3.363 \(\int (a+b \sec (c+d x))^n \sqrt {\tan (c+d x)} \, dx\)

Optimal. Leaf size=26 \[ \text {Int}\left (\sqrt {\tan (c+d x)} (a+b \sec (c+d x))^n,x\right ) \]

[Out]

Unintegrable((a+b*sec(d*x+c))^n*tan(d*x+c)^(1/2),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 = {} \[ \int (a+b \sec (c+d x))^n \sqrt {\tan (c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 5.60, size = 0, normalized size = 0.00 \[ \int (a+b \sec (c+d x))^n \sqrt {\tan (c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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