3.21 \(\int \sqrt {c \sin ^m(a+b x)} \, dx\)

Optimal. Leaf size=74 \[ \frac {2 \sin (a+b x) \cos (a+b x) \sqrt {c \sin ^m(a+b x)} \, _2F_1\left (\frac {1}{2},\frac {m+2}{4};\frac {m+6}{4};\sin ^2(a+b x)\right )}{b (m+2) \sqrt {\cos ^2(a+b x)}} \]

[Out]

2*cos(b*x+a)*hypergeom([1/2, 1/2+1/4*m],[3/2+1/4*m],sin(b*x+a)^2)*sin(b*x+a)*(c*sin(b*x+a)^m)^(1/2)/b/(2+m)/(c
os(b*x+a)^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3208, 2643} \[ \frac {2 \sin (a+b x) \cos (a+b x) \sqrt {c \sin ^m(a+b x)} \, _2F_1\left (\frac {1}{2},\frac {m+2}{4};\frac {m+6}{4};\sin ^2(a+b x)\right )}{b (m+2) \sqrt {\cos ^2(a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*Sin[a + b*x]^m],x]

[Out]

(2*Cos[a + b*x]*Hypergeometric2F1[1/2, (2 + m)/4, (6 + m)/4, Sin[a + b*x]^2]*Sin[a + b*x]*Sqrt[c*Sin[a + b*x]^
m])/(b*(2 + m)*Sqrt[Cos[a + b*x]^2])

Rule 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rule 3208

Int[(u_.)*((b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[(b^IntPart[p]*(b*(c*Sin[e + f*x
])^n)^FracPart[p])/(c*Sin[e + f*x])^(n*FracPart[p]), Int[ActivateTrig[u]*(c*Sin[e + f*x])^(n*p), x], x] /; Fre
eQ[{b, c, e, f, n, p}, x] &&  !IntegerQ[p] &&  !IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x]
)^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])

Rubi steps

\begin {align*} \int \sqrt {c \sin ^m(a+b x)} \, dx &=\left (\sin ^{-\frac {m}{2}}(a+b x) \sqrt {c \sin ^m(a+b x)}\right ) \int \sin ^{\frac {m}{2}}(a+b x) \, dx\\ &=\frac {2 \cos (a+b x) \, _2F_1\left (\frac {1}{2},\frac {2+m}{4};\frac {6+m}{4};\sin ^2(a+b x)\right ) \sin (a+b x) \sqrt {c \sin ^m(a+b x)}}{b (2+m) \sqrt {\cos ^2(a+b x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 68, normalized size = 0.92 \[ \frac {2 \sqrt {\cos ^2(a+b x)} \tan (a+b x) \sqrt {c \sin ^m(a+b x)} \, _2F_1\left (\frac {1}{2},\frac {m+2}{4};\frac {m+6}{4};\sin ^2(a+b x)\right )}{b (m+2)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c*Sin[a + b*x]^m],x]

[Out]

(2*Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[1/2, (2 + m)/4, (6 + m)/4, Sin[a + b*x]^2]*Sqrt[c*Sin[a + b*x]^m]*Ta
n[a + b*x])/(b*(2 + m))

________________________________________________________________________________________

fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(c*sin(b*x + a)^m), x)

________________________________________________________________________________________

maple [F]  time = 0.66, size = 0, normalized size = 0.00 \[ \int \sqrt {c \left (\sin ^{m}\left (b x +a \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*sin(b*x+a)^m)^(1/2),x)

[Out]

int((c*sin(b*x+a)^m)^(1/2),x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(c*sin(b*x + a)^m), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*sin(a + b*x)^m)^(1/2),x)

[Out]

int((c*sin(a + b*x)^m)^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {c \sin ^{m}{\left (a + b x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*sin(b*x+a)**m)**(1/2),x)

[Out]

Integral(sqrt(c*sin(a + b*x)**m), x)

________________________________________________________________________________________