3.280 \(\int \frac {(a+b \sec (c+d x))^n}{\sin ^{\frac {3}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {(a+b \sec (c+d x))^n}{\sin ^{\frac {3}{2}}(c+d x)},x\right ) \]

[Out]

Unintegrable((a+b*sec(d*x+c))^n/sin(d*x+c)^(3/2),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 = {} \[ \int \frac {(a+b \sec (c+d x))^n}{\sin ^{\frac {3}{2}}(c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Defer[Int][(a + b*Sec[c + d*x])^n/Sin[c + d*x]^(3/2), x]

Rubi steps

\begin {align*} \int \frac {(a+b \sec (c+d x))^n}{\sin ^{\frac {3}{2}}(c+d x)} \, dx &=\int \frac {(a+b \sec (c+d x))^n}{\sin ^{\frac {3}{2}}(c+d x)} \, dx\\ \end {align*}

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Mathematica [A]  time = 2.91, size = 0, normalized size = 0.00 \[ \int \frac {(a+b \sec (c+d x))^n}{\sin ^{\frac {3}{2}}(c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{n} \sqrt {\sin \left (d x + c\right )}}{\cos \left (d x + c\right )^{2} - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(b*sec(d*x + c) + a)^n*sqrt(sin(d*x + c))/(cos(d*x + c)^2 - 1), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{n}}{\sin \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.83, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \sec \left (d x +c \right )\right )^{n}}{\sin \left (d x +c \right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{n}}{\sin \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^n}{{\sin \left (c+d\,x\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/cos(c + d*x))^n/sin(c + d*x)^(3/2),x)

[Out]

int((a + b/cos(c + d*x))^n/sin(c + d*x)^(3/2), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \sec {\left (c + d x \right )}\right )^{n}}{\sin ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))**n/sin(d*x+c)**(3/2),x)

[Out]

Integral((a + b*sec(c + d*x))**n/sin(c + d*x)**(3/2), x)

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