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

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {\tan ^m(c+d x)}{a+b \sin ^n(c+d x)},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\tan ^m(c+d x)}{a+b \sin ^n(c+d x)} \, dx \end {gather*}

Verification is not applicable to the result.

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

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Mathematica [A]
time = 5.99, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tan ^m(c+d x)}{a+b \sin ^n(c+d x)} \, dx \end {gather*}

Verification is not applicable to the result.

[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]

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Maple [A]
time = 0.43, size = 0, normalized size = 0.00 \[\int \frac {\tan ^{m}\left (d x +c \right )}{a +b \left (\sin ^{n}\left (d x +c \right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]
time = 0.40, size = 25, normalized size = 0.96 \begin {gather*} {\rm integral}\left (\frac {\tan \left (d x + c\right )^{m}}{b \sin \left (d x + c\right )^{n} + a}, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tan ^{m}{\left (c + d x \right )}}{a + b \sin ^{n}{\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {tan}\left (c+d\,x\right )}^m}{a+b\,{\sin \left (c+d\,x\right )}^n} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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