\(\int \frac {\sin ^2(a+b x-c x^2)}{x} \, dx\) [23]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {\sin ^2\left (a+b x-c x^2\right )}{x} \, dx=\frac {\log (x)}{2}-\frac {1}{2} \text {Int}\left (\frac {\cos \left (2 a+2 b x-2 c x^2\right )}{x},x\right ) \]

[Out]

1/2*ln(x)-1/2*Unintegrable(cos(-2*c*x^2+2*b*x+2*a)/x,x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sin ^2\left (a+b x-c x^2\right )}{x} \, dx=\int \frac {\sin ^2\left (a+b x-c x^2\right )}{x} \, dx \]

[In]

Int[Sin[a + b*x - c*x^2]^2/x,x]

[Out]

Log[x]/2 - Defer[Int][Cos[2*a + 2*b*x - 2*c*x^2]/x, x]/2

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

Mathematica [N/A]

Not integrable

Time = 4.71 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\sin ^2\left (a+b x-c x^2\right )}{x} \, dx=\int \frac {\sin ^2\left (a+b x-c x^2\right )}{x} \, dx \]

[In]

Integrate[Sin[a + b*x - c*x^2]^2/x,x]

[Out]

Integrate[Sin[a + b*x - c*x^2]^2/x, x]

Maple [N/A] (verified)

Not integrable

Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

\[\int \frac {\sin ^{2}\left (-c \,x^{2}+b x +a \right )}{x}d x\]

[In]

int(sin(-c*x^2+b*x+a)^2/x,x)

[Out]

int(sin(-c*x^2+b*x+a)^2/x,x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39 \[ \int \frac {\sin ^2\left (a+b x-c x^2\right )}{x} \, dx=\int { \frac {\sin \left (-c x^{2} + b x + a\right )^{2}}{x} \,d x } \]

[In]

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

[Out]

integral(-(cos(c*x^2 - b*x - a)^2 - 1)/x, x)

Sympy [N/A]

Not integrable

Time = 0.80 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {\sin ^2\left (a+b x-c x^2\right )}{x} \, dx=\int \frac {\sin ^{2}{\left (a + b x - c x^{2} \right )}}{x}\, dx \]

[In]

integrate(sin(-c*x**2+b*x+a)**2/x,x)

[Out]

Integral(sin(a + b*x - c*x**2)**2/x, x)

Maxima [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.56 \[ \int \frac {\sin ^2\left (a+b x-c x^2\right )}{x} \, dx=\int { \frac {\sin \left (-c x^{2} + b x + a\right )^{2}}{x} \,d x } \]

[In]

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

[Out]

-1/2*integrate(cos(2*c*x^2 - 2*b*x - 2*a)/x, x) + 1/2*log(x)

Giac [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\sin ^2\left (a+b x-c x^2\right )}{x} \, dx=\int { \frac {\sin \left (-c x^{2} + b x + a\right )^{2}}{x} \,d x } \]

[In]

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

[Out]

integrate(sin(-c*x^2 + b*x + a)^2/x, x)

Mupad [N/A]

Not integrable

Time = 5.48 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\sin ^2\left (a+b x-c x^2\right )}{x} \, dx=\int \frac {{\sin \left (-c\,x^2+b\,x+a\right )}^2}{x} \,d x \]

[In]

int(sin(a + b*x - c*x^2)^2/x,x)

[Out]

int(sin(a + b*x - c*x^2)^2/x, x)