\(\int (a+b \sin ^n(c+d x))^p \tan (c+d x) \, dx\) [583]

Optimal result

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

[Out]

Unintegrable((a+b*sin(d*x+c)^n)^p*tan(d*x+c),x)

Rubi [N/A]

Not integrable

Time = 0.03 (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 = {} \[ \int \left (a+b \sin ^n(c+d x)\right )^p \tan (c+d x) \, dx=\int \left (a+b \sin ^n(c+d x)\right )^p \tan (c+d x) \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.51 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \left (a+b \sin ^n(c+d x)\right )^p \tan (c+d x) \, dx=\int \left (a+b \sin ^n(c+d x)\right )^p \tan (c+d x) \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.82 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00

\[\int {\left (a +b \left (\sin ^{n}\left (d x +c \right )\right )\right )}^{p} \tan \left (d x +c \right )d x\]

[In]

int((a+b*sin(d*x+c)^n)^p*tan(d*x+c),x)

[Out]

int((a+b*sin(d*x+c)^n)^p*tan(d*x+c),x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \left (a+b \sin ^n(c+d x)\right )^p \tan (c+d x) \, dx=\int { {\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \tan \left (d x + c\right ) \,d x } \]

[In]

integrate((a+b*sin(d*x+c)^n)^p*tan(d*x+c),x, algorithm="fricas")

[Out]

integral((b*sin(d*x + c)^n + a)^p*tan(d*x + c), x)

Sympy [N/A]

Not integrable

Time = 109.77 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \left (a+b \sin ^n(c+d x)\right )^p \tan (c+d x) \, dx=\int \left (a + b \sin ^{n}{\left (c + d x \right )}\right )^{p} \tan {\left (c + d x \right )}\, dx \]

[In]

integrate((a+b*sin(d*x+c)**n)**p*tan(d*x+c),x)

[Out]

Integral((a + b*sin(c + d*x)**n)**p*tan(c + d*x), x)

Maxima [N/A]

Not integrable

Time = 3.13 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \left (a+b \sin ^n(c+d x)\right )^p \tan (c+d x) \, dx=\int { {\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \tan \left (d x + c\right ) \,d x } \]

[In]

integrate((a+b*sin(d*x+c)^n)^p*tan(d*x+c),x, algorithm="maxima")

[Out]

integrate((b*sin(d*x + c)^n + a)^p*tan(d*x + c), x)

Giac [N/A]

Not integrable

Time = 6.17 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \left (a+b \sin ^n(c+d x)\right )^p \tan (c+d x) \, dx=\int { {\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \tan \left (d x + c\right ) \,d x } \]

[In]

integrate((a+b*sin(d*x+c)^n)^p*tan(d*x+c),x, algorithm="giac")

[Out]

integrate((b*sin(d*x + c)^n + a)^p*tan(d*x + c), x)

Mupad [N/A]

Not integrable

Time = 14.70 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \left (a+b \sin ^n(c+d x)\right )^p \tan (c+d x) \, dx=\int \mathrm {tan}\left (c+d\,x\right )\,{\left (a+b\,{\sin \left (c+d\,x\right )}^n\right )}^p \,d x \]

[In]

int(tan(c + d*x)*(a + b*sin(c + d*x)^n)^p,x)

[Out]

int(tan(c + d*x)*(a + b*sin(c + d*x)^n)^p, x)