Optimal. Leaf size=97 \[ -e^{2 i a} b^2 2^{-m} x^m (-i b x)^{-m} \text{Gamma}(m-2,-2 i b x)-e^{-2 i a} b^2 2^{-m} x^m (i b x)^{-m} \text{Gamma}(m-2,2 i b x)-\frac{x^{m-2}}{2 (2-m)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.173118, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3312, 3307, 2181} \[ -e^{2 i a} b^2 2^{-m} x^m (-i b x)^{-m} \text{Gamma}(m-2,-2 i b x)-e^{-2 i a} b^2 2^{-m} x^m (i b x)^{-m} \text{Gamma}(m-2,2 i b x)-\frac{x^{m-2}}{2 (2-m)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3312
Rule 3307
Rule 2181
Rubi steps
\begin{align*} \int x^{-3+m} \sin ^2(a+b x) \, dx &=\int \left (\frac{x^{-3+m}}{2}-\frac{1}{2} x^{-3+m} \cos (2 a+2 b x)\right ) \, dx\\ &=-\frac{x^{-2+m}}{2 (2-m)}-\frac{1}{2} \int x^{-3+m} \cos (2 a+2 b x) \, dx\\ &=-\frac{x^{-2+m}}{2 (2-m)}-\frac{1}{4} \int e^{-i (2 a+2 b x)} x^{-3+m} \, dx-\frac{1}{4} \int e^{i (2 a+2 b x)} x^{-3+m} \, dx\\ &=-\frac{x^{-2+m}}{2 (2-m)}-2^{-m} b^2 e^{2 i a} x^m (-i b x)^{-m} \Gamma (-2+m,-2 i b x)-2^{-m} b^2 e^{-2 i a} x^m (i b x)^{-m} \Gamma (-2+m,2 i b x)\\ \end{align*}
Mathematica [A] time = 0.364202, size = 121, normalized size = 1.25 \[ \frac{2^{-m-1} x^{m-2} \left (b^2 x^2\right )^{-m} \left (-2 b^2 (m-2) x^2 (\cos (a)-i \sin (a))^2 (-i b x)^m \text{Gamma}(m-2,2 i b x)+2 (m-2) (\cos (2 a)+i \sin (2 a)) (i b x)^{m+2} \text{Gamma}(m-2,-2 i b x)+2^m \left (b^2 x^2\right )^m\right )}{m-2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.072, size = 0, normalized size = 0. \begin{align*} \int{x}^{m-3} \left ( \sin \left ( bx+a \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (m - 2\right )} x^{2} \int \frac{x^{m} \cos \left (2 \, b x + 2 \, a\right )}{x^{3}}\,{d x} - x^{m}}{2 \,{\left (m - 2\right )} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.81052, size = 235, normalized size = 2.42 \begin{align*} \frac{4 \, b x x^{m - 3} +{\left (-i \, m + 2 i\right )} e^{\left (-{\left (m - 3\right )} \log \left (2 i \, b\right ) - 2 i \, a\right )} \Gamma \left (m - 2, 2 i \, b x\right ) +{\left (i \, m - 2 i\right )} e^{\left (-{\left (m - 3\right )} \log \left (-2 i \, b\right ) + 2 i \, a\right )} \Gamma \left (m - 2, -2 i \, b x\right )}{8 \,{\left (b m - 2 \, b\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m - 3} \sin \left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]