3.299 \(\int \frac{1}{(d \cos (a+b x))^{11/2} \sqrt{c \sin (a+b x)}} \, dx\)

Optimal. Leaf size=112 \[ \frac{64 \sqrt{c \sin (a+b x)}}{45 b c d^5 \sqrt{d \cos (a+b x)}}+\frac{16 \sqrt{c \sin (a+b x)}}{45 b c d^3 (d \cos (a+b x))^{5/2}}+\frac{2 \sqrt{c \sin (a+b x)}}{9 b c d (d \cos (a+b x))^{9/2}} \]

[Out]

(2*Sqrt[c*Sin[a + b*x]])/(9*b*c*d*(d*Cos[a + b*x])^(9/2)) + (16*Sqrt[c*Sin[a + b*x]])/(45*b*c*d^3*(d*Cos[a + b
*x])^(5/2)) + (64*Sqrt[c*Sin[a + b*x]])/(45*b*c*d^5*Sqrt[d*Cos[a + b*x]])

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Rubi [A]  time = 0.168754, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2571, 2563} \[ \frac{64 \sqrt{c \sin (a+b x)}}{45 b c d^5 \sqrt{d \cos (a+b x)}}+\frac{16 \sqrt{c \sin (a+b x)}}{45 b c d^3 (d \cos (a+b x))^{5/2}}+\frac{2 \sqrt{c \sin (a+b x)}}{9 b c d (d \cos (a+b x))^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((d*Cos[a + b*x])^(11/2)*Sqrt[c*Sin[a + b*x]]),x]

[Out]

(2*Sqrt[c*Sin[a + b*x]])/(9*b*c*d*(d*Cos[a + b*x])^(9/2)) + (16*Sqrt[c*Sin[a + b*x]])/(45*b*c*d^3*(d*Cos[a + b
*x])^(5/2)) + (64*Sqrt[c*Sin[a + b*x]])/(45*b*c*d^5*Sqrt[d*Cos[a + b*x]])

Rule 2571

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

Rule 2563

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

Rubi steps

\begin{align*} \int \frac{1}{(d \cos (a+b x))^{11/2} \sqrt{c \sin (a+b x)}} \, dx &=\frac{2 \sqrt{c \sin (a+b x)}}{9 b c d (d \cos (a+b x))^{9/2}}+\frac{8 \int \frac{1}{(d \cos (a+b x))^{7/2} \sqrt{c \sin (a+b x)}} \, dx}{9 d^2}\\ &=\frac{2 \sqrt{c \sin (a+b x)}}{9 b c d (d \cos (a+b x))^{9/2}}+\frac{16 \sqrt{c \sin (a+b x)}}{45 b c d^3 (d \cos (a+b x))^{5/2}}+\frac{32 \int \frac{1}{(d \cos (a+b x))^{3/2} \sqrt{c \sin (a+b x)}} \, dx}{45 d^4}\\ &=\frac{2 \sqrt{c \sin (a+b x)}}{9 b c d (d \cos (a+b x))^{9/2}}+\frac{16 \sqrt{c \sin (a+b x)}}{45 b c d^3 (d \cos (a+b x))^{5/2}}+\frac{64 \sqrt{c \sin (a+b x)}}{45 b c d^5 \sqrt{d \cos (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.244228, size = 67, normalized size = 0.6 \[ \frac{2 (20 \cos (2 (a+b x))+4 \cos (4 (a+b x))+21) \sec ^5(a+b x) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{45 b c d^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d*Cos[a + b*x])^(11/2)*Sqrt[c*Sin[a + b*x]]),x]

[Out]

(2*Sqrt[d*Cos[a + b*x]]*(21 + 20*Cos[2*(a + b*x)] + 4*Cos[4*(a + b*x)])*Sec[a + b*x]^5*Sqrt[c*Sin[a + b*x]])/(
45*b*c*d^6)

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Maple [A]  time = 0.088, size = 60, normalized size = 0.5 \begin{align*}{\frac{ \left ( 64\, \left ( \cos \left ( bx+a \right ) \right ) ^{4}+16\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}+10 \right ) \sin \left ( bx+a \right ) \cos \left ( bx+a \right ) }{45\,b} \left ( d\cos \left ( bx+a \right ) \right ) ^{-{\frac{11}{2}}}{\frac{1}{\sqrt{c\sin \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*cos(b*x+a))^(11/2)/(c*sin(b*x+a))^(1/2),x)

[Out]

2/45/b*(32*cos(b*x+a)^4+8*cos(b*x+a)^2+5)*sin(b*x+a)*cos(b*x+a)/(d*cos(b*x+a))^(11/2)/(c*sin(b*x+a))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d \cos \left (b x + a\right )\right )^{\frac{11}{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(1/(d*cos(b*x+a))^(11/2)/(c*sin(b*x+a))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((d*cos(b*x + a))^(11/2)*sqrt(c*sin(b*x + a))), x)

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Fricas [A]  time = 3.36594, size = 157, normalized size = 1.4 \begin{align*} \frac{2 \,{\left (32 \, \cos \left (b x + a\right )^{4} + 8 \, \cos \left (b x + a\right )^{2} + 5\right )} \sqrt{d \cos \left (b x + a\right )} \sqrt{c \sin \left (b x + a\right )}}{45 \, b c d^{6} \cos \left (b x + a\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*cos(b*x+a))^(11/2)/(c*sin(b*x+a))^(1/2),x, algorithm="fricas")

[Out]

2/45*(32*cos(b*x + a)^4 + 8*cos(b*x + a)^2 + 5)*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))/(b*c*d^6*cos(b*x + a
)^5)

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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(1/(d*cos(b*x+a))**(11/2)/(c*sin(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{1}{\left (d \cos \left (b x + a\right )\right )^{\frac{11}{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(1/(d*cos(b*x+a))^(11/2)/(c*sin(b*x+a))^(1/2),x, algorithm="giac")

[Out]

integrate(1/((d*cos(b*x + a))^(11/2)*sqrt(c*sin(b*x + a))), x)