3.4.49 \(\int \sqrt {a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx\) [349]

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 = 0.04, 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 \sqrt {a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx \end {gather*}

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 = 0.70, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx \end {gather*}

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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Maple [A]
time = 0.16, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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

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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