Integrand size = 18, antiderivative size = 199 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \, dx=\frac {b c F^{c (a+b x)} \cos ^3(d+e x) \log (F)}{9 e^2+b^2 c^2 \log ^2(F)}+\frac {6 b c e^2 F^{c (a+b x)} \cos (d+e x) \log (F)}{9 e^4+10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)}+\frac {3 e F^{c (a+b x)} \cos ^2(d+e x) \sin (d+e x)}{9 e^2+b^2 c^2 \log ^2(F)}+\frac {6 e^3 F^{c (a+b x)} \sin (d+e x)}{9 e^4+10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)} \]
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Time = 0.07 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4520, 4518} \[ \int F^{c (a+b x)} \cos ^3(d+e x) \, dx=\frac {b c \log (F) \cos ^3(d+e x) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)+9 e^2}+\frac {3 e \sin (d+e x) \cos ^2(d+e x) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)+9 e^2}+\frac {6 b c e^2 \log (F) \cos (d+e x) F^{c (a+b x)}}{b^4 c^4 \log ^4(F)+10 b^2 c^2 e^2 \log ^2(F)+9 e^4}+\frac {6 e^3 \sin (d+e x) F^{c (a+b x)}}{b^4 c^4 \log ^4(F)+10 b^2 c^2 e^2 \log ^2(F)+9 e^4} \]
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Rule 4518
Rule 4520
Rubi steps \begin{align*} \text {integral}& = \frac {b c F^{c (a+b x)} \cos ^3(d+e x) \log (F)}{9 e^2+b^2 c^2 \log ^2(F)}+\frac {3 e F^{c (a+b x)} \cos ^2(d+e x) \sin (d+e x)}{9 e^2+b^2 c^2 \log ^2(F)}+\frac {\left (6 e^2\right ) \int F^{c (a+b x)} \cos (d+e x) \, dx}{9 e^2+b^2 c^2 \log ^2(F)} \\ & = \frac {b c F^{c (a+b x)} \cos ^3(d+e x) \log (F)}{9 e^2+b^2 c^2 \log ^2(F)}+\frac {6 b c e^2 F^{c (a+b x)} \cos (d+e x) \log (F)}{9 e^4+10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)}+\frac {3 e F^{c (a+b x)} \cos ^2(d+e x) \sin (d+e x)}{9 e^2+b^2 c^2 \log ^2(F)}+\frac {6 e^3 F^{c (a+b x)} \sin (d+e x)}{9 e^4+10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.78 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \, dx=\frac {F^{c (a+b x)} \left (b c \cos (3 (d+e x)) \log (F) \left (e^2+b^2 c^2 \log ^2(F)\right )+3 b c \cos (d+e x) \log (F) \left (9 e^2+b^2 c^2 \log ^2(F)\right )+6 e \left (5 e^2+b^2 c^2 \log ^2(F)+\cos (2 (d+e x)) \left (e^2+b^2 c^2 \log ^2(F)\right )\right ) \sin (d+e x)\right )}{4 \left (9 e^4+10 b^2 c^2 e^2 \log ^2(F)+b^4 c^4 \log ^4(F)\right )} \]
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Time = 1.25 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.72
method | result | size |
parallelrisch | \(\frac {F^{c \left (x b +a \right )} \left (b c \ln \left (F \right ) \left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right ) \cos \left (3 e x +3 d \right )+\left (3 \ln \left (F \right )^{2} b^{2} c^{2} e +3 e^{3}\right ) \sin \left (3 e x +3 d \right )+3 \left (9 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right ) \left (\cos \left (e x +d \right ) \ln \left (F \right ) b c +e \sin \left (e x +d \right )\right )\right )}{4 b^{4} c^{4} \ln \left (F \right )^{4}+40 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+36 e^{4}}\) | \(144\) |
risch | \(\frac {3 b c \,F^{c \left (x b +a \right )} \cos \left (e x +d \right ) \ln \left (F \right )}{4 \left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}+\frac {3 e \,F^{c \left (x b +a \right )} \sin \left (e x +d \right )}{4 \left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}+\frac {F^{c \left (x b +a \right )} b c \ln \left (F \right ) \cos \left (3 e x +3 d \right )}{4 b^{2} c^{2} \ln \left (F \right )^{2}+36 e^{2}}+\frac {3 e \,F^{c \left (x b +a \right )} \sin \left (3 e x +3 d \right )}{4 \left (9 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}\) | \(158\) |
norman | \(\frac {\frac {\ln \left (F \right ) b c \left (b^{2} c^{2} \ln \left (F \right )^{2}+7 e^{2}\right ) {\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )}}{9 e^{4}+10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}-\frac {12 e \left (b^{2} c^{2} \ln \left (F \right )^{2}-e^{2}\right ) {\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{9 e^{4}+10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}+\frac {6 e \left (b^{2} c^{2} \ln \left (F \right )^{2}+3 e^{2}\right ) {\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{9 e^{4}+10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}+\frac {6 e \left (b^{2} c^{2} \ln \left (F \right )^{2}+3 e^{2}\right ) {\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{9 e^{4}+10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}-\frac {\ln \left (F \right ) b c \left (b^{2} c^{2} \ln \left (F \right )^{2}+7 e^{2}\right ) {\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{6}}{9 e^{4}+10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}-\frac {3 \ln \left (F \right ) b c \left (b^{2} c^{2} \ln \left (F \right )^{2}-e^{2}\right ) {\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{9 e^{4}+10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}+\frac {3 \ln \left (F \right ) b c \left (b^{2} c^{2} \ln \left (F \right )^{2}-e^{2}\right ) {\mathrm e}^{c \left (x b +a \right ) \ln \left (F \right )} \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{9 e^{4}+10 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+b^{4} c^{4} \ln \left (F \right )^{4}}}{\left (1+\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}\right )^{3}}\) | \(541\) |
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Time = 0.27 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.71 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \, dx=\frac {{\left (b^{3} c^{3} \cos \left (e x + d\right )^{3} \log \left (F\right )^{3} + {\left (b c e^{2} \cos \left (e x + d\right )^{3} + 6 \, b c e^{2} \cos \left (e x + d\right )\right )} \log \left (F\right ) + 3 \, {\left (b^{2} c^{2} e \cos \left (e x + d\right )^{2} \log \left (F\right )^{2} + e^{3} \cos \left (e x + d\right )^{2} + 2 \, e^{3}\right )} \sin \left (e x + d\right )\right )} F^{b c x + a c}}{b^{4} c^{4} \log \left (F\right )^{4} + 10 \, b^{2} c^{2} e^{2} \log \left (F\right )^{2} + 9 \, e^{4}} \]
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Result contains complex when optimal does not.
Time = 6.23 (sec) , antiderivative size = 1671, normalized size of antiderivative = 8.40 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 813 vs. \(2 (199) = 398\).
Time = 0.25 (sec) , antiderivative size = 813, normalized size of antiderivative = 4.09 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \, dx=\frac {{\left (F^{a c} b^{3} c^{3} \cos \left (3 \, d\right ) \log \left (F\right )^{3} + 3 \, F^{a c} b^{2} c^{2} e \log \left (F\right )^{2} \sin \left (3 \, d\right ) + F^{a c} b c e^{2} \cos \left (3 \, d\right ) \log \left (F\right ) + 3 \, F^{a c} e^{3} \sin \left (3 \, d\right )\right )} F^{b c x} \cos \left (3 \, e x\right ) + {\left (F^{a c} b^{3} c^{3} \cos \left (3 \, d\right ) \log \left (F\right )^{3} - 3 \, F^{a c} b^{2} c^{2} e \log \left (F\right )^{2} \sin \left (3 \, d\right ) + F^{a c} b c e^{2} \cos \left (3 \, d\right ) \log \left (F\right ) - 3 \, F^{a c} e^{3} \sin \left (3 \, d\right )\right )} F^{b c x} \cos \left (3 \, e x + 6 \, d\right ) + 3 \, {\left (F^{a c} b^{3} c^{3} \cos \left (3 \, d\right ) \log \left (F\right )^{3} - F^{a c} b^{2} c^{2} e \log \left (F\right )^{2} \sin \left (3 \, d\right ) + 9 \, F^{a c} b c e^{2} \cos \left (3 \, d\right ) \log \left (F\right ) - 9 \, F^{a c} e^{3} \sin \left (3 \, d\right )\right )} F^{b c x} \cos \left (e x + 4 \, d\right ) + 3 \, {\left (F^{a c} b^{3} c^{3} \cos \left (3 \, d\right ) \log \left (F\right )^{3} + F^{a c} b^{2} c^{2} e \log \left (F\right )^{2} \sin \left (3 \, d\right ) + 9 \, F^{a c} b c e^{2} \cos \left (3 \, d\right ) \log \left (F\right ) + 9 \, F^{a c} e^{3} \sin \left (3 \, d\right )\right )} F^{b c x} \cos \left (e x - 2 \, d\right ) - {\left (F^{a c} b^{3} c^{3} \log \left (F\right )^{3} \sin \left (3 \, d\right ) - 3 \, F^{a c} b^{2} c^{2} e \cos \left (3 \, d\right ) \log \left (F\right )^{2} + F^{a c} b c e^{2} \log \left (F\right ) \sin \left (3 \, d\right ) - 3 \, F^{a c} e^{3} \cos \left (3 \, d\right )\right )} F^{b c x} \sin \left (3 \, e x\right ) + {\left (F^{a c} b^{3} c^{3} \log \left (F\right )^{3} \sin \left (3 \, d\right ) + 3 \, F^{a c} b^{2} c^{2} e \cos \left (3 \, d\right ) \log \left (F\right )^{2} + F^{a c} b c e^{2} \log \left (F\right ) \sin \left (3 \, d\right ) + 3 \, F^{a c} e^{3} \cos \left (3 \, d\right )\right )} F^{b c x} \sin \left (3 \, e x + 6 \, d\right ) + 3 \, {\left (F^{a c} b^{3} c^{3} \log \left (F\right )^{3} \sin \left (3 \, d\right ) + F^{a c} b^{2} c^{2} e \cos \left (3 \, d\right ) \log \left (F\right )^{2} + 9 \, F^{a c} b c e^{2} \log \left (F\right ) \sin \left (3 \, d\right ) + 9 \, F^{a c} e^{3} \cos \left (3 \, d\right )\right )} F^{b c x} \sin \left (e x + 4 \, d\right ) - 3 \, {\left (F^{a c} b^{3} c^{3} \log \left (F\right )^{3} \sin \left (3 \, d\right ) - F^{a c} b^{2} c^{2} e \cos \left (3 \, d\right ) \log \left (F\right )^{2} + 9 \, F^{a c} b c e^{2} \log \left (F\right ) \sin \left (3 \, d\right ) - 9 \, F^{a c} e^{3} \cos \left (3 \, d\right )\right )} F^{b c x} \sin \left (e x - 2 \, d\right )}{8 \, {\left (b^{4} c^{4} \cos \left (3 \, d\right )^{2} \log \left (F\right )^{4} + b^{4} c^{4} \log \left (F\right )^{4} \sin \left (3 \, d\right )^{2} + 9 \, {\left (\cos \left (3 \, d\right )^{2} + \sin \left (3 \, d\right )^{2}\right )} e^{4} + 10 \, {\left (b^{2} c^{2} \cos \left (3 \, d\right )^{2} \log \left (F\right )^{2} + b^{2} c^{2} \log \left (F\right )^{2} \sin \left (3 \, d\right )^{2}\right )} e^{2}\right )}} \]
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Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 1271, normalized size of antiderivative = 6.39 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \, dx=\text {Too large to display} \]
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Time = 28.71 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.96 \[ \int F^{c (a+b x)} \cos ^3(d+e x) \, dx=-\frac {F^{c\,\left (a+b\,x\right )}\,\left (\cos \left (e\,x\right )+\sin \left (e\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (d\right )+\sin \left (d\right )\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{8\,\left (e-b\,c\,\ln \left (F\right )\,1{}\mathrm {i}\right )}-\frac {F^{c\,\left (a+b\,x\right )}\,\left (\cos \left (3\,e\,x\right )-\sin \left (3\,e\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (3\,d\right )-\sin \left (3\,d\right )\,1{}\mathrm {i}\right )}{8\,\left (-b\,c\,\ln \left (F\right )+e\,3{}\mathrm {i}\right )}-\frac {F^{c\,\left (a+b\,x\right )}\,\left (\cos \left (3\,e\,x\right )+\sin \left (3\,e\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (3\,d\right )+\sin \left (3\,d\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{8\,\left (3\,e-b\,c\,\ln \left (F\right )\,1{}\mathrm {i}\right )}-\frac {3\,F^{c\,\left (a+b\,x\right )}\,\left (\cos \left (e\,x\right )-\sin \left (e\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (d\right )-\sin \left (d\right )\,1{}\mathrm {i}\right )}{8\,\left (-b\,c\,\ln \left (F\right )+e\,1{}\mathrm {i}\right )} \]
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