3.274 \(\int (a+b \sec (c+d x))^n \sin ^2(c+d x) \, dx\)

Optimal. Leaf size=23 \[ \text{Unintegrable}\left (\sin ^2(c+d x) (a+b \sec (c+d x))^n,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0398864, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int (a+b \sec (c+d x))^n \sin ^2(c+d x) \, dx \]

Verification is Not applicable to the result.

[In]

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

Mathematica [A]  time = 3.77905, size = 0, normalized size = 0. \[ \int (a+b \sec (c+d x))^n \sin ^2(c+d x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.588, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\sec \left ( dx+c \right ) \right ) ^{n} \left ( \sin \left ( dx+c \right ) \right ) ^{2}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{2}\,{d x} \end{align*}

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., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (\cos \left (d x + c\right )^{2} - 1\right )}{\left (b \sec \left (d x + c\right ) + a\right )}^{n}, x\right ) \end{align*}

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 [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{2}\,{d x} \end{align*}

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)