3.1.26 \(\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-2-m} (A+B \sin (e+f x)+C \sin ^2(e+f x)) \, dx\) [26]

Optimal. Leaf size=410 \[ \frac {\left (c^2 C-B c d+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m}}{d \left (c^2-d^2\right ) f (1+m)}-\frac {2^{\frac {1}{2}+m} a \left (c d (A+C+A m+B m+C m)-c^2 (C+2 C m)-d^2 (A m+B (1+m)-C (1+m))\right ) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {(c-d) (1-\sin (e+f x))}{2 (c+d \sin (e+f x))}\right ) (a+a \sin (e+f x))^{-1+m} \left (\frac {(c+d) (1+\sin (e+f x))}{c+d \sin (e+f x)}\right )^{\frac {1}{2}-m} (c+d \sin (e+f x))^{-m}}{(c-d) d (c+d)^2 f (1+m)}+\frac {\sqrt {2} C F_1\left (\frac {3}{2}+m;\frac {1}{2},1+m;\frac {5}{2}+m;\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^{-m} \left (\frac {c+d \sin (e+f x)}{c-d}\right )^m}{a (c-d) d f (3+2 m) \sqrt {1-\sin (e+f x)}} \]

[Out]

(A*d^2-B*c*d+C*c^2)*cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^(-1-m)/d/(c^2-d^2)/f/(1+m)-2^(1/2+m)*a*(c*d
*(A*m+B*m+C*m+A+C)-c^2*(2*C*m+C)-d^2*(A*m+B*(1+m)-C*(1+m)))*cos(f*x+e)*hypergeom([1/2, 1/2-m],[3/2],1/2*(c-d)*
(1-sin(f*x+e))/(c+d*sin(f*x+e)))*(a+a*sin(f*x+e))^(-1+m)*((c+d)*(1+sin(f*x+e))/(c+d*sin(f*x+e)))^(1/2-m)/(c-d)
/d/(c+d)^2/f/(1+m)/((c+d*sin(f*x+e))^m)+C*AppellF1(3/2+m,1+m,1/2,5/2+m,-d*(1+sin(f*x+e))/(c-d),1/2+1/2*sin(f*x
+e))*cos(f*x+e)*(a+a*sin(f*x+e))^(1+m)*((c+d*sin(f*x+e))/(c-d))^m*2^(1/2)/a/(c-d)/d/f/(3+2*m)/((c+d*sin(f*x+e)
)^m)/(1-sin(f*x+e))^(1/2)

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Rubi [A]
time = 0.71, antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3122, 3066, 2867, 134, 145, 144, 143} \begin {gather*} -\frac {a 2^{m+\frac {1}{2}} \cos (e+f x) (a \sin (e+f x)+a)^{m-1} \left (c d (A m+A+B m+C m+C)-d^2 (A m+B (m+1)-C (m+1))-\left (c^2 (2 C m+C)\right )\right ) \left (\frac {(c+d) (\sin (e+f x)+1)}{c+d \sin (e+f x)}\right )^{\frac {1}{2}-m} (c+d \sin (e+f x))^{-m} \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {(c-d) (1-\sin (e+f x))}{2 (c+d \sin (e+f x))}\right )}{d f (m+1) (c-d) (c+d)^2}+\frac {\cos (e+f x) \left (A d^2-B c d+c^2 C\right ) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{-m-1}}{d f (m+1) \left (c^2-d^2\right )}+\frac {\sqrt {2} C \cos (e+f x) (a \sin (e+f x)+a)^{m+1} \left (\frac {c+d \sin (e+f x)}{c-d}\right )^m (c+d \sin (e+f x))^{-m} F_1\left (m+\frac {3}{2};\frac {1}{2},m+1;m+\frac {5}{2};\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{a d f (2 m+3) (c-d) \sqrt {1-\sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((c^2*C - B*c*d + A*d^2)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(-1 - m))/(d*(c^2 - d^2)*f*(
1 + m)) - (2^(1/2 + m)*a*(c*d*(A + C + A*m + B*m + C*m) - c^2*(C + 2*C*m) - d^2*(A*m + B*(1 + m) - C*(1 + m)))
*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, ((c - d)*(1 - Sin[e + f*x]))/(2*(c + d*Sin[e + f*x]))]*(a +
a*Sin[e + f*x])^(-1 + m)*(((c + d)*(1 + Sin[e + f*x]))/(c + d*Sin[e + f*x]))^(1/2 - m))/((c - d)*d*(c + d)^2*
f*(1 + m)*(c + d*Sin[e + f*x])^m) + (Sqrt[2]*C*AppellF1[3/2 + m, 1/2, 1 + m, 5/2 + m, (1 + Sin[e + f*x])/2, -(
(d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*((c + d*Sin[e + f*x])/(c - d))^m)/(
a*(c - d)*d*f*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]*(c + d*Sin[e + f*x])^m)

Rule 134

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x
)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((b*e - a*f)*(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c
*f))*((a + b*x)/((b*c - a*d)*(e + f*x)))])/((b*e - a*f)*((c + d*x)/((b*c - a*d)*(e + f*x))))^n, x] /; FreeQ[{a
, b, c, d, e, f, m, n, p}, x] && EqQ[m + n + p + 2, 0] &&  !IntegerQ[n]

Rule 143

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x)
^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n*(b/(b*e - a*f))^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(b*c
- a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !Inte
gerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !(GtQ[d/(d*a - c*b), 0] && GtQ[
d/(d*e - c*f), 0] && SimplerQ[c + d*x, a + b*x]) &&  !(GtQ[f/(f*a - e*b), 0] && GtQ[f/(f*c - e*d), 0] && Simpl
erQ[e + f*x, a + b*x])

Rule 144

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(e + f*x)^
FracPart[p]/((b/(b*e - a*f))^IntPart[p]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p]), Int[(a + b*x)^m*(c + d*x)^n*
(b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !GtQ[b/(b*e - a*f), 0]

Rule 145

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(c + d*x)^
FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n]), Int[(a + b*x)^m*(b*(c/(b*c -
a*d)) + b*d*(x/(b*c - a*d)))^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !IntegerQ[p] &&  !GtQ[b/(b*c - a*d), 0] &&  !SimplerQ[c + d*x, a + b*x] &&  !SimplerQ[e +
f*x, a + b*x]

Rule 2867

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dis
t[a^2*(Cos[e + f*x]/(f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[a - b*Sin[e + f*x]])), Subst[Int[(a + b*x)^(m - 1/2)*((c
+ d*x)^n/Sqrt[a - b*x]), x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] &
& EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !IntegerQ[m]

Rule 3066

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(A*b - a*B)/b, Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x
], x] + Dist[B/b, Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f,
A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A*b + a*B, 0]

Rule 3122

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

Rubi steps

\begin {align*} \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-2-m} \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx &=\frac {\left (c^2 C-B c d+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m}}{d \left (c^2-d^2\right ) f (1+m)}-\frac {\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m} \left (-a (A d (c+c m-d m)+(c C-B d) (d-c m+d m))-a C \left (c^2-d^2\right ) (1+m) \sin (e+f x)\right ) \, dx}{a d \left (c^2-d^2\right ) (1+m)}\\ &=\frac {\left (c^2 C-B c d+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m}}{d \left (c^2-d^2\right ) f (1+m)}+\frac {C \int (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^{-1-m} \, dx}{a d}+\frac {\left (c d (A+C+A m+B m+C m)-c^2 (C+2 C m)-d^2 (A m+B (1+m)-C (1+m))\right ) \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m} \, dx}{d \left (c^2-d^2\right ) (1+m)}\\ &=\frac {\left (c^2 C-B c d+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m}}{d \left (c^2-d^2\right ) f (1+m)}+\frac {(a C \cos (e+f x)) \text {Subst}\left (\int \frac {(a+a x)^{\frac {1}{2}+m} (c+d x)^{-1-m}}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{d f \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}+\frac {\left (a^2 \left (c d (A+C+A m+B m+C m)-c^2 (C+2 C m)-d^2 (A m+B (1+m)-C (1+m))\right ) \cos (e+f x)\right ) \text {Subst}\left (\int \frac {(a+a x)^{-\frac {1}{2}+m} (c+d x)^{-1-m}}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{d \left (c^2-d^2\right ) f (1+m) \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\\ &=\frac {\left (c^2 C-B c d+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m}}{d \left (c^2-d^2\right ) f (1+m)}-\frac {2^{\frac {1}{2}+m} a \left (c d (A+C+A m+B m+C m)-c^2 (C+2 C m)-d^2 (A m+B (1+m)-C (1+m))\right ) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {(c-d) (1-\sin (e+f x))}{2 (c+d \sin (e+f x))}\right ) (a+a \sin (e+f x))^{-1+m} \left (\frac {(c+d) (1+\sin (e+f x))}{c+d \sin (e+f x)}\right )^{\frac {1}{2}-m} (c+d \sin (e+f x))^{-m}}{(c-d) d (c+d)^2 f (1+m)}+\frac {\left (a C \cos (e+f x) \sqrt {\frac {a-a \sin (e+f x)}{a}}\right ) \text {Subst}\left (\int \frac {(a+a x)^{\frac {1}{2}+m} (c+d x)^{-1-m}}{\sqrt {\frac {1}{2}-\frac {x}{2}}} \, dx,x,\sin (e+f x)\right )}{\sqrt {2} d f (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}}\\ &=\frac {\left (c^2 C-B c d+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m}}{d \left (c^2-d^2\right ) f (1+m)}-\frac {2^{\frac {1}{2}+m} a \left (c d (A+C+A m+B m+C m)-c^2 (C+2 C m)-d^2 (A m+B (1+m)-C (1+m))\right ) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {(c-d) (1-\sin (e+f x))}{2 (c+d \sin (e+f x))}\right ) (a+a \sin (e+f x))^{-1+m} \left (\frac {(c+d) (1+\sin (e+f x))}{c+d \sin (e+f x)}\right )^{\frac {1}{2}-m} (c+d \sin (e+f x))^{-m}}{(c-d) d (c+d)^2 f (1+m)}+\frac {\left (a^2 C \cos (e+f x) \sqrt {\frac {a-a \sin (e+f x)}{a}} (c+d \sin (e+f x))^{-m} \left (\frac {a (c+d \sin (e+f x))}{a c-a d}\right )^m\right ) \text {Subst}\left (\int \frac {(a+a x)^{\frac {1}{2}+m} \left (\frac {a c}{a c-a d}+\frac {a d x}{a c-a d}\right )^{-1-m}}{\sqrt {\frac {1}{2}-\frac {x}{2}}} \, dx,x,\sin (e+f x)\right )}{\sqrt {2} d (a c-a d) f (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}}\\ &=\frac {\left (c^2 C-B c d+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{-1-m}}{d \left (c^2-d^2\right ) f (1+m)}-\frac {2^{\frac {1}{2}+m} a \left (c d (A+C+A m+B m+C m)-c^2 (C+2 C m)-d^2 (A m+B (1+m)-C (1+m))\right ) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {(c-d) (1-\sin (e+f x))}{2 (c+d \sin (e+f x))}\right ) (a+a \sin (e+f x))^{-1+m} \left (\frac {(c+d) (1+\sin (e+f x))}{c+d \sin (e+f x)}\right )^{\frac {1}{2}-m} (c+d \sin (e+f x))^{-m}}{(c-d) d (c+d)^2 f (1+m)}+\frac {\sqrt {2} C F_1\left (\frac {3}{2}+m;\frac {1}{2},1+m;\frac {5}{2}+m;\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) \sqrt {1-\sin (e+f x)} (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^{-m} \left (\frac {c+d \sin (e+f x)}{c-d}\right )^m}{(c-d) d f (3+2 m) (a-a \sin (e+f x))}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(5175\) vs. \(2(410)=820\).
time = 40.22, size = 5175, normalized size = 12.62 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [F]
time = 5.91, size = 0, normalized size = 0.00 \[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c +d \sin \left (f x +e \right )\right )^{-2-m} \left (A +B \sin \left (f x +e \right )+C \left (\sin ^{2}\left (f x +e \right )\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^(-2-m)*(A+B*sin(f*x+e)+C*sin(f*x+e)^2),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((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^(-2-m)*(A+B*sin(f*x+e)+C*sin(f*x+e)^2),x, algorithm="maxima")

[Out]

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

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**m*(c+d*sin(f*x+e))**(-2-m)*(A+B*sin(f*x+e)+C*sin(f*x+e)**2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 8011 deep

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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((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^(-2-m)*(A+B*sin(f*x+e)+C*sin(f*x+e)^2),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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