3.215 \(\int \sqrt{d \cos (a+b x)} \sin ^4(a+b x) \, dx\)

Optimal. Leaf size=99 \[ -\frac{2 \sin ^3(a+b x) (d \cos (a+b x))^{3/2}}{9 b d}-\frac{4 \sin (a+b x) (d \cos (a+b x))^{3/2}}{15 b d}+\frac{8 E\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{d \cos (a+b x)}}{15 b \sqrt{\cos (a+b x)}} \]

[Out]

(8*Sqrt[d*Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2])/(15*b*Sqrt[Cos[a + b*x]]) - (4*(d*Cos[a + b*x])^(3/2)*Sin[a
+ b*x])/(15*b*d) - (2*(d*Cos[a + b*x])^(3/2)*Sin[a + b*x]^3)/(9*b*d)

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Rubi [A]  time = 0.0964384, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2568, 2640, 2639} \[ -\frac{2 \sin ^3(a+b x) (d \cos (a+b x))^{3/2}}{9 b d}-\frac{4 \sin (a+b x) (d \cos (a+b x))^{3/2}}{15 b d}+\frac{8 E\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{d \cos (a+b x)}}{15 b \sqrt{\cos (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d*Cos[a + b*x]]*Sin[a + b*x]^4,x]

[Out]

(8*Sqrt[d*Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2])/(15*b*Sqrt[Cos[a + b*x]]) - (4*(d*Cos[a + b*x])^(3/2)*Sin[a
+ b*x])/(15*b*d) - (2*(d*Cos[a + b*x])^(3/2)*Sin[a + b*x]^3)/(9*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 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 \sqrt{d \cos (a+b x)} \sin ^4(a+b x) \, dx &=-\frac{2 (d \cos (a+b x))^{3/2} \sin ^3(a+b x)}{9 b d}+\frac{2}{3} \int \sqrt{d \cos (a+b x)} \sin ^2(a+b x) \, dx\\ &=-\frac{4 (d \cos (a+b x))^{3/2} \sin (a+b x)}{15 b d}-\frac{2 (d \cos (a+b x))^{3/2} \sin ^3(a+b x)}{9 b d}+\frac{4}{15} \int \sqrt{d \cos (a+b x)} \, dx\\ &=-\frac{4 (d \cos (a+b x))^{3/2} \sin (a+b x)}{15 b d}-\frac{2 (d \cos (a+b x))^{3/2} \sin ^3(a+b x)}{9 b d}+\frac{\left (4 \sqrt{d \cos (a+b x)}\right ) \int \sqrt{\cos (a+b x)} \, dx}{15 \sqrt{\cos (a+b x)}}\\ &=\frac{8 \sqrt{d \cos (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{15 b \sqrt{\cos (a+b x)}}-\frac{4 (d \cos (a+b x))^{3/2} \sin (a+b x)}{15 b d}-\frac{2 (d \cos (a+b x))^{3/2} \sin ^3(a+b x)}{9 b d}\\ \end{align*}

Mathematica [C]  time = 0.0652732, size = 58, normalized size = 0.59 \[ \frac{d \sin ^5(a+b x) \sqrt [4]{\cos ^2(a+b x)} \, _2F_1\left (\frac{1}{4},\frac{5}{2};\frac{7}{2};\sin ^2(a+b x)\right )}{5 b \sqrt{d \cos (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d*Cos[a + b*x]]*Sin[a + b*x]^4,x]

[Out]

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

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Maple [A]  time = 0.059, size = 221, normalized size = 2.2 \begin{align*} -{\frac{8\,d}{45\,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 ( 40\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{11}-120\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{9}+118\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{7}-36\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{5}-5\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{3}-3\,\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 ) +3\,\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))^(1/2)*sin(b*x+a)^4,x)

[Out]

-8/45*(d*(2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)*d*(40*cos(1/2*b*x+1/2*a)^11-120*cos(1/2*b*x+1/
2*a)^9+118*cos(1/2*b*x+1/2*a)^7-36*cos(1/2*b*x+1/2*a)^5-5*cos(1/2*b*x+1/2*a)^3-3*(sin(1/2*b*x+1/2*a)^2)^(1/2)*
(-2*cos(1/2*b*x+1/2*a)^2+1)^(1/2)*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2))+3*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 \sqrt{d \cos \left (b x + a\right )} \sin \left (b x + a\right )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^(1/2)*sin(b*x+a)^4,x, algorithm="maxima")

[Out]

integrate(sqrt(d*cos(b*x + a))*sin(b*x + a)^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\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))^(1/2)*sin(b*x+a)^4,x, algorithm="fricas")

[Out]

integral((cos(b*x + a)^4 - 2*cos(b*x + a)^2 + 1)*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))**(1/2)*sin(b*x+a)**4,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d \cos \left (b x + a\right )} \sin \left (b x + a\right )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^(1/2)*sin(b*x+a)^4,x, algorithm="giac")

[Out]

integrate(sqrt(d*cos(b*x + a))*sin(b*x + a)^4, x)