\(\int \frac {\tan ^m(c+d x)}{(a+b \sin ^n(c+d x))^2} \, dx\) [578]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx=\text {Int}\left (\frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2},x\right ) \]

[Out]

Unintegrable(tan(d*x+c)^m/(a+b*sin(d*x+c)^n)^2,x)

Rubi [N/A]

Not integrable

Time = 0.06 (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 = {} \[ \int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx=\int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 47.94 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx=\int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 1.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.87 \[ \int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx=\int { \frac {\tan \left (d x + c\right )^{m}}{{\left (b \sin \left (d x + c\right )^{n} + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

integral(tan(d*x + c)^m/(b^2*sin(d*x + c)^(2*n) + 2*a*b*sin(d*x + c)^n + a^2), x)

Sympy [N/A]

Not integrable

Time = 107.53 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx=\int \frac {\tan ^{m}{\left (c + d x \right )}}{\left (a + b \sin ^{n}{\left (c + d x \right )}\right )^{2}}\, dx \]

[In]

integrate(tan(d*x+c)**m/(a+b*sin(d*x+c)**n)**2,x)

[Out]

Integral(tan(c + d*x)**m/(a + b*sin(c + d*x)**n)**2, x)

Maxima [N/A]

Not integrable

Time = 3.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx=\int { \frac {\tan \left (d x + c\right )^{m}}{{\left (b \sin \left (d x + c\right )^{n} + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

integrate(tan(d*x + c)^m/(b*sin(d*x + c)^n + a)^2, x)

Giac [N/A]

Not integrable

Time = 1.80 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx=\int { \frac {\tan \left (d x + c\right )^{m}}{{\left (b \sin \left (d x + c\right )^{n} + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

integrate(tan(d*x + c)^m/(b*sin(d*x + c)^n + a)^2, x)

Mupad [N/A]

Not integrable

Time = 15.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {tan}\left (c+d\,x\right )}^m}{{\left (a+b\,{\sin \left (c+d\,x\right )}^n\right )}^2} \,d x \]

[In]

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

[Out]

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