\(\int \frac {(d \cos (a+b x))^{3/2}}{\sqrt {c \sin (a+b x)}} \, dx\) [292]

Optimal result

Integrand size = 25, antiderivative size = 92 \[ \int \frac {(d \cos (a+b x))^{3/2}}{\sqrt {c \sin (a+b x)}} \, dx=\frac {d \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}{b c}+\frac {d^2 \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {\sin (2 a+2 b x)}}{2 b \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}} \]

[Out]

d*(d*cos(b*x+a))^(1/2)*(c*sin(b*x+a))^(1/2)/b/c-1/2*d^2*(sin(a+1/4*Pi+b*x)^2)^(1/2)/sin(a+1/4*Pi+b*x)*Elliptic
F(cos(a+1/4*Pi+b*x),2^(1/2))*sin(2*b*x+2*a)^(1/2)/b/(d*cos(b*x+a))^(1/2)/(c*sin(b*x+a))^(1/2)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2649, 2653, 2720} \[ \int \frac {(d \cos (a+b x))^{3/2}}{\sqrt {c \sin (a+b x)}} \, dx=\frac {d^2 \sqrt {\sin (2 a+2 b x)} \operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right )}{2 b \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}+\frac {d \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}{b c} \]

[In]

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

[Out]

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

Rule 2649

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

Rule 2653

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 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {d \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}{b c}+\frac {1}{2} d^2 \int \frac {1}{\sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}} \, dx \\ & = \frac {d \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}{b c}+\frac {\left (d^2 \sqrt {\sin (2 a+2 b x)}\right ) \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx}{2 \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}} \\ & = \frac {d \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}{b c}+\frac {d^2 \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {\sin (2 a+2 b x)}}{2 b \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.74 \[ \int \frac {(d \cos (a+b x))^{3/2}}{\sqrt {c \sin (a+b x)}} \, dx=\frac {2 d^2 \cos ^2(a+b x)^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{4},\frac {5}{4},\sin ^2(a+b x)\right ) \tan (a+b x)}{b \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}} \]

[In]

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

[Out]

(2*d^2*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[-1/4, 1/4, 5/4, Sin[a + b*x]^2]*Tan[a + b*x])/(b*Sqrt[d*Cos[a
+ b*x]]*Sqrt[c*Sin[a + b*x]])

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 203, normalized size of antiderivative = 2.21

method	result	size
default	\(\frac {\sqrt {2}\, \sqrt {d \cos \left (b x +a \right )}\, d \left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )+\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \sec \left (b x +a \right )+\sqrt {2}\, \sin \left (b x +a \right )\right )}{2 b \sqrt {c \sin \left (b x +a \right )}}\)	\(203\)

[In]

int((d*cos(b*x+a))^(3/2)/(c*sin(b*x+a))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2/b*2^(1/2)*(d*cos(b*x+a))^(1/2)*d/(c*sin(b*x+a))^(1/2)*((-cot(b*x+a)+csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*
x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticF((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2*2^(1/2))+(-cot(b*x+
a)+csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticF((-cot(b*x+a)+cs
c(b*x+a)+1)^(1/2),1/2*2^(1/2))*sec(b*x+a)+2^(1/2)*sin(b*x+a))

Fricas [F]

\[ \int \frac {(d \cos (a+b x))^{3/2}}{\sqrt {c \sin (a+b x)}} \, dx=\int { \frac {\left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}}}{\sqrt {c \sin \left (b x + a\right )}} \,d x } \]

[In]

integrate((d*cos(b*x+a))^(3/2)/(c*sin(b*x+a))^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {(d \cos (a+b x))^{3/2}}{\sqrt {c \sin (a+b x)}} \, dx=\int \frac {\left (d \cos {\left (a + b x \right )}\right )^{\frac {3}{2}}}{\sqrt {c \sin {\left (a + b x \right )}}}\, dx \]

[In]

integrate((d*cos(b*x+a))**(3/2)/(c*sin(b*x+a))**(1/2),x)

[Out]

Integral((d*cos(a + b*x))**(3/2)/sqrt(c*sin(a + b*x)), x)

Maxima [F]

\[ \int \frac {(d \cos (a+b x))^{3/2}}{\sqrt {c \sin (a+b x)}} \, dx=\int { \frac {\left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}}}{\sqrt {c \sin \left (b x + a\right )}} \,d x } \]

[In]

integrate((d*cos(b*x+a))^(3/2)/(c*sin(b*x+a))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {(d \cos (a+b x))^{3/2}}{\sqrt {c \sin (a+b x)}} \, dx=\int { \frac {\left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}}}{\sqrt {c \sin \left (b x + a\right )}} \,d x } \]

[In]

integrate((d*cos(b*x+a))^(3/2)/(c*sin(b*x+a))^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(d \cos (a+b x))^{3/2}}{\sqrt {c \sin (a+b x)}} \, dx=\int \frac {{\left (d\,\cos \left (a+b\,x\right )\right )}^{3/2}}{\sqrt {c\,\sin \left (a+b\,x\right )}} \,d x \]

[In]

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

[Out]

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