.19, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x \right )^{2-n} \left (\sin ^{n}\left (a x \right )\right )}{\left (a c x \cos \left (a x \right )-c \sin \left (a x \right )\right )^{2}}\, dx \]
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 \[ \int \frac {\left (b x\right )^{-n + 2} \sin \left (a x\right )^{n}}{{\left (a c x \cos \left (a x\right ) - c \sin \left (a x\right )\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\sin \left (a\,x\right )}^n\,{\left (b\,x\right )}^{2-n}}{{\left (c\,\sin \left (a\,x\right )-a\,c\,x\,\cos \left (a\,x\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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