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

Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\frac{\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0580189, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[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*} \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\\ \end{align*}

Mathematica [A]  time = 23.4877, size = 0, normalized size = 0. \[ \int \frac{\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[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]

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{m}}{ \left ( a+b \left ( \sin \left ( dx+c \right ) \right ) ^{n} \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan \left (d x + c\right )^{m}}{{\left (b \sin \left (d x + c\right )^{n} + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\tan \left (d x + c\right )^{m}}{b^{2} \sin \left (d x + c\right )^{2 \, n} + 2 \, a b \sin \left (d x + c\right )^{n} + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan \left (d x + c\right )^{m}}{{\left (b \sin \left (d x + c\right )^{n} + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)