Optimal. Leaf size=99 \[ \frac {f F^{a c+b c x}}{b c \log (F)}-\frac {e f F^{a c+b c x} \cos (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac {b c f F^{a c+b c x} \log (F) \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.11, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6873, 12, 6874,
2225, 4517} \begin {gather*} \frac {b c f \log (F) \sin (d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+e^2}-\frac {e f \cos (d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+e^2}+\frac {f F^{a c+b c x}}{b c \log (F)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2225
Rule 4517
Rule 6873
Rule 6874
Rubi steps
\begin {align*} \int F^{c (a+b x)} (f+f \sin (d+e x)) \, dx &=\int f F^{a c+b c x} (1+\sin (d+e x)) \, dx\\ &=f \int F^{a c+b c x} (1+\sin (d+e x)) \, dx\\ &=f \int \left (F^{a c+b c x}+F^{a c+b c x} \sin (d+e x)\right ) \, dx\\ &=f \int F^{a c+b c x} \, dx+f \int F^{a c+b c x} \sin (d+e x) \, dx\\ &=\frac {f F^{a c+b c x}}{b c \log (F)}-\frac {e f F^{a c+b c x} \cos (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac {b c f F^{a c+b c x} \log (F) \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.23, size = 83, normalized size = 0.84 \begin {gather*} \frac {f F^{c (a+b x)} \left (e^2-b c e \cos (d+e x) \log (F)+b^2 c^2 \log ^2(F)+b^2 c^2 \log ^2(F) \sin (d+e x)\right )}{b c \log (F) \left (e^2+b^2 c^2 \log ^2(F)\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 97, normalized size = 0.98
method | result | size |
risch | \(\frac {f \,F^{c \left (b x +a \right )}}{b c \ln \left (F \right )}-\frac {e \,F^{c \left (b x +a \right )} f \cos \left (e x +d \right )}{e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}+\frac {\ln \left (F \right ) c b \,F^{c \left (b x +a \right )} f \sin \left (e x +d \right )}{e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}\) | \(97\) |
norman | \(\frac {\frac {f \left (b^{2} c^{2} \ln \left (F \right )^{2}-\ln \left (F \right ) b c e +e^{2}\right ) {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right ) b c \ln \left (F \right )}+\frac {f \left (b^{2} c^{2} \ln \left (F \right )^{2}+\ln \left (F \right ) b c e +e^{2}\right ) {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )} \left (\tan ^{2}\left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{\left (e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right ) b c \ln \left (F \right )}+\frac {2 f b c \ln \left (F \right ) {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )} \tan \left (\frac {d}{2}+\frac {e x}{2}\right )}{e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}}{1+\tan ^{2}\left (\frac {d}{2}+\frac {e x}{2}\right )}\) | \(193\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 221 vs.
\(2 (100) = 200\).
time = 0.29, size = 221, normalized size = 2.23 \begin {gather*} -\frac {{\left ({\left (F^{a c} b c \log \left (F\right ) \sin \left (d\right ) + F^{a c} \cos \left (d\right ) e\right )} F^{b c x} \cos \left (x e + 2 \, d\right ) - {\left (F^{a c} b c \log \left (F\right ) \sin \left (d\right ) - F^{a c} \cos \left (d\right ) e\right )} F^{b c x} \cos \left (x e\right ) - {\left (F^{a c} b c \cos \left (d\right ) \log \left (F\right ) - F^{a c} e \sin \left (d\right )\right )} F^{b c x} \sin \left (x e + 2 \, d\right ) - {\left (F^{a c} b c \cos \left (d\right ) \log \left (F\right ) + F^{a c} e \sin \left (d\right )\right )} F^{b c x} \sin \left (x e\right )\right )} f}{2 \, {\left ({\left (b^{2} c^{2} \log \left (F\right )^{2} + e^{2}\right )} \cos \left (d\right )^{2} + {\left (b^{2} c^{2} \log \left (F\right )^{2} + e^{2}\right )} \sin \left (d\right )^{2}\right )}} + \frac {F^{b c x + a c} f}{b c \log \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.65, size = 84, normalized size = 0.85 \begin {gather*} \frac {{\left (b^{2} c^{2} f \log \left (F\right )^{2} \sin \left (x e + d\right ) + b^{2} c^{2} f \log \left (F\right )^{2} - b c f \cos \left (x e + d\right ) e \log \left (F\right ) + f e^{2}\right )} F^{b c x + a c}}{b^{3} c^{3} \log \left (F\right )^{3} + b c e^{2} \log \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 2.49, size = 920, normalized size = 9.29 \begin {gather*} \begin {cases} f x - \frac {f \cos {\left (d + e x \right )}}{e} & \text {for}\: F = 1 \\\frac {b^{2} c^{2} f \left (e^{- \frac {i e}{b c}}\right )^{a c} \left (e^{- \frac {i e}{b c}}\right )^{b c x} \log {\left (e^{- \frac {i e}{b c}} \right )}^{2} \sin {\left (d + e x \right )}}{b^{3} c^{3} \log {\left (e^{- \frac {i e}{b c}} \right )}^{3} + b c e^{2} \log {\left (e^{- \frac {i e}{b c}} \right )}} + \frac {b^{2} c^{2} f \left (e^{- \frac {i e}{b c}}\right )^{a c} \left (e^{- \frac {i e}{b c}}\right )^{b c x} \log {\left (e^{- \frac {i e}{b c}} \right )}^{2}}{b^{3} c^{3} \log {\left (e^{- \frac {i e}{b c}} \right )}^{3} + b c e^{2} \log {\left (e^{- \frac {i e}{b c}} \right )}} - \frac {b c e f \left (e^{- \frac {i e}{b c}}\right )^{a c} \left (e^{- \frac {i e}{b c}}\right )^{b c x} \log {\left (e^{- \frac {i e}{b c}} \right )} \cos {\left (d + e x \right )}}{b^{3} c^{3} \log {\left (e^{- \frac {i e}{b c}} \right )}^{3} + b c e^{2} \log {\left (e^{- \frac {i e}{b c}} \right )}} + \frac {e^{2} f \left (e^{- \frac {i e}{b c}}\right )^{a c} \left (e^{- \frac {i e}{b c}}\right )^{b c x}}{b^{3} c^{3} \log {\left (e^{- \frac {i e}{b c}} \right )}^{3} + b c e^{2} \log {\left (e^{- \frac {i e}{b c}} \right )}} & \text {for}\: F = e^{- \frac {i e}{b c}} \\\frac {b^{2} c^{2} f \left (e^{\frac {i e}{b c}}\right )^{a c} \left (e^{\frac {i e}{b c}}\right )^{b c x} \log {\left (e^{\frac {i e}{b c}} \right )}^{2} \sin {\left (d + e x \right )}}{b^{3} c^{3} \log {\left (e^{\frac {i e}{b c}} \right )}^{3} + b c e^{2} \log {\left (e^{\frac {i e}{b c}} \right )}} + \frac {b^{2} c^{2} f \left (e^{\frac {i e}{b c}}\right )^{a c} \left (e^{\frac {i e}{b c}}\right )^{b c x} \log {\left (e^{\frac {i e}{b c}} \right )}^{2}}{b^{3} c^{3} \log {\left (e^{\frac {i e}{b c}} \right )}^{3} + b c e^{2} \log {\left (e^{\frac {i e}{b c}} \right )}} - \frac {b c e f \left (e^{\frac {i e}{b c}}\right )^{a c} \left (e^{\frac {i e}{b c}}\right )^{b c x} \log {\left (e^{\frac {i e}{b c}} \right )} \cos {\left (d + e x \right )}}{b^{3} c^{3} \log {\left (e^{\frac {i e}{b c}} \right )}^{3} + b c e^{2} \log {\left (e^{\frac {i e}{b c}} \right )}} + \frac {e^{2} f \left (e^{\frac {i e}{b c}}\right )^{a c} \left (e^{\frac {i e}{b c}}\right )^{b c x}}{b^{3} c^{3} \log {\left (e^{\frac {i e}{b c}} \right )}^{3} + b c e^{2} \log {\left (e^{\frac {i e}{b c}} \right )}} & \text {for}\: F = e^{\frac {i e}{b c}} \\F^{a c} \left (f x - \frac {f \cos {\left (d + e x \right )}}{e}\right ) & \text {for}\: b = 0 \\f x - \frac {f \cos {\left (d + e x \right )}}{e} & \text {for}\: c = 0 \\\frac {F^{a c} F^{b c x} b^{2} c^{2} f \log {\left (F \right )}^{2} \sin {\left (d + e x \right )}}{b^{3} c^{3} \log {\left (F \right )}^{3} + b c e^{2} \log {\left (F \right )}} + \frac {F^{a c} F^{b c x} b^{2} c^{2} f \log {\left (F \right )}^{2}}{b^{3} c^{3} \log {\left (F \right )}^{3} + b c e^{2} \log {\left (F \right )}} - \frac {F^{a c} F^{b c x} b c e f \log {\left (F \right )} \cos {\left (d + e x \right )}}{b^{3} c^{3} \log {\left (F \right )}^{3} + b c e^{2} \log {\left (F \right )}} + \frac {F^{a c} F^{b c x} e^{2} f}{b^{3} c^{3} \log {\left (F \right )}^{3} + b c e^{2} \log {\left (F \right )}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [C] Result contains complex when optimal does not.
time = 0.45, size = 923, normalized size = 9.32 \begin {gather*} 2 \, {\left (\frac {2 \, b c f \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 )} f \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 )} + {\left (\frac {2 \, b c f \log \left ({\left | F \right |}\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 + e x + 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 + 2 \, e\right )}^{2}} - \frac {{\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c + 2 \, e\right )} f \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 + e x + 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 + 2 \, 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 f \log \left ({\left | F \right |}\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 - e x - 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 - 2 \, e\right )}^{2}} - \frac {{\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c - 2 \, e\right )} f \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 - e x - 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 - 2 \, 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 {i \, f 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 + i \, e x + i \, d\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 ) + 4 i \, e} - \frac {i \, f 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 - i \, e x - i \, d\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 ) - 4 i \, e}\right )} e^{\left (b c x \log \left ({\left | F \right |}\right ) + a c \log \left ({\left | F \right |}\right )\right )} - {\left (\frac {i \, f 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 - i \, e x - i \, d\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 ) - 4 i \, e} + \frac {i \, f 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 + i \, e x + i \, d\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 ) + 4 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 \, f 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 )}}{i \, \pi b c \mathrm {sgn}\left (F\right ) - i \, \pi b c + 2 \, b c \log \left ({\left | F \right |}\right )} - \frac {i \, f 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 )}}{-i \, \pi b c \mathrm {sgn}\left (F\right ) + i \, \pi b c + 2 \, 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.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.59, size = 84, normalized size = 0.85 \begin {gather*} \frac {F^{a\,c+b\,c\,x}\,f\,\left (e^2+b^2\,c^2\,{\ln \left (F\right )}^2+b^2\,c^2\,\sin \left (d+e\,x\right )\,{\ln \left (F\right )}^2-b\,c\,e\,\cos \left (d+e\,x\right )\,\ln \left (F\right )\right )}{b\,c\,\ln \left (F\right )\,\left (b^2\,c^2\,{\ln \left (F\right )}^2+e^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________