3.2.39 \(\int F^{c (a+b x)} (f+f \cos (d+e x))^2 \, dx\) [139]
Optimal. Leaf size=245 \[ \frac {f^2 F^{a c+b c x}}{b c \log (F)}+\frac {2 b c f^2 F^{a c+b c x} \cos (d+e x) \log (F)}{e^2+b^2 c^2 \log ^2(F)}+\frac {2 e^2 f^2 F^{a c+b c x}}{b c \log (F) \left (4 e^2+b^2 c^2 \log ^2(F)\right )}+\frac {b c f^2 F^{a c+b c x} \cos ^2(d+e x) \log (F)}{4 e^2+b^2 c^2 \log ^2(F)}+\frac {2 e f^2 F^{a c+b c x} \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac {2 e f^2 F^{a c+b c x} \cos (d+e x) \sin (d+e x)}{4 e^2+b^2 c^2 \log ^2(F)} \]
[Out]
f^2*F^(b*c*x+a*c)/b/c/ln(F)+2*b*c*f^2*F^(b*c*x+a*c)*cos(e*x+d)*ln(F)/(e^2+b^2*c^2*ln(F)^2)+2*e^2*f^2*F^(b*c*x+
a*c)/b/c/ln(F)/(4*e^2+b^2*c^2*ln(F)^2)+b*c*f^2*F^(b*c*x+a*c)*cos(e*x+d)^2*ln(F)/(4*e^2+b^2*c^2*ln(F)^2)+2*e*f^
2*F^(b*c*x+a*c)*sin(e*x+d)/(e^2+b^2*c^2*ln(F)^2)+2*e*f^2*F^(b*c*x+a*c)*cos(e*x+d)*sin(e*x+d)/(4*e^2+b^2*c^2*ln
(F)^2)
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Rubi [A]
time = 0.23, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6873, 12, 6874,
2225, 4518, 4520} \begin {gather*} \frac {2 e f^2 \sin (d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+e^2}+\frac {b c f^2 \log (F) \cos ^2(d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+4 e^2}+\frac {2 b c f^2 \log (F) \cos (d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+e^2}+\frac {2 e f^2 \sin (d+e x) \cos (d+e x) F^{a c+b c x}}{b^2 c^2 \log ^2(F)+4 e^2}+\frac {2 e^2 f^2 F^{a c+b c x}}{b c \log (F) \left (b^2 c^2 \log ^2(F)+4 e^2\right )}+\frac {f^2 F^{a c+b c x}}{b c \log (F)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[F^(c*(a + b*x))*(f + f*Cos[d + e*x])^2,x]
[Out]
(f^2*F^(a*c + b*c*x))/(b*c*Log[F]) + (2*b*c*f^2*F^(a*c + b*c*x)*Cos[d + e*x]*Log[F])/(e^2 + b^2*c^2*Log[F]^2)
+ (2*e^2*f^2*F^(a*c + b*c*x))/(b*c*Log[F]*(4*e^2 + b^2*c^2*Log[F]^2)) + (b*c*f^2*F^(a*c + b*c*x)*Cos[d + e*x]^
2*Log[F])/(4*e^2 + b^2*c^2*Log[F]^2) + (2*e*f^2*F^(a*c + b*c*x)*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2) + (2*e*
f^2*F^(a*c + b*c*x)*Cos[d + e*x]*Sin[d + e*x])/(4*e^2 + b^2*c^2*Log[F]^2)
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 2225
Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]
Rule 4518
Int[Cos[(d_.) + (e_.)*(x_)]*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[b*c*Log[F]*F^(c*(a + b*x))*(C
os[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x] + Simp[e*F^(c*(a + b*x))*(Sin[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x]
/; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]
Rule 4520
Int[Cos[(d_.) + (e_.)*(x_)]^(m_)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[b*c*Log[F]*F^(c*(a + b*x
))*(Cos[d + e*x]^m/(e^2*m^2 + b^2*c^2*Log[F]^2)), x] + (Dist[(m*(m - 1)*e^2)/(e^2*m^2 + b^2*c^2*Log[F]^2), Int
[F^(c*(a + b*x))*Cos[d + e*x]^(m - 2), x], x] + Simp[e*m*F^(c*(a + b*x))*Sin[d + e*x]*(Cos[d + e*x]^(m - 1)/(e
^2*m^2 + b^2*c^2*Log[F]^2)), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*m^2 + b^2*c^2*Log[F]^2, 0] && GtQ[
m, 1]
Rule 6873
Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]
Rule 6874
Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]
Rubi steps
\begin {align*} \int F^{c (a+b x)} (f+f \cos (d+e x))^2 \, dx &=\int f^2 F^{a c+b c x} (1+\cos (d+e x))^2 \, dx\\ &=f^2 \int F^{a c+b c x} (1+\cos (d+e x))^2 \, dx\\ &=f^2 \int \left (F^{a c+b c x}+2 F^{a c+b c x} \cos (d+e x)+F^{a c+b c x} \cos ^2(d+e x)\right ) \, dx\\ &=f^2 \int F^{a c+b c x} \, dx+f^2 \int F^{a c+b c x} \cos ^2(d+e x) \, dx+\left (2 f^2\right ) \int F^{a c+b c x} \cos (d+e x) \, dx\\ &=\frac {f^2 F^{a c+b c x}}{b c \log (F)}+\frac {2 b c f^2 F^{a c+b c x} \cos (d+e x) \log (F)}{e^2+b^2 c^2 \log ^2(F)}+\frac {b c f^2 F^{a c+b c x} \cos ^2(d+e x) \log (F)}{4 e^2+b^2 c^2 \log ^2(F)}+\frac {2 e f^2 F^{a c+b c x} \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac {2 e f^2 F^{a c+b c x} \cos (d+e x) \sin (d+e x)}{4 e^2+b^2 c^2 \log ^2(F)}+\frac {\left (2 e^2 f^2\right ) \int F^{a c+b c x} \, dx}{4 e^2+b^2 c^2 \log ^2(F)}\\ &=\frac {f^2 F^{a c+b c x}}{b c \log (F)}+\frac {2 b c f^2 F^{a c+b c x} \cos (d+e x) \log (F)}{e^2+b^2 c^2 \log ^2(F)}+\frac {2 e^2 f^2 F^{a c+b c x}}{b c \log (F) \left (4 e^2+b^2 c^2 \log ^2(F)\right )}+\frac {b c f^2 F^{a c+b c x} \cos ^2(d+e x) \log (F)}{4 e^2+b^2 c^2 \log ^2(F)}+\frac {2 e f^2 F^{a c+b c x} \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac {2 e f^2 F^{a c+b c x} \cos (d+e x) \sin (d+e x)}{4 e^2+b^2 c^2 \log ^2(F)}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 228, normalized size = 0.93 \begin {gather*} \frac {f^2 F^{c (a+b x)} \left (12 e^4+15 b^2 c^2 e^2 \log ^2(F)+3 b^4 c^4 \log ^4(F)+b^2 c^2 \cos (2 (d+e x)) \log ^2(F) \left (e^2+b^2 c^2 \log ^2(F)\right )+4 b^2 c^2 \cos (d+e x) \log ^2(F) \left (4 e^2+b^2 c^2 \log ^2(F)\right )+16 b c e^3 \log (F) \sin (d+e x)+4 b^3 c^3 e \log ^3(F) \sin (d+e x)+2 b c e^3 \log (F) \sin (2 (d+e x))+2 b^3 c^3 e \log ^3(F) \sin (2 (d+e x))\right )}{2 \left (4 b c e^4 \log (F)+5 b^3 c^3 e^2 \log ^3(F)+b^5 c^5 \log ^5(F)\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[F^(c*(a + b*x))*(f + f*Cos[d + e*x])^2,x]
[Out]
(f^2*F^(c*(a + b*x))*(12*e^4 + 15*b^2*c^2*e^2*Log[F]^2 + 3*b^4*c^4*Log[F]^4 + b^2*c^2*Cos[2*(d + e*x)]*Log[F]^
2*(e^2 + b^2*c^2*Log[F]^2) + 4*b^2*c^2*Cos[d + e*x]*Log[F]^2*(4*e^2 + b^2*c^2*Log[F]^2) + 16*b*c*e^3*Log[F]*Si
n[d + e*x] + 4*b^3*c^3*e*Log[F]^3*Sin[d + e*x] + 2*b*c*e^3*Log[F]*Sin[2*(d + e*x)] + 2*b^3*c^3*e*Log[F]^3*Sin[
2*(d + e*x)]))/(2*(4*b*c*e^4*Log[F] + 5*b^3*c^3*e^2*Log[F]^3 + b^5*c^5*Log[F]^5))
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Maple [A]
time = 0.43, size = 274, normalized size = 1.12
| | |
| method |
result |
size |
| | |
| risch |
\(\frac {3 f^{2} F^{c \left (b x +a \right )}}{2 b c \ln \left (F \right )}+\frac {2 \ln \left (F \right ) c b \,f^{2} F^{c \left (b x +a \right )} \cos \left (e x +d \right )}{e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}+\frac {2 F^{c \left (b x +a \right )} e \,f^{2} \sin \left (e x +d \right )}{e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}+\frac {\ln \left (F \right ) c b \,f^{2} F^{c \left (b x +a \right )} \cos \left (2 e x +2 d \right )}{2 b^{2} c^{2} \ln \left (F \right )^{2}+8 e^{2}}+\frac {e \,f^{2} F^{c \left (b x +a \right )} \sin \left (2 e x +2 d \right )}{4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}\) |
\(193\) |
| default |
\(\frac {F^{a c} f^{2} \left (\frac {3 F^{b c x}}{b c \ln \left (F \right )}+\frac {\frac {8 e \,{\mathrm e}^{b c x \ln \left (F \right )} \tan \left (\frac {d}{2}+\frac {e x}{2}\right )}{e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}+\frac {4 b c \ln \left (F \right ) {\mathrm e}^{b c x \ln \left (F \right )}}{e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}-\frac {4 b c \ln \left (F \right ) {\mathrm e}^{b c x \ln \left (F \right )} \left (\tan ^{2}\left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}}{1+\tan ^{2}\left (\frac {d}{2}+\frac {e x}{2}\right )}+\frac {\frac {\ln \left (F \right ) b c \,{\mathrm e}^{b c x \ln \left (F \right )}}{4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}+\frac {4 e \,{\mathrm e}^{b c x \ln \left (F \right )} \tan \left (e x +d \right )}{4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}-\frac {\ln \left (F \right ) b c \,{\mathrm e}^{b c x \ln \left (F \right )} \left (\tan ^{2}\left (e x +d \right )\right )}{4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}}}{1+\tan ^{2}\left (e x +d \right )}\right )}{2}\) |
\(274\) |
| norman |
\(\frac {\frac {12 e^{3} f^{2} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )} \left (\tan ^{3}\left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{b^{4} c^{4} \ln \left (F \right )^{4}+5 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+4 e^{4}}+\frac {4 \left (2 b^{2} c^{2} \ln \left (F \right )^{2}+5 e^{2}\right ) e \,f^{2} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )} \tan \left (\frac {d}{2}+\frac {e x}{2}\right )}{b^{4} c^{4} \ln \left (F \right )^{4}+5 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+4 e^{4}}+\frac {2 f^{2} \left (2 b^{4} c^{4} \ln \left (F \right )^{4}+8 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+3 e^{4}\right ) {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{b c \ln \left (F \right ) \left (b^{4} c^{4} \ln \left (F \right )^{4}+5 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+4 e^{4}\right )}+\frac {6 e^{4} f^{2} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )} \left (\tan ^{4}\left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{b c \ln \left (F \right ) \left (b^{4} c^{4} \ln \left (F \right )^{4}+5 b^{2} c^{2} e^{2} \ln \left (F \right )^{2}+4 e^{4}\right )}+\frac {12 f^{2} e^{2} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )} \left (\tan ^{2}\left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{\left (4 e^{2}+b^{2} c^{2} \ln \left (F \right )^{2}\right ) b c \ln \left (F \right )}}{\left (1+\tan ^{2}\left (\frac {d}{2}+\frac {e x}{2}\right )\right )^{2}}\) |
\(381\) |
| | |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(F^(c*(b*x+a))*(f+f*cos(e*x+d))^2,x,method=_RETURNVERBOSE)
[Out]
1/2*F^(a*c)*f^2*(3*F^(b*c*x)/b/c/ln(F)+(8/(e^2+b^2*c^2*ln(F)^2)*e*exp(b*c*x*ln(F))*tan(1/2*d+1/2*e*x)+4*b*c*ln
(F)/(e^2+b^2*c^2*ln(F)^2)*exp(b*c*x*ln(F))-4*b*c*ln(F)/(e^2+b^2*c^2*ln(F)^2)*exp(b*c*x*ln(F))*tan(1/2*d+1/2*e*
x)^2)/(1+tan(1/2*d+1/2*e*x)^2)+(1/(4*e^2+b^2*c^2*ln(F)^2)*ln(F)*b*c*exp(b*c*x*ln(F))+4/(4*e^2+b^2*c^2*ln(F)^2)
*e*exp(b*c*x*ln(F))*tan(e*x+d)-1/(4*e^2+b^2*c^2*ln(F)^2)*ln(F)*b*c*exp(b*c*x*ln(F))*tan(e*x+d)^2)/(1+tan(e*x+d
)^2))
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 573 vs.
\(2 (246) = 492\).
time = 0.30, size = 573, normalized size = 2.34 \begin {gather*} \frac {{\left ({\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}\right )} f^{2}}{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 )}} + \frac {{\left ({\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} \cos \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} \cos \left (x e\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} \sin \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} \sin \left (x e\right )\right )} f^{2}}{{\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}} + \frac {F^{b c x + a c} f^{2}}{b c \log \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(c*(b*x+a))*(f+f*cos(e*x+d))^2,x, algorithm="maxima")
[Out]
1/4*((F^(a*c)*b^2*c^2*cos(2*d)*log(F)^2 + 2*F^(a*c)*b*c*e*log(F)*sin(2*d))*F^(b*c*x)*cos(2*x*e) + (F^(a*c)*b^2
*c^2*cos(2*d)*log(F)^2 - 2*F^(a*c)*b*c*e*log(F)*sin(2*d))*F^(b*c*x)*cos(2*x*e + 4*d) - (F^(a*c)*b^2*c^2*log(F)
^2*sin(2*d) - 2*F^(a*c)*b*c*cos(2*d)*e*log(F))*F^(b*c*x)*sin(2*x*e) + (F^(a*c)*b^2*c^2*log(F)^2*sin(2*d) + 2*F
^(a*c)*b*c*cos(2*d)*e*log(F))*F^(b*c*x)*sin(2*x*e + 4*d) + 2*((F^(a*c)*b^2*c^2*log(F)^2 + 4*F^(a*c)*e^2)*cos(2
*d)^2 + (F^(a*c)*b^2*c^2*log(F)^2 + 4*F^(a*c)*e^2)*sin(2*d)^2)*F^(b*c*x))*f^2/((b^3*c^3*log(F)^3 + 4*b*c*e^2*l
og(F))*cos(2*d)^2 + (b^3*c^3*log(F)^3 + 4*b*c*e^2*log(F))*sin(2*d)^2) + ((F^(a*c)*b*c*cos(d)*log(F) - F^(a*c)*
e*sin(d))*F^(b*c*x)*cos(x*e + 2*d) + (F^(a*c)*b*c*cos(d)*log(F) + F^(a*c)*e*sin(d))*F^(b*c*x)*cos(x*e) + (F^(a
*c)*b*c*log(F)*sin(d) + F^(a*c)*cos(d)*e)*F^(b*c*x)*sin(x*e + 2*d) - (F^(a*c)*b*c*log(F)*sin(d) - F^(a*c)*cos(
d)*e)*F^(b*c*x)*sin(x*e))*f^2/((b^2*c^2*log(F)^2 + e^2)*cos(d)^2 + (b^2*c^2*log(F)^2 + e^2)*sin(d)^2) + F^(b*c
*x + a*c)*f^2/(b*c*log(F))
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Fricas [A]
time = 2.39, size = 243, normalized size = 0.99 \begin {gather*} \frac {{\left ({\left (b^{4} c^{4} f^{2} \cos \left (x e + d\right )^{2} + 2 \, b^{4} c^{4} f^{2} \cos \left (x e + d\right ) + b^{4} c^{4} f^{2}\right )} \log \left (F\right )^{4} + 6 \, f^{2} e^{4} + {\left (b^{2} c^{2} f^{2} \cos \left (x e + d\right )^{2} e^{2} + 8 \, b^{2} c^{2} f^{2} \cos \left (x e + d\right ) e^{2} + 7 \, b^{2} c^{2} f^{2} e^{2}\right )} \log \left (F\right )^{2} + 2 \, {\left ({\left (b^{3} c^{3} f^{2} \cos \left (x e + d\right ) e + b^{3} c^{3} f^{2} e\right )} \log \left (F\right )^{3} + {\left (b c f^{2} \cos \left (x e + d\right ) e^{3} + 4 \, b c f^{2} e^{3}\right )} \log \left (F\right )\right )} \sin \left (x e + d\right )\right )} F^{b c x + a c}}{b^{5} c^{5} \log \left (F\right )^{5} + 5 \, b^{3} c^{3} e^{2} \log \left (F\right )^{3} + 4 \, b c e^{4} \log \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(c*(b*x+a))*(f+f*cos(e*x+d))^2,x, algorithm="fricas")
[Out]
((b^4*c^4*f^2*cos(x*e + d)^2 + 2*b^4*c^4*f^2*cos(x*e + d) + b^4*c^4*f^2)*log(F)^4 + 6*f^2*e^4 + (b^2*c^2*f^2*c
os(x*e + d)^2*e^2 + 8*b^2*c^2*f^2*cos(x*e + d)*e^2 + 7*b^2*c^2*f^2*e^2)*log(F)^2 + 2*((b^3*c^3*f^2*cos(x*e + d
)*e + b^3*c^3*f^2*e)*log(F)^3 + (b*c*f^2*cos(x*e + d)*e^3 + 4*b*c*f^2*e^3)*log(F))*sin(x*e + d))*F^(b*c*x + a*
c)/(b^5*c^5*log(F)^5 + 5*b^3*c^3*e^2*log(F)^3 + 4*b*c*e^4*log(F))
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 23.62, size = 8277, normalized size = 33.78 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F**(c*(b*x+a))*(f+f*cos(e*x+d))**2,x)
[Out]
Piecewise((f**2*x*sin(d + e*x)**2/2 + f**2*x*cos(d + e*x)**2/2 + f**2*x + f**2*sin(d + e*x)*cos(d + e*x)/(2*e)
+ 2*f**2*sin(d + e*x)/e, Eq(F, 1)), (b**4*c**4*f**2*exp(-2*I*e/(b*c))**(a*c)*exp(-2*I*e/(b*c))**(b*c*x)*log(e
xp(-2*I*e/(b*c)))**4*cos(d + e*x)**2/(b**5*c**5*log(exp(-2*I*e/(b*c)))**5 + 5*b**3*c**3*e**2*log(exp(-2*I*e/(b
*c)))**3 + 4*b*c*e**4*log(exp(-2*I*e/(b*c)))) + 2*b**4*c**4*f**2*exp(-2*I*e/(b*c))**(a*c)*exp(-2*I*e/(b*c))**(
b*c*x)*log(exp(-2*I*e/(b*c)))**4*cos(d + e*x)/(b**5*c**5*log(exp(-2*I*e/(b*c)))**5 + 5*b**3*c**3*e**2*log(exp(
-2*I*e/(b*c)))**3 + 4*b*c*e**4*log(exp(-2*I*e/(b*c)))) + b**4*c**4*f**2*exp(-2*I*e/(b*c))**(a*c)*exp(-2*I*e/(b
*c))**(b*c*x)*log(exp(-2*I*e/(b*c)))**4/(b**5*c**5*log(exp(-2*I*e/(b*c)))**5 + 5*b**3*c**3*e**2*log(exp(-2*I*e
/(b*c)))**3 + 4*b*c*e**4*log(exp(-2*I*e/(b*c)))) + 2*b**3*c**3*e*f**2*exp(-2*I*e/(b*c))**(a*c)*exp(-2*I*e/(b*c
))**(b*c*x)*log(exp(-2*I*e/(b*c)))**3*sin(d + e*x)*cos(d + e*x)/(b**5*c**5*log(exp(-2*I*e/(b*c)))**5 + 5*b**3*
c**3*e**2*log(exp(-2*I*e/(b*c)))**3 + 4*b*c*e**4*log(exp(-2*I*e/(b*c)))) + 2*b**3*c**3*e*f**2*exp(-2*I*e/(b*c)
)**(a*c)*exp(-2*I*e/(b*c))**(b*c*x)*log(exp(-2*I*e/(b*c)))**3*sin(d + e*x)/(b**5*c**5*log(exp(-2*I*e/(b*c)))**
5 + 5*b**3*c**3*e**2*log(exp(-2*I*e/(b*c)))**3 + 4*b*c*e**4*log(exp(-2*I*e/(b*c)))) + 2*b**2*c**2*e**2*f**2*ex
p(-2*I*e/(b*c))**(a*c)*exp(-2*I*e/(b*c))**(b*c*x)*log(exp(-2*I*e/(b*c)))**2*sin(d + e*x)**2/(b**5*c**5*log(exp
(-2*I*e/(b*c)))**5 + 5*b**3*c**3*e**2*log(exp(-2*I*e/(b*c)))**3 + 4*b*c*e**4*log(exp(-2*I*e/(b*c)))) + 3*b**2*
c**2*e**2*f**2*exp(-2*I*e/(b*c))**(a*c)*exp(-2*I*e/(b*c))**(b*c*x)*log(exp(-2*I*e/(b*c)))**2*cos(d + e*x)**2/(
b**5*c**5*log(exp(-2*I*e/(b*c)))**5 + 5*b**3*c**3*e**2*log(exp(-2*I*e/(b*c)))**3 + 4*b*c*e**4*log(exp(-2*I*e/(
b*c)))) + 8*b**2*c**2*e**2*f**2*exp(-2*I*e/(b*c))**(a*c)*exp(-2*I*e/(b*c))**(b*c*x)*log(exp(-2*I*e/(b*c)))**2*
cos(d + e*x)/(b**5*c**5*log(exp(-2*I*e/(b*c)))**5 + 5*b**3*c**3*e**2*log(exp(-2*I*e/(b*c)))**3 + 4*b*c*e**4*lo
g(exp(-2*I*e/(b*c)))) + 5*b**2*c**2*e**2*f**2*exp(-2*I*e/(b*c))**(a*c)*exp(-2*I*e/(b*c))**(b*c*x)*log(exp(-2*I
*e/(b*c)))**2/(b**5*c**5*log(exp(-2*I*e/(b*c)))**5 + 5*b**3*c**3*e**2*log(exp(-2*I*e/(b*c)))**3 + 4*b*c*e**4*l
og(exp(-2*I*e/(b*c)))) + 2*b*c*e**3*f**2*exp(-2*I*e/(b*c))**(a*c)*exp(-2*I*e/(b*c))**(b*c*x)*log(exp(-2*I*e/(b
*c)))*sin(d + e*x)*cos(d + e*x)/(b**5*c**5*log(exp(-2*I*e/(b*c)))**5 + 5*b**3*c**3*e**2*log(exp(-2*I*e/(b*c)))
**3 + 4*b*c*e**4*log(exp(-2*I*e/(b*c)))) + 8*b*c*e**3*f**2*exp(-2*I*e/(b*c))**(a*c)*exp(-2*I*e/(b*c))**(b*c*x)
*log(exp(-2*I*e/(b*c)))*sin(d + e*x)/(b**5*c**5*log(exp(-2*I*e/(b*c)))**5 + 5*b**3*c**3*e**2*log(exp(-2*I*e/(b
*c)))**3 + 4*b*c*e**4*log(exp(-2*I*e/(b*c)))) + 2*e**4*f**2*exp(-2*I*e/(b*c))**(a*c)*exp(-2*I*e/(b*c))**(b*c*x
)*sin(d + e*x)**2/(b**5*c**5*log(exp(-2*I*e/(b*c)))**5 + 5*b**3*c**3*e**2*log(exp(-2*I*e/(b*c)))**3 + 4*b*c*e*
*4*log(exp(-2*I*e/(b*c)))) + 2*e**4*f**2*exp(-2*I*e/(b*c))**(a*c)*exp(-2*I*e/(b*c))**(b*c*x)*cos(d + e*x)**2/(
b**5*c**5*log(exp(-2*I*e/(b*c)))**5 + 5*b**3*c**3*e**2*log(exp(-2*I*e/(b*c)))**3 + 4*b*c*e**4*log(exp(-2*I*e/(
b*c)))) + 4*e**4*f**2*exp(-2*I*e/(b*c))**(a*c)*exp(-2*I*e/(b*c))**(b*c*x)/(b**5*c**5*log(exp(-2*I*e/(b*c)))**5
+ 5*b**3*c**3*e**2*log(exp(-2*I*e/(b*c)))**3 + 4*b*c*e**4*log(exp(-2*I*e/(b*c)))), Eq(F, exp(-2*I*e/(b*c)))),
(b**4*c**4*f**2*exp(-I*e/(b*c))**(a*c)*exp(-I*e/(b*c))**(b*c*x)*log(exp(-I*e/(b*c)))**4*cos(d + e*x)**2/(b**5
*c**5*log(exp(-I*e/(b*c)))**5 + 5*b**3*c**3*e**2*log(exp(-I*e/(b*c)))**3 + 4*b*c*e**4*log(exp(-I*e/(b*c)))) +
2*b**4*c**4*f**2*exp(-I*e/(b*c))**(a*c)*exp(-I*e/(b*c))**(b*c*x)*log(exp(-I*e/(b*c)))**4*cos(d + e*x)/(b**5*c*
*5*log(exp(-I*e/(b*c)))**5 + 5*b**3*c**3*e**2*log(exp(-I*e/(b*c)))**3 + 4*b*c*e**4*log(exp(-I*e/(b*c)))) + b**
4*c**4*f**2*exp(-I*e/(b*c))**(a*c)*exp(-I*e/(b*c))**(b*c*x)*log(exp(-I*e/(b*c)))**4/(b**5*c**5*log(exp(-I*e/(b
*c)))**5 + 5*b**3*c**3*e**2*log(exp(-I*e/(b*c)))**3 + 4*b*c*e**4*log(exp(-I*e/(b*c)))) + 2*b**3*c**3*e*f**2*ex
p(-I*e/(b*c))**(a*c)*exp(-I*e/(b*c))**(b*c*x)*log(exp(-I*e/(b*c)))**3*sin(d + e*x)*cos(d + e*x)/(b**5*c**5*log
(exp(-I*e/(b*c)))**5 + 5*b**3*c**3*e**2*log(exp(-I*e/(b*c)))**3 + 4*b*c*e**4*log(exp(-I*e/(b*c)))) + 2*b**3*c*
*3*e*f**2*exp(-I*e/(b*c))**(a*c)*exp(-I*e/(b*c))**(b*c*x)*log(exp(-I*e/(b*c)))**3*sin(d + e*x)/(b**5*c**5*log(
exp(-I*e/(b*c)))**5 + 5*b**3*c**3*e**2*log(exp(-I*e/(b*c)))**3 + 4*b*c*e**4*log(exp(-I*e/(b*c)))) + 2*b**2*c**
2*e**2*f**2*exp(-I*e/(b*c))**(a*c)*exp(-I*e/(b*c))**(b*c*x)*log(exp(-I*e/(b*c)))**2*sin(d + e*x)**2/(b**5*c**5
*log(exp(-I*e/(b*c)))**5 + 5*b**3*c**3*e**2*log(exp(-I*e/(b*c)))**3 + 4*b*c*e**4*log(exp(-I*e/(b*c)))) + 3*b**
2*c**2*e**2*f**2*exp(-I*e/(b*c))**(a*c)*exp(-I*e/(b*c))**(b*c*x)*log(exp(-I*e/(b*c)))**2*cos(d + e*x)**2/(b**5
*c**5*log(exp(-I*e/(b*c)))**5 + 5*b**3*c**3*e**2*log(exp(-I*e/(b*c)))**3 + 4*b*c*e**4*log(exp(-I*e/(b*c)))) +
8*b**2*c**2*e**2*f**2*exp(-I*e/(b*c))**(a*c)*exp(-I*e/(b*c))**(b*c*x)*log(exp(-I*e/(b*c)))**2*cos(d + e*x)/(b*
*5*c**5*log(exp(-I*e/(b*c)))**5 + 5*b**3*c**3*e...
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Giac [C] Result contains complex when optimal does not.
time = 0.50, size = 1736, normalized size = 7.09 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(c*(b*x+a))*(f+f*cos(e*x+d))^2,x, algorithm="giac")
[Out]
1/2*(2*b*c*f^2*cos(1/2*pi*b*c*x*sgn(F) - 1/2*pi*b*c*x + 1/2*pi*a*c*sgn(F) - 1/2*pi*a*c + 2*e*x + 2*d)*log(abs(
F))/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c + 4*e)^2) + (pi*b*c*sgn(F) - pi*b*c + 4*e)*f^2*sin(1/2*
pi*b*c*x*sgn(F) - 1/2*pi*b*c*x + 1/2*pi*a*c*sgn(F) - 1/2*pi*a*c + 2*e*x + 2*d)/(4*b^2*c^2*log(abs(F))^2 + (pi*
b*c*sgn(F) - pi*b*c + 4*e)^2))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) + 2*(2*b*c*f^2*cos(1/2*pi*b*c*x*sgn(F)
- 1/2*pi*b*c*x + 1/2*pi*a*c*sgn(F) - 1/2*pi*a*c + e*x + d)*log(abs(F))/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(
F) - pi*b*c + 2*e)^2) + (pi*b*c*sgn(F) - pi*b*c + 2*e)*f^2*sin(1/2*pi*b*c*x*sgn(F) - 1/2*pi*b*c*x + 1/2*pi*a*c
*sgn(F) - 1/2*pi*a*c + e*x + d)/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c + 2*e)^2))*e^(b*c*x*log(abs
(F)) + a*c*log(abs(F))) + 2*(2*b*c*f^2*cos(1/2*pi*b*c*x*sgn(F) - 1/2*pi*b*c*x + 1/2*pi*a*c*sgn(F) - 1/2*pi*a*c
- e*x - d)*log(abs(F))/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c - 2*e)^2) + (pi*b*c*sgn(F) - pi*b*c
- 2*e)*f^2*sin(1/2*pi*b*c*x*sgn(F) - 1/2*pi*b*c*x + 1/2*pi*a*c*sgn(F) - 1/2*pi*a*c - e*x - d)/(4*b^2*c^2*log(
abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c - 2*e)^2))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) + 1/2*(2*b*c*f^2*cos(1/
2*pi*b*c*x*sgn(F) - 1/2*pi*b*c*x + 1/2*pi*a*c*sgn(F) - 1/2*pi*a*c - 2*e*x - 2*d)*log(abs(F))/(4*b^2*c^2*log(ab
s(F))^2 + (pi*b*c*sgn(F) - pi*b*c - 4*e)^2) + (pi*b*c*sgn(F) - pi*b*c - 4*e)*f^2*sin(1/2*pi*b*c*x*sgn(F) - 1/2
*pi*b*c*x + 1/2*pi*a*c*sgn(F) - 1/2*pi*a*c - 2*e*x - 2*d)/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c -
4*e)^2))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) + 3*(2*b*c*f^2*cos(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2
*pi*a*c*sgn(F) + 1/2*pi*a*c)*log(abs(F))/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c)^2) - (pi*b*c*sgn(
F) - pi*b*c)*f^2*sin(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*pi*a*c*sgn(F) + 1/2*pi*a*c)/(4*b^2*c^2*log(abs(
F))^2 + (pi*b*c*sgn(F) - pi*b*c)^2))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) + I*(I*f^2*e^(1/2*I*pi*b*c*x*sgn(
F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a*c*sgn(F) - 1/2*I*pi*a*c + 2*I*e*x + 2*I*d)/(4*I*pi*b*c*sgn(F) - 4*I*pi*b*c +
8*b*c*log(abs(F)) + 16*I*e) - I*f^2*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*pi*a*c*sgn(F) + 1/2*I*p
i*a*c - 2*I*e*x - 2*I*d)/(-4*I*pi*b*c*sgn(F) + 4*I*pi*b*c + 8*b*c*log(abs(F)) - 16*I*e))*e^(b*c*x*log(abs(F))
+ a*c*log(abs(F))) + I*(I*f^2*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a*c*sgn(F) - 1/2*I*pi*a*c +
I*e*x + I*d)/(I*pi*b*c*sgn(F) - I*pi*b*c + 2*b*c*log(abs(F)) + 2*I*e) - I*f^2*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2
*I*pi*b*c*x - 1/2*I*pi*a*c*sgn(F) + 1/2*I*pi*a*c - I*e*x - I*d)/(-I*pi*b*c*sgn(F) + I*pi*b*c + 2*b*c*log(abs(F
)) - 2*I*e))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) + I*(I*f^2*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/
2*I*pi*a*c*sgn(F) - 1/2*I*pi*a*c - I*e*x - I*d)/(I*pi*b*c*sgn(F) - I*pi*b*c + 2*b*c*log(abs(F)) - 2*I*e) - I*f
^2*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*pi*a*c*sgn(F) + 1/2*I*pi*a*c + I*e*x + I*d)/(-I*pi*b*c*s
gn(F) + I*pi*b*c + 2*b*c*log(abs(F)) + 2*I*e))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) + I*(I*f^2*e^(1/2*I*pi*
b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a*c*sgn(F) - 1/2*I*pi*a*c - 2*I*e*x - 2*I*d)/(4*I*pi*b*c*sgn(F) - 4*I
*pi*b*c + 8*b*c*log(abs(F)) - 16*I*e) - I*f^2*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*pi*a*c*sgn(F)
+ 1/2*I*pi*a*c + 2*I*e*x + 2*I*d)/(-4*I*pi*b*c*sgn(F) + 4*I*pi*b*c + 8*b*c*log(abs(F)) + 16*I*e))*e^(b*c*x*lo
g(abs(F)) + a*c*log(abs(F))) + I*(I*f^2*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a*c*sgn(F) - 1/2*
I*pi*a*c)/(2*I*pi*b*c*sgn(F) - 2*I*pi*b*c + 4*b*c*log(abs(F))) - I*f^2*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*
c*x - 1/2*I*pi*a*c*sgn(F) + 1/2*I*pi*a*c)/(-2*I*pi*b*c*sgn(F) + 2*I*pi*b*c + 4*b*c*log(abs(F))))*e^(b*c*x*log(
abs(F)) + a*c*log(abs(F))) + I*(I*f^2*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a*c*sgn(F) - 1/2*I*
pi*a*c)/(I*pi*b*c*sgn(F) - I*pi*b*c + 2*b*c*log(abs(F))) - I*f^2*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x -
1/2*I*pi*a*c*sgn(F) + 1/2*I*pi*a*c)/(-I*pi*b*c*sgn(F) + I*pi*b*c + 2*b*c*log(abs(F))))*e^(b*c*x*log(abs(F)) +
a*c*log(abs(F)))
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Mupad [B]
time = 3.31, size = 247, normalized size = 1.01 \begin {gather*} \frac {F^{a\,c+b\,c\,x}\,f^2\,\left (6\,e^4+\frac {3\,b^4\,c^4\,{\ln \left (F\right )}^4}{2}+2\,b^4\,c^4\,\cos \left (d+e\,x\right )\,{\ln \left (F\right )}^4+\frac {b^4\,c^4\,{\ln \left (F\right )}^4\,\cos \left (2\,d+2\,e\,x\right )}{2}+\frac {15\,b^2\,c^2\,e^2\,{\ln \left (F\right )}^2}{2}+8\,b\,c\,e^3\,\sin \left (d+e\,x\right )\,\ln \left (F\right )+8\,b^2\,c^2\,e^2\,\cos \left (d+e\,x\right )\,{\ln \left (F\right )}^2+b^3\,c^3\,e\,{\ln \left (F\right )}^3\,\sin \left (2\,d+2\,e\,x\right )+b\,c\,e^3\,\ln \left (F\right )\,\sin \left (2\,d+2\,e\,x\right )+\frac {b^2\,c^2\,e^2\,{\ln \left (F\right )}^2\,\cos \left (2\,d+2\,e\,x\right )}{2}+2\,b^3\,c^3\,e\,\sin \left (d+e\,x\right )\,{\ln \left (F\right )}^3\right )}{b\,c\,\ln \left (F\right )\,\left (b^4\,c^4\,{\ln \left (F\right )}^4+5\,b^2\,c^2\,e^2\,{\ln \left (F\right )}^2+4\,e^4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(F^(c*(a + b*x))*(f + f*cos(d + e*x))^2,x)
[Out]
(F^(a*c + b*c*x)*f^2*(6*e^4 + (3*b^4*c^4*log(F)^4)/2 + 2*b^4*c^4*cos(d + e*x)*log(F)^4 + (b^4*c^4*log(F)^4*cos
(2*d + 2*e*x))/2 + (15*b^2*c^2*e^2*log(F)^2)/2 + 8*b*c*e^3*sin(d + e*x)*log(F) + 8*b^2*c^2*e^2*cos(d + e*x)*lo
g(F)^2 + b^3*c^3*e*log(F)^3*sin(2*d + 2*e*x) + b*c*e^3*log(F)*sin(2*d + 2*e*x) + (b^2*c^2*e^2*log(F)^2*cos(2*d
+ 2*e*x))/2 + 2*b^3*c^3*e*sin(d + e*x)*log(F)^3))/(b*c*log(F)*(4*e^4 + b^4*c^4*log(F)^4 + 5*b^2*c^2*e^2*log(F
)^2))
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