3.291 \(\int \frac{(d \cos (a+b x))^{7/2}}{\sqrt{c \sin (a+b x)}} \, dx\)

Optimal. Leaf size=132 \[ \frac{5 d^3 \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{6 b c}+\frac{5 d^4 \sqrt{\sin (2 a+2 b x)} F\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{12 b \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}+\frac{d \sqrt{c \sin (a+b x)} (d \cos (a+b x))^{5/2}}{3 b c} \]

[Out]

(5*d^3*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]])/(6*b*c) + (d*(d*Cos[a + b*x])^(5/2)*Sqrt[c*Sin[a + b*x]])/(3
*b*c) + (5*d^4*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]])/(12*b*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a +
b*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.178004, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2569, 2573, 2641} \[ \frac{5 d^3 \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{6 b c}+\frac{5 d^4 \sqrt{\sin (2 a+2 b x)} F\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{12 b \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}+\frac{d \sqrt{c \sin (a+b x)} (d \cos (a+b x))^{5/2}}{3 b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*d^3*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]])/(6*b*c) + (d*(d*Cos[a + b*x])^(5/2)*Sqrt[c*Sin[a + b*x]])/(3
*b*c) + (5*d^4*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]])/(12*b*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a +
b*x]])

Rule 2569

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*(b*Sin[e +
f*x])^(n + 1)*(a*Cos[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Sin[e + f*x])^
n*(a*Cos[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 2573

Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[Sqrt[Sin[2*
e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b*Cos[e + f*x]]), Int[1/Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b,
e, f}, 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{(d \cos (a+b x))^{7/2}}{\sqrt{c \sin (a+b x)}} \, dx &=\frac{d (d \cos (a+b x))^{5/2} \sqrt{c \sin (a+b x)}}{3 b c}+\frac{1}{6} \left (5 d^2\right ) \int \frac{(d \cos (a+b x))^{3/2}}{\sqrt{c \sin (a+b x)}} \, dx\\ &=\frac{5 d^3 \sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}}{6 b c}+\frac{d (d \cos (a+b x))^{5/2} \sqrt{c \sin (a+b x)}}{3 b c}+\frac{1}{12} \left (5 d^4\right ) \int \frac{1}{\sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}} \, dx\\ &=\frac{5 d^3 \sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}}{6 b c}+\frac{d (d \cos (a+b x))^{5/2} \sqrt{c \sin (a+b x)}}{3 b c}+\frac{\left (5 d^4 \sqrt{\sin (2 a+2 b x)}\right ) \int \frac{1}{\sqrt{\sin (2 a+2 b x)}} \, dx}{12 \sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}}\\ &=\frac{5 d^3 \sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}}{6 b c}+\frac{d (d \cos (a+b x))^{5/2} \sqrt{c \sin (a+b x)}}{3 b c}+\frac{5 d^4 F\left (\left .a-\frac{\pi }{4}+b x\right |2\right ) \sqrt{\sin (2 a+2 b x)}}{12 b \sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}}\\ \end{align*}

Mathematica [C]  time = 0.111582, size = 70, normalized size = 0.53 \[ \frac{2 \cos ^2(a+b x)^{3/4} \sec ^5(a+b x) \sqrt{c \sin (a+b x)} (d \cos (a+b x))^{7/2} \, _2F_1\left (-\frac{5}{4},\frac{1}{4};\frac{5}{4};\sin ^2(a+b x)\right )}{b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.112, size = 216, normalized size = 1.6 \begin{align*} -{\frac{\sqrt{2}\sin \left ( bx+a \right ) }{12\,b \left ( -1+\cos \left ( bx+a \right ) \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{4}} \left ( 5\,\sin \left ( bx+a \right ) \sqrt{{\frac{1-\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}}\sqrt{{\frac{-1+\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}}\sqrt{{\frac{-1+\cos \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{1-\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}},1/2\,\sqrt{2} \right ) -2\, \left ( \cos \left ( bx+a \right ) \right ) ^{4}\sqrt{2}+2\, \left ( \cos \left ( bx+a \right ) \right ) ^{3}\sqrt{2}-5\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}\sqrt{2}+5\,\cos \left ( bx+a \right ) \sqrt{2} \right ) \left ( d\cos \left ( bx+a \right ) \right ) ^{{\frac{7}{2}}}{\frac{1}{\sqrt{c\sin \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cos(b*x+a))^(7/2)/(c*sin(b*x+a))^(1/2),x)

[Out]

-1/12/b*2^(1/2)*(5*sin(b*x+a)*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)+sin(b*x+a))/sin(b*x
+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticF(((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/
2))-2*cos(b*x+a)^4*2^(1/2)+2*cos(b*x+a)^3*2^(1/2)-5*cos(b*x+a)^2*2^(1/2)+5*cos(b*x+a)*2^(1/2))*(d*cos(b*x+a))^
(7/2)*sin(b*x+a)/(-1+cos(b*x+a))/cos(b*x+a)^4/(c*sin(b*x+a))^(1/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d \cos \left (b x + a\right )\right )^{\frac{7}{2}}}{\sqrt{c \sin \left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*cos(b*x + a))^(7/2)/sqrt(c*sin(b*x + a)), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d \cos \left (b x + a\right )} \sqrt{c \sin \left (b x + a\right )} d^{3} \cos \left (b x + a\right )^{3}}{c \sin \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))^(7/2)/(c*sin(b*x+a))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))*d^3*cos(b*x + a)^3/(c*sin(b*x + a)), x)

________________________________________________________________________________________

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))**(7/2)/(c*sin(b*x+a))**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d \cos \left (b x + a\right )\right )^{\frac{7}{2}}}{\sqrt{c \sin \left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*cos(b*x + a))^(7/2)/sqrt(c*sin(b*x + a)), x)