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

Optimal. Leaf size=131 \[ \frac{2^{m+\frac{3}{2}} \sqrt{a \sec (c+d x)+a} \left (\frac{1}{\sec (c+d x)+1}\right )^{m+\frac{3}{2}} (e \tan (c+d x))^{m+1} F_1\left (\frac{m+1}{2};m+\frac{1}{2},1;\frac{m+3}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{d e (m+1)} \]

[Out]

(2^(3/2 + m)*AppellF1[(1 + m)/2, 1/2 + m, 1, (3 + m)/2, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*S
ec[c + d*x])/(a + a*Sec[c + d*x])]*((1 + Sec[c + d*x])^(-1))^(3/2 + m)*Sqrt[a + a*Sec[c + d*x]]*(e*Tan[c + d*x
])^(1 + m))/(d*e*(1 + m))

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Rubi [A]  time = 0.0855779, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {3889} \[ \frac{2^{m+\frac{3}{2}} \sqrt{a \sec (c+d x)+a} \left (\frac{1}{\sec (c+d x)+1}\right )^{m+\frac{3}{2}} (e \tan (c+d x))^{m+1} F_1\left (\frac{m+1}{2};m+\frac{1}{2},1;\frac{m+3}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{d e (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2^(3/2 + m)*AppellF1[(1 + m)/2, 1/2 + m, 1, (3 + m)/2, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*S
ec[c + d*x])/(a + a*Sec[c + d*x])]*((1 + Sec[c + d*x])^(-1))^(3/2 + m)*Sqrt[a + a*Sec[c + d*x]]*(e*Tan[c + d*x
])^(1 + m))/(d*e*(1 + m))

Rule 3889

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> -Simp[(2^(m
+ n + 1)*(e*Cot[c + d*x])^(m + 1)*(a + b*Csc[c + d*x])^n*(a/(a + b*Csc[c + d*x]))^(m + n + 1)*AppellF1[(m + 1
)/2, m + n, 1, (m + 3)/2, -((a - b*Csc[c + d*x])/(a + b*Csc[c + d*x])), (a - b*Csc[c + d*x])/(a + b*Csc[c + d*
x])])/(d*e*(m + 1)), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[n]

Rubi steps

\begin{align*} \int \sqrt{a+a \sec (c+d x)} (e \tan (c+d x))^m \, dx &=\frac{2^{\frac{3}{2}+m} F_1\left (\frac{1+m}{2};\frac{1}{2}+m,1;\frac{3+m}{2};-\frac{a-a \sec (c+d x)}{a+a \sec (c+d x)},\frac{a-a \sec (c+d x)}{a+a \sec (c+d x)}\right ) \left (\frac{1}{1+\sec (c+d x)}\right )^{\frac{3}{2}+m} \sqrt{a+a \sec (c+d x)} (e \tan (c+d x))^{1+m}}{d e (1+m)}\\ \end{align*}

Mathematica [F]  time = 7.3573, size = 0, normalized size = 0. \[ \int \sqrt{a+a \sec (c+d x)} (e \tan (c+d x))^m \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.279, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+a\sec \left ( dx+c \right ) } \left ( e\tan \left ( dx+c \right ) \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \sec \left (d x + c\right ) + a} \left (e \tan \left (d x + c\right )\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{a \sec \left (d x + c\right ) + a} \left (e \tan \left (d x + c\right )\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \left (\sec{\left (c + d x \right )} + 1\right )} \left (e \tan{\left (c + d x \right )}\right )^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*(sec(c + d*x) + 1))*(e*tan(c + d*x))**m, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \sec \left (d x + c\right ) + a} \left (e \tan \left (d x + c\right )\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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