3.477 \(\int \frac{A+B \sec (c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx\)

Optimal. Leaf size=125 \[ A \text{Unintegrable}\left (\frac{1}{(a+b \sec (c+d x))^{2/3}},x\right )+\frac{\sqrt{2} B \tan (c+d x) \left (\frac{a+b \sec (c+d x)}{a+b}\right )^{2/3} F_1\left (\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right )}{d \sqrt{\sec (c+d x)+1} (a+b \sec (c+d x))^{2/3}} \]

[Out]

(Sqrt[2]*B*AppellF1[1/2, 1/2, 2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c +
d*x])/(a + b))^(2/3)*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(2/3)) + A*Unintegrable[(a +
b*Sec[c + d*x])^(-2/3), x]

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Rubi [A]  time = 0.16203, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{A+B \sec (c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(2/3),x]

[Out]

(Sqrt[2]*B*AppellF1[1/2, 1/2, 2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c +
d*x])/(a + b))^(2/3)*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(2/3)) + A*Defer[Int][(a + b
*Sec[c + d*x])^(-2/3), x]

Rubi steps

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

Mathematica [A]  time = 3.33016, size = 0, normalized size = 0. \[ \int \frac{A+B \sec (c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(2/3),x]

[Out]

Integrate[(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(2/3), x]

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Maple [A]  time = 0.181, size = 0, normalized size = 0. \begin{align*} \int{(A+B\sec \left ( dx+c \right ) ) \left ( a+b\sec \left ( dx+c \right ) \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*sec(d*x+c))/(a+b*sec(d*x+c))^(2/3),x)

[Out]

int((A+B*sec(d*x+c))/(a+b*sec(d*x+c))^(2/3),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sec(d*x+c))/(a+b*sec(d*x+c))^(2/3),x, algorithm="maxima")

[Out]

integrate((B*sec(d*x + c) + A)/(b*sec(d*x + c) + a)^(2/3), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sec(d*x+c))/(a+b*sec(d*x+c))^(2/3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B \sec{\left (c + d x \right )}}{\left (a + b \sec{\left (c + d x \right )}\right )^{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sec(d*x+c))/(a+b*sec(d*x+c))**(2/3),x)

[Out]

Integral((A + B*sec(c + d*x))/(a + b*sec(c + d*x))**(2/3), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sec(d*x+c))/(a+b*sec(d*x+c))^(2/3),x, algorithm="giac")

[Out]

integrate((B*sec(d*x + c) + A)/(b*sec(d*x + c) + a)^(2/3), x)