Integrand size = 28, antiderivative size = 75 \[ \int \frac {(e+f x)^2 \cos ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {(e+f x)^3}{3 a f}-\frac {2 f^2 \cos (c+d x)}{a d^3}+\frac {(e+f x)^2 \cos (c+d x)}{a d}-\frac {2 f (e+f x) \sin (c+d x)}{a d^2} \]
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Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4619, 32, 3377, 2718} \[ \int \frac {(e+f x)^2 \cos ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {2 f^2 \cos (c+d x)}{a d^3}-\frac {2 f (e+f x) \sin (c+d x)}{a d^2}+\frac {(e+f x)^2 \cos (c+d x)}{a d}+\frac {(e+f x)^3}{3 a f} \]
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Rule 32
Rule 2718
Rule 3377
Rule 4619
Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^2 \, dx}{a}-\frac {\int (e+f x)^2 \sin (c+d x) \, dx}{a} \\ & = \frac {(e+f x)^3}{3 a f}+\frac {(e+f x)^2 \cos (c+d x)}{a d}-\frac {(2 f) \int (e+f x) \cos (c+d x) \, dx}{a d} \\ & = \frac {(e+f x)^3}{3 a f}+\frac {(e+f x)^2 \cos (c+d x)}{a d}-\frac {2 f (e+f x) \sin (c+d x)}{a d^2}+\frac {\left (2 f^2\right ) \int \sin (c+d x) \, dx}{a d^2} \\ & = \frac {(e+f x)^3}{3 a f}-\frac {2 f^2 \cos (c+d x)}{a d^3}+\frac {(e+f x)^2 \cos (c+d x)}{a d}-\frac {2 f (e+f x) \sin (c+d x)}{a d^2} \\ \end{align*}
Time = 0.72 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.99 \[ \int \frac {(e+f x)^2 \cos ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {d^3 x \left (3 e^2+3 e f x+f^2 x^2\right )+3 \left (-2 f^2+d^2 (e+f x)^2\right ) \cos (c+d x)-6 d f (e+f x) \sin (c+d x)}{3 a d^3} \]
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Time = 0.24 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.17
method | result | size |
parallelrisch | \(\frac {\left (3 \left (f x +e \right )^{2} d^{2}-6 f^{2}\right ) \cos \left (d x +c \right )-6 d f \left (f x +e \right ) \sin \left (d x +c \right )+\left (f^{2} x^{3}+3 x^{2} e f +3 x \,e^{2}\right ) d^{3}+3 d^{2} e^{2}-6 f^{2}}{3 a \,d^{3}}\) | \(88\) |
risch | \(\frac {f^{2} x^{3}}{3 a}+\frac {e f \,x^{2}}{a}+\frac {e^{2} x}{a}+\frac {e^{3}}{3 a f}+\frac {\left (d^{2} x^{2} f^{2}+2 f e x \,d^{2}+d^{2} e^{2}-2 f^{2}\right ) \cos \left (d x +c \right )}{a \,d^{3}}-\frac {2 f \left (f x +e \right ) \sin \left (d x +c \right )}{a \,d^{2}}\) | \(105\) |
derivativedivides | \(-\frac {-\cos \left (d x +c \right ) c^{2} f^{2}+2 \cos \left (d x +c \right ) c d e f -2 c \,f^{2} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )-\cos \left (d x +c \right ) d^{2} e^{2}+2 d e f \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )+f^{2} \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )-c^{2} f^{2} \left (d x +c \right )+2 c d e f \left (d x +c \right )+c \,f^{2} \left (d x +c \right )^{2}-d^{2} e^{2} \left (d x +c \right )-d e f \left (d x +c \right )^{2}-\frac {f^{2} \left (d x +c \right )^{3}}{3}}{d^{3} a}\) | \(215\) |
default | \(-\frac {-\cos \left (d x +c \right ) c^{2} f^{2}+2 \cos \left (d x +c \right ) c d e f -2 c \,f^{2} \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )-\cos \left (d x +c \right ) d^{2} e^{2}+2 d e f \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )+f^{2} \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )-c^{2} f^{2} \left (d x +c \right )+2 c d e f \left (d x +c \right )+c \,f^{2} \left (d x +c \right )^{2}-d^{2} e^{2} \left (d x +c \right )-d e f \left (d x +c \right )^{2}-\frac {f^{2} \left (d x +c \right )^{3}}{3}}{d^{3} a}\) | \(215\) |
norman | \(\frac {\frac {2 d^{2} e^{2}+4 d e f -4 f^{2}}{a \,d^{3}}+\frac {4 e f \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d^{2} a}+\frac {\left (2 d^{2} e^{2}-4 f^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a \,d^{3}}+\frac {e \left (d e +2 f \right ) x}{d a}+\frac {f \left (d e +f \right ) x^{2}}{d a}+\frac {\left (d^{2} e^{2}-2 d e f -4 f^{2}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \,d^{2}}+\frac {\left (d^{2} e^{2}+2 d e f -4 f^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a \,d^{2}}+\frac {e \left (d e -2 f \right ) x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {f \left (d e -f \right ) x^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {f \left (d e -f \right ) x^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {f \left (d e +f \right ) x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}+\frac {f^{2} x^{3}}{3 a}+\frac {f^{2} x^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a}+\frac {2 f^{2} x^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a}+\frac {2 f^{2} x^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a}+\frac {f^{2} x^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a}+\frac {f^{2} x^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a}+\frac {2 \left (d^{2} e^{2}+2 d e f -2 f^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \,d^{3}}+\frac {2 \left (d^{2} e^{2}+2 d e f -2 f^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \,d^{3}}+\frac {2 e f \,x^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {2 e f \,x^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {2 \left (d^{2} e^{2}-2 f^{2}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \,d^{2}}+\frac {2 \left (d^{2} e^{2}-2 f^{2}\right ) x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \,d^{2}}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\) | \(623\) |
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Time = 0.26 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.28 \[ \int \frac {(e+f x)^2 \cos ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {d^{3} f^{2} x^{3} + 3 \, d^{3} e f x^{2} + 3 \, d^{3} e^{2} x + 3 \, {\left (d^{2} f^{2} x^{2} + 2 \, d^{2} e f x + d^{2} e^{2} - 2 \, f^{2}\right )} \cos \left (d x + c\right ) - 6 \, {\left (d f^{2} x + d e f\right )} \sin \left (d x + c\right )}{3 \, a d^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 605 vs. \(2 (65) = 130\).
Time = 1.74 (sec) , antiderivative size = 605, normalized size of antiderivative = 8.07 \[ \int \frac {(e+f x)^2 \cos ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\begin {cases} \frac {3 d^{3} e^{2} x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a d^{3} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d^{3}} + \frac {3 d^{3} e^{2} x}{3 a d^{3} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d^{3}} + \frac {3 d^{3} e f x^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a d^{3} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d^{3}} + \frac {3 d^{3} e f x^{2}}{3 a d^{3} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d^{3}} + \frac {d^{3} f^{2} x^{3} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a d^{3} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d^{3}} + \frac {d^{3} f^{2} x^{3}}{3 a d^{3} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d^{3}} + \frac {6 d^{2} e^{2}}{3 a d^{3} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d^{3}} - \frac {6 d^{2} e f x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a d^{3} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d^{3}} + \frac {6 d^{2} e f x}{3 a d^{3} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d^{3}} - \frac {3 d^{2} f^{2} x^{2} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a d^{3} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d^{3}} + \frac {3 d^{2} f^{2} x^{2}}{3 a d^{3} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d^{3}} - \frac {12 d e f \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a d^{3} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d^{3}} - \frac {12 d f^{2} x \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a d^{3} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d^{3}} - \frac {12 f^{2}}{3 a d^{3} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 a d^{3}} & \text {for}\: d \neq 0 \\\frac {\left (e^{2} x + e f x^{2} + \frac {f^{2} x^{3}}{3}\right ) \cos ^{2}{\left (c \right )}}{a \sin {\left (c \right )} + a} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 309 vs. \(2 (73) = 146\).
Time = 0.32 (sec) , antiderivative size = 309, normalized size of antiderivative = 4.12 \[ \int \frac {(e+f x)^2 \cos ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {6 \, c^{2} f^{2} {\left (\frac {1}{a d^{2} + \frac {a d^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}} + \frac {\arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a d^{2}}\right )} - 12 \, c e f {\left (\frac {1}{a d + \frac {a d \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}} + \frac {\arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a d}\right )} + 6 \, e^{2} {\left (\frac {\arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {1}{a + \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}\right )} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} + 2 \, {\left (d x + c\right )} \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right )\right )} e f}{a d} - \frac {3 \, {\left ({\left (d x + c\right )}^{2} + 2 \, {\left (d x + c\right )} \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right )\right )} c f^{2}}{a d^{2}} + \frac {{\left ({\left (d x + c\right )}^{3} + 3 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 6 \, {\left (d x + c\right )} \sin \left (d x + c\right )\right )} f^{2}}{a d^{2}}}{3 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 656 vs. \(2 (73) = 146\).
Time = 0.32 (sec) , antiderivative size = 656, normalized size of antiderivative = 8.75 \[ \int \frac {(e+f x)^2 \cos ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {d^{3} f^{2} x^{3} \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 3 \, d^{3} e f x^{2} \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + d^{3} f^{2} x^{3} \tan \left (\frac {1}{2} \, d x\right )^{2} + d^{3} f^{2} x^{3} \tan \left (\frac {1}{2} \, c\right )^{2} + 3 \, d^{3} e^{2} x \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 3 \, d^{2} f^{2} x^{2} \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 3 \, d^{3} e f x^{2} \tan \left (\frac {1}{2} \, d x\right )^{2} + 3 \, d^{3} e f x^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 6 \, d^{2} e f x \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + d^{3} f^{2} x^{3} + 3 \, d^{3} e^{2} x \tan \left (\frac {1}{2} \, d x\right )^{2} - 3 \, d^{2} f^{2} x^{2} \tan \left (\frac {1}{2} \, d x\right )^{2} - 12 \, d^{2} f^{2} x^{2} \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right ) + 3 \, d^{3} e^{2} x \tan \left (\frac {1}{2} \, c\right )^{2} - 3 \, d^{2} f^{2} x^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 3 \, d^{2} e^{2} \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 3 \, d^{3} e f x^{2} - 6 \, d^{2} e f x \tan \left (\frac {1}{2} \, d x\right )^{2} - 24 \, d^{2} e f x \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right ) + 12 \, d f^{2} x \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 6 \, d^{2} e f x \tan \left (\frac {1}{2} \, c\right )^{2} + 12 \, d f^{2} x \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 3 \, d^{3} e^{2} x + 3 \, d^{2} f^{2} x^{2} - 3 \, d^{2} e^{2} \tan \left (\frac {1}{2} \, d x\right )^{2} - 12 \, d^{2} e^{2} \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right ) + 12 \, d e f \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 3 \, d^{2} e^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 12 \, d e f \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - 6 \, f^{2} \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 6 \, d^{2} e f x - 12 \, d f^{2} x \tan \left (\frac {1}{2} \, d x\right ) - 12 \, d f^{2} x \tan \left (\frac {1}{2} \, c\right ) + 3 \, d^{2} e^{2} - 12 \, d e f \tan \left (\frac {1}{2} \, d x\right ) + 6 \, f^{2} \tan \left (\frac {1}{2} \, d x\right )^{2} - 12 \, d e f \tan \left (\frac {1}{2} \, c\right ) + 24 \, f^{2} \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right ) + 6 \, f^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - 6 \, f^{2}}{3 \, {\left (a d^{3} \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + a d^{3} \tan \left (\frac {1}{2} \, d x\right )^{2} + a d^{3} \tan \left (\frac {1}{2} \, c\right )^{2} + a d^{3}\right )}} \]
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Time = 2.69 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.47 \[ \int \frac {(e+f x)^2 \cos ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {e^2\,x+e\,f\,x^2+\frac {f^2\,x^3}{3}}{a}-\frac {2\,f^2\,\cos \left (c+d\,x\right )-d^2\,\left (e^2\,\cos \left (c+d\,x\right )+f^2\,x^2\,\cos \left (c+d\,x\right )+2\,e\,f\,x\,\cos \left (c+d\,x\right )\right )+d\,\left (2\,x\,\sin \left (c+d\,x\right )\,f^2+2\,e\,\sin \left (c+d\,x\right )\,f\right )}{a\,d^3} \]
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