3.79 \(\int \cos ^3(c+d x) (b \cos (c+d x))^{3/2} \, dx\)

Optimal. Leaf size=95 \[ \frac{2 \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b^2 d}+\frac{14 \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 d}+\frac{14 b E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}} \]

[Out]

(14*b*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(15*d*Sqrt[Cos[c + d*x]]) + (14*(b*Cos[c + d*x])^(3/2)*S
in[c + d*x])/(45*d) + (2*(b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b^2*d)

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Rubi [A]  time = 0.0601152, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {16, 2635, 2640, 2639} \[ \frac{2 \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b^2 d}+\frac{14 \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 d}+\frac{14 b E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^3*(b*Cos[c + d*x])^(3/2),x]

[Out]

(14*b*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(15*d*Sqrt[Cos[c + d*x]]) + (14*(b*Cos[c + d*x])^(3/2)*S
in[c + d*x])/(45*d) + (2*(b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b^2*d)

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

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]

Rule 2640

Int[Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[b*Sin[c + d*x]]/Sqrt[Sin[c + d*x]], Int[Sqrt[Si
n[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

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]

Rubi steps

\begin{align*} \int \cos ^3(c+d x) (b \cos (c+d x))^{3/2} \, dx &=\frac{\int (b \cos (c+d x))^{9/2} \, dx}{b^3}\\ &=\frac{2 (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^2 d}+\frac{7 \int (b \cos (c+d x))^{5/2} \, dx}{9 b}\\ &=\frac{14 (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}+\frac{2 (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^2 d}+\frac{1}{15} (7 b) \int \sqrt{b \cos (c+d x)} \, dx\\ &=\frac{14 (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}+\frac{2 (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^2 d}+\frac{\left (7 b \sqrt{b \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{15 \sqrt{\cos (c+d x)}}\\ &=\frac{14 b \sqrt{b \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d \sqrt{\cos (c+d x)}}+\frac{14 (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}+\frac{2 (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^2 d}\\ \end{align*}

Mathematica [A]  time = 0.100421, size = 75, normalized size = 0.79 \[ \frac{(b \cos (c+d x))^{3/2} \left (168 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+(38 \sin (2 (c+d x))+5 \sin (4 (c+d x))) \sqrt{\cos (c+d x)}\right )}{180 d \cos ^{\frac{3}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^3*(b*Cos[c + d*x])^(3/2),x]

[Out]

((b*Cos[c + d*x])^(3/2)*(168*EllipticE[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(38*Sin[2*(c + d*x)] + 5*Sin[4*(c
+ d*x)])))/(180*d*Cos[c + d*x]^(3/2))

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Maple [B]  time = 1.695, size = 223, normalized size = 2.4 \begin{align*} -{\frac{2\,{b}^{2}}{45\,d}\sqrt{b \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 160\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{11}-480\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{9}+616\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{7}-432\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}+160\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}-21\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -24\,\cos \left ( 1/2\,dx+c/2 \right ) \right ){\frac{1}{\sqrt{-b \left ( 2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}- \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) }}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{b \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^3*(b*cos(d*x+c))^(3/2),x)

[Out]

-2/45*(b*(2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*b^2*(160*cos(1/2*d*x+1/2*c)^11-480*cos(1/2*d*x
+1/2*c)^9+616*cos(1/2*d*x+1/2*c)^7-432*cos(1/2*d*x+1/2*c)^5+160*cos(1/2*d*x+1/2*c)^3-21*(sin(1/2*d*x+1/2*c)^2)
^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-24*cos(1/2*d*x+1/2*c))/(-b*(2*s
in(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2))^(1/2)/sin(1/2*d*x+1/2*c)/(b*(2*cos(1/2*d*x+1/2*c)^2-1))^(1/2)/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \cos \left (d x + c\right )\right )^{\frac{3}{2}} \cos \left (d x + c\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(b*cos(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*cos(d*x + c))^(3/2)*cos(d*x + c)^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{b \cos \left (d x + c\right )} b \cos \left (d x + c\right )^{4}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(b*cos(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*cos(d*x + c))*b*cos(d*x + c)^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**3*(b*cos(d*x+c))**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \cos \left (d x + c\right )\right )^{\frac{3}{2}} \cos \left (d x + c\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(b*cos(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate((b*cos(d*x + c))^(3/2)*cos(d*x + c)^3, x)