3.58 \(\int (c \cos ^m(a+b x))^{5/2} \, dx\)

Optimal. Leaf size=89 \[ -\frac {2 c^2 \sin (a+b x) \cos ^{2 m+1}(a+b x) \sqrt {c \cos ^m(a+b x)} \, _2F_1\left (\frac {1}{2},\frac {1}{4} (5 m+2);\frac {1}{4} (5 m+6);\cos ^2(a+b x)\right )}{b (5 m+2) \sqrt {\sin ^2(a+b x)}} \]

[Out]

-2*c^2*cos(b*x+a)^(1+2*m)*hypergeom([1/2, 1/2+5/4*m],[3/2+5/4*m],cos(b*x+a)^2)*sin(b*x+a)*(c*cos(b*x+a)^m)^(1/
2)/b/(2+5*m)/(sin(b*x+a)^2)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 89, 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} \[ -\frac {2 c^2 \sin (a+b x) \cos ^{2 m+1}(a+b x) \sqrt {c \cos ^m(a+b x)} \, _2F_1\left (\frac {1}{2},\frac {1}{4} (5 m+2);\frac {1}{4} (5 m+6);\cos ^2(a+b x)\right )}{b (5 m+2) \sqrt {\sin ^2(a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[(c*Cos[a + b*x]^m)^(5/2),x]

[Out]

(-2*c^2*Cos[a + b*x]^(1 + 2*m)*Sqrt[c*Cos[a + b*x]^m]*Hypergeometric2F1[1/2, (2 + 5*m)/4, (6 + 5*m)/4, Cos[a +
b*x]^2]*Sin[a + b*x])/(b*(2 + 5*m)*Sqrt[Sin[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 \left (c \cos ^m(a+b x)\right )^{5/2} \, dx &=\left (c^2 \cos ^{-\frac {m}{2}}(a+b x) \sqrt {c \cos ^m(a+b x)}\right ) \int \cos ^{\frac {5 m}{2}}(a+b x) \, dx\\ &=-\frac {2 c^2 \cos ^{1+2 m}(a+b x) \sqrt {c \cos ^m(a+b x)} \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2+5 m);\frac {1}{4} (6+5 m);\cos ^2(a+b x)\right ) \sin (a+b x)}{b (2+5 m) \sqrt {\sin ^2(a+b x)}}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 74, normalized size = 0.83 \[ -\frac {2 \sqrt {\sin ^2(a+b x)} \cot (a+b x) \left (c \cos ^m(a+b x)\right )^{5/2} \, _2F_1\left (\frac {1}{2},\frac {1}{4} (5 m+2);\frac {1}{4} (5 m+6);\cos ^2(a+b x)\right )}{b (5 m+2)} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Cos[a + b*x]^m)^(5/2),x]

[Out]

(-2*(c*Cos[a + b*x]^m)^(5/2)*Cot[a + b*x]*Hypergeometric2F1[1/2, (2 + 5*m)/4, (6 + 5*m)/4, Cos[a + b*x]^2]*Sqr
t[Sin[a + b*x]^2])/(b*(2 + 5*m))

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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*cos(b*x+a)^m)^(5/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \cos \left (b x + a\right )^{m}\right )^{\frac {5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*cos(b*x+a)^m)^(5/2),x, algorithm="giac")

[Out]

integrate((c*cos(b*x + a)^m)^(5/2), x)

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maple [F]  time = 0.41, size = 0, normalized size = 0.00 \[ \int \left (c \left (\cos ^{m}\left (b x +a \right )\right )\right )^{\frac {5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*cos(b*x+a)^m)^(5/2),x)

[Out]

int((c*cos(b*x+a)^m)^(5/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \cos \left (b x + a\right )^{m}\right )^{\frac {5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*cos(b*x+a)^m)^(5/2),x, algorithm="maxima")

[Out]

integrate((c*cos(b*x + a)^m)^(5/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (c\,{\cos \left (a+b\,x\right )}^m\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*cos(a + b*x)^m)^(5/2),x)

[Out]

int((c*cos(a + b*x)^m)^(5/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*cos(b*x+a)**m)**(5/2),x)

[Out]

Timed out

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