\(\int \frac {x \sin (a+b x)}{\cos ^{\frac {5}{2}}(a+b x)} \, dx\) [333]

Optimal result

Integrand size = 18, antiderivative size = 60 \[ \int \frac {x \sin (a+b x)}{\cos ^{\frac {5}{2}}(a+b x)} \, dx=\frac {2 x}{3 b \cos ^{\frac {3}{2}}(a+b x)}+\frac {4 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b^2}-\frac {4 \sin (a+b x)}{3 b^2 \sqrt {\cos (a+b x)}} \]

[Out]

2/3*x/b/cos(b*x+a)^(3/2)+4/3*(cos(1/2*a+1/2*b*x)^2)^(1/2)/cos(1/2*a+1/2*b*x)*EllipticE(sin(1/2*a+1/2*b*x),2^(1
/2))/b^2-4/3*sin(b*x+a)/b^2/cos(b*x+a)^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3525, 2716, 2719} \[ \int \frac {x \sin (a+b x)}{\cos ^{\frac {5}{2}}(a+b x)} \, dx=\frac {4 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b^2}-\frac {4 \sin (a+b x)}{3 b^2 \sqrt {\cos (a+b x)}}+\frac {2 x}{3 b \cos ^{\frac {3}{2}}(a+b x)} \]

[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 2716

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 2719

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

Rule 3525

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.90 \[ \int \frac {x \sin (a+b x)}{\cos ^{\frac {5}{2}}(a+b x)} \, dx=\frac {2 \left (b x+2 \cos ^{\frac {3}{2}}(a+b x) E\left (\left .\frac {1}{2} (a+b x)\right |2\right )-\sin (2 (a+b x))\right )}{3 b^2 \cos ^{\frac {3}{2}}(a+b x)} \]

[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))

Maple [F]

\[\int \frac {x \sin \left (x b +a \right )}{\cos \left (x b +a \right )^{\frac {5}{2}}}d x\]

[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)

Fricas [F(-2)]

Exception generated. \[ \int \frac {x \sin (a+b x)}{\cos ^{\frac {5}{2}}(a+b x)} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {x \sin (a+b x)}{\cos ^{\frac {5}{2}}(a+b x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {x \sin (a+b x)}{\cos ^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {x \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{\frac {5}{2}}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {x \sin (a+b x)}{\cos ^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {x \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{\frac {5}{2}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x \sin (a+b x)}{\cos ^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {x\,\sin \left (a+b\,x\right )}{{\cos \left (a+b\,x\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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