∫ cos7 _ 2 (c + dx)∘ _______

3.140 \(\int \cos ^{\frac {7}{2}}(c+d x) \sqrt {b \cos (c+d x)} \, dx\)

Optimal. Leaf size=98 \[ \frac {3 x \sqrt {b \cos (c+d x)}}{8 \sqrt {\cos (c+d x)}}+\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)}}{4 d}+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}}{8 d} \]

[Out]

1/4*cos(d*x+c)^(5/2)*sin(d*x+c)*(b*cos(d*x+c))^(1/2)/d+3/8*x*(b*cos(d*x+c))^(1/2)/cos(d*x+c)^(1/2)+3/8*sin(d*x

+c)*cos(d*x+c)^(1/2)*(b*cos(d*x+c))^(1/2)/d

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Rubi [A] time = 0.03, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {17, 2635, 8} \[ \frac {3 x \sqrt {b \cos (c+d x)}}{8 \sqrt {\cos (c+d x)}}+\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)}}{4 d}+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}}{8 d} \]

Antiderivative was successfully veriﬁed.

[In]

Int[Cos[c + d*x]^(7/2)*Sqrt[b*Cos[c + d*x]],x]

[Out]

(3*x*Sqrt[b*Cos[c + d*x]])/(8*Sqrt[Cos[c + d*x]]) + (3*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(

8*d) + (Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),

x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer

Q[2*n]

Rubi steps

\begin {align*} \int \cos ^{\frac {7}{2}}(c+d x) \sqrt {b \cos (c+d x)} \, dx &=\frac {\sqrt {b \cos (c+d x)} \int \cos ^4(c+d x) \, dx}{\sqrt {\cos (c+d x)}}\\ &=\frac {\cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{4 d}+\frac {\left (3 \sqrt {b \cos (c+d x)}\right ) \int \cos ^2(c+d x) \, dx}{4 \sqrt {\cos (c+d x)}}\\ &=\frac {3 \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \sin (c+d x)}{8 d}+\frac {\cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{4 d}+\frac {\left (3 \sqrt {b \cos (c+d x)}\right ) \int 1 \, dx}{8 \sqrt {\cos (c+d x)}}\\ &=\frac {3 x \sqrt {b \cos (c+d x)}}{8 \sqrt {\cos (c+d x)}}+\frac {3 \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \sin (c+d x)}{8 d}+\frac {\cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{4 d}\\ \end {align*}

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Mathematica [A] time = 0.08, size = 55, normalized size = 0.56 \[ \frac {(12 (c+d x)+8 \sin (2 (c+d x))+\sin (4 (c+d x))) \sqrt {b \cos (c+d x)}}{32 d \sqrt {\cos (c+d x)}} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[Cos[c + d*x]^(7/2)*Sqrt[b*Cos[c + d*x]],x]

[Out]

(Sqrt[b*Cos[c + d*x]]*(12*(c + d*x) + 8*Sin[2*(c + d*x)] + Sin[4*(c + d*x)]))/(32*d*Sqrt[Cos[c + d*x]])

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fricas [A] time = 0.68, size = 176, normalized size = 1.80 \[ \left [\frac {2 \, \sqrt {b \cos \left (d x + c\right )} {\left (2 \, \cos \left (d x + c\right )^{2} + 3\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 3 \, \sqrt {-b} \log \left (2 \, b \cos \left (d x + c\right )^{2} - 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right )}{16 \, d}, \frac {\sqrt {b \cos \left (d x + c\right )} {\left (2 \, \cos \left (d x + c\right )^{2} + 3\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 3 \, \sqrt {b} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt {b} \cos \left (d x + c\right )^{\frac {3}{2}}}\right )}{8 \, d}\right ] \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(7/2)*(b*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

[1/16*(2*sqrt(b*cos(d*x + c))*(2*cos(d*x + c)^2 + 3)*sqrt(cos(d*x + c))*sin(d*x + c) + 3*sqrt(-b)*log(2*b*cos(

d*x + c)^2 - 2*sqrt(b*cos(d*x + c))*sqrt(-b)*sqrt(cos(d*x + c))*sin(d*x + c) - b))/d, 1/8*(sqrt(b*cos(d*x + c)

)*(2*cos(d*x + c)^2 + 3)*sqrt(cos(d*x + c))*sin(d*x + c) + 3*sqrt(b)*arctan(sqrt(b*cos(d*x + c))*sin(d*x + c)/

(sqrt(b)*cos(d*x + c)^(3/2))))/d]

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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(7/2)*(b*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (2*pi/x/2)>(-2*pi/

x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check si