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

Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\tan ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^n,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0459208, 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 = {} \[ \int (a+b \sec (c+d x))^n \tan ^{\frac{3}{2}}(c+d x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 4.16643, size = 0, normalized size = 0. \[ \int (a+b \sec (c+d x))^n \tan ^{\frac{3}{2}}(c+d x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.303, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\sec \left ( dx+c \right ) \right ) ^{n} \left ( \tan \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a+b*sec(d*x+c))^n*tan(d*x+c)^(3/2),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} \tan \left (d x + c\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{\frac{3}{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*sec(d*x + c) + a)^n*tan(d*x + c)^(3/2), 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*tan(d*x+c)**(3/2),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} \tan \left (d x + c\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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