3.350 \(\int \frac{(e \tan (c+d x))^m}{\sqrt{a+b \sec (c+d x)}} \, dx\)

Optimal. Leaf size=27 \[ \text{Unintegrable}\left (\frac{(e \tan (c+d x))^m}{\sqrt{a+b \sec (c+d x)}},x\right ) \]

[Out]

Unintegrable[(e*Tan[c + d*x])^m/Sqrt[a + b*Sec[c + d*x]], x]

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Rubi [A]  time = 0.0624351, 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 = {} \[ \int \frac{(e \tan (c+d x))^m}{\sqrt{a+b \sec (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(e*Tan[c + d*x])^m/Sqrt[a + b*Sec[c + d*x]],x]

[Out]

Defer[Int][(e*Tan[c + d*x])^m/Sqrt[a + b*Sec[c + d*x]], x]

Rubi steps

\begin{align*} \int \frac{(e \tan (c+d x))^m}{\sqrt{a+b \sec (c+d x)}} \, dx &=\int \frac{(e \tan (c+d x))^m}{\sqrt{a+b \sec (c+d x)}} \, dx\\ \end{align*}

Mathematica [A]  time = 2.98802, size = 0, normalized size = 0. \[ \int \frac{(e \tan (c+d x))^m}{\sqrt{a+b \sec (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(e*Tan[c + d*x])^m/Sqrt[a + b*Sec[c + d*x]],x]

[Out]

Integrate[(e*Tan[c + d*x])^m/Sqrt[a + b*Sec[c + d*x]], x]

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Maple [A]  time = 0.299, size = 0, normalized size = 0. \begin{align*} \int{ \left ( e\tan \left ( dx+c \right ) \right ) ^{m}{\frac{1}{\sqrt{a+b\sec \left ( dx+c \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*tan(d*x+c))^m/(a+b*sec(d*x+c))^(1/2),x)

[Out]

int((e*tan(d*x+c))^m/(a+b*sec(d*x+c))^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*tan(d*x+c))^m/(a+b*sec(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate((e*tan(d*x + c))^m/sqrt(b*sec(d*x + c) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*tan(d*x+c))^m/(a+b*sec(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral((e*tan(d*x + c))^m/sqrt(b*sec(d*x + c) + a), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e \tan{\left (c + d x \right )}\right )^{m}}{\sqrt{a + b \sec{\left (c + d x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*tan(d*x+c))**m/(a+b*sec(d*x+c))**(1/2),x)

[Out]

Integral((e*tan(c + d*x))**m/sqrt(a + b*sec(c + d*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*tan(d*x+c))^m/(a+b*sec(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate((e*tan(d*x + c))^m/sqrt(b*sec(d*x + c) + a), x)