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

Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{\sin \left (a+b (c+d x)^2\right )}{(e+f x)^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0126098, 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 = {} \[ \int \frac{\sin \left (a+b (c+d x)^2\right )}{(e+f x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{\sin \left (a+b (c+d x)^2\right )}{(e+f x)^2} \, dx &=\int \frac{\sin \left (a+b (c+d x)^2\right )}{(e+f x)^2} \, dx\\ \end{align*}

Mathematica [A]  time = 25.9943, size = 0, normalized size = 0. \[ \int \frac{\sin \left (a+b (c+d x)^2\right )}{(e+f x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.178, size = 0, normalized size = 0. \begin{align*} \int{\frac{\sin \left ( a+ \left ( dx+c \right ) ^{2}b \right ) }{ \left ( fx+e \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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