Optimal. Leaf size=129 \[ \frac {3 d^2 \cos (2 a+2 b x)}{128 b^3}-\frac {d^2 \cos (6 a+6 b x)}{3456 b^3}+\frac {3 d (c+d x) \sin (2 a+2 b x)}{64 b^2}-\frac {d (c+d x) \sin (6 a+6 b x)}{576 b^2}-\frac {3 (c+d x)^2 \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^2 \cos (6 a+6 b x)}{192 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4406, 3296, 2638} \[ \frac {3 d (c+d x) \sin (2 a+2 b x)}{64 b^2}-\frac {d (c+d x) \sin (6 a+6 b x)}{576 b^2}+\frac {3 d^2 \cos (2 a+2 b x)}{128 b^3}-\frac {d^2 \cos (6 a+6 b x)}{3456 b^3}-\frac {3 (c+d x)^2 \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^2 \cos (6 a+6 b x)}{192 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2638
Rule 3296
Rule 4406
Rubi steps
\begin {align*} \int (c+d x)^2 \cos ^3(a+b x) \sin ^3(a+b x) \, dx &=\int \left (\frac {3}{32} (c+d x)^2 \sin (2 a+2 b x)-\frac {1}{32} (c+d x)^2 \sin (6 a+6 b x)\right ) \, dx\\ &=-\left (\frac {1}{32} \int (c+d x)^2 \sin (6 a+6 b x) \, dx\right )+\frac {3}{32} \int (c+d x)^2 \sin (2 a+2 b x) \, dx\\ &=-\frac {3 (c+d x)^2 \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^2 \cos (6 a+6 b x)}{192 b}-\frac {d \int (c+d x) \cos (6 a+6 b x) \, dx}{96 b}+\frac {(3 d) \int (c+d x) \cos (2 a+2 b x) \, dx}{32 b}\\ &=-\frac {3 (c+d x)^2 \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^2 \cos (6 a+6 b x)}{192 b}+\frac {3 d (c+d x) \sin (2 a+2 b x)}{64 b^2}-\frac {d (c+d x) \sin (6 a+6 b x)}{576 b^2}+\frac {d^2 \int \sin (6 a+6 b x) \, dx}{576 b^2}-\frac {\left (3 d^2\right ) \int \sin (2 a+2 b x) \, dx}{64 b^2}\\ &=\frac {3 d^2 \cos (2 a+2 b x)}{128 b^3}-\frac {3 (c+d x)^2 \cos (2 a+2 b x)}{64 b}-\frac {d^2 \cos (6 a+6 b x)}{3456 b^3}+\frac {(c+d x)^2 \cos (6 a+6 b x)}{192 b}+\frac {3 d (c+d x) \sin (2 a+2 b x)}{64 b^2}-\frac {d (c+d x) \sin (6 a+6 b x)}{576 b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.56, size = 91, normalized size = 0.71 \[ \frac {-81 \cos (2 (a+b x)) \left (2 b^2 (c+d x)^2-d^2\right )+\cos (6 (a+b x)) \left (18 b^2 (c+d x)^2-d^2\right )-6 b d (c+d x) (\sin (6 (a+b x))-27 \sin (2 (a+b x)))}{3456 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 194, normalized size = 1.50 \[ \frac {2 \, {\left (18 \, b^{2} d^{2} x^{2} + 36 \, b^{2} c d x + 18 \, b^{2} c^{2} - d^{2}\right )} \cos \left (b x + a\right )^{6} + 9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x - 3 \, {\left (18 \, b^{2} d^{2} x^{2} + 36 \, b^{2} c d x + 18 \, b^{2} c^{2} - d^{2}\right )} \cos \left (b x + a\right )^{4} + 9 \, d^{2} \cos \left (b x + a\right )^{2} - 6 \, {\left (2 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{5} - 2 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{3} - 3 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{216 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.41, size = 145, normalized size = 1.12 \[ \frac {{\left (18 \, b^{2} d^{2} x^{2} + 36 \, b^{2} c d x + 18 \, b^{2} c^{2} - d^{2}\right )} \cos \left (6 \, b x + 6 \, a\right )}{3456 \, b^{3}} - \frac {3 \, {\left (2 \, b^{2} d^{2} x^{2} + 4 \, b^{2} c d x + 2 \, b^{2} c^{2} - d^{2}\right )} \cos \left (2 \, b x + 2 \, a\right )}{128 \, b^{3}} - \frac {{\left (b d^{2} x + b c d\right )} \sin \left (6 \, b x + 6 \, a\right )}{576 \, b^{3}} + \frac {3 \, {\left (b d^{2} x + b c d\right )} \sin \left (2 \, b x + 2 \, a\right )}{64 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 498, normalized size = 3.86 \[ \frac {\frac {d^{2} \left (\frac {\left (b x +a \right )^{2} \left (\sin ^{4}\left (b x +a \right )\right )}{4}-\frac {\left (b x +a \right ) \left (-\frac {\left (\sin ^{3}\left (b x +a \right )+\frac {3 \sin \left (b x +a \right )}{2}\right ) \cos \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )}{2}+\frac {\left (b x +a \right )^{2}}{24}-\frac {\left (\sin ^{4}\left (b x +a \right )\right )}{72}-\frac {\left (\sin ^{2}\left (b x +a \right )\right )}{24}-\frac {\left (b x +a \right )^{2} \left (\sin ^{6}\left (b x +a \right )\right )}{6}+\frac {\left (b x +a \right ) \left (-\frac {\left (\sin ^{5}\left (b x +a \right )+\frac {5 \left (\sin ^{3}\left (b x +a \right )\right )}{4}+\frac {15 \sin \left (b x +a \right )}{8}\right ) \cos \left (b x +a \right )}{6}+\frac {5 b x}{16}+\frac {5 a}{16}\right )}{3}+\frac {\left (\sin ^{6}\left (b x +a \right )\right )}{108}\right )}{b^{2}}-\frac {2 a \,d^{2} \left (\frac {\left (b x +a \right ) \left (\sin ^{4}\left (b x +a \right )\right )}{4}+\frac {\left (\sin ^{3}\left (b x +a \right )+\frac {3 \sin \left (b x +a \right )}{2}\right ) \cos \left (b x +a \right )}{16}-\frac {b x}{24}-\frac {a}{24}-\frac {\left (b x +a \right ) \left (\sin ^{6}\left (b x +a \right )\right )}{6}-\frac {\left (\sin ^{5}\left (b x +a \right )+\frac {5 \left (\sin ^{3}\left (b x +a \right )\right )}{4}+\frac {15 \sin \left (b x +a \right )}{8}\right ) \cos \left (b x +a \right )}{36}\right )}{b^{2}}+\frac {2 c d \left (\frac {\left (b x +a \right ) \left (\sin ^{4}\left (b x +a \right )\right )}{4}+\frac {\left (\sin ^{3}\left (b x +a \right )+\frac {3 \sin \left (b x +a \right )}{2}\right ) \cos \left (b x +a \right )}{16}-\frac {b x}{24}-\frac {a}{24}-\frac {\left (b x +a \right ) \left (\sin ^{6}\left (b x +a \right )\right )}{6}-\frac {\left (\sin ^{5}\left (b x +a \right )+\frac {5 \left (\sin ^{3}\left (b x +a \right )\right )}{4}+\frac {15 \sin \left (b x +a \right )}{8}\right ) \cos \left (b x +a \right )}{36}\right )}{b}+\frac {d^{2} a^{2} \left (-\frac {\left (\sin ^{2}\left (b x +a \right )\right ) \left (\cos ^{4}\left (b x +a \right )\right )}{6}-\frac {\left (\cos ^{4}\left (b x +a \right )\right )}{12}\right )}{b^{2}}-\frac {2 c d a \left (-\frac {\left (\sin ^{2}\left (b x +a \right )\right ) \left (\cos ^{4}\left (b x +a \right )\right )}{6}-\frac {\left (\cos ^{4}\left (b x +a \right )\right )}{12}\right )}{b}+c^{2} \left (-\frac {\left (\sin ^{2}\left (b x +a \right )\right ) \left (\cos ^{4}\left (b x +a \right )\right )}{6}-\frac {\left (\cos ^{4}\left (b x +a \right )\right )}{12}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.41, size = 303, normalized size = 2.35 \[ -\frac {288 \, {\left (2 \, \sin \left (b x + a\right )^{6} - 3 \, \sin \left (b x + a\right )^{4}\right )} c^{2} - \frac {576 \, {\left (2 \, \sin \left (b x + a\right )^{6} - 3 \, \sin \left (b x + a\right )^{4}\right )} a c d}{b} + \frac {288 \, {\left (2 \, \sin \left (b x + a\right )^{6} - 3 \, \sin \left (b x + a\right )^{4}\right )} a^{2} d^{2}}{b^{2}} - \frac {6 \, {\left (6 \, {\left (b x + a\right )} \cos \left (6 \, b x + 6 \, a\right ) - 54 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (6 \, b x + 6 \, a\right ) + 27 \, \sin \left (2 \, b x + 2 \, a\right )\right )} c d}{b} + \frac {6 \, {\left (6 \, {\left (b x + a\right )} \cos \left (6 \, b x + 6 \, a\right ) - 54 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (6 \, b x + 6 \, a\right ) + 27 \, \sin \left (2 \, b x + 2 \, a\right )\right )} a d^{2}}{b^{2}} - \frac {{\left ({\left (18 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (6 \, b x + 6 \, a\right ) - 81 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) - 6 \, {\left (b x + a\right )} \sin \left (6 \, b x + 6 \, a\right ) + 162 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} d^{2}}{b^{2}}}{3456 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.81, size = 202, normalized size = 1.57 \[ \frac {81\,d^2\,\cos \left (2\,a+2\,b\,x\right )-d^2\,\cos \left (6\,a+6\,b\,x\right )-162\,b^2\,c^2\,\cos \left (2\,a+2\,b\,x\right )+18\,b^2\,c^2\,\cos \left (6\,a+6\,b\,x\right )+162\,b\,c\,d\,\sin \left (2\,a+2\,b\,x\right )-6\,b\,c\,d\,\sin \left (6\,a+6\,b\,x\right )-162\,b^2\,d^2\,x^2\,\cos \left (2\,a+2\,b\,x\right )+18\,b^2\,d^2\,x^2\,\cos \left (6\,a+6\,b\,x\right )+162\,b\,d^2\,x\,\sin \left (2\,a+2\,b\,x\right )-6\,b\,d^2\,x\,\sin \left (6\,a+6\,b\,x\right )-324\,b^2\,c\,d\,x\,\cos \left (2\,a+2\,b\,x\right )+36\,b^2\,c\,d\,x\,\cos \left (6\,a+6\,b\,x\right )}{3456\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 9.99, size = 461, normalized size = 3.57 \[ \begin {cases} \frac {c^{2} \sin ^{6}{\left (a + b x \right )}}{12 b} + \frac {c^{2} \sin ^{4}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4 b} + \frac {c d x \sin ^{6}{\left (a + b x \right )}}{12 b} + \frac {c d x \sin ^{4}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4 b} - \frac {c d x \sin ^{2}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{4 b} - \frac {c d x \cos ^{6}{\left (a + b x \right )}}{12 b} + \frac {d^{2} x^{2} \sin ^{6}{\left (a + b x \right )}}{24 b} + \frac {d^{2} x^{2} \sin ^{4}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{8 b} - \frac {d^{2} x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{8 b} - \frac {d^{2} x^{2} \cos ^{6}{\left (a + b x \right )}}{24 b} + \frac {c d \sin ^{5}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{12 b^{2}} + \frac {2 c d \sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {c d \sin {\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{12 b^{2}} + \frac {d^{2} x \sin ^{5}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{12 b^{2}} + \frac {2 d^{2} x \sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {d^{2} x \sin {\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{12 b^{2}} - \frac {7 d^{2} \sin ^{6}{\left (a + b x \right )}}{216 b^{3}} - \frac {d^{2} \sin ^{4}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{18 b^{3}} + \frac {d^{2} \cos ^{6}{\left (a + b x \right )}}{72 b^{3}} & \text {for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) \sin ^{3}{\relax (a )} \cos ^{3}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________