\(\int (c+d x)^m \sin (a+b x) \tan (a+b x) \, dx\) [215]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int (c+d x)^m \sin (a+b x) \tan (a+b x) \, dx=\frac {i e^{i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i b (c+d x)}{d}\right )}{2 b}-\frac {i e^{-i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i b (c+d x)}{d}\right )}{2 b}+\text {Int}\left ((c+d x)^m \sec (a+b x),x\right ) \]

[Out]

1/2*I*exp(I*(a-b*c/d))*(d*x+c)^m*GAMMA(1+m,-I*b*(d*x+c)/d)/b/((-I*b*(d*x+c)/d)^m)-1/2*I*(d*x+c)^m*GAMMA(1+m,I*
b*(d*x+c)/d)/b/exp(I*(a-b*c/d))/((I*b*(d*x+c)/d)^m)+Unintegrable((d*x+c)^m*sec(b*x+a),x)

Rubi [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 20, 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 = {} \[ \int (c+d x)^m \sin (a+b x) \tan (a+b x) \, dx=\int (c+d x)^m \sin (a+b x) \tan (a+b x) \, dx \]

[In]

Int[(c + d*x)^m*Sin[a + b*x]*Tan[a + b*x],x]

[Out]

((I/2)*E^(I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-I)*b*(c + d*x))/d])/(b*(((-I)*b*(c + d*x))/d)^m) - ((I/
2)*(c + d*x)^m*Gamma[1 + m, (I*b*(c + d*x))/d])/(b*E^(I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m) + Defer[Int][(c
+ d*x)^m*Sec[a + b*x], x]

Rubi steps \begin{align*} \text {integral}& = -\int (c+d x)^m \cos (a+b x) \, dx+\int (c+d x)^m \sec (a+b x) \, dx \\ & = -\left (\frac {1}{2} \int e^{-i (a+b x)} (c+d x)^m \, dx\right )-\frac {1}{2} \int e^{i (a+b x)} (c+d x)^m \, dx+\int (c+d x)^m \sec (a+b x) \, dx \\ & = \frac {i e^{i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i b (c+d x)}{d}\right )}{2 b}-\frac {i e^{-i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i b (c+d x)}{d}\right )}{2 b}+\int (c+d x)^m \sec (a+b x) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 11.88 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (c+d x)^m \sin (a+b x) \tan (a+b x) \, dx=\int (c+d x)^m \sin (a+b x) \tan (a+b x) \, dx \]

[In]

Integrate[(c + d*x)^m*Sin[a + b*x]*Tan[a + b*x],x]

[Out]

Integrate[(c + d*x)^m*Sin[a + b*x]*Tan[a + b*x], x]

Maple [N/A] (verified)

Not integrable

Time = 0.55 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int (c+d x)^m \sin (a+b x) \tan (a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \sec \left (b x + a\right ) \sin \left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

integral(-(cos(b*x + a)^2 - 1)*(d*x + c)^m*sec(b*x + a), x)

Sympy [N/A]

Not integrable

Time = 121.93 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int (c+d x)^m \sin (a+b x) \tan (a+b x) \, dx=\int \left (c + d x\right )^{m} \sin ^{2}{\left (a + b x \right )} \sec {\left (a + b x \right )}\, dx \]

[In]

integrate((d*x+c)**m*sec(b*x+a)*sin(b*x+a)**2,x)

[Out]

Integral((c + d*x)**m*sin(a + b*x)**2*sec(a + b*x), x)

Maxima [N/A]

Not integrable

Time = 0.68 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int (c+d x)^m \sin (a+b x) \tan (a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \sec \left (b x + a\right ) \sin \left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int (c+d x)^m \sin (a+b x) \tan (a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \sec \left (b x + a\right ) \sin \left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 25.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int (c+d x)^m \sin (a+b x) \tan (a+b x) \, dx=\int \frac {{\sin \left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^m}{\cos \left (a+b\,x\right )} \,d x \]

[In]

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

[Out]

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