3.351 \(\int \frac{(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=27 \[ \text{Unintegrable}\left (\frac{(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}},x\right ) \]

[Out]

Unintegrable[(e*Tan[c + d*x])^m/(a + b*Sec[c + d*x])^(3/2), x]

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Rubi [A]  time = 0.0700814, 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 = {} \[ \int \frac{(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(e*Tan[c + d*x])^m/(a + b*Sec[c + d*x])^(3/2),x]

[Out]

Defer[Int][(e*Tan[c + d*x])^m/(a + b*Sec[c + d*x])^(3/2), x]

Rubi steps

\begin{align*} \int \frac{(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx &=\int \frac{(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx\\ \end{align*}

Mathematica [A]  time = 3.70481, size = 0, normalized size = 0. \[ \int \frac{(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(e*Tan[c + d*x])^m/(a + b*Sec[c + d*x])^(3/2),x]

[Out]

Integrate[(e*Tan[c + d*x])^m/(a + b*Sec[c + d*x])^(3/2), x]

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Maple [A]  time = 0.268, size = 0, normalized size = 0. \begin{align*} \int{ \left ( e\tan \left ( dx+c \right ) \right ) ^{m} \left ( a+b\sec \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}}\, 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))^(3/2),x)

[Out]

int((e*tan(d*x+c))^m/(a+b*sec(d*x+c))^(3/2),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}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{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))^(3/2),x, algorithm="maxima")

[Out]

integrate((e*tan(d*x + c))^m/(b*sec(d*x + c) + a)^(3/2), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \sec \left (d x + c\right ) + a} \left (e \tan \left (d x + c\right )\right )^{m}}{b^{2} \sec \left (d x + c\right )^{2} + 2 \, a b \sec \left (d x + c\right ) + a^{2}}, 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))^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*sec(d*x + c) + a)*(e*tan(d*x + c))^m/(b^2*sec(d*x + c)^2 + 2*a*b*sec(d*x + c) + a^2), 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}}{\left (a + b \sec{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, 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))**(3/2),x)

[Out]

Integral((e*tan(c + d*x))**m/(a + b*sec(c + d*x))**(3/2), 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}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{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))^(3/2),x, algorithm="giac")

[Out]

integrate((e*tan(d*x + c))^m/(b*sec(d*x + c) + a)^(3/2), x)