3.3.74 \(\int (a+b \sec (c+d x))^n \sin ^2(c+d x) \, dx\) [274]

Optimal. Leaf size=24 \[ \text {Int}\left ((a+b \sec (c+d x))^n \sin ^2(c+d x),x\right ) \]

[Out]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 2.02, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b \sec (c+d x))^n \sin ^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]^2,x]

[Out]

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

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Maple [A]
time = 0.28, size = 0, normalized size = 0.00 \[\int \left (a +b \sec \left (d x +c \right )\right )^{n} \left (\sin ^{2}\left (d x +c \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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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} \sin ^{2}{\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)**2,x)

[Out]

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

[Out]

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

[Out]

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

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