[next] [prev] [prev-tail] [tail] [up]

3.2.66 \(\int \frac {\cos (a+b x)}{\sin ^{\frac {5}{2}}(2 a+2 b x)} \, dx\) [166]

Optimal. Leaf size=53 \[ -\frac {\cos (a+b x)}{3 b \sin ^{\frac {3}{2}}(2 a+2 b x)}+\frac {2 \sin (a+b x)}{3 b \sqrt {\sin (2 a+2 b x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4388, 4377} \begin {gather*} \frac {2 \sin (a+b x)}{3 b \sqrt {\sin (2 a+2 b x)}}-\frac {\cos (a+b x)}{3 b \sin ^{\frac {3}{2}}(2 a+2 b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4377

Int[((e_.)*sin[(a_.) + (b_.)*(x_)])^(m_.)*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> Simp[(e*Sin[a + b
*x])^m*((g*Sin[c + d*x])^(p + 1)/(b*g*m)), x] /; FreeQ[{a, b, c, d, e, g, m, p}, x] && EqQ[b*c - a*d, 0] && Eq
Q[d/b, 2] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0]

Rule 4388

Int[cos[(a_.) + (b_.)*(x_)]*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> Simp[Cos[a + b*x]*((g*Sin[c + d
*x])^(p + 1)/(2*b*g*(p + 1))), x] + Dist[(2*p + 3)/(2*g*(p + 1)), Int[Sin[a + b*x]*(g*Sin[c + d*x])^(p + 1), x
], x] /; FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] &&  !IntegerQ[p] && LtQ[p, -1] && Integ
erQ[2*p]

Rubi steps

\begin {align*} \int \frac {\cos (a+b x)}{\sin ^{\frac {5}{2}}(2 a+2 b x)} \, dx &=-\frac {\cos (a+b x)}{3 b \sin ^{\frac {3}{2}}(2 a+2 b x)}+\frac {2}{3} \int \frac {\sin (a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx\\ &=-\frac {\cos (a+b x)}{3 b \sin ^{\frac {3}{2}}(2 a+2 b x)}+\frac {2 \sin (a+b x)}{3 b \sqrt {\sin (2 a+2 b x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.12, size = 43, normalized size = 0.81 \begin {gather*} \frac {\left (-\frac {1}{12} \cot (a+b x) \csc (a+b x)+\frac {1}{4} \sec (a+b x)\right ) \sqrt {\sin (2 (a+b x))}}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1/12*(Cot[a + b*x]*Csc[a + b*x]) + Sec[a + b*x]/4)*Sqrt[Sin[2*(a + b*x)]])/b

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 17.64, size = 194, normalized size = 3.66

method result size
default \(-\frac {\sqrt {-\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{\tan ^{2}\left (\frac {a}{2}+\frac {x b}{2}\right )-1}}\, \left (\tan ^{2}\left (\frac {a}{2}+\frac {x b}{2}\right )-1\right ) \left (2 \sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )+2}\, \sqrt {-\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}\, \EllipticF \left (\sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )-\left (\tan ^{4}\left (\frac {a}{2}+\frac {x b}{2}\right )\right )+1\right )}{24 b \tan \left (\frac {a}{2}+\frac {x b}{2}\right ) \sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right ) \left (\tan ^{2}\left (\frac {a}{2}+\frac {x b}{2}\right )-1\right )}\, \sqrt {\tan ^{3}\left (\frac {a}{2}+\frac {x b}{2}\right )-\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}}\) \(194\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24/b*(-tan(1/2*a+1/2*x*b)/(tan(1/2*a+1/2*x*b)^2-1))^(1/2)*(tan(1/2*a+1/2*x*b)^2-1)/tan(1/2*a+1/2*x*b)*(2*(t
an(1/2*a+1/2*x*b)+1)^(1/2)*(-2*tan(1/2*a+1/2*x*b)+2)^(1/2)*(-tan(1/2*a+1/2*x*b))^(1/2)*EllipticF((tan(1/2*a+1/
2*x*b)+1)^(1/2),1/2*2^(1/2))*tan(1/2*a+1/2*x*b)-tan(1/2*a+1/2*x*b)^4+1)/(tan(1/2*a+1/2*x*b)*(tan(1/2*a+1/2*x*b
)^2-1))^(1/2)/(tan(1/2*a+1/2*x*b)^3-tan(1/2*a+1/2*x*b))^(1/2)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 1.54, size = 74, normalized size = 1.40 \begin {gather*} \frac {4 \, \cos \left (b x + a\right )^{3} + \sqrt {2} {\left (4 \, \cos \left (b x + a\right )^{2} - 3\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} - 4 \, \cos \left (b x + a\right )}{12 \, {\left (b \cos \left (b x + a\right )^{3} - b \cos \left (b x + a\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(4*cos(b*x + a)^3 + sqrt(2)*(4*cos(b*x + a)^2 - 3)*sqrt(cos(b*x + a)*sin(b*x + a)) - 4*cos(b*x + a))/(b*c
os(b*x + a)^3 - b*cos(b*x + a))

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 7875 vs. \(2 (45) = 90\).
time = 45.07, size = 7875, normalized size = 148.58 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*sqrt(2)*sqrt(-tan(1/2*b*x)^4*tan(1/2*a)^3 - tan(1/2*b*x)^3*tan(1/2*a)^4 + tan(1/2*b*x)^4*tan(1/2*a) + 6*t
an(1/2*b*x)^3*tan(1/2*a)^2 + 6*tan(1/2*b*x)^2*tan(1/2*a)^3 + tan(1/2*b*x)*tan(1/2*a)^4 - tan(1/2*b*x)^3 - 6*ta
n(1/2*b*x)^2*tan(1/2*a) - 6*tan(1/2*b*x)*tan(1/2*a)^2 - tan(1/2*a)^3 + tan(1/2*b*x) + tan(1/2*a))*(((((((sqrt(
2)*tan(1/2*a)^57 + 18*sqrt(2)*tan(1/2*a)^55 + 132*sqrt(2)*tan(1/2*a)^53 + 374*sqrt(2)*tan(1/2*a)^51 - 1375*sqr
t(2)*tan(1/2*a)^49 - 19620*sqrt(2)*tan(1/2*a)^47 - 108560*sqrt(2)*tan(1/2*a)^45 - 399740*sqrt(2)*tan(1/2*a)^43
 - 1096755*sqrt(2)*tan(1/2*a)^41 - 2340250*sqrt(2)*tan(1/2*a)^39 - 3941740*sqrt(2)*tan(1/2*a)^37 - 5204670*sqr
t(2)*tan(1/2*a)^35 - 5163155*sqrt(2)*tan(1/2*a)^33 - 3268760*sqrt(2)*tan(1/2*a)^31 + 3268760*sqrt(2)*tan(1/2*a
)^27 + 5163155*sqrt(2)*tan(1/2*a)^25 + 5204670*sqrt(2)*tan(1/2*a)^23 + 3941740*sqrt(2)*tan(1/2*a)^21 + 2340250
*sqrt(2)*tan(1/2*a)^19 + 1096755*sqrt(2)*tan(1/2*a)^17 + 399740*sqrt(2)*tan(1/2*a)^15 + 108560*sqrt(2)*tan(1/2
*a)^13 + 19620*sqrt(2)*tan(1/2*a)^11 + 1375*sqrt(2)*tan(1/2*a)^9 - 374*sqrt(2)*tan(1/2*a)^7 - 132*sqrt(2)*tan(
1/2*a)^5 - 18*sqrt(2)*tan(1/2*a)^3 - sqrt(2)*tan(1/2*a))*tan(1/2*b*x)/(tan(1/2*a)^51 + 23*tan(1/2*a)^49 + 252*
tan(1/2*a)^47 + 1748*tan(1/2*a)^45 + 8602*tan(1/2*a)^43 + 31878*tan(1/2*a)^41 + 92092*tan(1/2*a)^39 + 211508*t
an(1/2*a)^37 + 389367*tan(1/2*a)^35 + 572033*tan(1/2*a)^33 + 653752*tan(1/2*a)^31 + 534888*tan(1/2*a)^29 + 208
012*tan(1/2*a)^27 - 208012*tan(1/2*a)^25 - 534888*tan(1/2*a)^23 - 653752*tan(1/2*a)^21 - 572033*tan(1/2*a)^19
- 389367*tan(1/2*a)^17 - 211508*tan(1/2*a)^15 - 92092*tan(1/2*a)^13 - 31878*tan(1/2*a)^11 - 8602*tan(1/2*a)^9
- 1748*tan(1/2*a)^7 - 252*tan(1/2*a)^5 - 23*tan(1/2*a)^3 - tan(1/2*a)) - 24*(sqrt(2)*tan(1/2*a)^56 + 23*sqrt(2
)*tan(1/2*a)^54 + 251*sqrt(2)*tan(1/2*a)^52 + 1725*sqrt(2)*tan(1/2*a)^50 + 8350*sqrt(2)*tan(1/2*a)^48 + 30130*
sqrt(2)*tan(1/2*a)^46 + 83490*sqrt(2)*tan(1/2*a)^44 + 179630*sqrt(2)*tan(1/2*a)^42 + 297275*sqrt(2)*tan(1/2*a)
^40 + 360525*sqrt(2)*tan(1/2*a)^38 + 264385*sqrt(2)*tan(1/2*a)^36 - 37145*sqrt(2)*tan(1/2*a)^34 - 445740*sqrt(
2)*tan(1/2*a)^32 - 742900*sqrt(2)*tan(1/2*a)^30 - 742900*sqrt(2)*tan(1/2*a)^28 - 445740*sqrt(2)*tan(1/2*a)^26
- 37145*sqrt(2)*tan(1/2*a)^24 + 264385*sqrt(2)*tan(1/2*a)^22 + 360525*sqrt(2)*tan(1/2*a)^20 + 297275*sqrt(2)*t
an(1/2*a)^18 + 179630*sqrt(2)*tan(1/2*a)^16 + 83490*sqrt(2)*tan(1/2*a)^14 + 30130*sqrt(2)*tan(1/2*a)^12 + 8350
*sqrt(2)*tan(1/2*a)^10 + 1725*sqrt(2)*tan(1/2*a)^8 + 251*sqrt(2)*tan(1/2*a)^6 + 23*sqrt(2)*tan(1/2*a)^4 + sqrt
(2)*tan(1/2*a)^2)/(tan(1/2*a)^51 + 23*tan(1/2*a)^49 + 252*tan(1/2*a)^47 + 1748*tan(1/2*a)^45 + 8602*tan(1/2*a)
^43 + 31878*tan(1/2*a)^41 + 92092*tan(1/2*a)^39 + 211508*tan(1/2*a)^37 + 389367*tan(1/2*a)^35 + 572033*tan(1/2
*a)^33 + 653752*tan(1/2*a)^31 + 534888*tan(1/2*a)^29 + 208012*tan(1/2*a)^27 - 208012*tan(1/2*a)^25 - 534888*ta
n(1/2*a)^23 - 653752*tan(1/2*a)^21 - 572033*tan(1/2*a)^19 - 389367*tan(1/2*a)^17 - 211508*tan(1/2*a)^15 - 9209
2*tan(1/2*a)^13 - 31878*tan(1/2*a)^11 - 8602*tan(1/2*a)^9 - 1748*tan(1/2*a)^7 - 252*tan(1/2*a)^5 - 23*tan(1/2*
a)^3 - tan(1/2*a)))*tan(1/2*b*x) - 15*(sqrt(2)*tan(1/2*a)^57 + 18*sqrt(2)*tan(1/2*a)^55 + 132*sqrt(2)*tan(1/2*
a)^53 + 374*sqrt(2)*tan(1/2*a)^51 - 1375*sqrt(2)*tan(1/2*a)^49 - 19620*sqrt(2)*tan(1/2*a)^47 - 108560*sqrt(2)*
tan(1/2*a)^45 - 399740*sqrt(2)*tan(1/2*a)^43 - 1096755*sqrt(2)*tan(1/2*a)^41 - 2340250*sqrt(2)*tan(1/2*a)^39 -
 3941740*sqrt(2)*tan(1/2*a)^37 - 5204670*sqrt(2)*tan(1/2*a)^35 - 5163155*sqrt(2)*tan(1/2*a)^33 - 3268760*sqrt(
2)*tan(1/2*a)^31 + 3268760*sqrt(2)*tan(1/2*a)^27 + 5163155*sqrt(2)*tan(1/2*a)^25 + 5204670*sqrt(2)*tan(1/2*a)^
23 + 3941740*sqrt(2)*tan(1/2*a)^21 + 2340250*sqrt(2)*tan(1/2*a)^19 + 1096755*sqrt(2)*tan(1/2*a)^17 + 399740*sq
rt(2)*tan(1/2*a)^15 + 108560*sqrt(2)*tan(1/2*a)^13 + 19620*sqrt(2)*tan(1/2*a)^11 + 1375*sqrt(2)*tan(1/2*a)^9 -
 374*sqrt(2)*tan(1/2*a)^7 - 132*sqrt(2)*tan(1/2*a)^5 - 18*sqrt(2)*tan(1/2*a)^3 - sqrt(2)*tan(1/2*a))/(tan(1/2*
a)^51 + 23*tan(1/2*a)^49 + 252*tan(1/2*a)^47 + 1748*tan(1/2*a)^45 + 8602*tan(1/2*a)^43 + 31878*tan(1/2*a)^41 +
 92092*tan(1/2*a)^39 + 211508*tan(1/2*a)^37 + 389367*tan(1/2*a)^35 + 572033*tan(1/2*a)^33 + 653752*tan(1/2*a)^
31 + 534888*tan(1/2*a)^29 + 208012*tan(1/2*a)^27 - 208012*tan(1/2*a)^25 - 534888*tan(1/2*a)^23 - 653752*tan(1/
2*a)^21 - 572033*tan(1/2*a)^19 - 389367*tan(1/2*a)^17 - 211508*tan(1/2*a)^15 - 92092*tan(1/2*a)^13 - 31878*tan
(1/2*a)^11 - 8602*tan(1/2*a)^9 - 1748*tan(1/2*a)^7 - 252*tan(1/2*a)^5 - 23*tan(1/2*a)^3 - tan(1/2*a)))*tan(1/2
*b*x) + 80*(sqrt(2)*tan(1/2*a)^56 + 23*sqrt(2)*tan(1/2*a)^54 + 251*sqrt(2)*tan(1/2*a)^52 + 1725*sqrt(2)*tan(1/
2*a)^50 + 8350*sqrt(2)*tan(1/2*a)^48 + 30130*sqrt(2)*tan(1/2*a)^46 + 83490*sqrt(2)*tan(1/2*a)^44 + 179630*sqrt
(2)*tan(1/2*a)^42 + 297275*sqrt(2)*tan(1/2*a)^40 + 360525*sqrt(2)*tan(1/2*a)^38 + 264385*sqrt(2)*tan(1/2*a)^36
 - 37145*sqrt(2)*tan(1/2*a)^34 - 445740*sqrt(2)*tan(1/2*a)^32 - 742900*sqrt(2)*tan(1/2*a)^30 - 742900*sqrt(2)*
tan(1/2*a)^28 - 445740*sqrt(2)*tan(1/2*a)^26 - ...

________________________________________________________________________________________

Mupad [B]
time = 3.17, size = 104, normalized size = 1.96 \begin {gather*} -\frac {2\,\sqrt {\sin \left (2\,a+2\,b\,x\right )}\,\left (3\,\cos \left (a+b\,x\right )-6\,\cos \left (3\,a+3\,b\,x\right )+4\,\cos \left (5\,a+5\,b\,x\right )-\cos \left (7\,a+7\,b\,x\right )\right )}{3\,b\,\left (4\,\cos \left (2\,a+2\,b\,x\right )+4\,\cos \left (4\,a+4\,b\,x\right )-4\,\cos \left (6\,a+6\,b\,x\right )+\cos \left (8\,a+8\,b\,x\right )-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*sin(2*a + 2*b*x)^(1/2)*(3*cos(a + b*x) - 6*cos(3*a + 3*b*x) + 4*cos(5*a + 5*b*x) - cos(7*a + 7*b*x)))/(3*b
*(4*cos(2*a + 2*b*x) + 4*cos(4*a + 4*b*x) - 4*cos(6*a + 6*b*x) + cos(8*a + 8*b*x) - 5))

________________________________________________________________________________________

[next] [prev] [prev-tail] [front] [up]