\(\int (c+d x)^m \sin ^2(a+b x) \tan (a+b x) \, dx\) [221]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int (c+d x)^m \sin ^2(a+b x) \tan (a+b x) \, dx=\frac {2^{-3-m} e^{2 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 i b (c+d x)}{d}\right )}{b}+\frac {2^{-3-m} e^{-2 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 i b (c+d x)}{d}\right )}{b}+\text {Int}\left ((c+d x)^m \tan (a+b x),x\right ) \]

[Out]

2^(-3-m)*exp(2*I*(a-b*c/d))*(d*x+c)^m*GAMMA(1+m,-2*I*b*(d*x+c)/d)/b/((-I*b*(d*x+c)/d)^m)+2^(-3-m)*(d*x+c)^m*GA
MMA(1+m,2*I*b*(d*x+c)/d)/b/exp(2*I*(a-b*c/d))/((I*b*(d*x+c)/d)^m)+Unintegrable((d*x+c)^m*tan(b*x+a),x)

Rubi [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 22, 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 = {} \[ \int (c+d x)^m \sin ^2(a+b x) \tan (a+b x) \, dx=\int (c+d x)^m \sin ^2(a+b x) \tan (a+b x) \, dx \]

[In]

Int[(c + d*x)^m*Sin[a + b*x]^2*Tan[a + b*x],x]

[Out]

(2^(-3 - m)*E^((2*I)*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-2*I)*b*(c + d*x))/d])/(b*(((-I)*b*(c + d*x))/d
)^m) + (2^(-3 - m)*(c + d*x)^m*Gamma[1 + m, ((2*I)*b*(c + d*x))/d])/(b*E^((2*I)*(a - (b*c)/d))*((I*b*(c + d*x)
)/d)^m) + Defer[Int][(c + d*x)^m*Tan[a + b*x], x]

Rubi steps \begin{align*} \text {integral}& = -\int (c+d x)^m \cos (a+b x) \sin (a+b x) \, dx+\int (c+d x)^m \tan (a+b x) \, dx \\ & = -\int \frac {1}{2} (c+d x)^m \sin (2 a+2 b x) \, dx+\int (c+d x)^m \tan (a+b x) \, dx \\ & = -\left (\frac {1}{2} \int (c+d x)^m \sin (2 a+2 b x) \, dx\right )+\int (c+d x)^m \tan (a+b x) \, dx \\ & = -\left (\frac {1}{4} i \int e^{-i (2 a+2 b x)} (c+d x)^m \, dx\right )+\frac {1}{4} i \int e^{i (2 a+2 b x)} (c+d x)^m \, dx+\int (c+d x)^m \tan (a+b x) \, dx \\ & = \frac {2^{-3-m} e^{2 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 {2 i b (c+d x)}{d}\right )}{b}+\frac {2^{-3-m} e^{-2 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 {2 i b (c+d x)}{d}\right )}{b}+\int (c+d x)^m \tan (a+b x) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 13.66 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \sin ^2(a+b x) \tan (a+b x) \, dx=\int (c+d x)^m \sin ^2(a+b x) \tan (a+b x) \, dx \]

[In]

Integrate[(c + d*x)^m*Sin[a + b*x]^2*Tan[a + b*x],x]

[Out]

Integrate[(c + d*x)^m*Sin[a + b*x]^2*Tan[a + b*x], x]

Maple [N/A] (verified)

Not integrable

Time = 0.68 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

\[\int \left (d x +c \right )^{m} \sec \left (x b +a \right ) \sin \left (x b +a \right )^{3}d x\]

[In]

int((d*x+c)^m*sec(b*x+a)*sin(b*x+a)^3,x)

[Out]

int((d*x+c)^m*sec(b*x+a)*sin(b*x+a)^3,x)

Fricas [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int (c+d x)^m \sin ^2(a+b x) \tan (a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \sec \left (b x + a\right ) \sin \left (b x + a\right )^{3} \,d x } \]

[In]

integrate((d*x+c)^m*sec(b*x+a)*sin(b*x+a)^3,x, algorithm="fricas")

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int (c+d x)^m \sin ^2(a+b x) \tan (a+b x) \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

integrate((d*x+c)**m*sec(b*x+a)*sin(b*x+a)**3,x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [N/A]

Not integrable

Time = 0.68 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \sin ^2(a+b x) \tan (a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \sec \left (b x + a\right ) \sin \left (b x + a\right )^{3} \,d x } \]

[In]

integrate((d*x+c)^m*sec(b*x+a)*sin(b*x+a)^3,x, algorithm="maxima")

[Out]

integrate((d*x + c)^m*sec(b*x + a)*sin(b*x + a)^3, x)

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m \sin ^2(a+b x) \tan (a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \sec \left (b x + a\right ) \sin \left (b x + a\right )^{3} \,d x } \]

[In]

integrate((d*x+c)^m*sec(b*x+a)*sin(b*x+a)^3,x, algorithm="giac")

[Out]

integrate((d*x + c)^m*sec(b*x + a)*sin(b*x + a)^3, x)

Mupad [N/A]

Not integrable

Time = 25.79 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int (c+d x)^m \sin ^2(a+b x) \tan (a+b x) \, dx=\int \frac {{\sin \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^m}{\cos \left (a+b\,x\right )} \,d x \]

[In]

int((sin(a + b*x)^3*(c + d*x)^m)/cos(a + b*x),x)

[Out]

int((sin(a + b*x)^3*(c + d*x)^m)/cos(a + b*x), x)