3.3.77 \(\int (a+b \sec (c+d x))^n \sin ^{\frac {3}{2}}(c+d x) \, dx\) [277]

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

[Out]

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

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Rubi [A]
time = 0.03, 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 \sin ^{\frac {3}{2}}(c+d x) \, dx \end {gather*}

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

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Mathematica [A]
time = 2.67, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b \sec (c+d x))^n \sin ^{\frac {3}{2}}(c+d x) \, dx \end {gather*}

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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Maple [A]
time = 0.09, size = 0, normalized size = 0.00 \[\int \left (a +b \sec \left (d x +c \right )\right )^{n} \left (\sin ^{\frac {3}{2}}\left (d x +c \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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)**(3/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6437 deep

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

[Out]

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

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