Optimal. Leaf size=250 \[ \frac {a^2 (c+d x)^4}{4 d}+\frac {12 a b d^2 (c+d x) \cos (e+f x)}{f^3}+\frac {6 a b d (c+d x)^2 \sin (e+f x)}{f^2}-\frac {2 a b (c+d x)^3 \cos (e+f x)}{f}-\frac {12 a b d^3 \sin (e+f x)}{f^4}+\frac {3 b^2 d^2 (c+d x) \sin (e+f x) \cos (e+f x)}{4 f^3}-\frac {3 b^2 c d^2 x}{4 f^2}+\frac {3 b^2 d (c+d x)^2 \sin ^2(e+f x)}{4 f^2}-\frac {b^2 (c+d x)^3 \sin (e+f x) \cos (e+f x)}{2 f}+\frac {b^2 (c+d x)^4}{8 d}-\frac {3 b^2 d^3 \sin ^2(e+f x)}{8 f^4}-\frac {3 b^2 d^3 x^2}{8 f^2} \]
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Rubi [A] time = 0.27, antiderivative size = 250, 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, 2637, 3311, 32, 3310} \[ \frac {a^2 (c+d x)^4}{4 d}+\frac {12 a b d^2 (c+d x) \cos (e+f x)}{f^3}+\frac {6 a b d (c+d x)^2 \sin (e+f x)}{f^2}-\frac {2 a b (c+d x)^3 \cos (e+f x)}{f}-\frac {12 a b d^3 \sin (e+f x)}{f^4}+\frac {3 b^2 d^2 (c+d x) \sin (e+f x) \cos (e+f x)}{4 f^3}-\frac {3 b^2 c d^2 x}{4 f^2}+\frac {3 b^2 d (c+d x)^2 \sin ^2(e+f x)}{4 f^2}-\frac {b^2 (c+d x)^3 \sin (e+f x) \cos (e+f x)}{2 f}+\frac {b^2 (c+d x)^4}{8 d}-\frac {3 b^2 d^3 \sin ^2(e+f x)}{8 f^4}-\frac {3 b^2 d^3 x^2}{8 f^2} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2637
Rule 3296
Rule 3310
Rule 3311
Rule 3317
Rubi steps
\begin {align*} \int (c+d x)^3 (a+b \sin (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)^3+2 a b (c+d x)^3 \sin (e+f x)+b^2 (c+d x)^3 \sin ^2(e+f x)\right ) \, dx\\ &=\frac {a^2 (c+d x)^4}{4 d}+(2 a b) \int (c+d x)^3 \sin (e+f x) \, dx+b^2 \int (c+d x)^3 \sin ^2(e+f x) \, dx\\ &=\frac {a^2 (c+d x)^4}{4 d}-\frac {2 a b (c+d x)^3 \cos (e+f x)}{f}-\frac {b^2 (c+d x)^3 \cos (e+f x) \sin (e+f x)}{2 f}+\frac {3 b^2 d (c+d x)^2 \sin ^2(e+f x)}{4 f^2}+\frac {1}{2} b^2 \int (c+d x)^3 \, dx-\frac {\left (3 b^2 d^2\right ) \int (c+d x) \sin ^2(e+f x) \, dx}{2 f^2}+\frac {(6 a b d) \int (c+d x)^2 \cos (e+f x) \, dx}{f}\\ &=\frac {a^2 (c+d x)^4}{4 d}+\frac {b^2 (c+d x)^4}{8 d}-\frac {2 a b (c+d x)^3 \cos (e+f x)}{f}+\frac {6 a b d (c+d x)^2 \sin (e+f x)}{f^2}+\frac {3 b^2 d^2 (c+d x) \cos (e+f x) \sin (e+f x)}{4 f^3}-\frac {b^2 (c+d x)^3 \cos (e+f x) \sin (e+f x)}{2 f}-\frac {3 b^2 d^3 \sin ^2(e+f x)}{8 f^4}+\frac {3 b^2 d (c+d x)^2 \sin ^2(e+f x)}{4 f^2}-\frac {\left (12 a b d^2\right ) \int (c+d x) \sin (e+f x) \, dx}{f^2}-\frac {\left (3 b^2 d^2\right ) \int (c+d x) \, dx}{4 f^2}\\ &=-\frac {3 b^2 c d^2 x}{4 f^2}-\frac {3 b^2 d^3 x^2}{8 f^2}+\frac {a^2 (c+d x)^4}{4 d}+\frac {b^2 (c+d x)^4}{8 d}+\frac {12 a b d^2 (c+d x) \cos (e+f x)}{f^3}-\frac {2 a b (c+d x)^3 \cos (e+f x)}{f}+\frac {6 a b d (c+d x)^2 \sin (e+f x)}{f^2}+\frac {3 b^2 d^2 (c+d x) \cos (e+f x) \sin (e+f x)}{4 f^3}-\frac {b^2 (c+d x)^3 \cos (e+f x) \sin (e+f x)}{2 f}-\frac {3 b^2 d^3 \sin ^2(e+f x)}{8 f^4}+\frac {3 b^2 d (c+d x)^2 \sin ^2(e+f x)}{4 f^2}-\frac {\left (12 a b d^3\right ) \int \cos (e+f x) \, dx}{f^3}\\ &=-\frac {3 b^2 c d^2 x}{4 f^2}-\frac {3 b^2 d^3 x^2}{8 f^2}+\frac {a^2 (c+d x)^4}{4 d}+\frac {b^2 (c+d x)^4}{8 d}+\frac {12 a b d^2 (c+d x) \cos (e+f x)}{f^3}-\frac {2 a b (c+d x)^3 \cos (e+f x)}{f}-\frac {12 a b d^3 \sin (e+f x)}{f^4}+\frac {6 a b d (c+d x)^2 \sin (e+f x)}{f^2}+\frac {3 b^2 d^2 (c+d x) \cos (e+f x) \sin (e+f x)}{4 f^3}-\frac {b^2 (c+d x)^3 \cos (e+f x) \sin (e+f x)}{2 f}-\frac {3 b^2 d^3 \sin ^2(e+f x)}{8 f^4}+\frac {3 b^2 d (c+d x)^2 \sin ^2(e+f x)}{4 f^2}\\ \end {align*}
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Mathematica [A] time = 1.41, size = 232, normalized size = 0.93 \[ \frac {2 f^4 x \left (2 a^2+b^2\right ) \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )+96 a b d \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2-2\right )\right ) \sin (e+f x)-32 a b f (c+d x) \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2-6\right )\right ) \cos (e+f x)-2 b^2 f (c+d x) \left (2 c^2 f^2+4 c d f^2 x+d^2 \left (2 f^2 x^2-3\right )\right ) \sin (2 (e+f x))-3 b^2 d \left (2 c^2 f^2+4 c d f^2 x+d^2 \left (2 f^2 x^2-1\right )\right ) \cos (2 (e+f x))}{16 f^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 382, normalized size = 1.53 \[ \frac {{\left (2 \, a^{2} + b^{2}\right )} d^{3} f^{4} x^{4} + 4 \, {\left (2 \, a^{2} + b^{2}\right )} c d^{2} f^{4} x^{3} + 3 \, {\left (2 \, {\left (2 \, a^{2} + b^{2}\right )} c^{2} d f^{4} + b^{2} d^{3} f^{2}\right )} x^{2} - 3 \, {\left (2 \, b^{2} d^{3} f^{2} x^{2} + 4 \, b^{2} c d^{2} f^{2} x + 2 \, b^{2} c^{2} d f^{2} - b^{2} d^{3}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (2 \, {\left (2 \, a^{2} + b^{2}\right )} c^{3} f^{4} + 3 \, b^{2} c d^{2} f^{2}\right )} x - 16 \, {\left (a b d^{3} f^{3} x^{3} + 3 \, a b c d^{2} f^{3} x^{2} + a b c^{3} f^{3} - 6 \, a b c d^{2} f + 3 \, {\left (a b c^{2} d f^{3} - 2 \, a b d^{3} f\right )} x\right )} \cos \left (f x + e\right ) + 2 \, {\left (24 \, a b d^{3} f^{2} x^{2} + 48 \, a b c d^{2} f^{2} x + 24 \, a b c^{2} d f^{2} - 48 \, a b d^{3} - {\left (2 \, b^{2} d^{3} f^{3} x^{3} + 6 \, b^{2} c d^{2} f^{3} x^{2} + 2 \, b^{2} c^{3} f^{3} - 3 \, b^{2} c d^{2} f + 3 \, {\left (2 \, b^{2} c^{2} d f^{3} - b^{2} d^{3} f\right )} x\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{8 \, f^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 371, normalized size = 1.48 \[ \frac {1}{4} \, a^{2} d^{3} x^{4} + \frac {1}{8} \, b^{2} d^{3} x^{4} + a^{2} c d^{2} x^{3} + \frac {1}{2} \, b^{2} c d^{2} x^{3} + \frac {3}{2} \, a^{2} c^{2} d x^{2} + \frac {3}{4} \, b^{2} c^{2} d x^{2} + a^{2} c^{3} x + \frac {1}{2} \, b^{2} c^{3} x - \frac {3 \, {\left (2 \, b^{2} d^{3} f^{2} x^{2} + 4 \, b^{2} c d^{2} f^{2} x + 2 \, b^{2} c^{2} d f^{2} - b^{2} d^{3}\right )} \cos \left (2 \, f x + 2 \, e\right )}{16 \, f^{4}} - \frac {2 \, {\left (a b d^{3} f^{3} x^{3} + 3 \, a b c d^{2} f^{3} x^{2} + 3 \, a b c^{2} d f^{3} x + a b c^{3} f^{3} - 6 \, a b d^{3} f x - 6 \, a b c d^{2} f\right )} \cos \left (f x + e\right )}{f^{4}} - \frac {{\left (2 \, b^{2} d^{3} f^{3} x^{3} + 6 \, b^{2} c d^{2} f^{3} x^{2} + 6 \, b^{2} c^{2} d f^{3} x + 2 \, b^{2} c^{3} f^{3} - 3 \, b^{2} d^{3} f x - 3 \, b^{2} c d^{2} f\right )} \sin \left (2 \, f x + 2 \, e\right )}{8 \, f^{4}} + \frac {6 \, {\left (a b d^{3} f^{2} x^{2} + 2 \, a b c d^{2} f^{2} x + a b c^{2} d f^{2} - 2 \, a b d^{3}\right )} \sin \left (f x + e\right )}{f^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 1125, normalized size = 4.50 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 959, normalized size = 3.84 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.64, size = 497, normalized size = 1.99 \[ \frac {\frac {3\,b^2\,d^3\,\cos \left (2\,e+2\,f\,x\right )}{2}+8\,a^2\,c^3\,f^4\,x+4\,b^2\,c^3\,f^4\,x-96\,a\,b\,d^3\,\sin \left (e+f\,x\right )-2\,b^2\,c^3\,f^3\,\sin \left (2\,e+2\,f\,x\right )+2\,a^2\,d^3\,f^4\,x^4+b^2\,d^3\,f^4\,x^4-16\,a\,b\,c^3\,f^3\,\cos \left (e+f\,x\right )-3\,b^2\,d^3\,f^2\,x^2\,\cos \left (2\,e+2\,f\,x\right )-2\,b^2\,d^3\,f^3\,x^3\,\sin \left (2\,e+2\,f\,x\right )+3\,b^2\,c\,d^2\,f\,\sin \left (2\,e+2\,f\,x\right )+3\,b^2\,d^3\,f\,x\,\sin \left (2\,e+2\,f\,x\right )-3\,b^2\,c^2\,d\,f^2\,\cos \left (2\,e+2\,f\,x\right )+12\,a^2\,c^2\,d\,f^4\,x^2+8\,a^2\,c\,d^2\,f^4\,x^3+6\,b^2\,c^2\,d\,f^4\,x^2+4\,b^2\,c\,d^2\,f^4\,x^3+96\,a\,b\,c\,d^2\,f\,\cos \left (e+f\,x\right )+96\,a\,b\,d^3\,f\,x\,\cos \left (e+f\,x\right )-6\,b^2\,c\,d^2\,f^2\,x\,\cos \left (2\,e+2\,f\,x\right )-6\,b^2\,c^2\,d\,f^3\,x\,\sin \left (2\,e+2\,f\,x\right )+48\,a\,b\,c^2\,d\,f^2\,\sin \left (e+f\,x\right )-6\,b^2\,c\,d^2\,f^3\,x^2\,\sin \left (2\,e+2\,f\,x\right )-16\,a\,b\,d^3\,f^3\,x^3\,\cos \left (e+f\,x\right )+48\,a\,b\,d^3\,f^2\,x^2\,\sin \left (e+f\,x\right )-48\,a\,b\,c\,d^2\,f^3\,x^2\,\cos \left (e+f\,x\right )-48\,a\,b\,c^2\,d\,f^3\,x\,\cos \left (e+f\,x\right )+96\,a\,b\,c\,d^2\,f^2\,x\,\sin \left (e+f\,x\right )}{8\,f^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.96, size = 779, normalized size = 3.12 \[ \begin {cases} a^{2} c^{3} x + \frac {3 a^{2} c^{2} d x^{2}}{2} + a^{2} c d^{2} x^{3} + \frac {a^{2} d^{3} x^{4}}{4} - \frac {2 a b c^{3} \cos {\left (e + f x \right )}}{f} - \frac {6 a b c^{2} d x \cos {\left (e + f x \right )}}{f} + \frac {6 a b c^{2} d \sin {\left (e + f x \right )}}{f^{2}} - \frac {6 a b c d^{2} x^{2} \cos {\left (e + f x \right )}}{f} + \frac {12 a b c d^{2} x \sin {\left (e + f x \right )}}{f^{2}} + \frac {12 a b c d^{2} \cos {\left (e + f x \right )}}{f^{3}} - \frac {2 a b d^{3} x^{3} \cos {\left (e + f x \right )}}{f} + \frac {6 a b d^{3} x^{2} \sin {\left (e + f x \right )}}{f^{2}} + \frac {12 a b d^{3} x \cos {\left (e + f x \right )}}{f^{3}} - \frac {12 a b d^{3} \sin {\left (e + f x \right )}}{f^{4}} + \frac {b^{2} c^{3} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {b^{2} c^{3} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {b^{2} c^{3} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {3 b^{2} c^{2} d x^{2} \sin ^{2}{\left (e + f x \right )}}{4} + \frac {3 b^{2} c^{2} d x^{2} \cos ^{2}{\left (e + f x \right )}}{4} - \frac {3 b^{2} c^{2} d x \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {3 b^{2} c^{2} d \cos ^{2}{\left (e + f x \right )}}{4 f^{2}} + \frac {b^{2} c d^{2} x^{3} \sin ^{2}{\left (e + f x \right )}}{2} + \frac {b^{2} c d^{2} x^{3} \cos ^{2}{\left (e + f x \right )}}{2} - \frac {3 b^{2} c d^{2} x^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {3 b^{2} c d^{2} x \sin ^{2}{\left (e + f x \right )}}{4 f^{2}} - \frac {3 b^{2} c d^{2} x \cos ^{2}{\left (e + f x \right )}}{4 f^{2}} + \frac {3 b^{2} c d^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{4 f^{3}} + \frac {b^{2} d^{3} x^{4} \sin ^{2}{\left (e + f x \right )}}{8} + \frac {b^{2} d^{3} x^{4} \cos ^{2}{\left (e + f x \right )}}{8} - \frac {b^{2} d^{3} x^{3} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {3 b^{2} d^{3} x^{2} \sin ^{2}{\left (e + f x \right )}}{8 f^{2}} - \frac {3 b^{2} d^{3} x^{2} \cos ^{2}{\left (e + f x \right )}}{8 f^{2}} + \frac {3 b^{2} d^{3} x \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{4 f^{3}} + \frac {3 b^{2} d^{3} \cos ^{2}{\left (e + f x \right )}}{8 f^{4}} & \text {for}\: f \neq 0 \\\left (a + b \sin {\relax (e )}\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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