\(\int \frac {(d \cos (a+b x))^{7/2}}{\sqrt {c \sin (a+b x)}} \, dx\) [291]

Optimal result

Integrand size = 25, antiderivative size = 132 \[ \int \frac {(d \cos (a+b x))^{7/2}}{\sqrt {c \sin (a+b x)}} \, dx=\frac {5 d^3 \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}{6 b c}+\frac {d (d \cos (a+b x))^{5/2} \sqrt {c \sin (a+b x)}}{3 b c}+\frac {5 d^4 \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {\sin (2 a+2 b x)}}{12 b \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}} \]

[Out]

1/3*d*(d*cos(b*x+a))^(5/2)*(c*sin(b*x+a))^(1/2)/b/c+5/6*d^3*(d*cos(b*x+a))^(1/2)*(c*sin(b*x+a))^(1/2)/b/c-5/12
*d^4*(sin(a+1/4*Pi+b*x)^2)^(1/2)/sin(a+1/4*Pi+b*x)*EllipticF(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 = 0.11 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2649, 2653, 2720} \[ \int \frac {(d \cos (a+b x))^{7/2}}{\sqrt {c \sin (a+b x)}} \, dx=\frac {5 d^4 \sqrt {\sin (2 a+2 b x)} \operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right )}{12 b \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}+\frac {5 d^3 \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}{6 b c}+\frac {d \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{5/2}}{3 b c} \]

[In]

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

[Out]

(5*d^3*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]])/(6*b*c) + (d*(d*Cos[a + b*x])^(5/2)*Sqrt[c*Sin[a + b*x]])/(3
*b*c) + (5*d^4*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]])/(12*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 (d \cos (a+b x))^{5/2} \sqrt {c \sin (a+b x)}}{3 b c}+\frac {1}{6} \left (5 d^2\right ) \int \frac {(d \cos (a+b x))^{3/2}}{\sqrt {c \sin (a+b x)}} \, dx \\ & = \frac {5 d^3 \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}{6 b c}+\frac {d (d \cos (a+b x))^{5/2} \sqrt {c \sin (a+b x)}}{3 b c}+\frac {1}{12} \left (5 d^4\right ) \int \frac {1}{\sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}} \, dx \\ & = \frac {5 d^3 \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}{6 b c}+\frac {d (d \cos (a+b x))^{5/2} \sqrt {c \sin (a+b x)}}{3 b c}+\frac {\left (5 d^4 \sqrt {\sin (2 a+2 b x)}\right ) \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx}{12 \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}} \\ & = \frac {5 d^3 \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}{6 b c}+\frac {d (d \cos (a+b x))^{5/2} \sqrt {c \sin (a+b x)}}{3 b c}+\frac {5 d^4 \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {\sin (2 a+2 b x)}}{12 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 = 70, normalized size of antiderivative = 0.53 \[ \int \frac {(d \cos (a+b x))^{7/2}}{\sqrt {c \sin (a+b x)}} \, dx=\frac {2 (d \cos (a+b x))^{7/2} \cos ^2(a+b x)^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {1}{4},\frac {5}{4},\sin ^2(a+b x)\right ) \sec ^5(a+b x) \sqrt {c \sin (a+b x)}}{b c} \]

[In]

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

[Out]

(2*(d*Cos[a + b*x])^(7/2)*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[-5/4, 1/4, 5/4, Sin[a + b*x]^2]*Sec[a + b*x
]^5*Sqrt[c*Sin[a + b*x]])/(b*c)

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.51 (sec) , antiderivative size = 1727, normalized size of antiderivative = 13.08

method	result	size
default	\(\text {Expression too large to display}\)	\(1727\)

[In]

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

[Out]

1/48/b*2^(1/2)*(d*cos(b*x+a))^(1/2)*d^3/(c*sin(b*x+a))^(1/2)*(-6*I*(-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)*EllipticPi((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2-1/2*I,1/2*
2^(1/2))+6*I*(-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)*El
lipticPi((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))+6*I*(-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)*EllipticPi((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2+1/2*I,1/
2*2^(1/2))*sec(b*x+a)-6*I*(-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)*EllipticPi((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*sec(b*x+a)+8*2^(1/2)*cos(b*x+a)^
2*sin(b*x+a)-6*(-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)*
EllipticPi((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))+32*(-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))
-6*(-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)*EllipticPi((
-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))-6*(-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)*EllipticPi((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*s
ec(b*x+a)+32*(-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)*El
lipticF((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2*2^(1/2))*sec(b*x+a)-6*(-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)*EllipticPi((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2-1/2*I,1/2*
2^(1/2))*sec(b*x+a)-3*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)*ln(-2*2^(1/2)*(-sin(b*x+a)*cos(b*x+a)/(1
+cos(b*x+a))^2)^(1/2)*cot(b*x+a)-2*2^(1/2)*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)*csc(b*x+a)+2-2*cot(
b*x+a))+3*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)*ln(2*2^(1/2)*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^
2)^(1/2)*cot(b*x+a)+2*2^(1/2)*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)*csc(b*x+a)+2-2*cot(b*x+a))+6*(-s
in(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)*arctan((sin(b*x+a)*2^(1/2)*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))
^2)^(1/2)-cos(b*x+a)+1)/(cos(b*x+a)-1))+6*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)*arctan((sin(b*x+a)*2
^(1/2)*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)+cos(b*x+a)-1)/(cos(b*x+a)-1))+20*2^(1/2)*sin(b*x+a)-3*(
-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)*ln(-2*2^(1/2)*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)*c
ot(b*x+a)-2*2^(1/2)*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)*csc(b*x+a)+2-2*cot(b*x+a))*sec(b*x+a)+3*(-
sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)*ln(2*2^(1/2)*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)*cot
(b*x+a)+2*2^(1/2)*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)*csc(b*x+a)+2-2*cot(b*x+a))*sec(b*x+a)+6*(-si
n(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)*arctan((sin(b*x+a)*2^(1/2)*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^
2)^(1/2)-cos(b*x+a)+1)/(cos(b*x+a)-1))*sec(b*x+a)+6*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)*arctan((si
n(b*x+a)*2^(1/2)*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)+cos(b*x+a)-1)/(cos(b*x+a)-1))*sec(b*x+a))

Fricas [F]

\[ \int \frac {(d \cos (a+b x))^{7/2}}{\sqrt {c \sin (a+b x)}} \, dx=\int { \frac {\left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}}}{\sqrt {c \sin \left (b x + a\right )}} \,d x } \]

[In]

integrate((d*cos(b*x+a))^(7/2)/(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^3*cos(b*x + a)^3/(c*sin(b*x + a)), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {(d \cos (a+b x))^{7/2}}{\sqrt {c \sin (a+b x)}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {(d \cos (a+b x))^{7/2}}{\sqrt {c \sin (a+b x)}} \, dx=\int { \frac {\left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}}}{\sqrt {c \sin \left (b x + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {(d \cos (a+b x))^{7/2}}{\sqrt {c \sin (a+b x)}} \, dx=\int { \frac {\left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}}}{\sqrt {c \sin \left (b x + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(d \cos (a+b x))^{7/2}}{\sqrt {c \sin (a+b x)}} \, dx=\int \frac {{\left (d\,\cos \left (a+b\,x\right )\right )}^{7/2}}{\sqrt {c\,\sin \left (a+b\,x\right )}} \,d x \]

[In]

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

[Out]

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