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

Optimal. Leaf size=38 \[ \frac {4 \sqrt {\cos (a+b x)}}{b^2}+\frac {2 x \sin (a+b x)}{b \sqrt {\cos (a+b x)}} \]

[Out]

2*x*sin(b*x+a)/b/cos(b*x+a)^(1/2)+4*cos(b*x+a)^(1/2)/b^2

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Rubi [A]  time = 0.06, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {3315} \[ \frac {4 \sqrt {\cos (a+b x)}}{b^2}+\frac {2 x \sin (a+b x)}{b \sqrt {\cos (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[x/Cos[a + b*x]^(3/2) + x*Sqrt[Cos[a + b*x]],x]

[Out]

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

Rule 3315

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((c + d*x)*Cos[e + f*x]*(b*Si
n[e + f*x])^(n + 1))/(b*f*(n + 1)), x] + (Dist[(n + 2)/(b^2*(n + 1)), Int[(c + d*x)*(b*Sin[e + f*x])^(n + 2),
x], x] - Simp[(d*(b*Sin[e + f*x])^(n + 2))/(b^2*f^2*(n + 1)*(n + 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && LtQ[
n, -1] && NeQ[n, -2]

Rubi steps

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

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Mathematica [A]  time = 0.40, size = 33, normalized size = 0.87 \[ \frac {2 (b x \sin (a+b x)+2 \cos (a+b x))}{b^2 \sqrt {\cos (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Cos[a + b*x]^(3/2) + x*Sqrt[Cos[a + b*x]],x]

[Out]

(2*(2*Cos[a + b*x] + b*x*Sin[a + b*x]))/(b^2*Sqrt[Cos[a + b*x]])

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/cos(b*x+a)^(3/2)+x*cos(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {\cos \left (b x + a\right )} + \frac {x}{\cos \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/cos(b*x+a)^(3/2)+x*cos(b*x+a)^(1/2),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {x}{\cos \left (b x +a \right )^{\frac {3}{2}}}+x \left (\sqrt {\cos }\left (b x +a \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/cos(b*x+a)^(3/2)+x*cos(b*x+a)^(1/2),x)

[Out]

int(x/cos(b*x+a)^(3/2)+x*cos(b*x+a)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {\cos \left (b x + a\right )} + \frac {x}{\cos \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/cos(b*x+a)^(3/2)+x*cos(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.66, size = 51, normalized size = 1.34 \[ \frac {2\,\sqrt {\cos \left (a+b\,x\right )}\,\left (2\,\cos \left (2\,a+2\,b\,x\right )+b\,x\,\sin \left (2\,a+2\,b\,x\right )+2\right )}{b^2\,\left (\cos \left (2\,a+2\,b\,x\right )+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cos(a + b*x)^(1/2) + x/cos(a + b*x)^(3/2),x)

[Out]

(2*cos(a + b*x)^(1/2)*(2*cos(2*a + 2*b*x) + b*x*sin(2*a + 2*b*x) + 2))/(b^2*(cos(2*a + 2*b*x) + 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (\cos ^{2}{\left (a + b x \right )} + 1\right )}{\cos ^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/cos(b*x+a)**(3/2)+x*cos(b*x+a)**(1/2),x)

[Out]

Integral(x*(cos(a + b*x)**2 + 1)/cos(a + b*x)**(3/2), x)

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