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

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

[Out]

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

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Rubi [A]  time = 0.06, 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 = {} \[ \int \sqrt {a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.71, size = 0, normalized size = 0.00 \[ \int \sqrt {a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \sec \left (d x + c\right ) + a} \left (e \tan \left (d x + c\right )\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 1.65, size = 0, normalized size = 0.00 \[ \int \sqrt {a +b \sec \left (d x +c \right )}\, \left (e \tan \left (d x +c \right )\right )^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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