\(\int \frac {\sec ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx\) [285]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {\sec ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\text {Int}\left (\frac {\sec ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 28, 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 = {} \[ \int \frac {\sec ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\int \frac {\sec ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 23.34 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sec ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\int \frac {\sec ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

\[\int \frac {\sec ^{3}\left (d x +c \right )}{\left (f x +e \right ) \left (a +a \sin \left (d x +c \right )\right )}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {\sec ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\int { \frac {\sec \left (d x + c\right )^{3}}{{\left (f x + e\right )} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.80 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {\sec ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\frac {\int \frac {\sec ^{3}{\left (c + d x \right )}}{e \sin {\left (c + d x \right )} + e + f x \sin {\left (c + d x \right )} + f x}\, dx}{a} \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sec ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(sec(d*x+c)^3/(f*x+e)/(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 = 269.76 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sec ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\int { \frac {\sec \left (d x + c\right )^{3}}{{\left (f x + e\right )} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 4.58 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sec ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^3\,\left (e+f\,x\right )\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]

[In]

int(1/(cos(c + d*x)^3*(e + f*x)*(a + a*sin(c + d*x))),x)

[Out]

int(1/(cos(c + d*x)^3*(e + f*x)*(a + a*sin(c + d*x))), x)