Optimal. Leaf size=78 \[ \frac{b^2 (1-n) \text{Unintegrable}\left ((b x)^{-n} \sin ^{n-2}(a x),x\right )}{a^2 c^2}+\frac{b (b x)^{1-n} \sin ^{n-1}(a x)}{a^2 \left (a c^2 x \cos (a x)-c^2 \sin (a x)\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.155194, 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 = {} \[ \int \frac{(b x)^{2-n} \sin ^n(a x)}{(a c x \cos (a x)-c \sin (a x))^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
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*}
Mathematica [A] time = 5.51458, size = 0, normalized size = 0. \[ \int \frac{(b x)^{2-n} \sin ^n(a x)}{(a c x \cos (a x)-c \sin (a x))^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.951, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx \right ) ^{2-n} \left ( \sin \left ( ax \right ) \right ) ^{n}}{ \left ( acx\cos \left ( ax \right ) -c\sin \left ( ax \right ) \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]