Optimal. Leaf size=75 \[ \frac {1}{2} i e^{i a} b x^m (-i b x)^{-m} \Gamma (m-1,-i b x)-\frac {1}{2} i e^{-i a} b x^m (i b x)^{-m} \Gamma (m-1,i b x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3307, 2181} \[ \frac {1}{2} i e^{i a} b x^m (-i b x)^{-m} \text {Gamma}(m-1,-i b x)-\frac {1}{2} i e^{-i a} b x^m (i b x)^{-m} \text {Gamma}(m-1,i b x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2181
Rule 3307
Rubi steps
\begin {align*} \int x^{-2+m} \cos (a+b x) \, dx &=\frac {1}{2} \int e^{-i (a+b x)} x^{-2+m} \, dx+\frac {1}{2} \int e^{i (a+b x)} x^{-2+m} \, dx\\ &=\frac {1}{2} i b e^{i a} x^m (-i b x)^{-m} \Gamma (-1+m,-i b x)-\frac {1}{2} i b e^{-i a} x^m (i b x)^{-m} \Gamma (-1+m,i b x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 75, normalized size = 1.00 \[ \frac {1}{2} i e^{i a} b x^m (-i b x)^{-m} \Gamma (m-1,-i b x)-\frac {1}{2} i e^{-i a} b x^m (i b x)^{-m} \Gamma (m-1,i b x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.70, size = 54, normalized size = 0.72 \[ \frac {i \, e^{\left (-{\left (m - 2\right )} \log \left (i \, b\right ) - i \, a\right )} \Gamma \left (m - 1, i \, b x\right ) - i \, e^{\left (-{\left (m - 2\right )} \log \left (-i \, b\right ) + i \, a\right )} \Gamma \left (m - 1, -i \, b x\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m - 2} \cos \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.12, size = 530, normalized size = 7.07 \[ 2^{-2+m} b^{2} \left (b^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} \sqrt {\pi }\, \left (\frac {3 \,2^{1-m} x^{-2+m} \left (b^{2}\right )^{-\frac {1}{2}+\frac {m}{2}} \left (2 x^{2} b^{2}+2 m +2\right ) \sin \left (b x \right )}{\sqrt {\pi }\, \left (-1+m \right ) \left (3+3 m \right ) b}-\frac {2^{2-m} x^{-2+m} \left (b^{2}\right )^{-\frac {1}{2}+\frac {m}{2}} \left (x^{2} b^{2}-m^{2}-m \right ) \left (\cos \left (b x \right ) x b -\sin \left (b x \right )\right )}{\sqrt {\pi }\, \left (-1+m \right ) b \left (1+m \right ) m}-\frac {3 \,2^{2-m} x^{2+m} \left (b^{2}\right )^{-\frac {1}{2}+\frac {m}{2}} b^{3} \left (b x \right )^{-\frac {3}{2}-m} \LommelS 1 \left (m +\frac {1}{2}, \frac {3}{2}, b x \right ) \sin \left (b x \right )}{\sqrt {\pi }\, \left (-1+m \right ) \left (3+3 m \right )}+\frac {2^{2-m} x^{2+m} \left (b^{2}\right )^{-\frac {1}{2}+\frac {m}{2}} b^{3} \left (b x \right )^{-\frac {5}{2}-m} \left (\cos \left (b x \right ) x b -\sin \left (b x \right )\right ) \LommelS 1 \left (m +\frac {3}{2}, \frac {1}{2}, b x \right )}{\sqrt {\pi }\, \left (-1+m \right ) \left (1+m \right ) m}\right ) \cos \relax (a )-2^{-2+m} b^{1-m} \sqrt {\pi }\, \left (\frac {2^{1-m} x^{-1+m} b^{-1+m} \left (-2 x^{2} b^{2}+2 m^{2}+2 m -4\right ) \sin \left (b x \right )}{\sqrt {\pi }\, m \left (2+m \right ) \left (-1+m \right )}-\frac {3 \,2^{2-m} x^{-1+m} b^{-1+m} \left (\cos \left (b x \right ) x b -\sin \left (b x \right )\right )}{\sqrt {\pi }\, m \left (-3+3 m \right )}+\frac {2^{2-m} x^{2+m} b^{2+m} \left (b x \right )^{-\frac {3}{2}-m} \LommelS 1 \left (m +\frac {3}{2}, \frac {3}{2}, b x \right ) \sin \left (b x \right )}{\sqrt {\pi }\, m \left (2+m \right ) \left (-1+m \right )}+\frac {3 \,2^{2-m} x^{2+m} b^{2+m} \left (b x \right )^{-\frac {5}{2}-m} \left (\cos \left (b x \right ) x b -\sin \left (b x \right )\right ) \LommelS 1 \left (m +\frac {1}{2}, \frac {1}{2}, b x \right )}{\sqrt {\pi }\, m \left (-3+3 m \right )}\right ) \sin \relax (a ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m - 2} \cos \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^{m-2}\,\cos \left (a+b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m - 2} \cos {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________