3.1.35 \(\int F^{c (a+b x)} (e \cos (d+e x)+b c \log (F) \sin (d+e x)) \, dx\) [35]

Optimal. Leaf size=16 \[ F^{c (a+b x)} \sin (d+e x) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2326} \begin {gather*} \sin (d+e x) F^{c (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

F^(c*(a + b*x))*Sin[d + e*x]

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {align*} \int F^{c (a+b x)} (e \cos (d+e x)+b c \log (F) \sin (d+e x)) \, dx &=F^{c (a+b x)} \sin (d+e x)\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 16, normalized size = 1.00 \begin {gather*} F^{c (a+b x)} \sin (d+e x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

F^(c*(a + b*x))*Sin[d + e*x]

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Maple [A]
time = 0.11, size = 17, normalized size = 1.06

method	result	size
risch	\(F^{c \left (b x +a \right )} \sin \left (e x +d \right )\)	\(17\)
norman	\(\frac {2 \,{\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )} \tan \left (\frac {d}{2}+\frac {e x}{2}\right )}{1+\tan ^{2}\left (\frac {d}{2}+\frac {e x}{2}\right )}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 399 vs. \(2 (17) = 34\).
time = 0.30, size = 399, normalized size = 24.94 \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 )} b c \log \left (F\right )}{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 {{\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 )} e}{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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [A]
time = 1.87, size = 18, normalized size = 1.12 \begin {gather*} F^{b c x + a c} \sin \left (x e + d\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

F^(b*c*x + a*c)*sin(x*e + d)

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Sympy [A]
time = 0.60, size = 17, normalized size = 1.06 \begin {gather*} F^{a c} F^{b c x} \sin {\left (d + e x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

F**(a*c)*F**(b*c*x)*sin(d + e*x)

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Giac [C] Result contains complex when optimal does not.
time = 0.44, size = 639, normalized size = 39.94 \begin {gather*} -i \, {\left (\frac {{\left (b c \log \left (F\right ) - i \, e\right )} 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 {{\left (b c \log \left (F\right ) - i \, e\right )} 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 {{\left (-i \, b c \log \left (F\right ) - e\right )} 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 {{\left (-i \, b c \log \left (F\right ) - e\right )} 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 {{\left (b c \log \left (F\right ) + i \, e\right )} 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 {{\left (b c \log \left (F\right ) + i \, e\right )} 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 {{\left (i \, b c \log \left (F\right ) - e\right )} 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 {{\left (i \, b c \log \left (F\right ) - e\right )} 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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*((b*c*log(F) - I*e)*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) - (b*c*log(F) - I*e)*e^(-1/2*I*pi*b*c*x*sg
n(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*b*c*log(F) - e)*e^(1/2*I*pi*b*c*x*sg
n(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*b*c*log(F) - e)*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(ab
s(F)) + a*c*log(abs(F))) + I*((b*c*log(F) + I*e)*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) - (b*c*log(F) +
I*e)*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*b*c*log(F)
- e)*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*b*c*log(F) - e)*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)))

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Mupad [B]
time = 2.35, size = 16, normalized size = 1.00 \begin {gather*} F^{c\,\left (a+b\,x\right )}\,\sin \left (d+e\,x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

F^(c*(a + b*x))*sin(d + e*x)

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