\(\int \sec (e+f x) (a+b \sin ^n(e+f x))^p \, dx\) [434]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int \sec (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx=\text {Int}\left (\sec (e+f x) \left (a+b \sin ^n(e+f x)\right )^p,x\right ) \]

[Out]

Unintegrable(sec(f*x+e)*(a+b*sin(f*x+e)^n)^p,x)

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 21, 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 = {} \[ \int \sec (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx=\int \sec (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx \]

[In]

Int[Sec[e + f*x]*(a + b*Sin[e + f*x]^n)^p,x]

[Out]

Defer[Int][Sec[e + f*x]*(a + b*Sin[e + f*x]^n)^p, x]

Rubi steps \begin{align*} \text {integral}& = \int \sec (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 3.85 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \sec (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx=\int \sec (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx \]

[In]

Integrate[Sec[e + f*x]*(a + b*Sin[e + f*x]^n)^p,x]

[Out]

Integrate[Sec[e + f*x]*(a + b*Sin[e + f*x]^n)^p, x]

Maple [N/A] (verified)

Not integrable

Time = 0.59 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00

\[\int \sec \left (f x +e \right ) {\left (a +b \left (\sin ^{n}\left (f x +e \right )\right )\right )}^{p}d x\]

[In]

int(sec(f*x+e)*(a+b*sin(f*x+e)^n)^p,x)

[Out]

int(sec(f*x+e)*(a+b*sin(f*x+e)^n)^p,x)

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \sec (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{n} + a\right )}^{p} \sec \left (f x + e\right ) \,d x } \]

[In]

integrate(sec(f*x+e)*(a+b*sin(f*x+e)^n)^p,x, algorithm="fricas")

[Out]

integral((b*sin(f*x + e)^n + a)^p*sec(f*x + e), x)

Sympy [N/A]

Not integrable

Time = 132.94 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \sec (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx=\int \left (a + b \sin ^{n}{\left (e + f x \right )}\right )^{p} \sec {\left (e + f x \right )}\, dx \]

[In]

integrate(sec(f*x+e)*(a+b*sin(f*x+e)**n)**p,x)

[Out]

Integral((a + b*sin(e + f*x)**n)**p*sec(e + f*x), x)

Maxima [N/A]

Not integrable

Time = 2.46 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \sec (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{n} + a\right )}^{p} \sec \left (f x + e\right ) \,d x } \]

[In]

integrate(sec(f*x+e)*(a+b*sin(f*x+e)^n)^p,x, algorithm="maxima")

[Out]

integrate((b*sin(f*x + e)^n + a)^p*sec(f*x + e), x)

Giac [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \sec (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{n} + a\right )}^{p} \sec \left (f x + e\right ) \,d x } \]

[In]

integrate(sec(f*x+e)*(a+b*sin(f*x+e)^n)^p,x, algorithm="giac")

[Out]

integrate((b*sin(f*x + e)^n + a)^p*sec(f*x + e), x)

Mupad [N/A]

Not integrable

Time = 13.65 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \sec (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx=\int \frac {{\left (a+b\,{\sin \left (e+f\,x\right )}^n\right )}^p}{\cos \left (e+f\,x\right )} \,d x \]

[In]

int((a + b*sin(e + f*x)^n)^p/cos(e + f*x),x)

[Out]

int((a + b*sin(e + f*x)^n)^p/cos(e + f*x), x)