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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0672621, 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 = {} \[ \int \frac{(e+f x)^m \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 6.20772, size = 0, normalized size = 0. \[ \int \frac{(e+f x)^m \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.129, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{m} \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{a+b\sin \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [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((f*x+e)**m*sec(d*x+c)**2/(a+b*sin(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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