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

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

[Out]

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

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Rubi [A]  time = 0.16, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, 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*}

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Mathematica [C]  time = 0.13, 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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fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )}}{d^{4} \cos \left (b x + a\right )^{4}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c \sin \left (b x + a\right )}}{\left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}}}\,{d x} \]

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)

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maple [B]  time = 0.19, size = 528, normalized size = 3.94 \[ \frac {\left (4 \left (\cos ^{3}\left (b x +a \right )\right ) \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticE \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )-2 \left (\cos ^{3}\left (b x +a \right )\right ) \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )+4 \left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticE \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )-2 \left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )-2 \left (\cos ^{3}\left (b x +a \right )\right ) \sqrt {2}+\left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {2}+\sqrt {2}\right ) \sqrt {c \sin \left (b x +a \right )}\, \cos \left (b x +a \right ) \sqrt {2}}{5 b \left (d \cos \left (b x +a \right )\right )^{\frac {7}{2}} \sin \left (b x +a \right )} \]

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*(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*co
s(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))/si
n(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)*2^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c \sin \left (b x + a\right )}}{\left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {c\,\sin \left (a+b\,x\right )}}{{\left (d\,\cos \left (a+b\,x\right )\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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