Integrand size = 16, antiderivative size = 134 \[ \int (c+d x)^3 \cos ^2(a+b x) \, dx=-\frac {3 c d^2 x}{4 b^2}-\frac {3 d^3 x^2}{8 b^2}+\frac {(c+d x)^4}{8 d}-\frac {3 d^3 \cos ^2(a+b x)}{8 b^4}+\frac {3 d (c+d x)^2 \cos ^2(a+b x)}{4 b^2}-\frac {3 d^2 (c+d x) \cos (a+b x) \sin (a+b x)}{4 b^3}+\frac {(c+d x)^3 \cos (a+b x) \sin (a+b x)}{2 b} \]
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Time = 0.12 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3392, 32, 3391} \[ \int (c+d x)^3 \cos ^2(a+b x) \, dx=-\frac {3 d^3 \cos ^2(a+b x)}{8 b^4}-\frac {3 d^2 (c+d x) \sin (a+b x) \cos (a+b x)}{4 b^3}+\frac {3 d (c+d x)^2 \cos ^2(a+b x)}{4 b^2}+\frac {(c+d x)^3 \sin (a+b x) \cos (a+b x)}{2 b}-\frac {3 c d^2 x}{4 b^2}-\frac {3 d^3 x^2}{8 b^2}+\frac {(c+d x)^4}{8 d} \]
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Rule 32
Rule 3391
Rule 3392
Rubi steps \begin{align*} \text {integral}& = \frac {3 d (c+d x)^2 \cos ^2(a+b x)}{4 b^2}+\frac {(c+d x)^3 \cos (a+b x) \sin (a+b x)}{2 b}+\frac {1}{2} \int (c+d x)^3 \, dx-\frac {\left (3 d^2\right ) \int (c+d x) \cos ^2(a+b x) \, dx}{2 b^2} \\ & = \frac {(c+d x)^4}{8 d}-\frac {3 d^3 \cos ^2(a+b x)}{8 b^4}+\frac {3 d (c+d x)^2 \cos ^2(a+b x)}{4 b^2}-\frac {3 d^2 (c+d x) \cos (a+b x) \sin (a+b x)}{4 b^3}+\frac {(c+d x)^3 \cos (a+b x) \sin (a+b x)}{2 b}-\frac {\left (3 d^2\right ) \int (c+d x) \, dx}{4 b^2} \\ & = -\frac {3 c d^2 x}{4 b^2}-\frac {3 d^3 x^2}{8 b^2}+\frac {(c+d x)^4}{8 d}-\frac {3 d^3 \cos ^2(a+b x)}{8 b^4}+\frac {3 d (c+d x)^2 \cos ^2(a+b x)}{4 b^2}-\frac {3 d^2 (c+d x) \cos (a+b x) \sin (a+b x)}{4 b^3}+\frac {(c+d x)^3 \cos (a+b x) \sin (a+b x)}{2 b} \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.79 \[ \int (c+d x)^3 \cos ^2(a+b x) \, dx=\frac {2 b^4 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )+3 d \left (-d^2+2 b^2 (c+d x)^2\right ) \cos (2 (a+b x))+2 b (c+d x) \left (-3 d^2+2 b^2 (c+d x)^2\right ) \sin (2 (a+b x))}{16 b^4} \]
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Time = 1.14 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.90
method | result | size |
parallelrisch | \(\frac {4 b \sin \left (2 b x +2 a \right ) \left (d x +c \right ) \left (\left (d x +c \right )^{2} b^{2}-\frac {3 d^{2}}{2}\right )+6 d \left (\left (d x +c \right )^{2} b^{2}-\frac {d^{2}}{2}\right ) \cos \left (2 b x +2 a \right )+2 \left (d^{3} x^{4}+4 d^{2} c \,x^{3}+6 d \,c^{2} x^{2}+4 c^{3} x \right ) b^{4}-6 b^{2} c^{2} d +3 d^{3}}{16 b^{4}}\) | \(121\) |
risch | \(\frac {d^{3} x^{4}}{8}+\frac {d^{2} c \,x^{3}}{2}+\frac {3 d \,c^{2} x^{2}}{4}+\frac {c^{3} x}{2}+\frac {c^{4}}{8 d}+\frac {3 d \left (2 x^{2} d^{2} b^{2}+4 b^{2} c d x +2 b^{2} c^{2}-d^{2}\right ) \cos \left (2 b x +2 a \right )}{16 b^{4}}+\frac {\left (2 b^{2} d^{3} x^{3}+6 b^{2} c \,d^{2} x^{2}+6 b^{2} c^{2} d x +2 b^{2} c^{3}-3 d^{3} x -3 d^{2} c \right ) \sin \left (2 b x +2 a \right )}{8 b^{3}}\) | \(158\) |
norman | \(\frac {d^{2} c \,x^{3} \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\frac {d^{3} x^{3} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}+\frac {d^{3} x^{4}}{8}+\frac {d^{2} c \,x^{3}}{2}+\frac {d^{3} x^{4} \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4}+\frac {d^{3} x^{4} \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8}+\frac {\left (-6 b^{2} c^{2} d +3 d^{3}\right ) \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2 b^{4}}+\frac {c \left (2 b^{2} c^{2}-3 d^{2}\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{2 b^{3}}-\frac {c \left (2 b^{2} c^{2}-3 d^{2}\right ) \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2 b^{3}}+\frac {c \left (2 b^{2} c^{2}+3 d^{2}\right ) x}{4 b^{2}}+\frac {3 d \left (2 b^{2} c^{2}+d^{2}\right ) x^{2}}{8 b^{2}}+\frac {d^{2} c \,x^{3} \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2}-\frac {d^{3} x^{3} \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}+\frac {3 d^{2} c \,x^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}-\frac {3 d^{2} c \,x^{2} \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}+\frac {c \left (2 b^{2} c^{2}-9 d^{2}\right ) x \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2 b^{2}}+\frac {c \left (2 b^{2} c^{2}+3 d^{2}\right ) x \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4 b^{2}}+\frac {3 d \left (2 b^{2} c^{2}-3 d^{2}\right ) x^{2} \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4 b^{2}}+\frac {3 d \left (2 b^{2} c^{2}-d^{2}\right ) x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{2 b^{3}}-\frac {3 d \left (2 b^{2} c^{2}-d^{2}\right ) x \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2 b^{3}}+\frac {3 d \left (2 b^{2} c^{2}+d^{2}\right ) x^{2} \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8 b^{2}}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{2}}\) | \(522\) |
derivativedivides | \(\frac {-\frac {a^{3} d^{3} \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b^{3}}+\frac {3 a^{2} c \,d^{2} \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b^{2}}+\frac {3 a^{2} d^{3} \left (\left (b x +a \right ) \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )-\frac {\left (b x +a \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (b x +a \right )\right )}{4}\right )}{b^{3}}-\frac {3 a \,c^{2} d \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b}-\frac {6 a c \,d^{2} \left (\left (b x +a \right ) \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )-\frac {\left (b x +a \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (b x +a \right )\right )}{4}\right )}{b^{2}}-\frac {3 a \,d^{3} \left (\left (b x +a \right )^{2} \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )+\frac {\left (b x +a \right ) \left (\cos ^{2}\left (b x +a \right )\right )}{2}-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{4}-\frac {b x}{4}-\frac {a}{4}-\frac {\left (b x +a \right )^{3}}{3}\right )}{b^{3}}+c^{3} \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )+\frac {3 c^{2} d \left (\left (b x +a \right ) \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )-\frac {\left (b x +a \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (b x +a \right )\right )}{4}\right )}{b}+\frac {3 c \,d^{2} \left (\left (b x +a \right )^{2} \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )+\frac {\left (b x +a \right ) \left (\cos ^{2}\left (b x +a \right )\right )}{2}-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{4}-\frac {b x}{4}-\frac {a}{4}-\frac {\left (b x +a \right )^{3}}{3}\right )}{b^{2}}+\frac {d^{3} \left (\left (b x +a \right )^{3} \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )+\frac {3 \left (b x +a \right )^{2} \left (\cos ^{2}\left (b x +a \right )\right )}{4}-\frac {3 \left (b x +a \right ) \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{2}+\frac {3 \left (b x +a \right )^{2}}{8}+\frac {3 \left (\sin ^{2}\left (b x +a \right )\right )}{8}-\frac {3 \left (b x +a \right )^{4}}{8}\right )}{b^{3}}}{b}\) | \(587\) |
default | \(\frac {-\frac {a^{3} d^{3} \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b^{3}}+\frac {3 a^{2} c \,d^{2} \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b^{2}}+\frac {3 a^{2} d^{3} \left (\left (b x +a \right ) \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )-\frac {\left (b x +a \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (b x +a \right )\right )}{4}\right )}{b^{3}}-\frac {3 a \,c^{2} d \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b}-\frac {6 a c \,d^{2} \left (\left (b x +a \right ) \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )-\frac {\left (b x +a \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (b x +a \right )\right )}{4}\right )}{b^{2}}-\frac {3 a \,d^{3} \left (\left (b x +a \right )^{2} \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )+\frac {\left (b x +a \right ) \left (\cos ^{2}\left (b x +a \right )\right )}{2}-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{4}-\frac {b x}{4}-\frac {a}{4}-\frac {\left (b x +a \right )^{3}}{3}\right )}{b^{3}}+c^{3} \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )+\frac {3 c^{2} d \left (\left (b x +a \right ) \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )-\frac {\left (b x +a \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (b x +a \right )\right )}{4}\right )}{b}+\frac {3 c \,d^{2} \left (\left (b x +a \right )^{2} \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )+\frac {\left (b x +a \right ) \left (\cos ^{2}\left (b x +a \right )\right )}{2}-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{4}-\frac {b x}{4}-\frac {a}{4}-\frac {\left (b x +a \right )^{3}}{3}\right )}{b^{2}}+\frac {d^{3} \left (\left (b x +a \right )^{3} \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )+\frac {3 \left (b x +a \right )^{2} \left (\cos ^{2}\left (b x +a \right )\right )}{4}-\frac {3 \left (b x +a \right ) \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{2}+\frac {3 \left (b x +a \right )^{2}}{8}+\frac {3 \left (\sin ^{2}\left (b x +a \right )\right )}{8}-\frac {3 \left (b x +a \right )^{4}}{8}\right )}{b^{3}}}{b}\) | \(587\) |
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Time = 0.27 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.42 \[ \int (c+d x)^3 \cos ^2(a+b x) \, dx=\frac {b^{4} d^{3} x^{4} + 4 \, b^{4} c d^{2} x^{3} + 3 \, {\left (2 \, b^{4} c^{2} d - b^{2} d^{3}\right )} x^{2} + 3 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{2} + 2 \, {\left (2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 2 \, b^{3} c^{3} - 3 \, b c d^{2} + 3 \, {\left (2 \, b^{3} c^{2} d - b d^{3}\right )} x\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 2 \, {\left (2 \, b^{4} c^{3} - 3 \, b^{2} c d^{2}\right )} x}{8 \, b^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (131) = 262\).
Time = 0.40 (sec) , antiderivative size = 456, normalized size of antiderivative = 3.40 \[ \int (c+d x)^3 \cos ^2(a+b x) \, dx=\begin {cases} \frac {c^{3} x \sin ^{2}{\left (a + b x \right )}}{2} + \frac {c^{3} x \cos ^{2}{\left (a + b x \right )}}{2} + \frac {3 c^{2} d x^{2} \sin ^{2}{\left (a + b x \right )}}{4} + \frac {3 c^{2} d x^{2} \cos ^{2}{\left (a + b x \right )}}{4} + \frac {c d^{2} x^{3} \sin ^{2}{\left (a + b x \right )}}{2} + \frac {c d^{2} x^{3} \cos ^{2}{\left (a + b x \right )}}{2} + \frac {d^{3} x^{4} \sin ^{2}{\left (a + b x \right )}}{8} + \frac {d^{3} x^{4} \cos ^{2}{\left (a + b x \right )}}{8} + \frac {c^{3} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{2 b} + \frac {3 c^{2} d x \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{2 b} + \frac {3 c d^{2} x^{2} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{2 b} + \frac {d^{3} x^{3} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{2 b} + \frac {3 c^{2} d \cos ^{2}{\left (a + b x \right )}}{4 b^{2}} - \frac {3 c d^{2} x \sin ^{2}{\left (a + b x \right )}}{4 b^{2}} + \frac {3 c d^{2} x \cos ^{2}{\left (a + b x \right )}}{4 b^{2}} - \frac {3 d^{3} x^{2} \sin ^{2}{\left (a + b x \right )}}{8 b^{2}} + \frac {3 d^{3} x^{2} \cos ^{2}{\left (a + b x \right )}}{8 b^{2}} - \frac {3 c d^{2} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{4 b^{3}} - \frac {3 d^{3} x \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{4 b^{3}} - \frac {3 d^{3} \cos ^{2}{\left (a + b x \right )}}{8 b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \cos ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 428 vs. \(2 (120) = 240\).
Time = 0.32 (sec) , antiderivative size = 428, normalized size of antiderivative = 3.19 \[ \int (c+d x)^3 \cos ^2(a+b x) \, dx=\frac {4 \, {\left (2 \, b x + 2 \, a + \sin \left (2 \, b x + 2 \, a\right )\right )} c^{3} - \frac {12 \, {\left (2 \, b x + 2 \, a + \sin \left (2 \, b x + 2 \, a\right )\right )} a c^{2} d}{b} + \frac {12 \, {\left (2 \, b x + 2 \, a + \sin \left (2 \, b x + 2 \, a\right )\right )} a^{2} c d^{2}}{b^{2}} - \frac {4 \, {\left (2 \, b x + 2 \, a + \sin \left (2 \, b x + 2 \, a\right )\right )} a^{3} d^{3}}{b^{3}} + \frac {6 \, {\left (2 \, {\left (b x + a\right )}^{2} + 2 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right ) + \cos \left (2 \, b x + 2 \, a\right )\right )} c^{2} d}{b} - \frac {12 \, {\left (2 \, {\left (b x + a\right )}^{2} + 2 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right ) + \cos \left (2 \, b x + 2 \, a\right )\right )} a c d^{2}}{b^{2}} + \frac {6 \, {\left (2 \, {\left (b x + a\right )}^{2} + 2 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right ) + \cos \left (2 \, b x + 2 \, a\right )\right )} a^{2} d^{3}}{b^{3}} + \frac {2 \, {\left (4 \, {\left (b x + a\right )}^{3} + 6 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) + 3 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} c d^{2}}{b^{2}} - \frac {2 \, {\left (4 \, {\left (b x + a\right )}^{3} + 6 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) + 3 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} a d^{3}}{b^{3}} + \frac {{\left (2 \, {\left (b x + a\right )}^{4} + 3 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + 2 \, {\left (2 \, {\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} d^{3}}{b^{3}}}{16 \, b} \]
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Time = 0.30 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.14 \[ \int (c+d x)^3 \cos ^2(a+b x) \, dx=\frac {1}{8} \, d^{3} x^{4} + \frac {1}{2} \, c d^{2} x^{3} + \frac {3}{4} \, c^{2} d x^{2} + \frac {1}{2} \, c^{3} x + \frac {3 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (2 \, b x + 2 \, a\right )}{16 \, b^{4}} + \frac {{\left (2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 6 \, b^{3} c^{2} d x + 2 \, b^{3} c^{3} - 3 \, b d^{3} x - 3 \, b c d^{2}\right )} \sin \left (2 \, b x + 2 \, a\right )}{8 \, b^{4}} \]
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Time = 14.34 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.71 \[ \int (c+d x)^3 \cos ^2(a+b x) \, dx=\frac {4\,b^4\,c^3\,x-\frac {3\,d^3\,\cos \left (2\,a+2\,b\,x\right )}{2}+2\,b^3\,c^3\,\sin \left (2\,a+2\,b\,x\right )+b^4\,d^3\,x^4+3\,b^2\,c^2\,d\,\cos \left (2\,a+2\,b\,x\right )+6\,b^4\,c^2\,d\,x^2+4\,b^4\,c\,d^2\,x^3+3\,b^2\,d^3\,x^2\,\cos \left (2\,a+2\,b\,x\right )+2\,b^3\,d^3\,x^3\,\sin \left (2\,a+2\,b\,x\right )-3\,b\,c\,d^2\,\sin \left (2\,a+2\,b\,x\right )-3\,b\,d^3\,x\,\sin \left (2\,a+2\,b\,x\right )+6\,b^2\,c\,d^2\,x\,\cos \left (2\,a+2\,b\,x\right )+6\,b^3\,c^2\,d\,x\,\sin \left (2\,a+2\,b\,x\right )+6\,b^3\,c\,d^2\,x^2\,\sin \left (2\,a+2\,b\,x\right )}{8\,b^4} \]
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