\(\int \frac {\sin ^2(a+b x+c x^2)}{d+e x} \, dx\) [36]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int \frac {\sin ^2\left (a+b x+c x^2\right )}{d+e x} \, dx=\frac {\log (d+e x)}{2 e}-\frac {1}{2} \text {Int}\left (\frac {\cos \left (2 a+2 b x+2 c x^2\right )}{d+e x},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 21, 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 = {} \[ \int \frac {\sin ^2\left (a+b x+c x^2\right )}{d+e x} \, dx=\int \frac {\sin ^2\left (a+b x+c x^2\right )}{d+e x} \, dx \]

[In]

Int[Sin[a + b*x + c*x^2]^2/(d + e*x),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 5.41 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\sin ^2\left (a+b x+c x^2\right )}{d+e x} \, dx=\int \frac {\sin ^2\left (a+b x+c x^2\right )}{d+e x} \, dx \]

[In]

Integrate[Sin[a + b*x + c*x^2]^2/(d + e*x),x]

[Out]

Integrate[Sin[a + b*x + c*x^2]^2/(d + e*x), x]

Maple [N/A] (verified)

Not integrable

Time = 0.34 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \frac {\sin ^2\left (a+b x+c x^2\right )}{d+e x} \, dx=\int { \frac {\sin \left (c x^{2} + b x + a\right )^{2}}{e x + d} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.78 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {\sin ^2\left (a+b x+c x^2\right )}{d+e x} \, dx=\int \frac {\sin ^{2}{\left (a + b x + c x^{2} \right )}}{d + e x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 118, normalized size of antiderivative = 5.62 \[ \int \frac {\sin ^2\left (a+b x+c x^2\right )}{d+e x} \, dx=\int { \frac {\sin \left (c x^{2} + b x + a\right )^{2}}{e x + d} \,d x } \]

[In]

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

[Out]

1/2*(2*e*integrate(-1/4*(cos(2*c*x^2 + 2*b*x)*cos(2*a) - sin(2*c*x^2 + 2*b*x)*sin(2*a))/((cos(2*a)^2 + sin(2*a
)^2)*e*x + (cos(2*a)^2 + sin(2*a)^2)*d), x) - 2*e*integrate(1/4*cos(2*c*x^2 + 2*b*x + 2*a)/(e*x + d), x) + log
(e*x + d))/e

Giac [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\sin ^2\left (a+b x+c x^2\right )}{d+e x} \, dx=\int { \frac {\sin \left (c x^{2} + b x + a\right )^{2}}{e x + d} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 5.71 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\sin ^2\left (a+b x+c x^2\right )}{d+e x} \, dx=\int \frac {{\sin \left (c\,x^2+b\,x+a\right )}^2}{d+e\,x} \,d x \]

[In]

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

[Out]

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