3.1.86 \(\int \frac {x}{\cos ^{\frac {3}{2}}(a+b x)} \, dx\) [86]

Optimal. Leaf size=55 \[ \frac {4 \sqrt {\cos (a+b x)}}{b^2}+\frac {2 x \sin (a+b x)}{b \sqrt {\cos (a+b x)}}-\text {Int}\left (x \sqrt {\cos (a+b x)},x\right ) \]

[Out]

2*x*sin(b*x+a)/b/cos(b*x+a)^(1/2)+4*cos(b*x+a)^(1/2)/b^2-Unintegrable(x*cos(b*x+a)^(1/2),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 \frac {x}{\cos ^{\frac {3}{2}}(a+b x)} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 1.61, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\cos ^{\frac {3}{2}}(a+b x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[x/Cos[a + b*x]^(3/2),x]

[Out]

Integrate[x/Cos[a + b*x]^(3/2), x]

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Maple [A]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x}{\cos \left (b x +a \right )^{\frac {3}{2}}}\, dx\]

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 = 0.00, 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(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\cos ^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(x/cos(b*x + a)^(3/2), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x}{{\cos \left (a+b\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/cos(a + b*x)^(3/2),x)

[Out]

int(x/cos(a + b*x)^(3/2), x)

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