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

Optimal. Leaf size=134 \[ \frac{4 (c \sin (a+b x))^{3/2}}{5 b c d^3 \sqrt{d \cos (a+b x)}}-\frac{4 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right ) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{5 b d^4 \sqrt{\sin (2 a+2 b x)}}+\frac{2 (c \sin (a+b x))^{3/2}}{5 b c d (d \cos (a+b x))^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.162821, antiderivative size = 134, 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 = {2571, 2572, 2639} \[ \frac{4 (c \sin (a+b x))^{3/2}}{5 b c d^3 \sqrt{d \cos (a+b x)}}-\frac{4 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right ) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{5 b d^4 \sqrt{\sin (2 a+2 b x)}}+\frac{2 (c \sin (a+b x))^{3/2}}{5 b c d (d \cos (a+b x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c*Sin[a + b*x])^(3/2))/(5*b*c*d*(d*Cos[a + b*x])^(5/2)) + (4*(c*Sin[a + b*x])^(3/2))/(5*b*c*d^3*Sqrt[d*Cos
[a + b*x]]) - (4*Sqrt[d*Cos[a + b*x]]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[c*Sin[a + b*x]])/(5*b*d^4*Sqrt[Sin[2*a
+ 2*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 2572

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

Mathematica [C]  time = 0.149874, size = 70, normalized size = 0.52 \[ \frac{2 \sqrt [4]{\cos ^2(a+b x)} \tan (a+b x) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)} \, _2F_1\left (\frac{3}{4},\frac{9}{4};\frac{7}{4};\sin ^2(a+b x)\right )}{3 b d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d*Cos[a + b*x]]*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[3/4, 9/4, 7/4, Sin[a + b*x]^2]*Sqrt[c*Sin[a +
b*x]]*Tan[a + b*x])/(3*b*d^4)

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Maple [B]  time = 0.144, size = 528, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^{4} \cos \left (b x + a\right )^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))/(d^4*cos(b*x + a)^4), 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((c*sin(b*x+a))**(1/2)/(d*cos(b*x+a))**(7/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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