3.4.47 \(\int (c \sin (a+b x))^m \, dx\) [347]

Optimal. Leaf size=68 \[ \frac {\cos (a+b x) \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};\sin ^2(a+b x)\right ) (c \sin (a+b x))^{1+m}}{b c (1+m) \sqrt {\cos ^2(a+b x)}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2722} \begin {gather*} \frac {\cos (a+b x) (c \sin (a+b x))^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(a+b x)\right )}{b c (m+1) \sqrt {\cos ^2(a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x])^m,x]

[Out]

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

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]

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 63, normalized size = 0.93 \begin {gather*} \frac {\sqrt {\cos ^2(a+b x)} \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};\sin ^2(a+b x)\right ) (c \sin (a+b x))^m \tan (a+b x)}{b (1+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*Sin[a + b*x])^m,x]

[Out]

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

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (c \sin \left (b x +a \right )\right )^{m}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate((c*sin(b*x + a))^m, x)

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Fricas [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((c*sin(b*x+a))^m,x, algorithm="fricas")

[Out]

integral((c*sin(b*x + a))^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c*sin(a + b*x))**m, x)

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

[Out]

integrate((c*sin(b*x + a))^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*sin(a + b*x))^m,x)

[Out]

int((c*sin(a + b*x))^m, x)

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