Optimal. Leaf size=75 \[ \frac {2 d \cos (a+b x)}{3 b^2}+\frac {d \cos ^3(a+b x)}{9 b^2}+\frac {2 (c+d x) \sin (a+b x)}{3 b}+\frac {(c+d x) \cos ^2(a+b x) \sin (a+b x)}{3 b} \]
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Rubi [A]
time = 0.03, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3391, 3377,
2718} \begin {gather*} \frac {d \cos ^3(a+b x)}{9 b^2}+\frac {2 d \cos (a+b x)}{3 b^2}+\frac {2 (c+d x) \sin (a+b x)}{3 b}+\frac {(c+d x) \sin (a+b x) \cos ^2(a+b x)}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3377
Rule 3391
Rubi steps
\begin {align*} \int (c+d x) \cos ^3(a+b x) \, dx &=\frac {d \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x) \cos ^2(a+b x) \sin (a+b x)}{3 b}+\frac {2}{3} \int (c+d x) \cos (a+b x) \, dx\\ &=\frac {d \cos ^3(a+b x)}{9 b^2}+\frac {2 (c+d x) \sin (a+b x)}{3 b}+\frac {(c+d x) \cos ^2(a+b x) \sin (a+b x)}{3 b}-\frac {(2 d) \int \sin (a+b x) \, dx}{3 b}\\ &=\frac {2 d \cos (a+b x)}{3 b^2}+\frac {d \cos ^3(a+b x)}{9 b^2}+\frac {2 (c+d x) \sin (a+b x)}{3 b}+\frac {(c+d x) \cos ^2(a+b x) \sin (a+b x)}{3 b}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 52, normalized size = 0.69 \begin {gather*} \frac {27 d \cos (a+b x)+d \cos (3 (a+b x))+3 b (c+d x) (9 \sin (a+b x)+\sin (3 (a+b x)))}{36 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 95, normalized size = 1.27
method | result | size |
risch | \(\frac {3 d \cos \left (b x +a \right )}{4 b^{2}}+\frac {3 \left (d x +c \right ) \sin \left (b x +a \right )}{4 b}+\frac {d \cos \left (3 b x +3 a \right )}{36 b^{2}}+\frac {\left (d x +c \right ) \sin \left (3 b x +3 a \right )}{12 b}\) | \(64\) |
derivativedivides | \(\frac {-\frac {d a \left (2+\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{3 b}+\frac {c \left (2+\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{3}+\frac {d \left (\frac {\left (b x +a \right ) \left (2+\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{3}+\frac {\left (\cos ^{3}\left (b x +a \right )\right )}{9}+\frac {2 \cos \left (b x +a \right )}{3}\right )}{b}}{b}\) | \(95\) |
default | \(\frac {-\frac {d a \left (2+\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{3 b}+\frac {c \left (2+\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{3}+\frac {d \left (\frac {\left (b x +a \right ) \left (2+\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{3}+\frac {\left (\cos ^{3}\left (b x +a \right )\right )}{9}+\frac {2 \cos \left (b x +a \right )}{3}\right )}{b}}{b}\) | \(95\) |
norman | \(\frac {\frac {2 d \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b^{2}}+\frac {14 d}{9 b^{2}}+\frac {2 c \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}+\frac {4 c \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}+\frac {2 c \left (\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}+\frac {8 d \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b^{2}}+\frac {2 d x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}+\frac {4 d x \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}+\frac {2 d x \left (\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{3}}\) | \(159\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 103, normalized size = 1.37 \begin {gather*} -\frac {12 \, {\left (\sin \left (b x + a\right )^{3} - 3 \, \sin \left (b x + a\right )\right )} c - \frac {12 \, {\left (\sin \left (b x + a\right )^{3} - 3 \, \sin \left (b x + a\right )\right )} a d}{b} - \frac {{\left (3 \, {\left (b x + a\right )} \sin \left (3 \, b x + 3 \, a\right ) + 27 \, {\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (3 \, b x + 3 \, a\right ) + 27 \, \cos \left (b x + a\right )\right )} d}{b}}{36 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 60, normalized size = 0.80 \begin {gather*} \frac {d \cos \left (b x + a\right )^{3} + 6 \, d \cos \left (b x + a\right ) + 3 \, {\left (2 \, b d x + {\left (b d x + b c\right )} \cos \left (b x + a\right )^{2} + 2 \, b c\right )} \sin \left (b x + a\right )}{9 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 126, normalized size = 1.68 \begin {gather*} \begin {cases} \frac {2 c \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac {c \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b} + \frac {2 d x \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac {d x \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b} + \frac {2 d \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{3 b^{2}} + \frac {7 d \cos ^{3}{\left (a + b x \right )}}{9 b^{2}} & \text {for}\: b \neq 0 \\\left (c x + \frac {d x^{2}}{2}\right ) \cos ^{3}{\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 = 0.51, size = 69, normalized size = 0.92 \begin {gather*} \frac {d \cos \left (3 \, b x + 3 \, a\right )}{36 \, b^{2}} + \frac {3 \, d \cos \left (b x + a\right )}{4 \, b^{2}} + \frac {{\left (b d x + b c\right )} \sin \left (3 \, b x + 3 \, a\right )}{12 \, b^{2}} + \frac {3 \, {\left (b d x + b c\right )} \sin \left (b x + a\right )}{4 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.26, size = 77, normalized size = 1.03 \begin {gather*} \frac {\frac {3\,c\,\sin \left (a+b\,x\right )}{4}+\frac {c\,\sin \left (3\,a+3\,b\,x\right )}{12}+\frac {d\,x\,\sin \left (3\,a+3\,b\,x\right )}{12}+\frac {3\,d\,x\,\sin \left (a+b\,x\right )}{4}}{b}+\frac {d\,\cos \left (3\,a+3\,b\,x\right )}{36\,b^2}+\frac {3\,d\,\cos \left (a+b\,x\right )}{4\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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