Optimal. Leaf size=68 \[ \frac {a (c+d x)^3}{3 d}+\frac {2 b d^2 \cos (e+f x)}{f^3}-\frac {b (c+d x)^2 \cos (e+f x)}{f}+\frac {2 b d (c+d x) \sin (e+f x)}{f^2} \]
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Rubi [A]
time = 0.06, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3398, 3377,
2718} \begin {gather*} \frac {a (c+d x)^3}{3 d}+\frac {2 b d (c+d x) \sin (e+f x)}{f^2}-\frac {b (c+d x)^2 \cos (e+f x)}{f}+\frac {2 b d^2 \cos (e+f x)}{f^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3377
Rule 3398
Rubi steps
\begin {align*} \int (c+d x)^2 (a+b \sin (e+f x)) \, dx &=\int \left (a (c+d x)^2+b (c+d x)^2 \sin (e+f x)\right ) \, dx\\ &=\frac {a (c+d x)^3}{3 d}+b \int (c+d x)^2 \sin (e+f x) \, dx\\ &=\frac {a (c+d x)^3}{3 d}-\frac {b (c+d x)^2 \cos (e+f x)}{f}+\frac {(2 b d) \int (c+d x) \cos (e+f x) \, dx}{f}\\ &=\frac {a (c+d x)^3}{3 d}-\frac {b (c+d x)^2 \cos (e+f x)}{f}+\frac {2 b d (c+d x) \sin (e+f x)}{f^2}-\frac {\left (2 b d^2\right ) \int \sin (e+f x) \, dx}{f^2}\\ &=\frac {a (c+d x)^3}{3 d}+\frac {2 b d^2 \cos (e+f x)}{f^3}-\frac {b (c+d x)^2 \cos (e+f x)}{f}+\frac {2 b d (c+d x) \sin (e+f x)}{f^2}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 84, normalized size = 1.24 \begin {gather*} \frac {1}{3} a x \left (3 c^2+3 c d x+d^2 x^2\right )-\frac {b \left (c^2 f^2+2 c d f^2 x+d^2 \left (-2+f^2 x^2\right )\right ) \cos (e+f x)}{f^3}+\frac {2 b d (c+d x) \sin (e+f x)}{f^2} \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. \(240\) vs.
\(2(66)=132\).
time = 0.05, size = 241, normalized size = 3.54
method | result | size |
risch | \(\frac {a \,d^{2} x^{3}}{3}+a d c \,x^{2}+a \,c^{2} x +\frac {a \,c^{3}}{3 d}-\frac {b \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}-2 d^{2}\right ) \cos \left (f x +e \right )}{f^{3}}+\frac {2 b d \left (d x +c \right ) \sin \left (f x +e \right )}{f^{2}}\) | \(94\) |
norman | \(\frac {\frac {\left (2 b \,c^{2} f^{2}-4 b \,d^{2}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f^{3}}+\frac {c \left (a c f -2 b d \right ) x}{f}+\frac {d \left (a c f -b d \right ) x^{2}}{f}+\frac {c \left (a c f +2 b d \right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {d \left (a c f +b d \right ) x^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {a \,d^{2} x^{3}}{3}+\frac {a \,d^{2} x^{3} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+\frac {4 b c d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f^{2}}+\frac {4 b \,d^{2} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f^{2}}}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}\) | \(199\) |
derivativedivides | \(\frac {a \,c^{2} \left (f x +e \right )-\frac {2 a c d e \left (f x +e \right )}{f}+\frac {a c d \left (f x +e \right )^{2}}{f}+\frac {a \,d^{2} e^{2} \left (f x +e \right )}{f^{2}}-\frac {a \,d^{2} e \left (f x +e \right )^{2}}{f^{2}}+\frac {a \,d^{2} \left (f x +e \right )^{3}}{3 f^{2}}-b \,c^{2} \cos \left (f x +e \right )+\frac {2 b c d e \cos \left (f x +e \right )}{f}+\frac {2 b c d \left (\sin \left (f x +e \right )-\left (f x +e \right ) \cos \left (f x +e \right )\right )}{f}-\frac {b \,d^{2} e^{2} \cos \left (f x +e \right )}{f^{2}}-\frac {2 b \,d^{2} e \left (\sin \left (f x +e \right )-\left (f x +e \right ) \cos \left (f x +e \right )\right )}{f^{2}}+\frac {b \,d^{2} \left (-\left (f x +e \right )^{2} \cos \left (f x +e \right )+2 \cos \left (f x +e \right )+2 \left (f x +e \right ) \sin \left (f x +e \right )\right )}{f^{2}}}{f}\) | \(241\) |
default | \(\frac {a \,c^{2} \left (f x +e \right )-\frac {2 a c d e \left (f x +e \right )}{f}+\frac {a c d \left (f x +e \right )^{2}}{f}+\frac {a \,d^{2} e^{2} \left (f x +e \right )}{f^{2}}-\frac {a \,d^{2} e \left (f x +e \right )^{2}}{f^{2}}+\frac {a \,d^{2} \left (f x +e \right )^{3}}{3 f^{2}}-b \,c^{2} \cos \left (f x +e \right )+\frac {2 b c d e \cos \left (f x +e \right )}{f}+\frac {2 b c d \left (\sin \left (f x +e \right )-\left (f x +e \right ) \cos \left (f x +e \right )\right )}{f}-\frac {b \,d^{2} e^{2} \cos \left (f x +e \right )}{f^{2}}-\frac {2 b \,d^{2} e \left (\sin \left (f x +e \right )-\left (f x +e \right ) \cos \left (f x +e \right )\right )}{f^{2}}+\frac {b \,d^{2} \left (-\left (f x +e \right )^{2} \cos \left (f x +e \right )+2 \cos \left (f x +e \right )+2 \left (f x +e \right ) \sin \left (f x +e \right )\right )}{f^{2}}}{f}\) | \(241\) |
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. 260 vs.
\(2 (69) = 138\).
time = 0.37, size = 260, normalized size = 3.82 \begin {gather*} \frac {3 \, {\left (f x + e\right )} a c^{2} + \frac {{\left (f x + e\right )}^{3} a d^{2}}{f^{2}} + \frac {3 \, {\left (f x + e\right )}^{2} a c d}{f} - 3 \, b c^{2} \cos \left (f x + e\right ) - \frac {3 \, {\left (f x + e\right )}^{2} a d^{2} e}{f^{2}} - \frac {6 \, {\left (f x + e\right )} a c d e}{f} + \frac {6 \, b c d \cos \left (f x + e\right ) e}{f} - \frac {6 \, {\left ({\left (f x + e\right )} \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right )} b c d}{f} + \frac {3 \, {\left (f x + e\right )} a d^{2} e^{2}}{f^{2}} - \frac {3 \, b d^{2} \cos \left (f x + e\right ) e^{2}}{f^{2}} + \frac {6 \, {\left ({\left (f x + e\right )} \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right )} b d^{2} e}{f^{2}} - \frac {3 \, {\left ({\left ({\left (f x + e\right )}^{2} - 2\right )} \cos \left (f x + e\right ) - 2 \, {\left (f x + e\right )} \sin \left (f x + e\right )\right )} b d^{2}}{f^{2}}}{3 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 104, normalized size = 1.53 \begin {gather*} \frac {a d^{2} f^{3} x^{3} + 3 \, a c d f^{3} x^{2} + 3 \, a c^{2} f^{3} x - 3 \, {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + b c^{2} f^{2} - 2 \, b d^{2}\right )} \cos \left (f x + e\right ) + 6 \, {\left (b d^{2} f x + b c d f\right )} \sin \left (f x + e\right )}{3 \, f^{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. 151 vs.
\(2 (65) = 130\).
time = 0.15, size = 151, normalized size = 2.22 \begin {gather*} \begin {cases} a c^{2} x + a c d x^{2} + \frac {a d^{2} x^{3}}{3} - \frac {b c^{2} \cos {\left (e + f x \right )}}{f} - \frac {2 b c d x \cos {\left (e + f x \right )}}{f} + \frac {2 b c d \sin {\left (e + f x \right )}}{f^{2}} - \frac {b d^{2} x^{2} \cos {\left (e + f x \right )}}{f} + \frac {2 b d^{2} x \sin {\left (e + f x \right )}}{f^{2}} + \frac {2 b d^{2} \cos {\left (e + f x \right )}}{f^{3}} & \text {for}\: f \neq 0 \\\left (a + b \sin {\left (e \right )}\right ) \left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.39, size = 95, normalized size = 1.40 \begin {gather*} \frac {1}{3} \, a d^{2} x^{3} + a c d x^{2} + a c^{2} x - \frac {{\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + b c^{2} f^{2} - 2 \, b d^{2}\right )} \cos \left (f x + e\right )}{f^{3}} + \frac {2 \, {\left (b d^{2} f x + b c d f\right )} \sin \left (f x + e\right )}{f^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.67, size = 112, normalized size = 1.65 \begin {gather*} \frac {a\,d^2\,x^3}{3}+\frac {\cos \left (e+f\,x\right )\,\left (2\,b\,d^2-b\,c^2\,f^2\right )}{f^3}+a\,c^2\,x+a\,c\,d\,x^2+\frac {2\,b\,d^2\,x\,\sin \left (e+f\,x\right )}{f^2}-\frac {b\,d^2\,x^2\,\cos \left (e+f\,x\right )}{f}+\frac {2\,b\,c\,d\,\sin \left (e+f\,x\right )}{f^2}-\frac {2\,b\,c\,d\,x\,\cos \left (e+f\,x\right )}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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