3.3.66 \(\int \frac {(e \sin (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx\) [266]

Optimal. Leaf size=28 \[ \text {Int}\left (\frac {(e \sin (c+d x))^m}{(a+b \sec (c+d x))^{3/2}},x\right ) \]

[Out]

Unintegrable((e*sin(d*x+c))^m/(a+b*sec(d*x+c))^(3/2),x)

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Rubi [A]
time = 0.05, 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 \frac {(e \sin (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 3.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(e \sin (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {\left (e \sin \left (d x +c \right )\right )^{m}}{\left (a +b \sec \left (d x +c \right )\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

[Out]

integral(sqrt(b*sec(d*x + c) + a)*(e*sin(d*x + c))^m/(b^2*sec(d*x + c)^2 + 2*a*b*sec(d*x + c) + a^2), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e \sin {\left (c + d x \right )}\right )^{m}}{\left (a + b \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\left (e\,\sin \left (c+d\,x\right )\right )}^m}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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