3.2.41 \(\int \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)} \, dx\) [141]
Optimal. Leaf size=70 \[ \frac {\sqrt {b \cos (c+d x)} \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {\sqrt {b \cos (c+d x)} \sin ^3(c+d x)}{3 d \sqrt {\cos (c+d x)}} \]
[Out]
sin(d*x+c)*(b*cos(d*x+c))^(1/2)/d/cos(d*x+c)^(1/2)-1/3*sin(d*x+c)^3*(b*cos(d*x+c))^(1/2)/d/cos(d*x+c)^(1/2)
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Rubi [A]
time = 0.01, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {17, 2713}
\begin {gather*} \frac {\sin (c+d x) \sqrt {b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-\frac {\sin ^3(c+d x) \sqrt {b \cos (c+d x)}}{3 d \sqrt {\cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]],x]
[Out]
(Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[C
os[c + d*x]])
Rule 17
Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[a^(m + 1/2)*b^(n - 1/2)*(Sqrt[b*v]/Sqrt[a*v])
, Int[u*v^(m + n), x], x] /; FreeQ[{a, b, m}, x] && !IntegerQ[m] && IGtQ[n + 1/2, 0] && IntegerQ[m + n]
Rule 2713
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Rubi steps
\begin {align*} \int \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)} \, dx &=\frac {\sqrt {b \cos (c+d x)} \int \cos ^3(c+d x) \, dx}{\sqrt {\cos (c+d x)}}\\ &=-\frac {\sqrt {b \cos (c+d x)} \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d \sqrt {\cos (c+d x)}}\\ &=\frac {\sqrt {b \cos (c+d x)} \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {\sqrt {b \cos (c+d x)} \sin ^3(c+d x)}{3 d \sqrt {\cos (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 45, normalized size = 0.64 \begin {gather*} \frac {\sqrt {b \cos (c+d x)} (5+\cos (2 (c+d x))) \sin (c+d x)}{6 d \sqrt {\cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]],x]
[Out]
(Sqrt[b*Cos[c + d*x]]*(5 + Cos[2*(c + d*x)])*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]])
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Maple [A]
time = 0.16, size = 40, normalized size = 0.57
| | |
| method |
result |
size |
| | |
| default |
\(\frac {\left (\cos ^{2}\left (d x +c \right )+2\right ) \sin \left (d x +c \right ) \sqrt {b \cos \left (d x +c \right )}}{3 d \sqrt {\cos \left (d x +c \right )}}\) |
\(40\) |
| risch |
\(-\frac {i \sqrt {b \cos \left (d x +c \right )}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) {\mathrm e}^{4 i \left (d x +c \right )}}{12 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}-\frac {3 i \sqrt {b \cos \left (d x +c \right )}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) {\mathrm e}^{2 i \left (d x +c \right )}}{4 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}+\frac {3 i \sqrt {b \cos \left (d x +c \right )}\, \left (\sqrt {\cos }\left (d x +c \right )\right )}{4 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}+\frac {i \sqrt {b \cos \left (d x +c \right )}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) {\mathrm e}^{-2 i \left (d x +c \right )}}{12 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}\) |
\(177\) |
| | |
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|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^(5/2)*(b*cos(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
[Out]
1/3/d*(cos(d*x+c)^2+2)*sin(d*x+c)*(b*cos(d*x+c))^(1/2)/cos(d*x+c)^(1/2)
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Maxima [A]
time = 0.59, size = 42, normalized size = 0.60 \begin {gather*} \frac {\sqrt {b} {\left (\sin \left (3 \, d x + 3 \, c\right ) + 9 \, \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (3 \, d x + 3 \, c\right ), \cos \left (3 \, d x + 3 \, c\right )\right )\right )\right )}}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^(5/2)*(b*cos(d*x+c))^(1/2),x, algorithm="maxima")
[Out]
1/12*sqrt(b)*(sin(3*d*x + 3*c) + 9*sin(1/3*arctan2(sin(3*d*x + 3*c), cos(3*d*x + 3*c))))/d
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Fricas [A]
time = 0.38, size = 39, normalized size = 0.56 \begin {gather*} \frac {\sqrt {b \cos \left (d x + c\right )} {\left (\cos \left (d x + c\right )^{2} + 2\right )} \sin \left (d x + c\right )}{3 \, d \sqrt {\cos \left (d x + c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^(5/2)*(b*cos(d*x+c))^(1/2),x, algorithm="fricas")
[Out]
1/3*sqrt(b*cos(d*x + c))*(cos(d*x + c)^2 + 2)*sin(d*x + c)/(d*sqrt(cos(d*x + c)))
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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(d*x+c)**(5/2)*(b*cos(d*x+c))**(1/2),x)
[Out]
Timed out
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 71047 vs.
\(2 (60) = 120\).
time = 6.75, size = 71047, normalized size = 1014.96 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^(5/2)*(b*cos(d*x+c))^(1/2),x, algorithm="giac")
[Out]
-1/96*(3*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^4*tan(1/2*d*x + 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^4*tan(1/3*c)^6*tan(
c)^2 - 3*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^4*tan(1/2*d*x + 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^4*tan(1/3*c)^6 - 24
*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^4*tan(1/2*d*x + 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^3*tan(1/3*c)^6*tan(c) - 24*
sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^3*tan(1/2*d*x + 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^4*tan(1/3*c)^6*tan(c) + 9*sq
rt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^4*tan(1/2*d*x + 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^4*tan(1/3*c)^4*tan(c)^2 - 18*s
qrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^4*tan(1/2*d*x + 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^2*tan(1/3*c)^6*tan(c)^2 - 48*
sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^3*tan(1/2*d*x + 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^3*tan(1/3*c)^6*tan(c)^2 + 9*
sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^4*tan(1/2*d*x + 1/6*c)^4*tan(-1/2*d*x + 1/2*c)^4*tan(1/3*c)^6*tan(c)^2 - 18
*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^2*tan(1/2*d*x + 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^4*tan(1/3*c)^6*tan(c)^2 - 9
*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^4*tan(1/2*d*x + 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^4*tan(1/3*c)^4 + 18*sqrt(b)
*d*x^4*tan(1/2*d*x + 1/2*c)^4*tan(1/2*d*x + 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^2*tan(1/3*c)^6 + 48*sqrt(b)*d*x^4*t
an(1/2*d*x + 1/2*c)^3*tan(1/2*d*x + 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^3*tan(1/3*c)^6 - 9*sqrt(b)*d*x^4*tan(1/2*d*
x + 1/2*c)^4*tan(1/2*d*x + 1/6*c)^4*tan(-1/2*d*x + 1/2*c)^4*tan(1/3*c)^6 + 18*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*
c)^2*tan(1/2*d*x + 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^4*tan(1/3*c)^6 - 72*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^4*tan
(1/2*d*x + 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^3*tan(1/3*c)^4*tan(c) - 72*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^3*tan(
1/2*d*x + 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^4*tan(1/3*c)^4*tan(c) + 24*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^4*tan(1
/2*d*x + 1/6*c)^6*tan(-1/2*d*x + 1/2*c)*tan(1/3*c)^6*tan(c) + 144*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^3*tan(1/2
*d*x + 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^2*tan(1/3*c)^6*tan(c) - 72*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^4*tan(1/2*
d*x + 1/6*c)^4*tan(-1/2*d*x + 1/2*c)^3*tan(1/3*c)^6*tan(c) + 144*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^2*tan(1/2*
d*x + 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^3*tan(1/3*c)^6*tan(c) - 72*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^3*tan(1/2*d
*x + 1/6*c)^4*tan(-1/2*d*x + 1/2*c)^4*tan(1/3*c)^6*tan(c) + 24*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)*tan(1/2*d*x
+ 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^4*tan(1/3*c)^6*tan(c) + 9*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^4*tan(1/2*d*x +
1/6*c)^6*tan(-1/2*d*x + 1/2*c)^4*tan(1/3*c)^2*tan(c)^2 - 54*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^4*tan(1/2*d*x +
1/6*c)^6*tan(-1/2*d*x + 1/2*c)^2*tan(1/3*c)^4*tan(c)^2 - 144*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^3*tan(1/2*d*x
+ 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^3*tan(1/3*c)^4*tan(c)^2 + 27*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^4*tan(1/2*d*
x + 1/6*c)^4*tan(-1/2*d*x + 1/2*c)^4*tan(1/3*c)^4*tan(c)^2 - 54*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^2*tan(1/2*d
*x + 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^4*tan(1/3*c)^4*tan(c)^2 + 3*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^4*tan(1/2*d
*x + 1/6*c)^6*tan(1/3*c)^6*tan(c)^2 + 48*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^3*tan(1/2*d*x + 1/6*c)^6*tan(-1/2*
d*x + 1/2*c)*tan(1/3*c)^6*tan(c)^2 - 54*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^4*tan(1/2*d*x + 1/6*c)^4*tan(-1/2*d
*x + 1/2*c)^2*tan(1/3*c)^6*tan(c)^2 + 108*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^2*tan(1/2*d*x + 1/6*c)^6*tan(-1/2
*d*x + 1/2*c)^2*tan(1/3*c)^6*tan(c)^2 - 144*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^3*tan(1/2*d*x + 1/6*c)^4*tan(-1
/2*d*x + 1/2*c)^3*tan(1/3*c)^6*tan(c)^2 + 48*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)*tan(1/2*d*x + 1/6*c)^6*tan(-1/
2*d*x + 1/2*c)^3*tan(1/3*c)^6*tan(c)^2 + 9*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^4*tan(1/2*d*x + 1/6*c)^2*tan(-1/
2*d*x + 1/2*c)^4*tan(1/3*c)^6*tan(c)^2 - 54*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^2*tan(1/2*d*x + 1/6*c)^4*tan(-1
/2*d*x + 1/2*c)^4*tan(1/3*c)^6*tan(c)^2 + 3*sqrt(b)*d*x^4*tan(1/2*d*x + 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^4*tan(1
/3*c)^6*tan(c)^2 - 9*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^4*tan(1/2*d*x + 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^4*tan(1
/3*c)^2 + 54*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^4*tan(1/2*d*x + 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^2*tan(1/3*c)^4
+ 144*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^3*tan(1/2*d*x + 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^3*tan(1/3*c)^4 - 27*sq
rt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^4*tan(1/2*d*x + 1/6*c)^4*tan(-1/2*d*x + 1/2*c)^4*tan(1/3*c)^4 + 54*sqrt(b)*d*
x^4*tan(1/2*d*x + 1/2*c)^2*tan(1/2*d*x + 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^4*tan(1/3*c)^4 - 3*sqrt(b)*d*x^4*tan(1
/2*d*x + 1/2*c)^4*tan(1/2*d*x + 1/6*c)^6*tan(1/3*c)^6 - 48*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^3*tan(1/2*d*x +
1/6*c)^6*tan(-1/2*d*x + 1/2*c)*tan(1/3*c)^6 + 54*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^4*tan(1/2*d*x + 1/6*c)^4*t
an(-1/2*d*x + 1/2*c)^2*tan(1/3*c)^6 - 108*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^2*tan(1/2*d*x + 1/6*c)^6*tan(-1/2
*d*x + 1/2*c)^2*tan(1/3*c)^6 + 144*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)^3*tan(1/2*d*x + 1/6*c)^4*tan(-1/2*d*x +
1/2*c)^3*tan(1/3*c)^6 - 48*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2*c)*tan(1/2*d*x + 1/6*c)^6*tan(-1/2*d*x + 1/2*c)^3*t
an(1/3*c)^6 - 9*sqrt(b)*d*x^4*tan(1/2*d*x + 1/2...
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Mupad [B]
time = 0.74, size = 57, normalized size = 0.81 \begin {gather*} \frac {\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (10\,\sin \left (2\,c+2\,d\,x\right )+\sin \left (4\,c+4\,d\,x\right )\right )}{12\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(c + d*x)^(5/2)*(b*cos(c + d*x))^(1/2),x)
[Out]
(cos(c + d*x)^(1/2)*(b*cos(c + d*x))^(1/2)*(10*sin(2*c + 2*d*x) + sin(4*c + 4*d*x)))/(12*d*(cos(2*c + 2*d*x) +
1))
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