3.269 \(\int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{5/2}} \, dx\)

Optimal. Leaf size=98 \[ \frac {2 c \sqrt {c \sin (a+b x)}}{3 b d (d \cos (a+b x))^{3/2}}-\frac {c^2 \sqrt {\sin (2 a+2 b x)} F\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{3 b d^2 \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}} \]

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2566, 2573, 2641} \[ \frac {2 c \sqrt {c \sin (a+b x)}}{3 b d (d \cos (a+b x))^{3/2}}-\frac {c^2 \sqrt {\sin (2 a+2 b x)} F\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{3 b d^2 \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2566

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

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 {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{5/2}} \, dx &=\frac {2 c \sqrt {c \sin (a+b x)}}{3 b d (d \cos (a+b x))^{3/2}}-\frac {c^2 \int \frac {1}{\sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}} \, dx}{3 d^2}\\ &=\frac {2 c \sqrt {c \sin (a+b x)}}{3 b d (d \cos (a+b x))^{3/2}}-\frac {\left (c^2 \sqrt {\sin (2 a+2 b x)}\right ) \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx}{3 d^2 \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}\\ &=\frac {2 c \sqrt {c \sin (a+b x)}}{3 b d (d \cos (a+b x))^{3/2}}-\frac {c^2 F\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {\sin (2 a+2 b x)}}{3 b d^2 \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}\\ \end {align*}

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Mathematica [C]  time = 0.16, size = 67, normalized size = 0.68 \[ \frac {2 \cos ^2(a+b x)^{3/4} (c \sin (a+b x))^{5/2} \, _2F_1\left (\frac {5}{4},\frac {7}{4};\frac {9}{4};\sin ^2(a+b x)\right )}{5 b c d (d \cos (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[5/4, 7/4, 9/4, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(5/2))/(5*b*c*d*(d
*Cos[a + b*x])^(3/2))

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fricas [F]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )} c \sin \left (b x + a\right )}{d^{3} \cos \left (b x + a\right )^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*sin(b*x+a))^(3/2)/(d*cos(b*x+a))^(5/2),x, algorithm="fricas")

[Out]

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

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giac [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))^(3/2)/(d*cos(b*x+a))^(5/2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.12, size = 186, normalized size = 1.90 \[ \frac {\left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right ) \sin \left (b x +a \right ) \cos \left (b x +a \right )+\cos \left (b x +a \right ) \sqrt {2}-\sqrt {2}\right ) \left (c \sin \left (b x +a \right )\right )^{\frac {3}{2}} \cos \left (b x +a \right ) \sqrt {2}}{3 b \left (-1+\cos \left (b x +a \right )\right ) \left (d \cos \left (b x +a \right )\right )^{\frac {5}{2}} \sin \left (b x +a \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*sin(b*x+a))^(3/2)/(d*cos(b*x+a))^(5/2),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}}}{\left (d \cos \left (b x + a\right )\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*sin(b*x+a))^(3/2)/(d*cos(b*x+a))^(5/2),x, algorithm="maxima")

[Out]

integrate((c*sin(b*x + a))^(3/2)/(d*cos(b*x + a))^(5/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c\,\sin \left (a+b\,x\right )\right )}^{3/2}}{{\left (d\,\cos \left (a+b\,x\right )\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*sin(a + b*x))^(3/2)/(d*cos(a + b*x))^(5/2),x)

[Out]

int((c*sin(a + b*x))^(3/2)/(d*cos(a + b*x))^(5/2), 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))**(3/2)/(d*cos(b*x+a))**(5/2),x)

[Out]

Timed out

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