Optimal. Leaf size=237 \[ -\frac {12 a^2 d^2 (c+d x) \sin (e+f x)}{f^3}-\frac {3 a^2 d^2 (c+d x) \sin (e+f x) \cos (e+f x)}{4 f^3}-\frac {3 a^2 c d^2 x}{4 f^2}+\frac {3 a^2 d (c+d x)^2 \cos ^2(e+f x)}{4 f^2}+\frac {6 a^2 d (c+d x)^2 \cos (e+f x)}{f^2}+\frac {2 a^2 (c+d x)^3 \sin (e+f x)}{f}+\frac {a^2 (c+d x)^3 \sin (e+f x) \cos (e+f x)}{2 f}+\frac {3 a^2 (c+d x)^4}{8 d}-\frac {3 a^2 d^3 \cos ^2(e+f x)}{8 f^4}-\frac {12 a^2 d^3 \cos (e+f x)}{f^4}-\frac {3 a^2 d^3 x^2}{8 f^2} \]
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Rubi [A] time = 0.26, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3317, 3296, 2638, 3311, 32, 3310} \[ -\frac {12 a^2 d^2 (c+d x) \sin (e+f x)}{f^3}-\frac {3 a^2 d^2 (c+d x) \sin (e+f x) \cos (e+f x)}{4 f^3}-\frac {3 a^2 c d^2 x}{4 f^2}+\frac {3 a^2 d (c+d x)^2 \cos ^2(e+f x)}{4 f^2}+\frac {6 a^2 d (c+d x)^2 \cos (e+f x)}{f^2}+\frac {2 a^2 (c+d x)^3 \sin (e+f x)}{f}+\frac {a^2 (c+d x)^3 \sin (e+f x) \cos (e+f x)}{2 f}+\frac {3 a^2 (c+d x)^4}{8 d}-\frac {3 a^2 d^3 \cos ^2(e+f x)}{8 f^4}-\frac {12 a^2 d^3 \cos (e+f x)}{f^4}-\frac {3 a^2 d^3 x^2}{8 f^2} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2638
Rule 3296
Rule 3310
Rule 3311
Rule 3317
Rubi steps
\begin {align*} \int (c+d x)^3 (a+a \cos (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)^3+2 a^2 (c+d x)^3 \cos (e+f x)+a^2 (c+d x)^3 \cos ^2(e+f x)\right ) \, dx\\ &=\frac {a^2 (c+d x)^4}{4 d}+a^2 \int (c+d x)^3 \cos ^2(e+f x) \, dx+\left (2 a^2\right ) \int (c+d x)^3 \cos (e+f x) \, dx\\ &=\frac {a^2 (c+d x)^4}{4 d}+\frac {3 a^2 d (c+d x)^2 \cos ^2(e+f x)}{4 f^2}+\frac {2 a^2 (c+d x)^3 \sin (e+f x)}{f}+\frac {a^2 (c+d x)^3 \cos (e+f x) \sin (e+f x)}{2 f}+\frac {1}{2} a^2 \int (c+d x)^3 \, dx-\frac {\left (3 a^2 d^2\right ) \int (c+d x) \cos ^2(e+f x) \, dx}{2 f^2}-\frac {\left (6 a^2 d\right ) \int (c+d x)^2 \sin (e+f x) \, dx}{f}\\ &=\frac {3 a^2 (c+d x)^4}{8 d}+\frac {6 a^2 d (c+d x)^2 \cos (e+f x)}{f^2}-\frac {3 a^2 d^3 \cos ^2(e+f x)}{8 f^4}+\frac {3 a^2 d (c+d x)^2 \cos ^2(e+f x)}{4 f^2}+\frac {2 a^2 (c+d x)^3 \sin (e+f x)}{f}-\frac {3 a^2 d^2 (c+d x) \cos (e+f x) \sin (e+f x)}{4 f^3}+\frac {a^2 (c+d x)^3 \cos (e+f x) \sin (e+f x)}{2 f}-\frac {\left (3 a^2 d^2\right ) \int (c+d x) \, dx}{4 f^2}-\frac {\left (12 a^2 d^2\right ) \int (c+d x) \cos (e+f x) \, dx}{f^2}\\ &=-\frac {3 a^2 c d^2 x}{4 f^2}-\frac {3 a^2 d^3 x^2}{8 f^2}+\frac {3 a^2 (c+d x)^4}{8 d}+\frac {6 a^2 d (c+d x)^2 \cos (e+f x)}{f^2}-\frac {3 a^2 d^3 \cos ^2(e+f x)}{8 f^4}+\frac {3 a^2 d (c+d x)^2 \cos ^2(e+f x)}{4 f^2}-\frac {12 a^2 d^2 (c+d x) \sin (e+f x)}{f^3}+\frac {2 a^2 (c+d x)^3 \sin (e+f x)}{f}-\frac {3 a^2 d^2 (c+d x) \cos (e+f x) \sin (e+f x)}{4 f^3}+\frac {a^2 (c+d x)^3 \cos (e+f x) \sin (e+f x)}{2 f}+\frac {\left (12 a^2 d^3\right ) \int \sin (e+f x) \, dx}{f^3}\\ &=-\frac {3 a^2 c d^2 x}{4 f^2}-\frac {3 a^2 d^3 x^2}{8 f^2}+\frac {3 a^2 (c+d x)^4}{8 d}-\frac {12 a^2 d^3 \cos (e+f x)}{f^4}+\frac {6 a^2 d (c+d x)^2 \cos (e+f x)}{f^2}-\frac {3 a^2 d^3 \cos ^2(e+f x)}{8 f^4}+\frac {3 a^2 d (c+d x)^2 \cos ^2(e+f x)}{4 f^2}-\frac {12 a^2 d^2 (c+d x) \sin (e+f x)}{f^3}+\frac {2 a^2 (c+d x)^3 \sin (e+f x)}{f}-\frac {3 a^2 d^2 (c+d x) \cos (e+f x) \sin (e+f x)}{4 f^3}+\frac {a^2 (c+d x)^3 \cos (e+f x) \sin (e+f x)}{2 f}\\ \end {align*}
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Mathematica [A] time = 1.44, size = 217, normalized size = 0.92 \[ \frac {a^2 \left (96 d \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2-2\right )\right ) \cos (e+f x)+3 d \left (2 c^2 f^2+4 c d f^2 x+d^2 \left (2 f^2 x^2-1\right )\right ) \cos (2 (e+f x))+2 f \left (16 (c+d x) \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2-6\right )\right ) \sin (e+f x)+(c+d x) \left (2 c^2 f^2+4 c d f^2 x+d^2 \left (2 f^2 x^2-3\right )\right ) \sin (2 (e+f x))+3 f^3 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )\right )\right )}{16 f^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 369, normalized size = 1.56 \[ \frac {3 \, a^{2} d^{3} f^{4} x^{4} + 12 \, a^{2} c d^{2} f^{4} x^{3} + 3 \, {\left (6 \, a^{2} c^{2} d f^{4} - a^{2} d^{3} f^{2}\right )} x^{2} + 3 \, {\left (2 \, a^{2} d^{3} f^{2} x^{2} + 4 \, a^{2} c d^{2} f^{2} x + 2 \, a^{2} c^{2} d f^{2} - a^{2} d^{3}\right )} \cos \left (f x + e\right )^{2} + 6 \, {\left (2 \, a^{2} c^{3} f^{4} - a^{2} c d^{2} f^{2}\right )} x + 48 \, {\left (a^{2} d^{3} f^{2} x^{2} + 2 \, a^{2} c d^{2} f^{2} x + a^{2} c^{2} d f^{2} - 2 \, a^{2} d^{3}\right )} \cos \left (f x + e\right ) + 2 \, {\left (8 \, a^{2} d^{3} f^{3} x^{3} + 24 \, a^{2} c d^{2} f^{3} x^{2} + 8 \, a^{2} c^{3} f^{3} - 48 \, a^{2} c d^{2} f + 24 \, {\left (a^{2} c^{2} d f^{3} - 2 \, a^{2} d^{3} f\right )} x + {\left (2 \, a^{2} d^{3} f^{3} x^{3} + 6 \, a^{2} c d^{2} f^{3} x^{2} + 2 \, a^{2} c^{3} f^{3} - 3 \, a^{2} c d^{2} f + 3 \, {\left (2 \, a^{2} c^{2} d f^{3} - a^{2} d^{3} f\right )} x\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{8 \, f^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.72, size = 339, normalized size = 1.43 \[ \frac {3}{8} \, a^{2} d^{3} x^{4} + \frac {3}{2} \, a^{2} c d^{2} x^{3} + \frac {9}{4} \, a^{2} c^{2} d x^{2} + \frac {3}{2} \, a^{2} c^{3} x + \frac {3 \, {\left (2 \, a^{2} d^{3} f^{2} x^{2} + 4 \, a^{2} c d^{2} f^{2} x + 2 \, a^{2} c^{2} d f^{2} - a^{2} d^{3}\right )} \cos \left (2 \, f x + 2 \, e\right )}{16 \, f^{4}} + \frac {6 \, {\left (a^{2} d^{3} f^{2} x^{2} + 2 \, a^{2} c d^{2} f^{2} x + a^{2} c^{2} d f^{2} - 2 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )}{f^{4}} + \frac {{\left (2 \, a^{2} d^{3} f^{3} x^{3} + 6 \, a^{2} c d^{2} f^{3} x^{2} + 6 \, a^{2} c^{2} d f^{3} x + 2 \, a^{2} c^{3} f^{3} - 3 \, a^{2} d^{3} f x - 3 \, a^{2} c d^{2} f\right )} \sin \left (2 \, f x + 2 \, e\right )}{8 \, f^{4}} + \frac {2 \, {\left (a^{2} d^{3} f^{3} x^{3} + 3 \, a^{2} c d^{2} f^{3} x^{2} + 3 \, a^{2} c^{2} d f^{3} x + a^{2} c^{3} f^{3} - 6 \, a^{2} d^{3} f x - 6 \, a^{2} c d^{2} f\right )} \sin \left (f x + e\right )}{f^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 1129, normalized size = 4.76 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.97, size = 949, normalized size = 4.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.91, size = 452, normalized size = 1.91 \[ \frac {16\,a^2\,c^3\,f^3\,\sin \left (e+f\,x\right )-\frac {3\,a^2\,d^3\,\cos \left (2\,e+2\,f\,x\right )}{2}-96\,a^2\,d^3\,\cos \left (e+f\,x\right )+12\,a^2\,c^3\,f^4\,x+2\,a^2\,c^3\,f^3\,\sin \left (2\,e+2\,f\,x\right )+3\,a^2\,d^3\,f^4\,x^4-96\,a^2\,c\,d^2\,f\,\sin \left (e+f\,x\right )-96\,a^2\,d^3\,f\,x\,\sin \left (e+f\,x\right )+3\,a^2\,d^3\,f^2\,x^2\,\cos \left (2\,e+2\,f\,x\right )+2\,a^2\,d^3\,f^3\,x^3\,\sin \left (2\,e+2\,f\,x\right )+48\,a^2\,c^2\,d\,f^2\,\cos \left (e+f\,x\right )-3\,a^2\,c\,d^2\,f\,\sin \left (2\,e+2\,f\,x\right )-3\,a^2\,d^3\,f\,x\,\sin \left (2\,e+2\,f\,x\right )+3\,a^2\,c^2\,d\,f^2\,\cos \left (2\,e+2\,f\,x\right )+18\,a^2\,c^2\,d\,f^4\,x^2+12\,a^2\,c\,d^2\,f^4\,x^3+48\,a^2\,d^3\,f^2\,x^2\,\cos \left (e+f\,x\right )+16\,a^2\,d^3\,f^3\,x^3\,\sin \left (e+f\,x\right )+6\,a^2\,c\,d^2\,f^2\,x\,\cos \left (2\,e+2\,f\,x\right )+6\,a^2\,c^2\,d\,f^3\,x\,\sin \left (2\,e+2\,f\,x\right )+48\,a^2\,c\,d^2\,f^3\,x^2\,\sin \left (e+f\,x\right )+6\,a^2\,c\,d^2\,f^3\,x^2\,\sin \left (2\,e+2\,f\,x\right )+96\,a^2\,c\,d^2\,f^2\,x\,\cos \left (e+f\,x\right )+48\,a^2\,c^2\,d\,f^3\,x\,\sin \left (e+f\,x\right )}{8\,f^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.45, size = 779, normalized size = 3.29 \[ \begin {cases} \frac {a^{2} c^{3} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {a^{2} c^{3} x \cos ^{2}{\left (e + f x \right )}}{2} + a^{2} c^{3} x + \frac {a^{2} c^{3} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {2 a^{2} c^{3} \sin {\left (e + f x \right )}}{f} + \frac {3 a^{2} c^{2} d x^{2} \sin ^{2}{\left (e + f x \right )}}{4} + \frac {3 a^{2} c^{2} d x^{2} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {3 a^{2} c^{2} d x^{2}}{2} + \frac {3 a^{2} c^{2} d x \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {6 a^{2} c^{2} d x \sin {\left (e + f x \right )}}{f} - \frac {3 a^{2} c^{2} d \sin ^{2}{\left (e + f x \right )}}{4 f^{2}} + \frac {6 a^{2} c^{2} d \cos {\left (e + f x \right )}}{f^{2}} + \frac {a^{2} c d^{2} x^{3} \sin ^{2}{\left (e + f x \right )}}{2} + \frac {a^{2} c d^{2} x^{3} \cos ^{2}{\left (e + f x \right )}}{2} + a^{2} c d^{2} x^{3} + \frac {3 a^{2} c d^{2} x^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {6 a^{2} c d^{2} x^{2} \sin {\left (e + f x \right )}}{f} - \frac {3 a^{2} c d^{2} x \sin ^{2}{\left (e + f x \right )}}{4 f^{2}} + \frac {3 a^{2} c d^{2} x \cos ^{2}{\left (e + f x \right )}}{4 f^{2}} + \frac {12 a^{2} c d^{2} x \cos {\left (e + f x \right )}}{f^{2}} - \frac {3 a^{2} c d^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{4 f^{3}} - \frac {12 a^{2} c d^{2} \sin {\left (e + f x \right )}}{f^{3}} + \frac {a^{2} d^{3} x^{4} \sin ^{2}{\left (e + f x \right )}}{8} + \frac {a^{2} d^{3} x^{4} \cos ^{2}{\left (e + f x \right )}}{8} + \frac {a^{2} d^{3} x^{4}}{4} + \frac {a^{2} d^{3} x^{3} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {2 a^{2} d^{3} x^{3} \sin {\left (e + f x \right )}}{f} - \frac {3 a^{2} d^{3} x^{2} \sin ^{2}{\left (e + f x \right )}}{8 f^{2}} + \frac {3 a^{2} d^{3} x^{2} \cos ^{2}{\left (e + f x \right )}}{8 f^{2}} + \frac {6 a^{2} d^{3} x^{2} \cos {\left (e + f x \right )}}{f^{2}} - \frac {3 a^{2} d^{3} x \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{4 f^{3}} - \frac {12 a^{2} d^{3} x \sin {\left (e + f x \right )}}{f^{3}} + \frac {3 a^{2} d^{3} \sin ^{2}{\left (e + f x \right )}}{8 f^{4}} - \frac {12 a^{2} d^{3} \cos {\left (e + f x \right )}}{f^{4}} & \text {for}\: f \neq 0 \\\left (a \cos {\relax (e )} + a\right )^{2} \left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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