\(\int \frac {(e+f x)^m \sec (c+d x)}{a+a \sin (c+d x)} \, dx\) [292]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {(e+f x)^m \sec (c+d x)}{a+a \sin (c+d x)} \, dx=\text {Int}\left (\frac {(e+f x)^m \sec (c+d x)}{a+a \sin (c+d x)},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(e+f x)^m \sec (c+d x)}{a+a \sin (c+d x)} \, dx=\int \frac {(e+f x)^m \sec (c+d x)}{a+a \sin (c+d x)} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 91.96 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {(e+f x)^m \sec (c+d x)}{a+a \sin (c+d x)} \, dx=\int \frac {(e+f x)^m \sec (c+d x)}{a+a \sin (c+d x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00

\[\int \frac {\left (f x +e \right )^{m} \sec \left (d x +c \right )}{a +a \sin \left (d x +c \right )}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {(e+f x)^m \sec (c+d x)}{a+a \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{m} \sec \left (d x + c\right )}{a \sin \left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 18.43 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {(e+f x)^m \sec (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {\left (e + f x\right )^{m} \sec {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]

[In]

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

[Out]

Integral((e + f*x)**m*sec(c + d*x)/(sin(c + d*x) + 1), x)/a

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x)^m \sec (c+d x)}{a+a \sin (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {(e+f x)^m \sec (c+d x)}{a+a \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{m} \sec \left (d x + c\right )}{a \sin \left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 3.59 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {(e+f x)^m \sec (c+d x)}{a+a \sin (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^m}{\cos \left (c+d\,x\right )\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]

[In]

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

[Out]

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