\(\int F^{c (a+b x)} (f x)^m \csc ^2(d+e x) \, dx\) [30]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int F^{c (a+b x)} (f x)^m \csc ^2(d+e x) \, dx=\text {Int}\left (F^{a c+b c x} (f x)^m \csc ^2(d+e x),x\right ) \]

[Out]

CannotIntegrate(F^(b*c*x+a*c)*(f*x)^m*csc(e*x+d)^2,x)

Rubi [N/A]

Not integrable

Time = 1.03 (sec) , antiderivative size = 23, 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 = {} \[ \int F^{c (a+b x)} (f x)^m \csc ^2(d+e x) \, dx=\int F^{c (a+b x)} (f x)^m \csc ^2(d+e x) \, dx \]

[In]

Int[F^(c*(a + b*x))*(f*x)^m*Csc[d + e*x]^2,x]

[Out]

Defer[Int][F^(a*c + b*c*x)*(f*x)^m*Csc[d + e*x]^2, x]

Rubi steps \begin{align*} \text {integral}& = \int F^{a c+b c x} (f x)^m \csc ^2(d+e x) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 14.70 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int F^{c (a+b x)} (f x)^m \csc ^2(d+e x) \, dx=\int F^{c (a+b x)} (f x)^m \csc ^2(d+e x) \, dx \]

[In]

Integrate[F^(c*(a + b*x))*(f*x)^m*Csc[d + e*x]^2,x]

[Out]

Integrate[F^(c*(a + b*x))*(f*x)^m*Csc[d + e*x]^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

\[\int \frac {F^{c \left (x b +a \right )} \left (f x \right )^{m}}{\sin \left (e x +d \right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int F^{c (a+b x)} (f x)^m \csc ^2(d+e x) \, dx=\int { \frac {\left (f x\right )^{m} F^{{\left (b x + a\right )} c}}{\sin \left (e x + d\right )^{2}} \,d x } \]

[In]

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

[Out]

integral(-(f*x)^m*F^(b*c*x + a*c)/(cos(e*x + d)^2 - 1), x)

Sympy [N/A]

Not integrable

Time = 2.83 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int F^{c (a+b x)} (f x)^m \csc ^2(d+e x) \, dx=\int \frac {F^{c \left (a + b x\right )} \left (f x\right )^{m}}{\sin ^{2}{\left (d + e x \right )}}\, dx \]

[In]

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

[Out]

Integral(F**(c*(a + b*x))*(f*x)**m/sin(d + e*x)**2, x)

Maxima [N/A]

Not integrable

Time = 0.63 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int F^{c (a+b x)} (f x)^m \csc ^2(d+e x) \, dx=\int { \frac {\left (f x\right )^{m} F^{{\left (b x + a\right )} c}}{\sin \left (e x + d\right )^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int F^{c (a+b x)} (f x)^m \csc ^2(d+e x) \, dx=\int { \frac {\left (f x\right )^{m} F^{{\left (b x + a\right )} c}}{\sin \left (e x + d\right )^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 28.57 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int F^{c (a+b x)} (f x)^m \csc ^2(d+e x) \, dx=\int \frac {F^{c\,\left (a+b\,x\right )}\,{\left (f\,x\right )}^m}{{\sin \left (d+e\,x\right )}^2} \,d x \]

[In]

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

[Out]

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