3.2.27 \(\int \csc ^2(a+b x) \sin ^m(2 a+2 b x) \, dx\) [127]

Optimal. Leaf size=85 \[ -\frac {\cos ^2(a+b x)^{\frac {1-m}{2}} \csc (a+b x) \, _2F_1\left (\frac {1-m}{2},\frac {1}{2} (-1+m);\frac {1+m}{2};\sin ^2(a+b x)\right ) \sec (a+b x) \sin ^m(2 a+2 b x)}{b (1-m)} \]

[Out]

-(cos(b*x+a)^2)^(1/2-1/2*m)*csc(b*x+a)*hypergeom([-1/2+1/2*m, 1/2-1/2*m],[1/2+1/2*m],sin(b*x+a)^2)*sec(b*x+a)*
sin(2*b*x+2*a)^m/b/(1-m)

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Rubi [A]
time = 0.06, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4395, 2657} \begin {gather*} -\frac {\csc (a+b x) \sec (a+b x) \sin ^m(2 a+2 b x) \cos ^2(a+b x)^{\frac {1-m}{2}} \, _2F_1\left (\frac {1-m}{2},\frac {m-1}{2};\frac {m+1}{2};\sin ^2(a+b x)\right )}{b (1-m)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^2*Sin[2*a + 2*b*x]^m,x]

[Out]

-(((Cos[a + b*x]^2)^((1 - m)/2)*Csc[a + b*x]*Hypergeometric2F1[(1 - m)/2, (-1 + m)/2, (1 + m)/2, Sin[a + b*x]^
2]*Sec[a + b*x]*Sin[2*a + 2*b*x]^m)/(b*(1 - m)))

Rule 2657

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b^(2*IntPart[
(n - 1)/2] + 1)*(b*Cos[e + f*x])^(2*FracPart[(n - 1)/2])*((a*Sin[e + f*x])^(m + 1)/(a*f*(m + 1)*(Cos[e + f*x]^
2)^FracPart[(n - 1)/2]))*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Sin[e + f*x]^2], x] /; FreeQ[{a, b
, e, f, m, n}, x]

Rule 4395

Int[((f_.)*sin[(a_.) + (b_.)*(x_)])^(n_.)*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> Dist[(g*Sin[c + d
*x])^p/(Cos[a + b*x]^p*(f*Sin[a + b*x])^p), Int[Cos[a + b*x]^p*(f*Sin[a + b*x])^(n + p), x], x] /; FreeQ[{a, b
, c, d, f, g, n, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] &&  !IntegerQ[p]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
time = 7.92, size = 938, normalized size = 11.04 \begin {gather*} \frac {2 \left ((-1+m) F_1\left (\frac {1+m}{2};-m,2 m;\frac {3+m}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+(1+m) F_1\left (\frac {1}{2} (-1+m);-m,2 m;\frac {1+m}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right ) \cot ^2\left (\frac {1}{2} (a+b x)\right )\right ) \csc ^2(a+b x) \sin ^m(2 (a+b x)) \tan \left (\frac {1}{2} (a+b x)\right )}{b \left (m (1+m) F_1\left (\frac {1}{2} (-1+m);-m,2 m;\frac {1+m}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )-(1+m) F_1\left (\frac {1}{2} (-1+m);-m,2 m;\frac {1+m}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right ) \csc ^2\left (\frac {1}{2} (a+b x)\right )+(-1+m) F_1\left (\frac {1+m}{2};-m,2 m;\frac {3+m}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right ) \sec ^2\left (\frac {1}{2} (a+b x)\right )-2 (-1+m) m \left (F_1\left (\frac {1+m}{2};1-m,2 m;\frac {3+m}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+2 F_1\left (\frac {1+m}{2};-m,1+2 m;\frac {3+m}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )\right ) \sec ^2\left (\frac {1}{2} (a+b x)\right )+(-1+m) m F_1\left (\frac {1+m}{2};-m,2 m;\frac {3+m}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right ) (-2+3 \cos (a+b x)) \sec (a+b x)+m (1+m) F_1\left (\frac {1}{2} (-1+m);-m,2 m;\frac {1+m}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right ) (-2+3 \cos (a+b x)) \cot ^2\left (\frac {1}{2} (a+b x)\right ) \sec (a+b x)+(-1+m) m F_1\left (\frac {1+m}{2};-m,2 m;\frac {3+m}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right ) \tan ^2\left (\frac {1}{2} (a+b x)\right )-\frac {2 (-1+m) m (1+m) \left (F_1\left (\frac {3+m}{2};1-m,2 m;\frac {5+m}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+2 F_1\left (\frac {3+m}{2};-m,1+2 m;\frac {5+m}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )\right ) \sec ^2\left (\frac {1}{2} (a+b x)\right ) \tan ^2\left (\frac {1}{2} (a+b x)\right )}{3+m}+2 m (1+m) F_1\left (\frac {1}{2} (-1+m);-m,2 m;\frac {1+m}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right ) \cot \left (\frac {1}{2} (a+b x)\right ) \tan (a+b x)+2 (-1+m) m F_1\left (\frac {1+m}{2};-m,2 m;\frac {3+m}{2};\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right ) \tan \left (\frac {1}{2} (a+b x)\right ) \tan (a+b x)\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Csc[a + b*x]^2*Sin[2*a + 2*b*x]^m,x]

[Out]

(2*((-1 + m)*AppellF1[(1 + m)/2, -m, 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + (1 + m)*Appell
F1[(-1 + m)/2, -m, 2*m, (1 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cot[(a + b*x)/2]^2)*Csc[a + b*x]^2
*Sin[2*(a + b*x)]^m*Tan[(a + b*x)/2])/(b*(m*(1 + m)*AppellF1[(-1 + m)/2, -m, 2*m, (1 + m)/2, Tan[(a + b*x)/2]^
2, -Tan[(a + b*x)/2]^2] - (1 + m)*AppellF1[(-1 + m)/2, -m, 2*m, (1 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/
2]^2]*Csc[(a + b*x)/2]^2 + (-1 + m)*AppellF1[(1 + m)/2, -m, 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)
/2]^2]*Sec[(a + b*x)/2]^2 - 2*(-1 + m)*m*(AppellF1[(1 + m)/2, 1 - m, 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[
(a + b*x)/2]^2] + 2*AppellF1[(1 + m)/2, -m, 1 + 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*Sec[
(a + b*x)/2]^2 + (-1 + m)*m*AppellF1[(1 + m)/2, -m, 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*(
-2 + 3*Cos[a + b*x])*Sec[a + b*x] + m*(1 + m)*AppellF1[(-1 + m)/2, -m, 2*m, (1 + m)/2, Tan[(a + b*x)/2]^2, -Ta
n[(a + b*x)/2]^2]*(-2 + 3*Cos[a + b*x])*Cot[(a + b*x)/2]^2*Sec[a + b*x] + (-1 + m)*m*AppellF1[(1 + m)/2, -m, 2
*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Tan[(a + b*x)/2]^2 - (2*(-1 + m)*m*(1 + m)*(AppellF1[(
3 + m)/2, 1 - m, 2*m, (5 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 2*AppellF1[(3 + m)/2, -m, 1 + 2*m,
(5 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*Sec[(a + b*x)/2]^2*Tan[(a + b*x)/2]^2)/(3 + m) + 2*m*(1
+ m)*AppellF1[(-1 + m)/2, -m, 2*m, (1 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cot[(a + b*x)/2]*Tan[a
+ b*x] + 2*(-1 + m)*m*AppellF1[(1 + m)/2, -m, 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Tan[(a
+ b*x)/2]*Tan[a + b*x]))

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Maple [F]
time = 0.23, size = 0, normalized size = 0.00 \[\int \left (\csc ^{2}\left (x b +a \right )\right ) \left (\sin ^{m}\left (2 x b +2 a \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^2*sin(2*b*x+2*a)^m,x)

[Out]

int(csc(b*x+a)^2*sin(2*b*x+2*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(csc(b*x+a)^2*sin(2*b*x+2*a)^m,x, algorithm="maxima")

[Out]

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

[Out]

integral(sin(2*b*x + 2*a)^m*csc(b*x + a)^2, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sin ^{m}{\left (2 a + 2 b x \right )} \csc ^{2}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)**2*sin(2*b*x+2*a)**m,x)

[Out]

Integral(sin(2*a + 2*b*x)**m*csc(a + b*x)**2, 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(csc(b*x+a)^2*sin(2*b*x+2*a)^m,x, algorithm="giac")

[Out]

integrate(sin(2*b*x + 2*a)^m*csc(b*x + a)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\sin \left (2\,a+2\,b\,x\right )}^m}{{\sin \left (a+b\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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