3.25 \(\int \sin (\log (a+b x)) \, dx\)

Optimal. Leaf size=39 \[ \frac{(a+b x) \sin (\log (a+b x))}{2 b}-\frac{(a+b x) \cos (\log (a+b x))}{2 b} \]

[Out]

-((a + b*x)*Cos[Log[a + b*x]])/(2*b) + ((a + b*x)*Sin[Log[a + b*x]])/(2*b)

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Rubi [A]  time = 0.0142934, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4475} \[ \frac{(a+b x) \sin (\log (a+b x))}{2 b}-\frac{(a+b x) \cos (\log (a+b x))}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[Sin[Log[a + b*x]],x]

[Out]

-((a + b*x)*Cos[Log[a + b*x]])/(2*b) + ((a + b*x)*Sin[Log[a + b*x]])/(2*b)

Rule 4475

Int[Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)], x_Symbol] :> Simp[(x*Sin[d*(a + b*Log[c*x^n])])/(b^2*d^2
*n^2 + 1), x] - Simp[(b*d*n*x*Cos[d*(a + b*Log[c*x^n])])/(b^2*d^2*n^2 + 1), x] /; FreeQ[{a, b, c, d, n}, x] &&
NeQ[b^2*d^2*n^2 + 1, 0]

Rubi steps

\begin{align*} \int \sin (\log (a+b x)) \, dx &=\frac{\operatorname{Subst}(\int \sin (\log (x)) \, dx,x,a+b x)}{b}\\ &=-\frac{(a+b x) \cos (\log (a+b x))}{2 b}+\frac{(a+b x) \sin (\log (a+b x))}{2 b}\\ \end{align*}

Mathematica [A]  time = 0.0155208, size = 29, normalized size = 0.74 \[ -\frac{(a+b x) (\cos (\log (a+b x))-\sin (\log (a+b x)))}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[Log[a + b*x]],x]

[Out]

-((a + b*x)*(Cos[Log[a + b*x]] - Sin[Log[a + b*x]]))/(2*b)

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Maple [B]  time = 0.024, size = 76, normalized size = 2. \begin{align*}{ \left ( x\tan \left ({\frac{\ln \left ( bx+a \right ) }{2}} \right ) +{\frac{a}{b}\tan \left ({\frac{\ln \left ( bx+a \right ) }{2}} \right ) }+{\frac{a}{b} \left ( \tan \left ({\frac{\ln \left ( bx+a \right ) }{2}} \right ) \right ) ^{2}}-{\frac{x}{2}}+{\frac{x}{2} \left ( \tan \left ({\frac{\ln \left ( bx+a \right ) }{2}} \right ) \right ) ^{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{\ln \left ( bx+a \right ) }{2}} \right ) \right ) ^{2} \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(ln(b*x+a)),x)

[Out]

(x*tan(1/2*ln(b*x+a))+a/b*tan(1/2*ln(b*x+a))+a/b*tan(1/2*ln(b*x+a))^2-1/2*x+1/2*x*tan(1/2*ln(b*x+a))^2)/(1+tan
(1/2*ln(b*x+a))^2)

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Maxima [A]  time = 1.11677, size = 36, normalized size = 0.92 \begin{align*} -\frac{{\left (b x + a\right )}{\left (\cos \left (\log \left (b x + a\right )\right ) - \sin \left (\log \left (b x + a\right )\right )\right )}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(log(b*x+a)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.483304, size = 92, normalized size = 2.36 \begin{align*} -\frac{{\left (b x + a\right )} \cos \left (\log \left (b x + a\right )\right ) -{\left (b x + a\right )} \sin \left (\log \left (b x + a\right )\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(log(b*x+a)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.01977, size = 56, normalized size = 1.44 \begin{align*} \begin{cases} \frac{a \sin{\left (\log{\left (a + b x \right )} \right )}}{2 b} - \frac{a \cos{\left (\log{\left (a + b x \right )} \right )}}{2 b} + \frac{x \sin{\left (\log{\left (a + b x \right )} \right )}}{2} - \frac{x \cos{\left (\log{\left (a + b x \right )} \right )}}{2} & \text{for}\: b \neq 0 \\x \sin{\left (\log{\left (a \right )} \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(ln(b*x+a)),x)

[Out]

Piecewise((a*sin(log(a + b*x))/(2*b) - a*cos(log(a + b*x))/(2*b) + x*sin(log(a + b*x))/2 - x*cos(log(a + b*x))
/2, Ne(b, 0)), (x*sin(log(a)), True))

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Giac [A]  time = 1.14293, size = 47, normalized size = 1.21 \begin{align*} -\frac{{\left (b x + a\right )} \cos \left (\log \left (b x + a\right )\right )}{2 \, b} + \frac{{\left (b x + a\right )} \sin \left (\log \left (b x + a\right )\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(log(b*x+a)),x, algorithm="giac")

[Out]

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