3.59 \(\int (c \cos ^m(a+b x))^{3/2} \, dx\)

Optimal. Leaf size=83 \[ -\frac{2 c \sin (a+b x) \cos ^{m+1}(a+b x) \sqrt{c \cos ^m(a+b x)} \, _2F_1\left (\frac{1}{2},\frac{1}{4} (3 m+2);\frac{3 (m+2)}{4};\cos ^2(a+b x)\right )}{b (3 m+2) \sqrt{\sin ^2(a+b x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.039055, antiderivative size = 83, normalized size of antiderivative = 1., 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} \[ -\frac{2 c \sin (a+b x) \cos ^{m+1}(a+b x) \sqrt{c \cos ^m(a+b x)} \, _2F_1\left (\frac{1}{2},\frac{1}{4} (3 m+2);\frac{3 (m+2)}{4};\cos ^2(a+b x)\right )}{b (3 m+2) \sqrt{\sin ^2(a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[(c*Cos[a + b*x]^m)^(3/2),x]

[Out]

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

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]])

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]

Rubi steps

\begin{align*} \int \left (c \cos ^m(a+b x)\right )^{3/2} \, dx &=\left (c \cos ^{-\frac{m}{2}}(a+b x) \sqrt{c \cos ^m(a+b x)}\right ) \int \cos ^{\frac{3 m}{2}}(a+b x) \, dx\\ &=-\frac{2 c \cos ^{1+m}(a+b x) \sqrt{c \cos ^m(a+b x)} \, _2F_1\left (\frac{1}{2},\frac{1}{4} (2+3 m);\frac{3 (2+m)}{4};\cos ^2(a+b x)\right ) \sin (a+b x)}{b (2+3 m) \sqrt{\sin ^2(a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.118886, size = 72, normalized size = 0.87 \[ -\frac{2 \sqrt{\sin ^2(a+b x)} \cot (a+b x) \left (c \cos ^m(a+b x)\right )^{3/2} \, _2F_1\left (\frac{1}{2},\frac{1}{4} (3 m+2);\frac{3 (m+2)}{4};\cos ^2(a+b x)\right )}{b (3 m+2)} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Cos[a + b*x]^m)^(3/2),x]

[Out]

(-2*(c*Cos[a + b*x]^m)^(3/2)*Cot[a + b*x]*Hypergeometric2F1[1/2, (2 + 3*m)/4, (3*(2 + m))/4, Cos[a + b*x]^2]*S
qrt[Sin[a + b*x]^2])/(b*(2 + 3*m))

________________________________________________________________________________________

Maple [F]  time = 0.2, size = 0, normalized size = 0. \begin{align*} \int \left ( c \left ( \cos \left ( bx+a \right ) \right ) ^{m} \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*cos(b*x+a)^m)^(3/2),x)

[Out]

int((c*cos(b*x+a)^m)^(3/2),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \cos \left (b x + a\right )^{m}\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*cos(b*x + a)^m)^(3/2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \cos ^{m}{\left (a + b x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*cos(b*x+a)**m)**(3/2),x)

[Out]

Integral((c*cos(a + b*x)**m)**(3/2), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \cos \left (b x + a\right )^{m}\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*cos(b*x + a)^m)^(3/2), x)