3.1.22 \(\int \frac {1}{\sqrt {c \sin ^m(a+b x)}} \, dx\) [22]

Optimal. Leaf size=80 \[ \frac {2 \cos (a+b x) \, _2F_1\left (\frac {1}{2},\frac {2-m}{4};\frac {6-m}{4};\sin ^2(a+b x)\right ) \sin (a+b x)}{b (2-m) \sqrt {\cos ^2(a+b x)} \sqrt {c \sin ^m(a+b x)}} \]

[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)/b/(2-m)/(cos(b*x+a)^2)^(1/2)/(c*s
in(b*x+a)^m)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 80, 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 = {3287, 2722} \begin {gather*} \frac {2 \sin (a+b x) \cos (a+b x) \, _2F_1\left (\frac {1}{2},\frac {2-m}{4};\frac {6-m}{4};\sin ^2(a+b x)\right )}{b (2-m) \sqrt {\cos ^2(a+b x)} \sqrt {c \sin ^m(a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/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])/(b*(2 - m)*Sqrt[Cos
[a + b*x]^2]*Sqrt[c*Sin[a + b*x]^m])

Rule 2722

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1
)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rule 3287

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 \frac {1}{\sqrt {c \sin ^m(a+b x)}} \, dx &=\frac {\sin ^{\frac {m}{2}}(a+b x) \int \sin ^{-\frac {m}{2}}(a+b x) \, dx}{\sqrt {c \sin ^m(a+b x)}}\\ &=\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)}{b (2-m) \sqrt {\cos ^2(a+b x)} \sqrt {c \sin ^m(a+b x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.05, size = 72, normalized size = 0.90 \begin {gather*} -\frac {2 \sqrt {\cos ^2(a+b x)} \, _2F_1\left (\frac {1}{2},\frac {2-m}{4};\frac {6-m}{4};\sin ^2(a+b x)\right ) \tan (a+b x)}{b (-2+m) \sqrt {c \sin ^m(a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/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]*Tan[a + b*x])/(b*(-2 + m
)*Sqrt[c*Sin[a + b*x]^m])

________________________________________________________________________________________

Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {c \left (\sin ^{m}\left (b x +a \right )\right )}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {c \sin ^{m}{\left (a + b x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(c*sin(a + b*x)**m), x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*sin(a + b*x)^m)^(1/2),x)

[Out]

int(1/(c*sin(a + b*x)^m)^(1/2), x)

________________________________________________________________________________________