3.86 \(\int \frac{x}{\cos ^{\frac{3}{2}}(a+b x)} \, dx\)

Optimal. Leaf size=54 \[ -\text{Unintegrable}\left (x \sqrt{\cos (a+b x)},x\right )+\frac{4 \sqrt{\cos (a+b x)}}{b^2}+\frac{2 x \sin (a+b x)}{b \sqrt{\cos (a+b x)}} \]

[Out]

(4*Sqrt[Cos[a + b*x]])/b^2 + (2*x*Sin[a + b*x])/(b*Sqrt[Cos[a + b*x]]) - Unintegrable[x*Sqrt[Cos[a + b*x]], x]

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Rubi [A]  time = 0.0363469, 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 = {} \[ \int \frac{x}{\cos ^{\frac{3}{2}}(a+b x)} \, dx \]

Verification is Not applicable to the result.

[In]

Int[x/Cos[a + b*x]^(3/2),x]

[Out]

(4*Sqrt[Cos[a + b*x]])/b^2 + (2*x*Sin[a + b*x])/(b*Sqrt[Cos[a + b*x]]) - Defer[Int][x*Sqrt[Cos[a + b*x]], x]

Rubi steps

\begin{align*} \int \frac{x}{\cos ^{\frac{3}{2}}(a+b x)} \, dx &=\frac{4 \sqrt{\cos (a+b x)}}{b^2}+\frac{2 x \sin (a+b x)}{b \sqrt{\cos (a+b x)}}-\int x \sqrt{\cos (a+b x)} \, dx\\ \end{align*}

Mathematica [F]  time = 0, size = 0, normalized size = 0. \[ \text{\$Aborted} \]

Verification is Not applicable to the result.

[In]

Integrate[x/Cos[a + b*x]^(3/2),x]

[Out]

$Aborted

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Maple [A]  time = 0.123, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \cos \left ( bx+a \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/cos(b*x+a)^(3/2),x)

[Out]

int(x/cos(b*x+a)^(3/2),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\cos \left (b x + a\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/cos(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

integrate(x/cos(b*x + a)^(3/2), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/cos(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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(x/cos(b*x+a)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\cos \left (b x + a\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/cos(b*x+a)^(3/2),x, algorithm="giac")

[Out]

integrate(x/cos(b*x + a)^(3/2), x)