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

Optimal. Leaf size=99 \[ -\frac{2 \sin ^3(a+b x) \sqrt{d \cos (a+b x)}}{7 b d}-\frac{4 \sin (a+b x) \sqrt{d \cos (a+b x)}}{7 b d}+\frac{8 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{7 b \sqrt{d \cos (a+b x)}} \]

[Out]

(8*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2])/(7*b*Sqrt[d*Cos[a + b*x]]) - (4*Sqrt[d*Cos[a + b*x]]*Sin[a +
b*x])/(7*b*d) - (2*Sqrt[d*Cos[a + b*x]]*Sin[a + b*x]^3)/(7*b*d)

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Rubi [A]  time = 0.0981128, 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, 2642, 2641} \[ -\frac{2 \sin ^3(a+b x) \sqrt{d \cos (a+b x)}}{7 b d}-\frac{4 \sin (a+b x) \sqrt{d \cos (a+b x)}}{7 b d}+\frac{8 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{7 b \sqrt{d \cos (a+b x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2])/(7*b*Sqrt[d*Cos[a + b*x]]) - (4*Sqrt[d*Cos[a + b*x]]*Sin[a +
b*x])/(7*b*d) - (2*Sqrt[d*Cos[a + b*x]]*Sin[a + b*x]^3)/(7*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 2642

Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\sin ^4(a+b x)}{\sqrt{d \cos (a+b x)}} \, dx &=-\frac{2 \sqrt{d \cos (a+b x)} \sin ^3(a+b x)}{7 b d}+\frac{6}{7} \int \frac{\sin ^2(a+b x)}{\sqrt{d \cos (a+b x)}} \, dx\\ &=-\frac{4 \sqrt{d \cos (a+b x)} \sin (a+b x)}{7 b d}-\frac{2 \sqrt{d \cos (a+b x)} \sin ^3(a+b x)}{7 b d}+\frac{4}{7} \int \frac{1}{\sqrt{d \cos (a+b x)}} \, dx\\ &=-\frac{4 \sqrt{d \cos (a+b x)} \sin (a+b x)}{7 b d}-\frac{2 \sqrt{d \cos (a+b x)} \sin ^3(a+b x)}{7 b d}+\frac{\left (4 \sqrt{\cos (a+b x)}\right ) \int \frac{1}{\sqrt{\cos (a+b x)}} \, dx}{7 \sqrt{d \cos (a+b x)}}\\ &=\frac{8 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{7 b \sqrt{d \cos (a+b x)}}-\frac{4 \sqrt{d \cos (a+b x)} \sin (a+b x)}{7 b d}-\frac{2 \sqrt{d \cos (a+b x)} \sin ^3(a+b x)}{7 b d}\\ \end{align*}

Mathematica [C]  time = 0.0695362, size = 58, normalized size = 0.59 \[ \frac{d \sin ^5(a+b x) \cos ^2(a+b x)^{3/4} \, _2F_1\left (\frac{3}{4},\frac{5}{2};\frac{7}{2};\sin ^2(a+b x)\right )}{5 b (d \cos (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.063, size = 208, normalized size = 2.1 \begin{align*} -{\frac{8}{7\,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 ( 4\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{8}\cos \left ( 1/2\,bx+a/2 \right ) -6\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{6}\cos \left ( 1/2\,bx+a/2 \right ) +\sqrt{ \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) ,\sqrt{2} \right ) + \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}\cos \left ({\frac{bx}{2}}+{\frac{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(sin(b*x+a)^4/(d*cos(b*x+a))^(1/2),x)

[Out]

-8/7*(d*(2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)*(4*sin(1/2*b*x+1/2*a)^8*cos(1/2*b*x+1/2*a)-6*si
n(1/2*b*x+1/2*a)^6*cos(1/2*b*x+1/2*a)+(sin(1/2*b*x+1/2*a)^2)^(1/2)*(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*EllipticF(
cos(1/2*b*x+1/2*a),2^(1/2))+sin(1/2*b*x+1/2*a)^2*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 \frac{\sin \left (b x + a\right )^{4}}{\sqrt{d \cos \left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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