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

Optimal. Leaf size=43 \[ \frac{\log (d+e x)}{2 e}-\frac{1}{2} \text{Unintegrable}\left (\frac{\cos \left (2 a+2 b x+2 c x^2\right )}{d+e x},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.037116, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sin ^2\left (a+b x+c x^2\right )}{d+e x} \, dx \]

Verification is Not applicable to the result.

[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*} \int \frac{\sin ^2\left (a+b x+c x^2\right )}{d+e x} \, dx &=\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 [A]  time = 7.40887, size = 0, normalized size = 0. \[ \int \frac{\sin ^2\left (a+b x+c x^2\right )}{d+e x} \, dx \]

Verification is Not applicable to the result.

[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]

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Maple [A]  time = 0.23, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \sin \left ( c{x}^{2}+bx+a \right ) \right ) ^{2}}{ex+d}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{-\frac{1}{2} \, e \int \frac{\cos \left (2 \, c x^{2} + 2 \, b x\right ) \cos \left (2 \, a\right ) - \sin \left (2 \, c x^{2} + 2 \, b x\right ) \sin \left (2 \, a\right )}{{\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right )} e x +{\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right )} d}\,{d x} - \frac{1}{2} \, e \int \frac{\cos \left (2 \, c x^{2} + 2 \, b x + 2 \, a\right )}{e x + d}\,{d x} + \log \left (e x + d\right )}{2 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\cos \left (c x^{2} + b x + a\right )^{2} - 1}{e x + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin ^{2}{\left (a + b x + c x^{2} \right )}}{d + e x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (c x^{2} + b x + a\right )^{2}}{e x + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)