3.1.81 \(\int (-\frac {x}{3 \sqrt {\cos (a+b x)}}+x \cos ^{\frac {3}{2}}(a+b x)) \, dx\) [81]

Optimal. Leaf size=42 \[ \frac {4 \cos ^{\frac {3}{2}}(a+b x)}{9 b^2}+\frac {2 x \sqrt {\cos (a+b x)} \sin (a+b x)}{3 b} \]

[Out]

4/9*cos(b*x+a)^(3/2)/b^2+2/3*x*sin(b*x+a)*cos(b*x+a)^(1/2)/b

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Rubi [A]
time = 0.04, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {3391} \begin {gather*} \frac {4 \cos ^{\frac {3}{2}}(a+b x)}{9 b^2}+\frac {2 x \sin (a+b x) \sqrt {\cos (a+b x)}}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-1/3*x/Sqrt[Cos[a + b*x]] + x*Cos[a + b*x]^(3/2),x]

[Out]

(4*Cos[a + b*x]^(3/2))/(9*b^2) + (2*x*Sqrt[Cos[a + b*x]]*Sin[a + b*x])/(3*b)

Rule 3391

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

Rubi steps

\begin {align*} \int \left (-\frac {x}{3 \sqrt {\cos (a+b x)}}+x \cos ^{\frac {3}{2}}(a+b x)\right ) \, dx &=-\left (\frac {1}{3} \int \frac {x}{\sqrt {\cos (a+b x)}} \, dx\right )+\int x \cos ^{\frac {3}{2}}(a+b x) \, dx\\ &=\frac {4 \cos ^{\frac {3}{2}}(a+b x)}{9 b^2}+\frac {2 x \sqrt {\cos (a+b x)} \sin (a+b x)}{3 b}\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 40, normalized size = 0.95 \begin {gather*} \frac {\sqrt {\cos (a+b x)} \left (\frac {8 \cos (a+b x)}{3 b}+4 x \sin (a+b x)\right )}{6 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-1/3*x/Sqrt[Cos[a + b*x]] + x*Cos[a + b*x]^(3/2),x]

[Out]

(Sqrt[Cos[a + b*x]]*((8*Cos[a + b*x])/(3*b) + 4*x*Sin[a + b*x]))/(6*b)

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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int x \left (\cos ^{\frac {3}{2}}\left (b x +a \right )\right )-\frac {x}{3 \sqrt {\cos \left (b x +a \right )}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
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)-1/3*x/cos(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(x*cos(b*x + a)^(3/2) - 1/3*x/sqrt(cos(b*x + a)), 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)-1/3*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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \left (- \frac {x}{\sqrt {\cos {\left (a + b x \right )}}}\right )\, dx + \int 3 x \cos ^{\frac {3}{2}}{\left (a + b x \right )}\, dx}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
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)-1/3*x/cos(b*x+a)^(1/2),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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