3.335 \(\int \frac {x \sin (a+b x)}{\cos ^{\frac {9}{2}}(a+b x)} \, dx\)

Optimal. Leaf size=83 \[ \frac {12 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{35 b^2}-\frac {4 \sin (a+b x)}{35 b^2 \cos ^{\frac {5}{2}}(a+b x)}-\frac {12 \sin (a+b x)}{35 b^2 \sqrt {\cos (a+b x)}}+\frac {2 x}{7 b \cos ^{\frac {7}{2}}(a+b x)} \]

[Out]

2/7*x/b/cos(b*x+a)^(7/2)+12/35*(cos(1/2*b*x+1/2*a)^2)^(1/2)/cos(1/2*b*x+1/2*a)*EllipticE(sin(1/2*b*x+1/2*a),2^
(1/2))/b^2-4/35*sin(b*x+a)/b^2/cos(b*x+a)^(5/2)-12/35*sin(b*x+a)/b^2/cos(b*x+a)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3444, 2636, 2639} \[ \frac {12 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{35 b^2}-\frac {4 \sin (a+b x)}{35 b^2 \cos ^{\frac {5}{2}}(a+b x)}-\frac {12 \sin (a+b x)}{35 b^2 \sqrt {\cos (a+b x)}}+\frac {2 x}{7 b \cos ^{\frac {7}{2}}(a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(x*Sin[a + b*x])/Cos[a + b*x]^(9/2),x]

[Out]

(2*x)/(7*b*Cos[a + b*x]^(7/2)) + (12*EllipticE[(a + b*x)/2, 2])/(35*b^2) - (4*Sin[a + b*x])/(35*b^2*Cos[a + b*
x]^(5/2)) - (12*Sin[a + b*x])/(35*b^2*Sqrt[Cos[a + b*x]])

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]

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]

Rubi steps

\begin {align*} \int \frac {x \sin (a+b x)}{\cos ^{\frac {9}{2}}(a+b x)} \, dx &=\frac {2 x}{7 b \cos ^{\frac {7}{2}}(a+b x)}-\frac {2 \int \frac {1}{\cos ^{\frac {7}{2}}(a+b x)} \, dx}{7 b}\\ &=\frac {2 x}{7 b \cos ^{\frac {7}{2}}(a+b x)}-\frac {4 \sin (a+b x)}{35 b^2 \cos ^{\frac {5}{2}}(a+b x)}-\frac {6 \int \frac {1}{\cos ^{\frac {3}{2}}(a+b x)} \, dx}{35 b}\\ &=\frac {2 x}{7 b \cos ^{\frac {7}{2}}(a+b x)}-\frac {4 \sin (a+b x)}{35 b^2 \cos ^{\frac {5}{2}}(a+b x)}-\frac {12 \sin (a+b x)}{35 b^2 \sqrt {\cos (a+b x)}}+\frac {6 \int \sqrt {\cos (a+b x)} \, dx}{35 b}\\ &=\frac {2 x}{7 b \cos ^{\frac {7}{2}}(a+b x)}+\frac {12 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{35 b^2}-\frac {4 \sin (a+b x)}{35 b^2 \cos ^{\frac {5}{2}}(a+b x)}-\frac {12 \sin (a+b x)}{35 b^2 \sqrt {\cos (a+b x)}}\\ \end {align*}

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Mathematica [A]  time = 0.26, size = 65, normalized size = 0.78 \[ \frac {-10 \sin (2 (a+b x))-3 \sin (4 (a+b x))+24 \cos ^{\frac {7}{2}}(a+b x) E\left (\left .\frac {1}{2} (a+b x)\right |2\right )+20 b x}{70 b^2 \cos ^{\frac {7}{2}}(a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*Sin[a + b*x])/Cos[a + b*x]^(9/2),x]

[Out]

(20*b*x + 24*Cos[a + b*x]^(7/2)*EllipticE[(a + b*x)/2, 2] - 10*Sin[2*(a + b*x)] - 3*Sin[4*(a + b*x)])/(70*b^2*
Cos[a + b*x]^(7/2))

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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*sin(b*x+a)/cos(b*x+a)^(9/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{\frac {9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sin(b*x+a)/cos(b*x+a)^(9/2),x, algorithm="giac")

[Out]

integrate(x*sin(b*x + a)/cos(b*x + a)^(9/2), x)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {x \sin \left (b x +a \right )}{\cos \left (b x +a \right )^{\frac {9}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sin(b*x+a)/cos(b*x+a)^(9/2),x)

[Out]

int(x*sin(b*x+a)/cos(b*x+a)^(9/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{\frac {9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sin(b*x+a)/cos(b*x+a)^(9/2),x, algorithm="maxima")

[Out]

integrate(x*sin(b*x + a)/cos(b*x + a)^(9/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\sin \left (a+b\,x\right )}{{\cos \left (a+b\,x\right )}^{9/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*sin(a + b*x))/cos(a + b*x)^(9/2),x)

[Out]

int((x*sin(a + b*x))/cos(a + b*x)^(9/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sin(b*x+a)/cos(b*x+a)**(9/2),x)

[Out]

Timed out

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