\(\int (a+b \sin ^2(x)) \, dx\) [74]

Optimal result

Integrand size = 8, antiderivative size = 19 \[ \int \left (a+b \sin ^2(x)\right ) \, dx=a x+\frac {b x}{2}-\frac {1}{2} b \cos (x) \sin (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2715, 8} \[ \int \left (a+b \sin ^2(x)\right ) \, dx=a x+\frac {b x}{2}-\frac {1}{2} b \sin (x) \cos (x) \]

[In]

Int[a + b*Sin[x]^2,x]

[Out]

a*x + (b*x)/2 - (b*Cos[x]*Sin[x])/2

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rubi steps \begin{align*} \text {integral}& = a x+b \int \sin ^2(x) \, dx \\ & = a x-\frac {1}{2} b \cos (x) \sin (x)+\frac {1}{2} b \int 1 \, dx \\ & = a x+\frac {b x}{2}-\frac {1}{2} b \cos (x) \sin (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \left (a+b \sin ^2(x)\right ) \, dx=a x+\frac {b x}{2}-\frac {1}{4} b \sin (2 x) \]

[In]

Integrate[a + b*Sin[x]^2,x]

[Out]

a*x + (b*x)/2 - (b*Sin[2*x])/4

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84

method	result	size
risch	\(a x +\frac {b x}{2}-\frac {b \sin \left (2 x \right )}{4}\)	\(16\)
default	\(a x +b \left (-\frac {\cos \left (x \right ) \sin \left (x \right )}{2}+\frac {x}{2}\right )\)	\(17\)
parallelrisch	\(b \left (\frac {x}{2}-\frac {\sin \left (2 x \right )}{4}\right )+a x\)	\(17\)
parts	\(a x +b \left (-\frac {\cos \left (x \right ) \sin \left (x \right )}{2}+\frac {x}{2}\right )\)	\(17\)
norman	\(\frac {b \left (\tan ^{3}\left (\frac {x}{2}\right )\right )+\left (a +\frac {b}{2}\right ) x +\left (a +\frac {b}{2}\right ) x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )+\left (2 a +b \right ) x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-b \tan \left (\frac {x}{2}\right )}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{2}}\)	\(61\)

[In]

int(a+b*sin(x)^2,x,method=_RETURNVERBOSE)

[Out]

a*x+1/2*b*x-1/4*b*sin(2*x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \left (a+b \sin ^2(x)\right ) \, dx=-\frac {1}{2} \, b \cos \left (x\right ) \sin \left (x\right ) + \frac {1}{2} \, {\left (2 \, a + b\right )} x \]

[In]

integrate(a+b*sin(x)^2,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \left (a+b \sin ^2(x)\right ) \, dx=a x + b \left (\frac {x}{2} - \frac {\sin {\left (x \right )} \cos {\left (x \right )}}{2}\right ) \]

[In]

integrate(a+b*sin(x)**2,x)

[Out]

a*x + b*(x/2 - sin(x)*cos(x)/2)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \left (a+b \sin ^2(x)\right ) \, dx=\frac {1}{4} \, b {\left (2 \, x - \sin \left (2 \, x\right )\right )} + a x \]

[In]

integrate(a+b*sin(x)^2,x, algorithm="maxima")

[Out]

1/4*b*(2*x - sin(2*x)) + a*x

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \left (a+b \sin ^2(x)\right ) \, dx=\frac {1}{4} \, b {\left (2 \, x - \sin \left (2 \, x\right )\right )} + a x \]

[In]

integrate(a+b*sin(x)^2,x, algorithm="giac")

[Out]

1/4*b*(2*x - sin(2*x)) + a*x

Mupad [B] (verification not implemented)

Time = 13.52 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \left (a+b \sin ^2(x)\right ) \, dx=x\,\left (a+\frac {b}{2}\right )-\frac {b\,\sin \left (2\,x\right )}{4} \]

[In]

int(a + b*sin(x)^2,x)

[Out]

x*(a + b/2) - (b*sin(2*x))/4