Optimal. Leaf size=49 \[ \frac {2 d (c+d x) \cos (a+b x)}{b^2}-\frac {2 d^2 \sin (a+b x)}{b^3}+\frac {(c+d x)^2 \sin (a+b x)}{b} \]
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Rubi [A]
time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3377, 2717}
\begin {gather*} -\frac {2 d^2 \sin (a+b x)}{b^3}+\frac {2 d (c+d x) \cos (a+b x)}{b^2}+\frac {(c+d x)^2 \sin (a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 3377
Rubi steps
\begin {align*} \int (c+d x)^2 \cos (a+b x) \, dx &=\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {(2 d) \int (c+d x) \sin (a+b x) \, dx}{b}\\ &=\frac {2 d (c+d x) \cos (a+b x)}{b^2}+\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {\left (2 d^2\right ) \int \cos (a+b x) \, dx}{b^2}\\ &=\frac {2 d (c+d x) \cos (a+b x)}{b^2}-\frac {2 d^2 \sin (a+b x)}{b^3}+\frac {(c+d x)^2 \sin (a+b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 44, normalized size = 0.90 \begin {gather*} \frac {2 b d (c+d x) \cos (a+b x)+\left (-2 d^2+b^2 (c+d x)^2\right ) \sin (a+b x)}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(142\) vs.
\(2(49)=98\).
time = 0.07, size = 143, normalized size = 2.92
method | result | size |
risch | \(\frac {2 d \left (d x +c \right ) \cos \left (b x +a \right )}{b^{2}}+\frac {\left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}-2 d^{2}\right ) \sin \left (b x +a \right )}{b^{3}}\) | \(60\) |
norman | \(\frac {\frac {4 c d}{b^{2}}+\frac {2 d^{2} x}{b^{2}}+\frac {2 \left (b^{2} c^{2}-2 d^{2}\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b^{3}}+\frac {2 d^{2} x^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}-\frac {2 d^{2} x \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b^{2}}+\frac {4 c d x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}}{1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\) | \(118\) |
derivativedivides | \(\frac {\frac {a^{2} d^{2} \sin \left (b x +a \right )}{b^{2}}-\frac {2 a c d \sin \left (b x +a \right )}{b}-\frac {2 a \,d^{2} \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )}{b^{2}}+c^{2} \sin \left (b x +a \right )+\frac {2 c d \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )}{b}+\frac {d^{2} \left (\left (b x +a \right )^{2} \sin \left (b x +a \right )-2 \sin \left (b x +a \right )+2 \left (b x +a \right ) \cos \left (b x +a \right )\right )}{b^{2}}}{b}\) | \(143\) |
default | \(\frac {\frac {a^{2} d^{2} \sin \left (b x +a \right )}{b^{2}}-\frac {2 a c d \sin \left (b x +a \right )}{b}-\frac {2 a \,d^{2} \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )}{b^{2}}+c^{2} \sin \left (b x +a \right )+\frac {2 c d \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )}{b}+\frac {d^{2} \left (\left (b x +a \right )^{2} \sin \left (b x +a \right )-2 \sin \left (b x +a \right )+2 \left (b x +a \right ) \cos \left (b x +a \right )\right )}{b^{2}}}{b}\) | \(143\) |
meijerg | \(\frac {4 d^{2} \cos \left (a \right ) \sqrt {\pi }\, \left (\frac {x \left (b^{2}\right )^{\frac {3}{2}} \cos \left (b x \right )}{2 \sqrt {\pi }\, b^{2}}-\frac {\left (b^{2}\right )^{\frac {3}{2}} \left (-\frac {3 x^{2} b^{2}}{2}+3\right ) \sin \left (b x \right )}{6 \sqrt {\pi }\, b^{3}}\right )}{b^{2} \sqrt {b^{2}}}-\frac {4 d^{2} \sin \left (a \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (-\frac {x^{2} b^{2}}{2}+1\right ) \cos \left (b x \right )}{2 \sqrt {\pi }}+\frac {x b \sin \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{3}}+\frac {4 d c \cos \left (a \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cos \left (b x \right )}{2 \sqrt {\pi }}+\frac {x b \sin \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{2}}-\frac {4 d c \sin \left (a \right ) \sqrt {\pi }\, \left (-\frac {x b \cos \left (b x \right )}{2 \sqrt {\pi }}+\frac {\sin \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{2}}+\frac {c^{2} \cos \left (a \right ) \sin \left (b x \right )}{b}-\frac {c^{2} \sin \left (a \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (b x \right )}{\sqrt {\pi }}\right )}{b}\) | \(225\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 136 vs.
\(2 (49) = 98\).
time = 0.33, size = 136, normalized size = 2.78 \begin {gather*} \frac {c^{2} \sin \left (b x + a\right ) - \frac {2 \, a c d \sin \left (b x + a\right )}{b} + \frac {a^{2} d^{2} \sin \left (b x + a\right )}{b^{2}} + \frac {2 \, {\left ({\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (b x + a\right )\right )} c d}{b} - \frac {2 \, {\left ({\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (b x + a\right )\right )} a d^{2}}{b^{2}} + \frac {{\left (2 \, {\left (b x + a\right )} \cos \left (b x + a\right ) + {\left ({\left (b x + a\right )}^{2} - 2\right )} \sin \left (b x + a\right )\right )} d^{2}}{b^{2}}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 62, normalized size = 1.27 \begin {gather*} \frac {2 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, d^{2}\right )} \sin \left (b x + a\right )}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs.
\(2 (48) = 96\).
time = 0.15, size = 112, normalized size = 2.29 \begin {gather*} \begin {cases} \frac {c^{2} \sin {\left (a + b x \right )}}{b} + \frac {2 c d x \sin {\left (a + b x \right )}}{b} + \frac {d^{2} x^{2} \sin {\left (a + b x \right )}}{b} + \frac {2 c d \cos {\left (a + b x \right )}}{b^{2}} + \frac {2 d^{2} x \cos {\left (a + b x \right )}}{b^{2}} - \frac {2 d^{2} \sin {\left (a + b x \right )}}{b^{3}} & \text {for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) \cos {\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.52, size = 64, normalized size = 1.31 \begin {gather*} \frac {2 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )}{b^{3}} + \frac {{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, d^{2}\right )} \sin \left (b x + a\right )}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 84, normalized size = 1.71 \begin {gather*} \frac {d^2\,x^2\,\sin \left (a+b\,x\right )}{b}-\frac {\sin \left (a+b\,x\right )\,\left (2\,d^2-b^2\,c^2\right )}{b^3}+\frac {2\,c\,d\,\cos \left (a+b\,x\right )}{b^2}+\frac {2\,d^2\,x\,\cos \left (a+b\,x\right )}{b^2}+\frac {2\,c\,d\,x\,\sin \left (a+b\,x\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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