3.2.36 \(\int F^{c (a+b x)} (f+f \sin (d+e x)) \, dx\) [136]

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]

f*F^(b*c*x+a*c)/b/c/ln(F)-e*f*F^(b*c*x+a*c)*cos(e*x+d)/(e^2+b^2*c^2*ln(F)^2)+b*c*f*F^(b*c*x+a*c)*ln(F)*sin(e*x
+d)/(e^2+b^2*c^2*ln(F)^2)

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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]

Int[F^(c*(a + b*x))*(f + f*Sin[d + e*x]),x]

[Out]

(f*F^(a*c + b*c*x))/(b*c*Log[F]) - (e*f*F^(a*c + b*c*x)*Cos[d + e*x])/(e^2 + b^2*c^2*Log[F]^2) + (b*c*f*F^(a*c
+ b*c*x)*Log[F]*Sin[d + e*x])/(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 4517

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)], x_Symbol] :> Simp[b*c*Log[F]*F^(c*(a + b*x))*(S
in[d + e*x]/(e^2 + b^2*c^2*Log[F]^2)), x] - Simp[e*F^(c*(a + b*x))*(Cos[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 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 \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*}

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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]

Integrate[F^(c*(a + b*x))*(f + f*Sin[d + e*x]),x]

[Out]

(f*F^(c*(a + b*x))*(e^2 - b*c*e*Cos[d + e*x]*Log[F] + b^2*c^2*Log[F]^2 + b^2*c^2*Log[F]^2*Sin[d + e*x]))/(b*c*
Log[F]*(e^2 + b^2*c^2*Log[F]^2))

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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]

int(F^(c*(b*x+a))*(f+f*sin(e*x+d)),x,method=_RETURNVERBOSE)

[Out]

1/b/c/ln(F)*f*F^(c*(b*x+a))-e*F^(c*(b*x+a))*f/(e^2+b^2*c^2*ln(F)^2)*cos(e*x+d)+ln(F)*c*b*F^(c*(b*x+a))*f/(e^2+
b^2*c^2*ln(F)^2)*sin(e*x+d)

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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]

integrate(F^(c*(b*x+a))*(f+f*sin(e*x+d)),x, algorithm="maxima")

[Out]

-1/2*((F^(a*c)*b*c*log(F)*sin(d) + F^(a*c)*cos(d)*e)*F^(b*c*x)*cos(x*e + 2*d) - (F^(a*c)*b*c*log(F)*sin(d) - F
^(a*c)*cos(d)*e)*F^(b*c*x)*cos(x*e) - (F^(a*c)*b*c*cos(d)*log(F) - F^(a*c)*e*sin(d))*F^(b*c*x)*sin(x*e + 2*d)
- (F^(a*c)*b*c*cos(d)*log(F) + F^(a*c)*e*sin(d))*F^(b*c*x)*sin(x*e))*f/((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/(b*c*log(F))

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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]

integrate(F^(c*(b*x+a))*(f+f*sin(e*x+d)),x, algorithm="fricas")

[Out]

(b^2*c^2*f*log(F)^2*sin(x*e + d) + b^2*c^2*f*log(F)^2 - b*c*f*cos(x*e + d)*e*log(F) + f*e^2)*F^(b*c*x + a*c)/(
b^3*c^3*log(F)^3 + b*c*e^2*log(F))

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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]

integrate(F**(c*(b*x+a))*(f+f*sin(e*x+d)),x)

[Out]

Piecewise((f*x - f*cos(d + e*x)/e, Eq(F, 1)), (b**2*c**2*f*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)/(b**3*c**3*log(exp(-I*e/(b*c)))**3 + b*c*e**2*log(exp(-I*e/(b*c)))) + b**2*c
**2*f*exp(-I*e/(b*c))**(a*c)*exp(-I*e/(b*c))**(b*c*x)*log(exp(-I*e/(b*c)))**2/(b**3*c**3*log(exp(-I*e/(b*c)))*
*3 + b*c*e**2*log(exp(-I*e/(b*c)))) - b*c*e*f*exp(-I*e/(b*c))**(a*c)*exp(-I*e/(b*c))**(b*c*x)*log(exp(-I*e/(b*
c)))*cos(d + e*x)/(b**3*c**3*log(exp(-I*e/(b*c)))**3 + b*c*e**2*log(exp(-I*e/(b*c)))) + e**2*f*exp(-I*e/(b*c))
**(a*c)*exp(-I*e/(b*c))**(b*c*x)/(b**3*c**3*log(exp(-I*e/(b*c)))**3 + b*c*e**2*log(exp(-I*e/(b*c)))), Eq(F, ex
p(-I*e/(b*c)))), (b**2*c**2*f*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
)/(b**3*c**3*log(exp(I*e/(b*c)))**3 + b*c*e**2*log(exp(I*e/(b*c)))) + b**2*c**2*f*exp(I*e/(b*c))**(a*c)*exp(I*
e/(b*c))**(b*c*x)*log(exp(I*e/(b*c)))**2/(b**3*c**3*log(exp(I*e/(b*c)))**3 + b*c*e**2*log(exp(I*e/(b*c)))) - b
*c*e*f*exp(I*e/(b*c))**(a*c)*exp(I*e/(b*c))**(b*c*x)*log(exp(I*e/(b*c)))*cos(d + e*x)/(b**3*c**3*log(exp(I*e/(
b*c)))**3 + b*c*e**2*log(exp(I*e/(b*c)))) + e**2*f*exp(I*e/(b*c))**(a*c)*exp(I*e/(b*c))**(b*c*x)/(b**3*c**3*lo
g(exp(I*e/(b*c)))**3 + b*c*e**2*log(exp(I*e/(b*c)))), Eq(F, exp(I*e/(b*c)))), (F**(a*c)*(f*x - f*cos(d + e*x)/
e), Eq(b, 0)), (f*x - f*cos(d + e*x)/e, Eq(c, 0)), (F**(a*c)*F**(b*c*x)*b**2*c**2*f*log(F)**2*sin(d + e*x)/(b*
*3*c**3*log(F)**3 + b*c*e**2*log(F)) + F**(a*c)*F**(b*c*x)*b**2*c**2*f*log(F)**2/(b**3*c**3*log(F)**3 + b*c*e*
*2*log(F)) - F**(a*c)*F**(b*c*x)*b*c*e*f*log(F)*cos(d + e*x)/(b**3*c**3*log(F)**3 + b*c*e**2*log(F)) + F**(a*c
)*F**(b*c*x)*e**2*f/(b**3*c**3*log(F)**3 + b*c*e**2*log(F)), True))

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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]

integrate(F^(c*(b*x+a))*(f+f*sin(e*x+d)),x, algorithm="giac")

[Out]

2*(2*b*c*f*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*lo
g(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c)^2) - (pi*b*c*sgn(F) - pi*b*c)*f*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))) + (2*b*c*f*log(abs(F))*sin(1/2*pi*b*c*x*sgn(F) - 1/2*pi*b*c*x + 1/2*pi*a*c*sgn(F) - 1/2*p
i*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) - (pi*b*c*sgn(F) - pi*b*c + 2*e)
*f*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)/(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*b*c*f*log(abs(F))*sin(1/2*p
i*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*s
gn(F) - pi*b*c - 2*e)^2) - (pi*b*c*sgn(F) - pi*b*c - 2*e)*f*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)/(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(ab
s(F)) + a*c*log(abs(F))) - (-I*f*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)/(2*I*pi*b*c*sgn(F) - 2*I*pi*b*c + 4*b*c*log(abs(F)) + 4*I*e) - I*f*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)/(-2*I*pi*b*c*sgn(F) + 2*I*pi*b*c + 4*b*c*
log(abs(F)) - 4*I*e))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) - (I*f*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)/(2*I*pi*b*c*sgn(F) - 2*I*pi*b*c + 4*b*c*log(abs(F)) - 4*I
*e) + I*f*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)/(-2*I
*pi*b*c*sgn(F) + 2*I*pi*b*c + 4*b*c*log(abs(F)) + 4*I*e))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) + I*(I*f*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*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 = 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]

int(F^(c*(a + b*x))*(f + f*sin(d + e*x)),x)

[Out]

(F^(a*c + b*c*x)*f*(e^2 + b^2*c^2*log(F)^2 + b^2*c^2*sin(d + e*x)*log(F)^2 - b*c*e*cos(d + e*x)*log(F)))/(b*c*
log(F)*(e^2 + b^2*c^2*log(F)^2))

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