3.79 \(\int a^x \cos (x) \, dx\)

Optimal. Leaf size=31 \[ \frac {a^x \sin (x)}{\log ^2(a)+1}+\frac {a^x \log (a) \cos (x)}{\log ^2(a)+1} \]

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Rubi [A]  time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4433} \[ \frac {a^x \sin (x)}{\log ^2(a)+1}+\frac {a^x \log (a) \cos (x)}{\log ^2(a)+1} \]

Antiderivative was successfully verified.

[In]

Int[a^x*Cos[x],x]

[Out]

(a^x*Cos[x]*Log[a])/(1 + Log[a]^2) + (a^x*Sin[x])/(1 + Log[a]^2)

Rule 4433

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]

Rubi steps

\begin {align*} \int a^x \cos (x) \, dx &=\frac {a^x \cos (x) \log (a)}{1+\log ^2(a)}+\frac {a^x \sin (x)}{1+\log ^2(a)}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 20, normalized size = 0.65 \[ \frac {a^x (\log (a) \cos (x)+\sin (x))}{\log ^2(a)+1} \]

Antiderivative was successfully verified.

[In]

Integrate[a^x*Cos[x],x]

[Out]

(a^x*(Cos[x]*Log[a] + Sin[x]))/(1 + Log[a]^2)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int a^x \cos (x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[a^x*Cos[x],x]

[Out]

Could not integrate

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fricas [A]  time = 1.40, size = 20, normalized size = 0.65 \[ \frac {{\left (\cos \relax (x) \log \relax (a) + \sin \relax (x)\right )} a^{x}}{\log \relax (a)^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^x*cos(x),x, algorithm="fricas")

[Out]

(cos(x)*log(a) + sin(x))*a^x/(log(a)^2 + 1)

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giac [C]  time = 1.14, size = 329, normalized size = 10.61 \[ {\left | a \right |}^{x} {\left (\frac {2 \, \cos \left (\frac {1}{2} \, \pi x \mathrm {sgn}\relax (a) - \frac {1}{2} \, \pi x + x\right ) \log \left ({\left | a \right |}\right )}{{\left (\pi - \pi \mathrm {sgn}\relax (a) - 2\right )}^{2} + 4 \, \log \left ({\left | a \right |}\right )^{2}} - \frac {{\left (\pi - \pi \mathrm {sgn}\relax (a) - 2\right )} \sin \left (\frac {1}{2} \, \pi x \mathrm {sgn}\relax (a) - \frac {1}{2} \, \pi x + x\right )}{{\left (\pi - \pi \mathrm {sgn}\relax (a) - 2\right )}^{2} + 4 \, \log \left ({\left | a \right |}\right )^{2}}\right )} + {\left | a \right |}^{x} {\left (\frac {2 \, \cos \left (\frac {1}{2} \, \pi x \mathrm {sgn}\relax (a) - \frac {1}{2} \, \pi x - x\right ) \log \left ({\left | a \right |}\right )}{{\left (\pi - \pi \mathrm {sgn}\relax (a) + 2\right )}^{2} + 4 \, \log \left ({\left | a \right |}\right )^{2}} - \frac {{\left (\pi - \pi \mathrm {sgn}\relax (a) + 2\right )} \sin \left (\frac {1}{2} \, \pi x \mathrm {sgn}\relax (a) - \frac {1}{2} \, \pi x - x\right )}{{\left (\pi - \pi \mathrm {sgn}\relax (a) + 2\right )}^{2} + 4 \, \log \left ({\left | a \right |}\right )^{2}}\right )} - \frac {1}{2} i \, {\left | a \right |}^{x} {\left (-\frac {2 i \, e^{\left (\frac {1}{2} i \, \pi x \mathrm {sgn}\relax (a) - \frac {1}{2} i \, \pi x + i \, x\right )}}{-2 i \, \pi + 2 i \, \pi \mathrm {sgn}\relax (a) + 4 \, \log \left ({\left | a \right |}\right ) + 4 i} + \frac {2 i \, e^{\left (-\frac {1}{2} i \, \pi x \mathrm {sgn}\relax (a) + \frac {1}{2} i \, \pi x - i \, x\right )}}{2 i \, \pi - 2 i \, \pi \mathrm {sgn}\relax (a) + 4 \, \log \left ({\left | a \right |}\right ) - 4 i}\right )} - \frac {1}{2} i \, {\left | a \right |}^{x} {\left (-\frac {2 i \, e^{\left (\frac {1}{2} i \, \pi x \mathrm {sgn}\relax (a) - \frac {1}{2} i \, \pi x - i \, x\right )}}{-2 i \, \pi + 2 i \, \pi \mathrm {sgn}\relax (a) + 4 \, \log \left ({\left | a \right |}\right ) - 4 i} + \frac {2 i \, e^{\left (-\frac {1}{2} i \, \pi x \mathrm {sgn}\relax (a) + \frac {1}{2} i \, \pi x + i \, x\right )}}{2 i \, \pi - 2 i \, \pi \mathrm {sgn}\relax (a) + 4 \, \log \left ({\left | a \right |}\right ) + 4 i}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^x*cos(x),x, algorithm="giac")

[Out]

abs(a)^x*(2*cos(1/2*pi*x*sgn(a) - 1/2*pi*x + x)*log(abs(a))/((pi - pi*sgn(a) - 2)^2 + 4*log(abs(a))^2) - (pi -
pi*sgn(a) - 2)*sin(1/2*pi*x*sgn(a) - 1/2*pi*x + x)/((pi - pi*sgn(a) - 2)^2 + 4*log(abs(a))^2)) + abs(a)^x*(2*
cos(1/2*pi*x*sgn(a) - 1/2*pi*x - x)*log(abs(a))/((pi - pi*sgn(a) + 2)^2 + 4*log(abs(a))^2) - (pi - pi*sgn(a) +
2)*sin(1/2*pi*x*sgn(a) - 1/2*pi*x - x)/((pi - pi*sgn(a) + 2)^2 + 4*log(abs(a))^2)) - 1/2*I*abs(a)^x*(-2*I*e^(
1/2*I*pi*x*sgn(a) - 1/2*I*pi*x + I*x)/(-2*I*pi + 2*I*pi*sgn(a) + 4*log(abs(a)) + 4*I) + 2*I*e^(-1/2*I*pi*x*sgn
(a) + 1/2*I*pi*x - I*x)/(2*I*pi - 2*I*pi*sgn(a) + 4*log(abs(a)) - 4*I)) - 1/2*I*abs(a)^x*(-2*I*e^(1/2*I*pi*x*s
gn(a) - 1/2*I*pi*x - I*x)/(-2*I*pi + 2*I*pi*sgn(a) + 4*log(abs(a)) - 4*I) + 2*I*e^(-1/2*I*pi*x*sgn(a) + 1/2*I*
pi*x + I*x)/(2*I*pi - 2*I*pi*sgn(a) + 4*log(abs(a)) + 4*I))

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maple [A]  time = 0.06, size = 32, normalized size = 1.03

method	result	size
risch	\(\frac {a^{x} \cos \relax (x ) \ln \relax (a )}{1+\ln \relax (a )^{2}}+\frac {a^{x} \sin \relax (x )}{1+\ln \relax (a )^{2}}\)	\(32\)
norman	\(\frac {\frac {\ln \relax (a ) {\mathrm e}^{x \ln \relax (a )}}{1+\ln \relax (a )^{2}}+\frac {2 \,{\mathrm e}^{x \ln \relax (a )} \tan \left (\frac {x}{2}\right )}{1+\ln \relax (a )^{2}}-\frac {\ln \relax (a ) {\mathrm e}^{x \ln \relax (a )} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{1+\ln \relax (a )^{2}}}{1+\tan ^{2}\left (\frac {x}{2}\right )}\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a^x*cos(x),x,method=_RETURNVERBOSE)

[Out]

a^x*cos(x)*ln(a)/(1+ln(a)^2)+a^x*sin(x)/(1+ln(a)^2)

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maxima [A]  time = 0.44, size = 24, normalized size = 0.77 \[ \frac {a^{x} \cos \relax (x) \log \relax (a) + a^{x} \sin \relax (x)}{\log \relax (a)^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^x*cos(x),x, algorithm="maxima")

[Out]

(a^x*cos(x)*log(a) + a^x*sin(x))/(log(a)^2 + 1)

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mupad [B]  time = 0.03, size = 20, normalized size = 0.65 \[ \frac {a^x\,\left (\sin \relax (x)+\ln \relax (a)\,\cos \relax (x)\right )}{{\ln \relax (a)}^2+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a^x*cos(x),x)

[Out]

(a^x*(sin(x) + log(a)*cos(x)))/(log(a)^2 + 1)

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sympy [A]  time = 1.06, size = 107, normalized size = 3.45 \[ \begin {cases} \frac {i x e^{- i x} \sin {\relax (x )}}{2} + \frac {x e^{- i x} \cos {\relax (x )}}{2} + \frac {i e^{- i x} \cos {\relax (x )}}{2} & \text {for}\: a = e^{- i} \\- \frac {i x e^{i x} \sin {\relax (x )}}{2} + \frac {x e^{i x} \cos {\relax (x )}}{2} - \frac {i e^{i x} \cos {\relax (x )}}{2} & \text {for}\: a = e^{i} \\\frac {a^{x} \log {\relax (a )} \cos {\relax (x )}}{\log {\relax (a )}^{2} + 1} + \frac {a^{x} \sin {\relax (x )}}{\log {\relax (a )}^{2} + 1} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a**x*cos(x),x)

[Out]

Piecewise((I*x*exp(-I*x)*sin(x)/2 + x*exp(-I*x)*cos(x)/2 + I*exp(-I*x)*cos(x)/2, Eq(a, exp(-I))), (-I*x*exp(I*
x)*sin(x)/2 + x*exp(I*x)*cos(x)/2 - I*exp(I*x)*cos(x)/2, Eq(a, exp(I))), (a**x*log(a)*cos(x)/(log(a)**2 + 1) +
a**x*sin(x)/(log(a)**2 + 1), True))

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