3.3.78 \(\int (a+b \sec (c+d x))^n \sqrt {\sin (c+d x)} \, dx\) [278]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, 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 (a+b \sec (c+d x))^n \sqrt {\sin (c+d x)} \, dx \end {gather*}

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

________________________________________________________________________________________

Mathematica [A]
time = 4.46, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b \sec (c+d x))^n \sqrt {\sin (c+d x)} \, dx \end {gather*}

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]

________________________________________________________________________________________

Maple [A]
time = 0.09, size = 0, normalized size = 0.00 \[\int \left (a +b \sec \left (d x +c \right )\right )^{n} \left (\sqrt {\sin }\left (d x +c \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sec(d*x+c))^n*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 \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))^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)

________________________________________________________________________________________

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))^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)

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \sec {\left (c + d x \right )}\right )^{n} \sqrt {\sin {\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))**n*sin(d*x+c)**(1/2),x)

[Out]

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

________________________________________________________________________________________

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))^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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________