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

Optimal. Leaf size=28 \[ \text {Int}\left (\frac {(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.07, 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 = {} \[ \int \frac {(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx \]

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*}

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Mathematica [A]  time = 3.89, size = 0, normalized size = 0.00 \[ \int \frac {(e \tan (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx \]

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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fricas [A]  time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sec \left (d x + c\right ) + a} \left (e \tan \left (d x + c\right )\right )^{m}}{b^{2} \sec \left (d x + c\right )^{2} + 2 \, a b \sec \left (d x + c\right ) + a^{2}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \tan \left (d x + c\right )\right )^{m}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]

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)

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maple [A]  time = 1.57, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \tan \left (d x +c \right )\right )^{m}}{\left (a +b \sec \left (d x +c \right )\right )^{\frac {3}{2}}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \tan \left (d x + c\right )\right )^{m}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^m}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \tan {\left (c + d x \right )}\right )^{m}}{\left (a + b \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]

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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