\(\int \frac {\tan ^m(c+d x)}{a+b \sin ^n(c+d x)} \, dx\) [577]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.07 (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 = {} \[ \int \frac {\tan ^m(c+d x)}{a+b \sin ^n(c+d x)} \, dx=\int \frac {\tan ^m(c+d x)}{a+b \sin ^n(c+d x)} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 5.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\tan ^m(c+d x)}{a+b \sin ^n(c+d x)} \, dx=\int \frac {\tan ^m(c+d x)}{a+b \sin ^n(c+d x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\tan ^m(c+d x)}{a+b \sin ^n(c+d x)} \, dx=\int { \frac {\tan \left (d x + c\right )^{m}}{b \sin \left (d x + c\right )^{n} + a} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 5.17 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {\tan ^m(c+d x)}{a+b \sin ^n(c+d x)} \, dx=\int \frac {\tan ^{m}{\left (c + d x \right )}}{a + b \sin ^{n}{\left (c + d x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.49 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\tan ^m(c+d x)}{a+b \sin ^n(c+d x)} \, dx=\int { \frac {\tan \left (d x + c\right )^{m}}{b \sin \left (d x + c\right )^{n} + a} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.38 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\tan ^m(c+d x)}{a+b \sin ^n(c+d x)} \, dx=\int { \frac {\tan \left (d x + c\right )^{m}}{b \sin \left (d x + c\right )^{n} + a} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 13.93 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\tan ^m(c+d x)}{a+b \sin ^n(c+d x)} \, dx=\int \frac {{\mathrm {tan}\left (c+d\,x\right )}^m}{a+b\,{\sin \left (c+d\,x\right )}^n} \,d x \]

[In]

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

[Out]

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