3.298 \(\int \frac{1}{(d \cos (a+b x))^{7/2} \sqrt{c \sin (a+b x)}} \, dx\)

Optimal. Leaf size=75 \[ \frac{8 \sqrt{c \sin (a+b x)}}{5 b c d^3 \sqrt{d \cos (a+b x)}}+\frac{2 \sqrt{c \sin (a+b x)}}{5 b c d (d \cos (a+b x))^{5/2}} \]

[Out]

(2*Sqrt[c*Sin[a + b*x]])/(5*b*c*d*(d*Cos[a + b*x])^(5/2)) + (8*Sqrt[c*Sin[a + b*x]])/(5*b*c*d^3*Sqrt[d*Cos[a +
b*x]])

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Rubi [A]  time = 0.11123, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2571, 2563} \[ \frac{8 \sqrt{c \sin (a+b x)}}{5 b c d^3 \sqrt{d \cos (a+b x)}}+\frac{2 \sqrt{c \sin (a+b x)}}{5 b c d (d \cos (a+b x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((d*Cos[a + b*x])^(7/2)*Sqrt[c*Sin[a + b*x]]),x]

[Out]

(2*Sqrt[c*Sin[a + b*x]])/(5*b*c*d*(d*Cos[a + b*x])^(5/2)) + (8*Sqrt[c*Sin[a + b*x]])/(5*b*c*d^3*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))^{7/2} \sqrt{c \sin (a+b x)}} \, dx &=\frac{2 \sqrt{c \sin (a+b x)}}{5 b c d (d \cos (a+b x))^{5/2}}+\frac{4 \int \frac{1}{(d \cos (a+b x))^{3/2} \sqrt{c \sin (a+b x)}} \, dx}{5 d^2}\\ &=\frac{2 \sqrt{c \sin (a+b x)}}{5 b c d (d \cos (a+b x))^{5/2}}+\frac{8 \sqrt{c \sin (a+b x)}}{5 b c d^3 \sqrt{d \cos (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.172872, size = 52, normalized size = 0.69 \[ \frac{2 (2 \cos (2 (a+b x))+3) \tan (a+b x)}{5 b d^2 \sqrt{c \sin (a+b x)} (d \cos (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d*Cos[a + b*x])^(7/2)*Sqrt[c*Sin[a + b*x]]),x]

[Out]

(2*(3 + 2*Cos[2*(a + b*x)])*Tan[a + b*x])/(5*b*d^2*(d*Cos[a + b*x])^(3/2)*Sqrt[c*Sin[a + b*x]])

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Maple [A]  time = 0.079, size = 50, normalized size = 0.7 \begin{align*}{\frac{ \left ( 8\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}+2 \right ) \sin \left ( bx+a \right ) \cos \left ( bx+a \right ) }{5\,b} \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(1/(d*cos(b*x+a))^(7/2)/(c*sin(b*x+a))^(1/2),x)

[Out]

2/5/b*(4*cos(b*x+a)^2+1)*sin(b*x+a)*cos(b*x+a)/(d*cos(b*x+a))^(7/2)/(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{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(1/(d*cos(b*x+a))^(7/2)/(c*sin(b*x+a))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.68717, size = 128, normalized size = 1.71 \begin{align*} \frac{2 \, \sqrt{d \cos \left (b x + a\right )}{\left (4 \, \cos \left (b x + a\right )^{2} + 1\right )} \sqrt{c \sin \left (b x + a\right )}}{5 \, b c d^{4} \cos \left (b x + a\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*sqrt(d*cos(b*x + a))*(4*cos(b*x + a)^2 + 1)*sqrt(c*sin(b*x + a))/(b*c*d^4*cos(b*x + a)^3)

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

[Out]

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