3.49 \(\int \frac{1}{(a+b \sin (c+d x^2))^2} \, dx\)

Optimal. Leaf size=16 \[ \text{Unintegrable}\left (\frac{1}{\left (a+b \sin \left (c+d x^2\right )\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0065561, 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 = {} \[ \int \frac{1}{\left (a+b \sin \left (c+d x^2\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Sin[c + d*x^2])^(-2),x]

[Out]

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

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b \sin \left (c+d x^2\right )\right )^2} \, dx &=\int \frac{1}{\left (a+b \sin \left (c+d x^2\right )\right )^2} \, dx\\ \end{align*}

Mathematica [A]  time = 4.80739, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a+b \sin \left (c+d x^2\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.509, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\sin \left ( d{x}^{2}+c \right ) \right ) ^{-2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sin(d*x^2+c))^2,x)

[Out]

int(1/(a+b*sin(d*x^2+c))^2,x)

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(d*x^2+c))^2,x, algorithm="maxima")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(d*x^2+c))^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(d*x**2+c))**2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(d*x^2+c))^2,x, algorithm="giac")

[Out]

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