3.194 \(\int (d \cos (a+b x))^{9/2} \sin ^2(a+b x) \, dx\)

Optimal. Leaf size=126 \[ \frac{28 d^3 \sin (a+b x) (d \cos (a+b x))^{3/2}}{585 b}+\frac{28 d^4 E\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{d \cos (a+b x)}}{195 b \sqrt{\cos (a+b x)}}-\frac{2 \sin (a+b x) (d \cos (a+b x))^{11/2}}{13 b d}+\frac{4 d \sin (a+b x) (d \cos (a+b x))^{7/2}}{117 b} \]

[Out]

(28*d^4*Sqrt[d*Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2])/(195*b*Sqrt[Cos[a + b*x]]) + (28*d^3*(d*Cos[a + b*x])^
(3/2)*Sin[a + b*x])/(585*b) + (4*d*(d*Cos[a + b*x])^(7/2)*Sin[a + b*x])/(117*b) - (2*(d*Cos[a + b*x])^(11/2)*S
in[a + b*x])/(13*b*d)

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Rubi [A]  time = 0.100993, antiderivative size = 126, 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 = {2568, 2635, 2640, 2639} \[ \frac{28 d^3 \sin (a+b x) (d \cos (a+b x))^{3/2}}{585 b}+\frac{28 d^4 E\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{d \cos (a+b x)}}{195 b \sqrt{\cos (a+b x)}}-\frac{2 \sin (a+b x) (d \cos (a+b x))^{11/2}}{13 b d}+\frac{4 d \sin (a+b x) (d \cos (a+b x))^{7/2}}{117 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(28*d^4*Sqrt[d*Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2])/(195*b*Sqrt[Cos[a + b*x]]) + (28*d^3*(d*Cos[a + b*x])^
(3/2)*Sin[a + b*x])/(585*b) + (4*d*(d*Cos[a + b*x])^(7/2)*Sin[a + b*x])/(117*b) - (2*(d*Cos[a + b*x])^(11/2)*S
in[a + b*x])/(13*b*d)

Rule 2568

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(a*(b*Cos[e
+ f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Cos[e + f*x])
^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*
m, 2*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]

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 (d \cos (a+b x))^{9/2} \sin ^2(a+b x) \, dx &=-\frac{2 (d \cos (a+b x))^{11/2} \sin (a+b x)}{13 b d}+\frac{2}{13} \int (d \cos (a+b x))^{9/2} \, dx\\ &=\frac{4 d (d \cos (a+b x))^{7/2} \sin (a+b x)}{117 b}-\frac{2 (d \cos (a+b x))^{11/2} \sin (a+b x)}{13 b d}+\frac{1}{117} \left (14 d^2\right ) \int (d \cos (a+b x))^{5/2} \, dx\\ &=\frac{28 d^3 (d \cos (a+b x))^{3/2} \sin (a+b x)}{585 b}+\frac{4 d (d \cos (a+b x))^{7/2} \sin (a+b x)}{117 b}-\frac{2 (d \cos (a+b x))^{11/2} \sin (a+b x)}{13 b d}+\frac{1}{195} \left (14 d^4\right ) \int \sqrt{d \cos (a+b x)} \, dx\\ &=\frac{28 d^3 (d \cos (a+b x))^{3/2} \sin (a+b x)}{585 b}+\frac{4 d (d \cos (a+b x))^{7/2} \sin (a+b x)}{117 b}-\frac{2 (d \cos (a+b x))^{11/2} \sin (a+b x)}{13 b d}+\frac{\left (14 d^4 \sqrt{d \cos (a+b x)}\right ) \int \sqrt{\cos (a+b x)} \, dx}{195 \sqrt{\cos (a+b x)}}\\ &=\frac{28 d^4 \sqrt{d \cos (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{195 b \sqrt{\cos (a+b x)}}+\frac{28 d^3 (d \cos (a+b x))^{3/2} \sin (a+b x)}{585 b}+\frac{4 d (d \cos (a+b x))^{7/2} \sin (a+b x)}{117 b}-\frac{2 (d \cos (a+b x))^{11/2} \sin (a+b x)}{13 b d}\\ \end{align*}

Mathematica [C]  time = 0.135806, size = 60, normalized size = 0.48 \[ \frac{d^2 \sqrt [4]{\cos ^2(a+b x)} \tan ^3(a+b x) (d \cos (a+b x))^{5/2} \, _2F_1\left (-\frac{7}{4},\frac{3}{2};\frac{5}{2};\sin ^2(a+b x)\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(d*Cos[a + b*x])^(5/2)*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[-7/4, 3/2, 5/2, Sin[a + b*x]^2]*Tan[a + b
*x]^3)/(3*b)

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Maple [A]  time = 0.095, size = 249, normalized size = 2. \begin{align*}{\frac{4\,{d}^{5}}{585\,b}\sqrt{d \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}} \left ( 2880\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{15}-11520\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{13}+19280\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{11}-17520\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{9}+9284\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{7}-2808\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{5}+425\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{3}+21\,\sqrt{ \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+1}{\it EllipticE} \left ( \cos \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) -21\,\cos \left ( 1/2\,bx+a/2 \right ) \right ){\frac{1}{\sqrt{-d \left ( 2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}- \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2} \right ) }}} \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{d \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/585*(d*(2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)*d^5*(2880*cos(1/2*b*x+1/2*a)^15-11520*cos(1/2*
b*x+1/2*a)^13+19280*cos(1/2*b*x+1/2*a)^11-17520*cos(1/2*b*x+1/2*a)^9+9284*cos(1/2*b*x+1/2*a)^7-2808*cos(1/2*b*
x+1/2*a)^5+425*cos(1/2*b*x+1/2*a)^3+21*(sin(1/2*b*x+1/2*a)^2)^(1/2)*(-2*cos(1/2*b*x+1/2*a)^2+1)^(1/2)*Elliptic
E(cos(1/2*b*x+1/2*a),2^(1/2))-21*cos(1/2*b*x+1/2*a))/(-d*(2*sin(1/2*b*x+1/2*a)^4-sin(1/2*b*x+1/2*a)^2))^(1/2)/
sin(1/2*b*x+1/2*a)/(d*(2*cos(1/2*b*x+1/2*a)^2-1))^(1/2)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cos \left (b x + a\right )\right )^{\frac{9}{2}} \sin \left (b x + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (d^{4} \cos \left (b x + a\right )^{6} - d^{4} \cos \left (b x + a\right )^{4}\right )} \sqrt{d \cos \left (b x + a\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(d^4*cos(b*x + a)^6 - d^4*cos(b*x + a)^4)*sqrt(d*cos(b*x + a)), 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((d*cos(b*x+a))**(9/2)*sin(b*x+a)**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cos \left (b x + a\right )\right )^{\frac{9}{2}} \sin \left (b x + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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