Optimal. Leaf size=128 \[ \frac {2 e^2 F^{c (a+b x)}}{b c \log (F) \left (4 e^2+b^2 c^2 \log ^2(F)\right )}-\frac {2 e F^{c (a+b x)} \cos (d+e x) \sin (d+e x)}{4 e^2+b^2 c^2 \log ^2(F)}+\frac {b c F^{c (a+b x)} \log (F) \sin ^2(d+e x)}{4 e^2+b^2 c^2 \log ^2(F)} \]
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Rubi [A]
time = 0.03, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4519, 2225}
\begin {gather*} \frac {b c \log (F) \sin ^2(d+e x) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)+4 e^2}-\frac {2 e \sin (d+e x) \cos (d+e x) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)+4 e^2}+\frac {2 e^2 F^{c (a+b x)}}{b c \log (F) \left (b^2 c^2 \log ^2(F)+4 e^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2225
Rule 4519
Rubi steps
\begin {align*} \int F^{c (a+b x)} \sin ^2(d+e x) \, dx &=-\frac {2 e F^{c (a+b x)} \cos (d+e x) \sin (d+e x)}{4 e^2+b^2 c^2 \log ^2(F)}+\frac {b c F^{c (a+b x)} \log (F) \sin ^2(d+e x)}{4 e^2+b^2 c^2 \log ^2(F)}+\frac {\left (2 e^2\right ) \int F^{c (a+b x)} \, dx}{4 e^2+b^2 c^2 \log ^2(F)}\\ &=\frac {2 e^2 F^{c (a+b x)}}{b c \log (F) \left (4 e^2+b^2 c^2 \log ^2(F)\right )}-\frac {2 e F^{c (a+b x)} \cos (d+e x) \sin (d+e x)}{4 e^2+b^2 c^2 \log ^2(F)}+\frac {b c F^{c (a+b x)} \log (F) \sin ^2(d+e x)}{4 e^2+b^2 c^2 \log ^2(F)}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 86, normalized size = 0.67 \begin {gather*} \frac {F^{c (a+b x)} \left (4 e^2+b^2 c^2 \log ^2(F)-b^2 c^2 \cos (2 (d+e x)) \log ^2(F)-2 b c e \log (F) \sin (2 (d+e x))\right )}{8 b c e^2 \log (F)+2 b^3 c^3 \log ^3(F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 106, normalized size = 0.83
method | result | size |
risch | \(\frac {F^{c \left (b x +a \right )}}{2 b c \ln \left (F \right )}-\frac {\ln \left (F \right ) c b \,F^{c \left (b x +a \right )} \cos \left (2 e x +2 d \right )}{2 \left (4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}-\frac {e \,F^{c \left (b x +a \right )} \sin \left (2 e x +2 d \right )}{4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}\) | \(106\) |
norman | \(\frac {-\frac {4 e \,{\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )} \tan \left (\frac {d}{2}+\frac {e x}{2}\right )}{4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}+\frac {4 e \,{\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )} \left (\tan ^{3}\left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}+\frac {2 e^{2} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{b c \ln \left (F \right ) \left (4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}+\frac {2 e^{2} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )} \left (\tan ^{4}\left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{b c \ln \left (F \right ) \left (4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}+\frac {4 \left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right ) {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )} \left (\tan ^{2}\left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{b c \ln \left (F \right ) \left (4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right )}}{\left (1+\tan ^{2}\left (\frac {d}{2}+\frac {e x}{2}\right )\right )^{2}}\) | \(268\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 348 vs.
\(2 (128) = 256\).
time = 0.29, size = 348, normalized size = 2.72 \begin {gather*} -\frac {{\left (F^{a c} b^{2} c^{2} \cos \left (2 \, d\right ) \log \left (F\right )^{2} + 2 \, F^{a c} b c e \log \left (F\right ) \sin \left (2 \, d\right )\right )} F^{b c x} \cos \left (2 \, x e\right ) + {\left (F^{a c} b^{2} c^{2} \cos \left (2 \, d\right ) \log \left (F\right )^{2} - 2 \, F^{a c} b c e \log \left (F\right ) \sin \left (2 \, d\right )\right )} F^{b c x} \cos \left (2 \, x e + 4 \, d\right ) - {\left (F^{a c} b^{2} c^{2} \log \left (F\right )^{2} \sin \left (2 \, d\right ) - 2 \, F^{a c} b c \cos \left (2 \, d\right ) e \log \left (F\right )\right )} F^{b c x} \sin \left (2 \, x e\right ) + {\left (F^{a c} b^{2} c^{2} \log \left (F\right )^{2} \sin \left (2 \, d\right ) + 2 \, F^{a c} b c \cos \left (2 \, d\right ) e \log \left (F\right )\right )} F^{b c x} \sin \left (2 \, x e + 4 \, d\right ) - 2 \, {\left ({\left (F^{a c} b^{2} c^{2} \log \left (F\right )^{2} + 4 \, F^{a c} e^{2}\right )} \cos \left (2 \, d\right )^{2} + {\left (F^{a c} b^{2} c^{2} \log \left (F\right )^{2} + 4 \, F^{a c} e^{2}\right )} \sin \left (2 \, d\right )^{2}\right )} F^{b c x}}{4 \, {\left ({\left (b^{3} c^{3} \log \left (F\right )^{3} + 4 \, b c e^{2} \log \left (F\right )\right )} \cos \left (2 \, d\right )^{2} + {\left (b^{3} c^{3} \log \left (F\right )^{3} + 4 \, b c e^{2} \log \left (F\right )\right )} \sin \left (2 \, d\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.77, size = 91, normalized size = 0.71 \begin {gather*} -\frac {{\left (2 \, b c \cos \left (x e + d\right ) e \log \left (F\right ) \sin \left (x e + d\right ) + {\left (b^{2} c^{2} \cos \left (x e + d\right )^{2} - b^{2} c^{2}\right )} \log \left (F\right )^{2} - 2 \, e^{2}\right )} F^{b c x + a c}}{b^{3} c^{3} \log \left (F\right )^{3} + 4 \, b c e^{2} \log \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 15.47, size = 1117, normalized size = 8.73 \begin {gather*} \begin {cases} \frac {x \sin ^{2}{\left (d + e x \right )}}{2} + \frac {x \cos ^{2}{\left (d + e x \right )}}{2} - \frac {\sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{2 e} & \text {for}\: F = 1 \\\frac {b^{2} c^{2} \left (e^{- \frac {2 i e}{b c}}\right )^{a c} \left (e^{- \frac {2 i e}{b c}}\right )^{b c x} \log {\left (e^{- \frac {2 i e}{b c}} \right )}^{2} \sin ^{2}{\left (d + e x \right )}}{b^{3} c^{3} \log {\left (e^{- \frac {2 i e}{b c}} \right )}^{3} + 4 b c e^{2} \log {\left (e^{- \frac {2 i e}{b c}} \right )}} - \frac {2 b c e \left (e^{- \frac {2 i e}{b c}}\right )^{a c} \left (e^{- \frac {2 i e}{b c}}\right )^{b c x} \log {\left (e^{- \frac {2 i e}{b c}} \right )} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{b^{3} c^{3} \log {\left (e^{- \frac {2 i e}{b c}} \right )}^{3} + 4 b c e^{2} \log {\left (e^{- \frac {2 i e}{b c}} \right )}} + \frac {2 e^{2} \left (e^{- \frac {2 i e}{b c}}\right )^{a c} \left (e^{- \frac {2 i e}{b c}}\right )^{b c x} \sin ^{2}{\left (d + e x \right )}}{b^{3} c^{3} \log {\left (e^{- \frac {2 i e}{b c}} \right )}^{3} + 4 b c e^{2} \log {\left (e^{- \frac {2 i e}{b c}} \right )}} + \frac {2 e^{2} \left (e^{- \frac {2 i e}{b c}}\right )^{a c} \left (e^{- \frac {2 i e}{b c}}\right )^{b c x} \cos ^{2}{\left (d + e x \right )}}{b^{3} c^{3} \log {\left (e^{- \frac {2 i e}{b c}} \right )}^{3} + 4 b c e^{2} \log {\left (e^{- \frac {2 i e}{b c}} \right )}} & \text {for}\: F = e^{- \frac {2 i e}{b c}} \\\frac {b^{2} c^{2} \left (e^{\frac {2 i e}{b c}}\right )^{a c} \left (e^{\frac {2 i e}{b c}}\right )^{b c x} \log {\left (e^{\frac {2 i e}{b c}} \right )}^{2} \sin ^{2}{\left (d + e x \right )}}{b^{3} c^{3} \log {\left (e^{\frac {2 i e}{b c}} \right )}^{3} + 4 b c e^{2} \log {\left (e^{\frac {2 i e}{b c}} \right )}} - \frac {2 b c e \left (e^{\frac {2 i e}{b c}}\right )^{a c} \left (e^{\frac {2 i e}{b c}}\right )^{b c x} \log {\left (e^{\frac {2 i e}{b c}} \right )} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{b^{3} c^{3} \log {\left (e^{\frac {2 i e}{b c}} \right )}^{3} + 4 b c e^{2} \log {\left (e^{\frac {2 i e}{b c}} \right )}} + \frac {2 e^{2} \left (e^{\frac {2 i e}{b c}}\right )^{a c} \left (e^{\frac {2 i e}{b c}}\right )^{b c x} \sin ^{2}{\left (d + e x \right )}}{b^{3} c^{3} \log {\left (e^{\frac {2 i e}{b c}} \right )}^{3} + 4 b c e^{2} \log {\left (e^{\frac {2 i e}{b c}} \right )}} + \frac {2 e^{2} \left (e^{\frac {2 i e}{b c}}\right )^{a c} \left (e^{\frac {2 i e}{b c}}\right )^{b c x} \cos ^{2}{\left (d + e x \right )}}{b^{3} c^{3} \log {\left (e^{\frac {2 i e}{b c}} \right )}^{3} + 4 b c e^{2} \log {\left (e^{\frac {2 i e}{b c}} \right )}} & \text {for}\: F = e^{\frac {2 i e}{b c}} \\F^{a c} \left (\frac {x \sin ^{2}{\left (d + e x \right )}}{2} + \frac {x \cos ^{2}{\left (d + e x \right )}}{2} - \frac {\sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{2 e}\right ) & \text {for}\: b = 0 \\\frac {x \sin ^{2}{\left (d + e x \right )}}{2} + \frac {x \cos ^{2}{\left (d + e x \right )}}{2} - \frac {\sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{2 e} & \text {for}\: c = 0 \\\frac {F^{a c} F^{b c x} b^{2} c^{2} \log {\left (F \right )}^{2} \sin ^{2}{\left (d + e x \right )}}{b^{3} c^{3} \log {\left (F \right )}^{3} + 4 b c e^{2} \log {\left (F \right )}} - \frac {2 F^{a c} F^{b c x} b c e \log {\left (F \right )} \sin {\left (d + e x \right )} \cos {\left (d + e x \right )}}{b^{3} c^{3} \log {\left (F \right )}^{3} + 4 b c e^{2} \log {\left (F \right )}} + \frac {2 F^{a c} F^{b c x} e^{2} \sin ^{2}{\left (d + e x \right )}}{b^{3} c^{3} \log {\left (F \right )}^{3} + 4 b c e^{2} \log {\left (F \right )}} + \frac {2 F^{a c} F^{b c x} e^{2} \cos ^{2}{\left (d + e x \right )}}{b^{3} c^{3} \log {\left (F \right )}^{3} + 4 b c e^{2} \log {\left (F \right )}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.45, size = 915, normalized size = 7.15 \begin {gather*} -\frac {1}{2} \, {\left (\frac {2 \, b c \cos \left (\frac {1}{2} \, \pi b c x \mathrm {sgn}\left (F\right ) - \frac {1}{2} \, \pi b c x + \frac {1}{2} \, \pi a c \mathrm {sgn}\left (F\right ) - \frac {1}{2} \, \pi a c + 2 \, e x + 2 \, d\right ) \log \left ({\left | F \right |}\right )}{4 \, b^{2} c^{2} \log \left ({\left | F \right |}\right )^{2} + {\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c + 4 \, e\right )}^{2}} + \frac {{\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c + 4 \, e\right )} \sin \left (\frac {1}{2} \, \pi b c x \mathrm {sgn}\left (F\right ) - \frac {1}{2} \, \pi b c x + \frac {1}{2} \, \pi a c \mathrm {sgn}\left (F\right ) - \frac {1}{2} \, \pi a c + 2 \, e x + 2 \, d\right )}{4 \, b^{2} c^{2} \log \left ({\left | F \right |}\right )^{2} + {\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c + 4 \, e\right )}^{2}}\right )} e^{\left (b c x \log \left ({\left | F \right |}\right ) + a c \log \left ({\left | F \right |}\right )\right )} - \frac {1}{2} \, {\left (\frac {2 \, b c \cos \left (\frac {1}{2} \, \pi b c x \mathrm {sgn}\left (F\right ) - \frac {1}{2} \, \pi b c x + \frac {1}{2} \, \pi a c \mathrm {sgn}\left (F\right ) - \frac {1}{2} \, \pi a c - 2 \, e x - 2 \, d\right ) \log \left ({\left | F \right |}\right )}{4 \, b^{2} c^{2} \log \left ({\left | F \right |}\right )^{2} + {\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c - 4 \, e\right )}^{2}} + \frac {{\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c - 4 \, e\right )} \sin \left (\frac {1}{2} \, \pi b c x \mathrm {sgn}\left (F\right ) - \frac {1}{2} \, \pi b c x + \frac {1}{2} \, \pi a c \mathrm {sgn}\left (F\right ) - \frac {1}{2} \, \pi a c - 2 \, e x - 2 \, d\right )}{4 \, b^{2} c^{2} \log \left ({\left | F \right |}\right )^{2} + {\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c - 4 \, e\right )}^{2}}\right )} e^{\left (b c x \log \left ({\left | F \right |}\right ) + a c \log \left ({\left | F \right |}\right )\right )} + {\left (\frac {2 \, b c \cos \left (-\frac {1}{2} \, \pi b c x \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi b c x - \frac {1}{2} \, \pi a c \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi a c\right ) \log \left ({\left | F \right |}\right )}{4 \, b^{2} c^{2} \log \left ({\left | F \right |}\right )^{2} + {\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c\right )}^{2}} - \frac {{\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c\right )} \sin \left (-\frac {1}{2} \, \pi b c x \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi b c x - \frac {1}{2} \, \pi a c \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi a c\right )}{4 \, b^{2} c^{2} \log \left ({\left | F \right |}\right )^{2} + {\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c\right )}^{2}}\right )} e^{\left (b c x \log \left ({\left | F \right |}\right ) + a c \log \left ({\left | F \right |}\right )\right )} + i \, {\left (-\frac {i \, e^{\left (\frac {1}{2} i \, \pi b c x \mathrm {sgn}\left (F\right ) - \frac {1}{2} i \, \pi b c x + \frac {1}{2} i \, \pi a c \mathrm {sgn}\left (F\right ) - \frac {1}{2} i \, \pi a c + 2 i \, e x + 2 i \, d\right )}}{4 i \, \pi b c \mathrm {sgn}\left (F\right ) - 4 i \, \pi b c + 8 \, b c \log \left ({\left | F \right |}\right ) + 16 i \, e} + \frac {i \, e^{\left (-\frac {1}{2} i \, \pi b c x \mathrm {sgn}\left (F\right ) + \frac {1}{2} i \, \pi b c x - \frac {1}{2} i \, \pi a c \mathrm {sgn}\left (F\right ) + \frac {1}{2} i \, \pi a c - 2 i \, e x - 2 i \, d\right )}}{-4 i \, \pi b c \mathrm {sgn}\left (F\right ) + 4 i \, \pi b c + 8 \, b c \log \left ({\left | F \right |}\right ) - 16 i \, e}\right )} e^{\left (b c x \log \left ({\left | F \right |}\right ) + a c \log \left ({\left | F \right |}\right )\right )} + i \, {\left (-\frac {i \, e^{\left (\frac {1}{2} i \, \pi b c x \mathrm {sgn}\left (F\right ) - \frac {1}{2} i \, \pi b c x + \frac {1}{2} i \, \pi a c \mathrm {sgn}\left (F\right ) - \frac {1}{2} i \, \pi a c - 2 i \, e x - 2 i \, d\right )}}{4 i \, \pi b c \mathrm {sgn}\left (F\right ) - 4 i \, \pi b c + 8 \, b c \log \left ({\left | F \right |}\right ) - 16 i \, e} + \frac {i \, e^{\left (-\frac {1}{2} i \, \pi b c x \mathrm {sgn}\left (F\right ) + \frac {1}{2} i \, \pi b c x - \frac {1}{2} i \, \pi a c \mathrm {sgn}\left (F\right ) + \frac {1}{2} i \, \pi a c + 2 i \, e x + 2 i \, d\right )}}{-4 i \, \pi b c \mathrm {sgn}\left (F\right ) + 4 i \, \pi b c + 8 \, b c \log \left ({\left | F \right |}\right ) + 16 i \, e}\right )} e^{\left (b c x \log \left ({\left | F \right |}\right ) + a c \log \left ({\left | F \right |}\right )\right )} + i \, {\left (\frac {i \, e^{\left (\frac {1}{2} i \, \pi b c x \mathrm {sgn}\left (F\right ) - \frac {1}{2} i \, \pi b c x + \frac {1}{2} i \, \pi a c \mathrm {sgn}\left (F\right ) - \frac {1}{2} i \, \pi a c\right )}}{2 i \, \pi b c \mathrm {sgn}\left (F\right ) - 2 i \, \pi b c + 4 \, b c \log \left ({\left | F \right |}\right )} - \frac {i \, e^{\left (-\frac {1}{2} i \, \pi b c x \mathrm {sgn}\left (F\right ) + \frac {1}{2} i \, \pi b c x - \frac {1}{2} i \, \pi a c \mathrm {sgn}\left (F\right ) + \frac {1}{2} i \, \pi a c\right )}}{-2 i \, \pi b c \mathrm {sgn}\left (F\right ) + 2 i \, \pi b c + 4 \, b c \log \left ({\left | F \right |}\right )}\right )} e^{\left (b c x \log \left ({\left | F \right |}\right ) + a c \log \left ({\left | F \right |}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.02, size = 95, normalized size = 0.74 \begin {gather*} \frac {F^{a\,c+b\,c\,x}\,\left (2\,e^2+\frac {b^2\,c^2\,{\ln \left (F\right )}^2}{2}-\frac {b^2\,c^2\,{\ln \left (F\right )}^2\,\cos \left (2\,d+2\,e\,x\right )}{2}-b\,c\,e\,\ln \left (F\right )\,\sin \left (2\,d+2\,e\,x\right )\right )}{b\,c\,\ln \left (F\right )\,\left (b^2\,c^2\,{\ln \left (F\right )}^2+4\,e^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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