3.279 \(\int \frac {(a+b \sec (c+d x))^n}{\sqrt {\sin (c+d x)}} \, dx\)

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

[Out]

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

________________________________________________________________________________________

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 = {} \[ \int \frac {(a+b \sec (c+d x))^n}{\sqrt {\sin (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 2.75, size = 0, normalized size = 0.00 \[ \int \frac {(a+b \sec (c+d x))^n}{\sqrt {\sin (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\sin \left (d x + c\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.82, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \sec \left (d x +c \right )\right )^{n}}{\sqrt {\sin \left (d x +c \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________