3.3.14 \(\int (d \sin (e+f x))^m (a+b \sin (e+f x))^2 \, dx\) [214]

3.3.14.1 Optimal result

Integrand size = 23, antiderivative size = 194 \[ \int (d \sin (e+f x))^m (a+b \sin (e+f x))^2 \, dx=-\frac {b^2 \cos (e+f x) (d \sin (e+f x))^{1+m}}{d f (2+m)}+\frac {\left (b^2 (1+m)+a^2 (2+m)\right ) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(e+f x)\right ) (d \sin (e+f x))^{1+m}}{d f (1+m) (2+m) \sqrt {\cos ^2(e+f x)}}+\frac {2 a b \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},\sin ^2(e+f x)\right ) (d \sin (e+f x))^{2+m}}{d^2 f (2+m) \sqrt {\cos ^2(e+f x)}} \]

output

-b^2*cos(f*x+e)*(d*sin(f*x+e))^(1+m)/d/f/(2+m)+(b^2*(1+m)+a^2*(2+m))*cos(f 
*x+e)*hypergeom([1/2, 1/2+1/2*m],[3/2+1/2*m],sin(f*x+e)^2)*(d*sin(f*x+e))^ 
(1+m)/d/f/(1+m)/(2+m)/(cos(f*x+e)^2)^(1/2)+2*a*b*cos(f*x+e)*hypergeom([1/2 
, 1+1/2*m],[2+1/2*m],sin(f*x+e)^2)*(d*sin(f*x+e))^(2+m)/d^2/f/(2+m)/(cos(f 
*x+e)^2)^(1/2)
 

3.3.14.2 Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.86 \[ \int (d \sin (e+f x))^m (a+b \sin (e+f x))^2 \, dx=\frac {\sqrt {\cos ^2(e+f x)} (d \sin (e+f x))^m \left (a^2 \left (6+5 m+m^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(e+f x)\right )+b (1+m) \sin (e+f x) \left (2 a (3+m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},\sin ^2(e+f x)\right )+b (2+m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},\sin ^2(e+f x)\right ) \sin (e+f x)\right )\right ) \tan (e+f x)}{f (1+m) (2+m) (3+m)} \]

input

Integrate[(d*Sin[e + f*x])^m*(a + b*Sin[e + f*x])^2,x]
 

output

(Sqrt[Cos[e + f*x]^2]*(d*Sin[e + f*x])^m*(a^2*(6 + 5*m + m^2)*Hypergeometr 
ic2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[e + f*x]^2] + b*(1 + m)*Sin[e + f*x]* 
(2*a*(3 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Sin[e + f*x]^2] 
+ b*(2 + m)*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, Sin[e + f*x]^2]*S 
in[e + f*x]))*Tan[e + f*x])/(f*(1 + m)*(2 + m)*(3 + m))
 

3.3.14.3 Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 3268, 3042, 3122, 3493, 3042, 3122}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (a+b \sin (e+f x))^2 (d \sin (e+f x))^m \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a+b \sin (e+f x))^2 (d \sin (e+f x))^mdx\)

\(\Big \downarrow \) 3268

\(\displaystyle \int (d \sin (e+f x))^m \left (a^2+b^2 \sin ^2(e+f x)\right )dx+\frac {2 a b \int (d \sin (e+f x))^{m+1}dx}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (d \sin (e+f x))^m \left (a^2+b^2 \sin (e+f x)^2\right )dx+\frac {2 a b \int (d \sin (e+f x))^{m+1}dx}{d}\)

\(\Big \downarrow \) 3122

\(\displaystyle \int (d \sin (e+f x))^m \left (a^2+b^2 \sin (e+f x)^2\right )dx+\frac {2 a b \cos (e+f x) (d \sin (e+f x))^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},\sin ^2(e+f x)\right )}{d^2 f (m+2) \sqrt {\cos ^2(e+f x)}}\)

\(\Big \downarrow \) 3493

\(\displaystyle \left (a^2+\frac {b^2 (m+1)}{m+2}\right ) \int (d \sin (e+f x))^mdx+\frac {2 a b \cos (e+f x) (d \sin (e+f x))^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},\sin ^2(e+f x)\right )}{d^2 f (m+2) \sqrt {\cos ^2(e+f x)}}-\frac {b^2 \cos (e+f x) (d \sin (e+f x))^{m+1}}{d f (m+2)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \left (a^2+\frac {b^2 (m+1)}{m+2}\right ) \int (d \sin (e+f x))^mdx+\frac {2 a b \cos (e+f x) (d \sin (e+f x))^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},\sin ^2(e+f x)\right )}{d^2 f (m+2) \sqrt {\cos ^2(e+f x)}}-\frac {b^2 \cos (e+f x) (d \sin (e+f x))^{m+1}}{d f (m+2)}\)

\(\Big \downarrow \) 3122

\(\displaystyle \frac {\left (a^2+\frac {b^2 (m+1)}{m+2}\right ) \cos (e+f x) (d \sin (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(e+f x)\right )}{d f (m+1) \sqrt {\cos ^2(e+f x)}}+\frac {2 a b \cos (e+f x) (d \sin (e+f x))^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},\sin ^2(e+f x)\right )}{d^2 f (m+2) \sqrt {\cos ^2(e+f x)}}-\frac {b^2 \cos (e+f x) (d \sin (e+f x))^{m+1}}{d f (m+2)}\)

input

Int[(d*Sin[e + f*x])^m*(a + b*Sin[e + f*x])^2,x]
 

output

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

3.3.14.3.1 Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3122

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]))*Hypergeometric2 
F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n}, x] 
 &&  !IntegerQ[2*n]
 

rule 3268

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x 
_)])^2, x_Symbol] :> Simp[2*c*(d/b)   Int[(b*Sin[e + f*x])^(m + 1), x], x] 
+ Int[(b*Sin[e + f*x])^m*(c^2 + d^2*Sin[e + f*x]^2), x] /; FreeQ[{b, c, d, 
e, f, m}, x]
 

rule 3493

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*( 
x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((b*Sin[e + f*x])^(m + 1)/(b*f 
*(m + 2))), x] + Simp[(A*(m + 2) + C*(m + 1))/(m + 2)   Int[(b*Sin[e + f*x] 
)^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] &&  !LtQ[m, -1]
 

3.3.14.4 Maple [F]

\[\int \left (d \sin \left (f x +e \right )\right )^{m} \left (a +b \sin \left (f x +e \right )\right )^{2}d x\]

input

int((d*sin(f*x+e))^m*(a+b*sin(f*x+e))^2,x)
 

output

int((d*sin(f*x+e))^m*(a+b*sin(f*x+e))^2,x)
 

3.3.14.5 Fricas [F]

\[ \int (d \sin (e+f x))^m (a+b \sin (e+f x))^2 \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{2} \left (d \sin \left (f x + e\right )\right )^{m} \,d x } \]

input

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

output

integral(-(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)*(d*sin(f*x 
 + e))^m, x)
 

3.3.14.6 Sympy [F(-1)]

Timed out. \[ \int (d \sin (e+f x))^m (a+b \sin (e+f x))^2 \, dx=\text {Timed out} \]

input

integrate((d*sin(f*x+e))**m*(a+b*sin(f*x+e))**2,x)
 

output

Timed out
 

3.3.14.7 Maxima [F]

\[ \int (d \sin (e+f x))^m (a+b \sin (e+f x))^2 \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{2} \left (d \sin \left (f x + e\right )\right )^{m} \,d x } \]

input

integrate((d*sin(f*x+e))^m*(a+b*sin(f*x+e))^2,x, algorithm="maxima")
 

output

integrate((b*sin(f*x + e) + a)^2*(d*sin(f*x + e))^m, x)
 

3.3.14.8 Giac [F]

\[ \int (d \sin (e+f x))^m (a+b \sin (e+f x))^2 \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{2} \left (d \sin \left (f x + e\right )\right )^{m} \,d x } \]

input

integrate((d*sin(f*x+e))^m*(a+b*sin(f*x+e))^2,x, algorithm="giac")
 

output

integrate((b*sin(f*x + e) + a)^2*(d*sin(f*x + e))^m, x)
 

3.3.14.9 Mupad [F(-1)]

Timed out. \[ \int (d \sin (e+f x))^m (a+b \sin (e+f x))^2 \, dx=\int {\left (d\,\sin \left (e+f\,x\right )\right )}^m\,{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2 \,d x \]

input

int((d*sin(e + f*x))^m*(a + b*sin(e + f*x))^2,x)
 

output

int((d*sin(e + f*x))^m*(a + b*sin(e + f*x))^2, x)