3.286 \(\int \frac{\sec ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx\)

Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{\sec ^3(c+d x)}{(e+f x)^2 (a \sin (c+d x)+a)},x\right ) \]

[Out]

Unintegrable[Sec[c + d*x]^3/((e + f*x)^2*(a + a*Sin[c + d*x])), x]

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Rubi [A]  time = 0.0721515, 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 = {} \[ \int \frac{\sec ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sec[c + d*x]^3/((e + f*x)^2*(a + a*Sin[c + d*x])),x]

[Out]

Defer[Int][Sec[c + d*x]^3/((e + f*x)^2*(a + a*Sin[c + d*x])), x]

Rubi steps

\begin{align*} \int \frac{\sec ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx &=\int \frac{\sec ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx\\ \end{align*}

Mathematica [A]  time = 51.9916, size = 0, normalized size = 0. \[ \int \frac{\sec ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sec[c + d*x]^3/((e + f*x)^2*(a + a*Sin[c + d*x])),x]

[Out]

Integrate[Sec[c + d*x]^3/((e + f*x)^2*(a + a*Sin[c + d*x])), x]

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Maple [A]  time = 1.749, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{ \left ( fx+e \right ) ^{2} \left ( a+a\sin \left ( dx+c \right ) \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^3/(f*x+e)^2/(a+a*sin(d*x+c)),x)

[Out]

int(sec(d*x+c)^3/(f*x+e)^2/(a+a*sin(d*x+c)),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(sec(d*x+c)^3/(f*x+e)^2/(a+a*sin(d*x+c)),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{\sec \left (d x + c\right )^{3}}{a f^{2} x^{2} + 2 \, a e f x + a e^{2} +{\left (a f^{2} x^{2} + 2 \, a e f x + a e^{2}\right )} \sin \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3/(f*x+e)^2/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

integral(sec(d*x + c)^3/(a*f^2*x^2 + 2*a*e*f*x + a*e^2 + (a*f^2*x^2 + 2*a*e*f*x + a*e^2)*sin(d*x + c)), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\sec ^{3}{\left (c + d x \right )}}{e^{2} \sin{\left (c + d x \right )} + e^{2} + 2 e f x \sin{\left (c + d x \right )} + 2 e f x + f^{2} x^{2} \sin{\left (c + d x \right )} + f^{2} x^{2}}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**3/(f*x+e)**2/(a+a*sin(d*x+c)),x)

[Out]

Integral(sec(c + d*x)**3/(e**2*sin(c + d*x) + e**2 + 2*e*f*x*sin(c + d*x) + 2*e*f*x + f**2*x**2*sin(c + d*x) +
f**2*x**2), x)/a

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (d x + c\right )^{3}}{{\left (f x + e\right )}^{2}{\left (a \sin \left (d x + c\right ) + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3/(f*x+e)^2/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

integrate(sec(d*x + c)^3/((f*x + e)^2*(a*sin(d*x + c) + a)), x)