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

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

[Out]

(2*c^2*Cos[a + b*x]*Hypergeometric2F1[1/2, (2 + 5*m)/4, (6 + 5*m)/4, Sin[a + b*x]^2]*Sin[a + b*x]^(1 + 2*m)*Sq
rt[c*Sin[a + b*x]^m])/(b*(2 + 5*m)*Sqrt[Cos[a + b*x]^2])

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Rubi [A]  time = 0.0390368, antiderivative size = 89, 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} \[ \frac{2 c^2 \cos (a+b x) \sin ^{2 m+1}(a+b x) \sqrt{c \sin ^m(a+b x)} \, _2F_1\left (\frac{1}{2},\frac{1}{4} (5 m+2);\frac{1}{4} (5 m+6);\sin ^2(a+b x)\right )}{b (5 m+2) \sqrt{\cos ^2(a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x]^m)^(5/2),x]

[Out]

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

Mathematica [A]  time = 0.178953, size = 74, normalized size = 0.83 \[ \frac{2 \sqrt{\cos ^2(a+b x)} \tan (a+b x) \left (c \sin ^m(a+b x)\right )^{5/2} \, _2F_1\left (\frac{1}{2},\frac{1}{4} (5 m+2);\frac{1}{4} (5 m+6);\sin ^2(a+b x)\right )}{b (5 m+2)} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Sin[a + b*x]^m)^(5/2),x]

[Out]

(2*Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[1/2, (2 + 5*m)/4, (6 + 5*m)/4, Sin[a + b*x]^2]*(c*Sin[a + b*x]^m)^(5
/2)*Tan[a + b*x])/(b*(2 + 5*m))

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Maple [F]  time = 0.468, size = 0, normalized size = 0. \begin{align*} \int \left ( c \left ( \sin \left ( bx+a \right ) \right ) ^{m} \right ) ^{{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \sin \left (b x + a\right )^{m}\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \sin \left (b x + a\right )^{m}\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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