3.333 \(\int \frac{x \sin (a+b x)}{\cos ^{\frac{5}{2}}(a+b x)} \, dx\)

Optimal. Leaf size=60 \[ \frac{4 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{3 b^2}-\frac{4 \sin (a+b x)}{3 b^2 \sqrt{\cos (a+b x)}}+\frac{2 x}{3 b \cos ^{\frac{3}{2}}(a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.0360545, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3444, 2636, 2639} \[ \frac{4 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{3 b^2}-\frac{4 \sin (a+b x)}{3 b^2 \sqrt{\cos (a+b x)}}+\frac{2 x}{3 b \cos ^{\frac{3}{2}}(a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(x*Sin[a + b*x])/Cos[a + b*x]^(5/2),x]

[Out]

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

Rule 3444

Int[Cos[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*(x_)^(m_.)*Sin[(a_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> -Simp[(x^(m - n
+ 1)*Cos[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] + Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Cos[a + b*x^n]
^(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]

Rule 2636

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

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

\begin{align*} \int \frac{x \sin (a+b x)}{\cos ^{\frac{5}{2}}(a+b x)} \, dx &=\frac{2 x}{3 b \cos ^{\frac{3}{2}}(a+b x)}-\frac{2 \int \frac{1}{\cos ^{\frac{3}{2}}(a+b x)} \, dx}{3 b}\\ &=\frac{2 x}{3 b \cos ^{\frac{3}{2}}(a+b x)}-\frac{4 \sin (a+b x)}{3 b^2 \sqrt{\cos (a+b x)}}+\frac{2 \int \sqrt{\cos (a+b x)} \, dx}{3 b}\\ &=\frac{2 x}{3 b \cos ^{\frac{3}{2}}(a+b x)}+\frac{4 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{3 b^2}-\frac{4 \sin (a+b x)}{3 b^2 \sqrt{\cos (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.198494, size = 54, normalized size = 0.9 \[ \frac{2 \left (-\sin (2 (a+b x))+2 \cos ^{\frac{3}{2}}(a+b x) E\left (\left .\frac{1}{2} (a+b x)\right |2\right )+b x\right )}{3 b^2 \cos ^{\frac{3}{2}}(a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*Sin[a + b*x])/Cos[a + b*x]^(5/2),x]

[Out]

(2*(b*x + 2*Cos[a + b*x]^(3/2)*EllipticE[(a + b*x)/2, 2] - Sin[2*(a + b*x)]))/(3*b^2*Cos[a + b*x]^(3/2))

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Maple [F]  time = 0.087, size = 0, normalized size = 0. \begin{align*} \int{x\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sin(b*x+a)/cos(b*x+a)^(5/2),x)

[Out]

int(x*sin(b*x+a)/cos(b*x+a)^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sin(b*x+a)/cos(b*x+a)^(5/2),x, algorithm="maxima")

[Out]

integrate(x*sin(b*x + a)/cos(b*x + a)^(5/2), 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*sin(b*x+a)/cos(b*x+a)^(5/2),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*sin(b*x+a)/cos(b*x+a)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sin(b*x+a)/cos(b*x+a)^(5/2),x, algorithm="giac")

[Out]

integrate(x*sin(b*x + a)/cos(b*x + a)^(5/2), x)