\(\int (a+a \sin (e+f x))^m (A+C \sin ^2(e+f x)) \, dx\) [13]

Optimal result

Integrand size = 25, antiderivative size = 171 \[ \int (a+a \sin (e+f x))^m \left (A+C \sin ^2(e+f x)\right ) \, dx=\frac {C \cos (e+f x) (a+a \sin (e+f x))^m}{f \left (2+3 m+m^2\right )}-\frac {2^{\frac {1}{2}+m} \left (C \left (1+m+m^2\right )+A \left (2+3 m+m^2\right )\right ) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{-\frac {1}{2}-m} (a+a \sin (e+f x))^m}{f (1+m) (2+m)}-\frac {C \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{a f (2+m)} \]

[Out]

C*cos(f*x+e)*(a+a*sin(f*x+e))^m/f/(m^2+3*m+2)-2^(1/2+m)*(C*(m^2+m+1)+A*(m^2+3*m+2))*cos(f*x+e)*hypergeom([1/2,
1/2-m],[3/2],1/2-1/2*sin(f*x+e))*(1+sin(f*x+e))^(-1/2-m)*(a+a*sin(f*x+e))^m/f/(m^2+3*m+2)-C*cos(f*x+e)*(a+a*s
in(f*x+e))^(1+m)/a/f/(2+m)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3103, 2830, 2731, 2730} \[ \int (a+a \sin (e+f x))^m \left (A+C \sin ^2(e+f x)\right ) \, dx=-\frac {2^{m+\frac {1}{2}} \left (A \left (m^2+3 m+2\right )+C \left (m^2+m+1\right )\right ) \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac {1}{2}} (a \sin (e+f x)+a)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x))\right )}{f (m+1) (m+2)}+\frac {C \cos (e+f x) (a \sin (e+f x)+a)^m}{f \left (m^2+3 m+2\right )}-\frac {C \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{a f (m+2)} \]

[In]

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

[Out]

(C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(2 + 3*m + m^2)) - (2^(1/2 + m)*(C*(1 + m + m^2) + A*(2 + 3*m + m^2
))*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/2 - m)*(a +
a*Sin[e + f*x])^m)/(f*(1 + m)*(2 + m)) - (C*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(2 + m))

Rule 2730

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-2^(n + 1/2))*a^(n - 1/2)*b*(Cos[c + d*x]/
(d*Sqrt[a + b*Sin[c + d*x]]))*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - b*(Sin[c + d*x]/a))], x] /; Free
Q[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] && GtQ[a, 0]

Rule 2731

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[a^IntPart[n]*((a + b*Sin[c + d*x])^FracPart
[n]/(1 + (b/a)*Sin[c + d*x])^FracPart[n]), Int[(1 + (b/a)*Sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, n}, x]
&& EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] &&  !GtQ[a, 0]

Rule 2830

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d
)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/(f*(m + 1))), x] + Dist[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*S
in[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&  !LtQ[m
, -2^(-1)]

Rule 3103

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

Rubi steps \begin{align*} \text {integral}& = -\frac {C \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{a f (2+m)}+\frac {\int (a+a \sin (e+f x))^m (a (C (1+m)+A (2+m))-a C \sin (e+f x)) \, dx}{a (2+m)} \\ & = \frac {C \cos (e+f x) (a+a \sin (e+f x))^m}{f \left (2+3 m+m^2\right )}-\frac {C \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{a f (2+m)}+\frac {\left (C \left (1+m+m^2\right )+A \left (2+3 m+m^2\right )\right ) \int (a+a \sin (e+f x))^m \, dx}{(1+m) (2+m)} \\ & = \frac {C \cos (e+f x) (a+a \sin (e+f x))^m}{f \left (2+3 m+m^2\right )}-\frac {C \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{a f (2+m)}+\frac {\left (\left (C \left (1+m+m^2\right )+A \left (2+3 m+m^2\right )\right ) (1+\sin (e+f x))^{-m} (a+a \sin (e+f x))^m\right ) \int (1+\sin (e+f x))^m \, dx}{(1+m) (2+m)} \\ & = \frac {C \cos (e+f x) (a+a \sin (e+f x))^m}{f \left (2+3 m+m^2\right )}-\frac {2^{\frac {1}{2}+m} \left (C \left (1+m+m^2\right )+A \left (2+3 m+m^2\right )\right ) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{-\frac {1}{2}-m} (a+a \sin (e+f x))^m}{f (1+m) (2+m)}-\frac {C \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{a f (2+m)} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1794\) vs. \(2(171)=342\).

Time = 26.44 (sec) , antiderivative size = 1794, normalized size of antiderivative = 10.49 \[ \int (a+a \sin (e+f x))^m \left (A+C \sin ^2(e+f x)\right ) \, dx=-\frac {2 \left (A (a+a \sin (e+f x))^m+\frac {1}{2} C (a+a \sin (e+f x))^m-\frac {1}{2} C \cos (2 (e+f x)) (a+a \sin (e+f x))^m\right ) \left (a+\frac {a \tan (e+f x)}{\sqrt {\sec ^2(e+f x)}}\right )^m \left (\sec ^2(e+f x)+\sqrt {\sec ^2(e+f x)} \tan (e+f x)\right ) \left (1+\left (\sqrt {\sec ^2(e+f x)}+\tan (e+f x)\right )^2\right )^m \left (-\left ((A+C) (3+2 m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2}+m,1+m,\frac {3}{2}+m,-\left (\sqrt {\sec ^2(e+f x)}+\tan (e+f x)\right )^2\right )\right )+4 C (1+2 m) \operatorname {Hypergeometric2F1}\left (\frac {3}{2}+m,3+m,\frac {5}{2}+m,-\left (\sqrt {\sec ^2(e+f x)}+\tan (e+f x)\right )^2\right ) \left (1+2 \sqrt {\sec ^2(e+f x)} \tan (e+f x)+2 \tan ^2(e+f x)\right )\right )}{f \left (3+8 m+4 m^2\right ) \sqrt {\sec ^2(e+f x)} \left (-\frac {4 m \left (\sqrt {\sec ^2(e+f x)}+\tan (e+f x)\right ) \left (a+\frac {a \tan (e+f x)}{\sqrt {\sec ^2(e+f x)}}\right )^m \left (\sec ^2(e+f x)+\sqrt {\sec ^2(e+f x)} \tan (e+f x)\right )^2 \left (1+\left (\sqrt {\sec ^2(e+f x)}+\tan (e+f x)\right )^2\right )^{-1+m} \left (-\left ((A+C) (3+2 m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2}+m,1+m,\frac {3}{2}+m,-\left (\sqrt {\sec ^2(e+f x)}+\tan (e+f x)\right )^2\right )\right )+4 C (1+2 m) \operatorname {Hypergeometric2F1}\left (\frac {3}{2}+m,3+m,\frac {5}{2}+m,-\left (\sqrt {\sec ^2(e+f x)}+\tan (e+f x)\right )^2\right ) \left (1+2 \sqrt {\sec ^2(e+f x)} \tan (e+f x)+2 \tan ^2(e+f x)\right )\right )}{\left (3+8 m+4 m^2\right ) \sqrt {\sec ^2(e+f x)}}+\frac {2 \tan (e+f x) \left (a+\frac {a \tan (e+f x)}{\sqrt {\sec ^2(e+f x)}}\right )^m \left (\sec ^2(e+f x)+\sqrt {\sec ^2(e+f x)} \tan (e+f x)\right ) \left (1+\left (\sqrt {\sec ^2(e+f x)}+\tan (e+f x)\right )^2\right )^m \left (-\left ((A+C) (3+2 m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2}+m,1+m,\frac {3}{2}+m,-\left (\sqrt {\sec ^2(e+f x)}+\tan (e+f x)\right )^2\right )\right )+4 C (1+2 m) \operatorname {Hypergeometric2F1}\left (\frac {3}{2}+m,3+m,\frac {5}{2}+m,-\left (\sqrt {\sec ^2(e+f x)}+\tan (e+f x)\right )^2\right ) \left (1+2 \sqrt {\sec ^2(e+f x)} \tan (e+f x)+2 \tan ^2(e+f x)\right )\right )}{\left (3+8 m+4 m^2\right ) \sqrt {\sec ^2(e+f x)}}-\frac {2 m \left (a+\frac {a \tan (e+f x)}{\sqrt {\sec ^2(e+f x)}}\right )^{-1+m} \left (\sec ^2(e+f x)+\sqrt {\sec ^2(e+f x)} \tan (e+f x)\right ) \left (a \sqrt {\sec ^2(e+f x)}-\frac {a \tan ^2(e+f x)}{\sqrt {\sec ^2(e+f x)}}\right ) \left (1+\left (\sqrt {\sec ^2(e+f x)}+\tan (e+f x)\right )^2\right )^m \left (-\left ((A+C) (3+2 m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2}+m,1+m,\frac {3}{2}+m,-\left (\sqrt {\sec ^2(e+f x)}+\tan (e+f x)\right )^2\right )\right )+4 C (1+2 m) \operatorname {Hypergeometric2F1}\left (\frac {3}{2}+m,3+m,\frac {5}{2}+m,-\left (\sqrt {\sec ^2(e+f x)}+\tan (e+f x)\right )^2\right ) \left (1+2 \sqrt {\sec ^2(e+f x)} \tan (e+f x)+2 \tan ^2(e+f x)\right )\right )}{\left (3+8 m+4 m^2\right ) \sqrt {\sec ^2(e+f x)}}-\frac {2 \left (a+\frac {a \tan (e+f x)}{\sqrt {\sec ^2(e+f x)}}\right )^m \left (\sec ^2(e+f x)^{3/2}+2 \sec ^2(e+f x) \tan (e+f x)+\sqrt {\sec ^2(e+f x)} \tan ^2(e+f x)\right ) \left (1+\left (\sqrt {\sec ^2(e+f x)}+\tan (e+f x)\right )^2\right )^m \left (-\left ((A+C) (3+2 m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2}+m,1+m,\frac {3}{2}+m,-\left (\sqrt {\sec ^2(e+f x)}+\tan (e+f x)\right )^2\right )\right )+4 C (1+2 m) \operatorname {Hypergeometric2F1}\left (\frac {3}{2}+m,3+m,\frac {5}{2}+m,-\left (\sqrt {\sec ^2(e+f x)}+\tan (e+f x)\right )^2\right ) \left (1+2 \sqrt {\sec ^2(e+f x)} \tan (e+f x)+2 \tan ^2(e+f x)\right )\right )}{\left (3+8 m+4 m^2\right ) \sqrt {\sec ^2(e+f x)}}-\frac {2 \left (a+\frac {a \tan (e+f x)}{\sqrt {\sec ^2(e+f x)}}\right )^m \left (\sec ^2(e+f x)+\sqrt {\sec ^2(e+f x)} \tan (e+f x)\right ) \left (1+\left (\sqrt {\sec ^2(e+f x)}+\tan (e+f x)\right )^2\right )^m \left (4 C (1+2 m) \operatorname {Hypergeometric2F1}\left (\frac {3}{2}+m,3+m,\frac {5}{2}+m,-\left (\sqrt {\sec ^2(e+f x)}+\tan (e+f x)\right )^2\right ) \left (2 \sec ^2(e+f x)^{3/2}+4 \sec ^2(e+f x) \tan (e+f x)+2 \sqrt {\sec ^2(e+f x)} \tan ^2(e+f x)\right )+\frac {8 C \left (\frac {3}{2}+m\right ) (1+2 m) \left (\sec ^2(e+f x)+\sqrt {\sec ^2(e+f x)} \tan (e+f x)\right ) \left (1+2 \sqrt {\sec ^2(e+f x)} \tan (e+f x)+2 \tan ^2(e+f x)\right ) \left (-\operatorname {Hypergeometric2F1}\left (\frac {3}{2}+m,3+m,\frac {5}{2}+m,-\left (\sqrt {\sec ^2(e+f x)}+\tan (e+f x)\right )^2\right )+\left (1+\left (\sqrt {\sec ^2(e+f x)}+\tan (e+f x)\right )^2\right )^{-3-m}\right )}{\sqrt {\sec ^2(e+f x)}+\tan (e+f x)}-\frac {2 (A+C) \left (\frac {1}{2}+m\right ) (3+2 m) \left (\sec ^2(e+f x)+\sqrt {\sec ^2(e+f x)} \tan (e+f x)\right ) \left (-\operatorname {Hypergeometric2F1}\left (\frac {1}{2}+m,1+m,\frac {3}{2}+m,-\left (\sqrt {\sec ^2(e+f x)}+\tan (e+f x)\right )^2\right )+\left (1+\left (\sqrt {\sec ^2(e+f x)}+\tan (e+f x)\right )^2\right )^{-1-m}\right )}{\sqrt {\sec ^2(e+f x)}+\tan (e+f x)}\right )}{\left (3+8 m+4 m^2\right ) \sqrt {\sec ^2(e+f x)}}\right )} \]

[In]

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

[Out]

(-2*(A*(a + a*Sin[e + f*x])^m + (C*(a + a*Sin[e + f*x])^m)/2 - (C*Cos[2*(e + f*x)]*(a + a*Sin[e + f*x])^m)/2)*
(a + (a*Tan[e + f*x])/Sqrt[Sec[e + f*x]^2])^m*(Sec[e + f*x]^2 + Sqrt[Sec[e + f*x]^2]*Tan[e + f*x])*(1 + (Sqrt[
Sec[e + f*x]^2] + Tan[e + f*x])^2)^m*(-((A + C)*(3 + 2*m)*Hypergeometric2F1[1/2 + m, 1 + m, 3/2 + m, -(Sqrt[Se
c[e + f*x]^2] + Tan[e + f*x])^2]) + 4*C*(1 + 2*m)*Hypergeometric2F1[3/2 + m, 3 + m, 5/2 + m, -(Sqrt[Sec[e + f*
x]^2] + Tan[e + f*x])^2]*(1 + 2*Sqrt[Sec[e + f*x]^2]*Tan[e + f*x] + 2*Tan[e + f*x]^2)))/(f*(3 + 8*m + 4*m^2)*S
qrt[Sec[e + f*x]^2]*((-4*m*(Sqrt[Sec[e + f*x]^2] + Tan[e + f*x])*(a + (a*Tan[e + f*x])/Sqrt[Sec[e + f*x]^2])^m
*(Sec[e + f*x]^2 + Sqrt[Sec[e + f*x]^2]*Tan[e + f*x])^2*(1 + (Sqrt[Sec[e + f*x]^2] + Tan[e + f*x])^2)^(-1 + m)
*(-((A + C)*(3 + 2*m)*Hypergeometric2F1[1/2 + m, 1 + m, 3/2 + m, -(Sqrt[Sec[e + f*x]^2] + Tan[e + f*x])^2]) +
4*C*(1 + 2*m)*Hypergeometric2F1[3/2 + m, 3 + m, 5/2 + m, -(Sqrt[Sec[e + f*x]^2] + Tan[e + f*x])^2]*(1 + 2*Sqrt
[Sec[e + f*x]^2]*Tan[e + f*x] + 2*Tan[e + f*x]^2)))/((3 + 8*m + 4*m^2)*Sqrt[Sec[e + f*x]^2]) + (2*Tan[e + f*x]
*(a + (a*Tan[e + f*x])/Sqrt[Sec[e + f*x]^2])^m*(Sec[e + f*x]^2 + Sqrt[Sec[e + f*x]^2]*Tan[e + f*x])*(1 + (Sqrt
[Sec[e + f*x]^2] + Tan[e + f*x])^2)^m*(-((A + C)*(3 + 2*m)*Hypergeometric2F1[1/2 + m, 1 + m, 3/2 + m, -(Sqrt[S
ec[e + f*x]^2] + Tan[e + f*x])^2]) + 4*C*(1 + 2*m)*Hypergeometric2F1[3/2 + m, 3 + m, 5/2 + m, -(Sqrt[Sec[e + f
*x]^2] + Tan[e + f*x])^2]*(1 + 2*Sqrt[Sec[e + f*x]^2]*Tan[e + f*x] + 2*Tan[e + f*x]^2)))/((3 + 8*m + 4*m^2)*Sq
rt[Sec[e + f*x]^2]) - (2*m*(a + (a*Tan[e + f*x])/Sqrt[Sec[e + f*x]^2])^(-1 + m)*(Sec[e + f*x]^2 + Sqrt[Sec[e +
f*x]^2]*Tan[e + f*x])*(a*Sqrt[Sec[e + f*x]^2] - (a*Tan[e + f*x]^2)/Sqrt[Sec[e + f*x]^2])*(1 + (Sqrt[Sec[e + f
*x]^2] + Tan[e + f*x])^2)^m*(-((A + C)*(3 + 2*m)*Hypergeometric2F1[1/2 + m, 1 + m, 3/2 + m, -(Sqrt[Sec[e + f*x
]^2] + Tan[e + f*x])^2]) + 4*C*(1 + 2*m)*Hypergeometric2F1[3/2 + m, 3 + m, 5/2 + m, -(Sqrt[Sec[e + f*x]^2] + T
an[e + f*x])^2]*(1 + 2*Sqrt[Sec[e + f*x]^2]*Tan[e + f*x] + 2*Tan[e + f*x]^2)))/((3 + 8*m + 4*m^2)*Sqrt[Sec[e +
f*x]^2]) - (2*(a + (a*Tan[e + f*x])/Sqrt[Sec[e + f*x]^2])^m*((Sec[e + f*x]^2)^(3/2) + 2*Sec[e + f*x]^2*Tan[e
+ f*x] + Sqrt[Sec[e + f*x]^2]*Tan[e + f*x]^2)*(1 + (Sqrt[Sec[e + f*x]^2] + Tan[e + f*x])^2)^m*(-((A + C)*(3 +
2*m)*Hypergeometric2F1[1/2 + m, 1 + m, 3/2 + m, -(Sqrt[Sec[e + f*x]^2] + Tan[e + f*x])^2]) + 4*C*(1 + 2*m)*Hyp
ergeometric2F1[3/2 + m, 3 + m, 5/2 + m, -(Sqrt[Sec[e + f*x]^2] + Tan[e + f*x])^2]*(1 + 2*Sqrt[Sec[e + f*x]^2]*
Tan[e + f*x] + 2*Tan[e + f*x]^2)))/((3 + 8*m + 4*m^2)*Sqrt[Sec[e + f*x]^2]) - (2*(a + (a*Tan[e + f*x])/Sqrt[Se
c[e + f*x]^2])^m*(Sec[e + f*x]^2 + Sqrt[Sec[e + f*x]^2]*Tan[e + f*x])*(1 + (Sqrt[Sec[e + f*x]^2] + Tan[e + f*x
])^2)^m*(4*C*(1 + 2*m)*Hypergeometric2F1[3/2 + m, 3 + m, 5/2 + m, -(Sqrt[Sec[e + f*x]^2] + Tan[e + f*x])^2]*(2
*(Sec[e + f*x]^2)^(3/2) + 4*Sec[e + f*x]^2*Tan[e + f*x] + 2*Sqrt[Sec[e + f*x]^2]*Tan[e + f*x]^2) + (8*C*(3/2 +
m)*(1 + 2*m)*(Sec[e + f*x]^2 + Sqrt[Sec[e + f*x]^2]*Tan[e + f*x])*(1 + 2*Sqrt[Sec[e + f*x]^2]*Tan[e + f*x] +
2*Tan[e + f*x]^2)*(-Hypergeometric2F1[3/2 + m, 3 + m, 5/2 + m, -(Sqrt[Sec[e + f*x]^2] + Tan[e + f*x])^2] + (1
+ (Sqrt[Sec[e + f*x]^2] + Tan[e + f*x])^2)^(-3 - m)))/(Sqrt[Sec[e + f*x]^2] + Tan[e + f*x]) - (2*(A + C)*(1/2
+ m)*(3 + 2*m)*(Sec[e + f*x]^2 + Sqrt[Sec[e + f*x]^2]*Tan[e + f*x])*(-Hypergeometric2F1[1/2 + m, 1 + m, 3/2 +
m, -(Sqrt[Sec[e + f*x]^2] + Tan[e + f*x])^2] + (1 + (Sqrt[Sec[e + f*x]^2] + Tan[e + f*x])^2)^(-1 - m)))/(Sqrt[
Sec[e + f*x]^2] + Tan[e + f*x])))/((3 + 8*m + 4*m^2)*Sqrt[Sec[e + f*x]^2])))

Maple [F]

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

[In]

int((a+a*sin(f*x+e))^m*(A+C*sin(f*x+e)^2),x)

[Out]

int((a+a*sin(f*x+e))^m*(A+C*sin(f*x+e)^2),x)

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

\[ \int (a+a \sin (e+f x))^m \left (A+C \sin ^2(e+f x)\right ) \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \left (A + C \sin ^{2}{\left (e + f x \right )}\right )\, dx \]

[In]

integrate((a+a*sin(f*x+e))**m*(A+C*sin(f*x+e)**2),x)

[Out]

Integral((a*(sin(e + f*x) + 1))**m*(A + C*sin(e + f*x)**2), x)

Maxima [F]

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

[In]

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

[Out]

integrate((C*sin(f*x + e)^2 + A)*(a*sin(f*x + e) + a)^m, x)

Giac [F]

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

[In]

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

[Out]

integrate((C*sin(f*x + e)^2 + A)*(a*sin(f*x + e) + a)^m, x)

Mupad [F(-1)]

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

[In]

int((A + C*sin(e + f*x)^2)*(a + a*sin(e + f*x))^m,x)

[Out]

int((A + C*sin(e + f*x)^2)*(a + a*sin(e + f*x))^m, x)