\(\int \sin ^{-2-m}(c+d x) (a+a \sin (c+d x))^m (1+m-m \sin (c+d x)) \, dx\) [15]
Optimal result
Integrand size = 37, antiderivative size = 35 \[
\int \sin ^{-2-m}(c+d x) (a+a \sin (c+d x))^m (1+m-m \sin (c+d x)) \, dx=-\frac {\cos (c+d x) \sin ^{-1-m}(c+d x) (a+a \sin (c+d x))^m}{d}
\]
[Out]
-cos(d*x+c)*sin(d*x+c)^(-1-m)*(a+a*sin(d*x+c))^m/d
Rubi [A] (verified)
Time = 0.06 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00,
number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {3053}
\[
\int \sin ^{-2-m}(c+d x) (a+a \sin (c+d x))^m (1+m-m \sin (c+d x)) \, dx=-\frac {\cos (c+d x) \sin ^{-m-1}(c+d x) (a \sin (c+d x)+a)^m}{d}
\]
[In]
Int[Sin[c + d*x]^(-2 - m)*(a + a*Sin[c + d*x])^m*(1 + m - m*Sin[c + d*x]),x]
[Out]
-((Cos[c + d*x]*Sin[c + d*x]^(-1 - m)*(a + a*Sin[c + d*x])^m)/d)
Rule 3053
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x]
)^(n + 1)/(f*(n + 1)*(c^2 - d^2))), 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] && EqQ[m + n + 2, 0] && EqQ[A*(a*d*m + b*c*(n + 1)) - B*(a*c*m + b*d*(n +
1)), 0]
Rubi steps \begin{align*}
\text {integral}& = -\frac {\cos (c+d x) \sin ^{-1-m}(c+d x) (a+a \sin (c+d x))^m}{d} \\
\end{align*}
Mathematica [A] (verified)
Time = 1.45 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00
\[
\int \sin ^{-2-m}(c+d x) (a+a \sin (c+d x))^m (1+m-m \sin (c+d x)) \, dx=-\frac {\cos (c+d x) \sin ^{-1-m}(c+d x) (a (1+\sin (c+d x)))^m}{d}
\]
[In]
Integrate[Sin[c + d*x]^(-2 - m)*(a + a*Sin[c + d*x])^m*(1 + m - m*Sin[c + d*x]),x]
[Out]
-((Cos[c + d*x]*Sin[c + d*x]^(-1 - m)*(a*(1 + Sin[c + d*x]))^m)/d)
Maple [F]
\[\int \left (\sin ^{-2-m}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{m} \left (1+m -m \sin \left (d x +c \right )\right )d x\]
[In]
int(sin(d*x+c)^(-2-m)*(a+a*sin(d*x+c))^m*(1+m-m*sin(d*x+c)),x)
[Out]
int(sin(d*x+c)^(-2-m)*(a+a*sin(d*x+c))^m*(1+m-m*sin(d*x+c)),x)
Fricas [A] (verification not implemented)
none
Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.17
\[
\int \sin ^{-2-m}(c+d x) (a+a \sin (c+d x))^m (1+m-m \sin (c+d x)) \, dx=-\frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{m} \sin \left (d x + c\right )^{-m - 2} \cos \left (d x + c\right ) \sin \left (d x + c\right )}{d}
\]
[In]
integrate(sin(d*x+c)^(-2-m)*(a+a*sin(d*x+c))^m*(1+m-m*sin(d*x+c)),x, algorithm="fricas")
[Out]
-(a*sin(d*x + c) + a)^m*sin(d*x + c)^(-m - 2)*cos(d*x + c)*sin(d*x + c)/d
Sympy [F]
\[
\int \sin ^{-2-m}(c+d x) (a+a \sin (c+d x))^m (1+m-m \sin (c+d x)) \, dx=- \int \left (- \left (a \sin {\left (c + d x \right )} + a\right )^{m} \sin ^{- m - 2}{\left (c + d x \right )}\right )\, dx - \int \left (- m \left (a \sin {\left (c + d x \right )} + a\right )^{m} \sin ^{- m - 2}{\left (c + d x \right )}\right )\, dx - \int m \left (a \sin {\left (c + d x \right )} + a\right )^{m} \sin {\left (c + d x \right )} \sin ^{- m - 2}{\left (c + d x \right )}\, dx
\]
[In]
integrate(sin(d*x+c)**(-2-m)*(a+a*sin(d*x+c))**m*(1+m-m*sin(d*x+c)),x)
[Out]
-Integral(-(a*sin(c + d*x) + a)**m*sin(c + d*x)**(-m - 2), x) - Integral(-m*(a*sin(c + d*x) + a)**m*sin(c + d*
x)**(-m - 2), x) - Integral(m*(a*sin(c + d*x) + a)**m*sin(c + d*x)*sin(c + d*x)**(-m - 2), x)
Maxima [F]
\[
\int \sin ^{-2-m}(c+d x) (a+a \sin (c+d x))^m (1+m-m \sin (c+d x)) \, dx=\int { -{\left (m \sin \left (d x + c\right ) - m - 1\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \sin \left (d x + c\right )^{-m - 2} \,d x }
\]
[In]
integrate(sin(d*x+c)^(-2-m)*(a+a*sin(d*x+c))^m*(1+m-m*sin(d*x+c)),x, algorithm="maxima")
[Out]
-integrate((m*sin(d*x + c) - m - 1)*(a*sin(d*x + c) + a)^m*sin(d*x + c)^(-m - 2), x)
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 5502 vs. \(2 (35) = 70\).
Time = 40.10 (sec) , antiderivative size = 5502, normalized size of antiderivative = 157.20
\[
\int \sin ^{-2-m}(c+d x) (a+a \sin (c+d x))^m (1+m-m \sin (c+d x)) \, dx=\text {Too large to display}
\]
[In]
integrate(sin(d*x+c)^(-2-m)*(a+a*sin(d*x+c))^m*(1+m-m*sin(d*x+c)),x, algorithm="giac")
[Out]
-8*(cos(2*pi*m*floor(-1/8*sgn(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)
+ 4*tan(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) + 2) + 5/8) + 1/4*pi*m*sgn(4*tan(d*x +
c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c
)^2 + 8*tan(1/2*d*x + 1/2*c) + 2) - 1/4*pi*m)*e^(m*log(sqrt(2)*sqrt(abs(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^
2 + 8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*
c) + 2)*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + abs(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)^2
*tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) + 2)*tan(d*x + c)
^2 + abs(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c)^2 +
2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) + 2)*tan(1/2*d*x + 1/2*c)^2 + abs(4*tan(d*x + c)^2*tan(1/2*d
*x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/
2*d*x + 1/2*c) + 2))*abs(a)/(tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + tan(d*x + c)^2 + tan(1/2*d*x + 1/2*c)^2 +
1)) - m*log(4*abs(tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)) - 2*log(4*abs(tan(1/2*d*x + 1/2*c))/(ta
n(1/2*d*x + 1/2*c)^2 + 1)))*tan(-1/2*pi + 1/4*pi*m*sgn(2*a*tan(1/2*d*x + 1/2*c)^4 + 4*a*tan(1/2*d*x + 1/2*c)^3
- 4*a*tan(1/2*d*x + 1/2*c) - 2*a)*sgn(4*a*tan(1/2*d*x + 1/2*c)^3 + 8*a*tan(1/2*d*x + 1/2*c)^2 + 4*a*tan(1/2*d
*x + 1/2*c)) - 1/4*pi*m*sgn(tan(1/2*d*x + 1/2*c)^2 - 1)*sgn(tan(1/2*d*x + 1/2*c)) + 1/4*pi*m*sgn(4*a*tan(1/2*d
*x + 1/2*c)^3 + 8*a*tan(1/2*d*x + 1/2*c)^2 + 4*a*tan(1/2*d*x + 1/2*c)) - 1/2*pi*sgn(tan(1/2*d*x + 1/2*c)^2 - 1
)*sgn(tan(1/2*d*x + 1/2*c)) - 1/4*pi*m + pi*floor(d*x/pi + c/pi + 1/2))^2*tan(1/2*d*x + 1/2*c)^3 - 2*e^(m*log(
sqrt(2)*sqrt(abs(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x +
c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) + 2)*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + abs(4*t
an(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*d*
x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) + 2)*tan(d*x + c)^2 + abs(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*ta
n(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) + 2)*
tan(1/2*d*x + 1/2*c)^2 + abs(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) +
4*tan(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) + 2))*abs(a)/(tan(d*x + c)^2*tan(1/2*d*x
+ 1/2*c)^2 + tan(d*x + c)^2 + tan(1/2*d*x + 1/2*c)^2 + 1)) - m*log(4*abs(tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x +
1/2*c)^2 + 1)) - 2*log(4*abs(tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)))*sin(2*pi*m*floor(-1/8*sgn(4
*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*
d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) + 2) + 5/8) + 1/4*pi*m*sgn(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8
*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) +
2) - 1/4*pi*m)*tan(-1/2*pi + 1/4*pi*m*sgn(2*a*tan(1/2*d*x + 1/2*c)^4 + 4*a*tan(1/2*d*x + 1/2*c)^3 - 4*a*tan(1/
2*d*x + 1/2*c) - 2*a)*sgn(4*a*tan(1/2*d*x + 1/2*c)^3 + 8*a*tan(1/2*d*x + 1/2*c)^2 + 4*a*tan(1/2*d*x + 1/2*c))
- 1/4*pi*m*sgn(tan(1/2*d*x + 1/2*c)^2 - 1)*sgn(tan(1/2*d*x + 1/2*c)) + 1/4*pi*m*sgn(4*a*tan(1/2*d*x + 1/2*c)^3
+ 8*a*tan(1/2*d*x + 1/2*c)^2 + 4*a*tan(1/2*d*x + 1/2*c)) - 1/2*pi*sgn(tan(1/2*d*x + 1/2*c)^2 - 1)*sgn(tan(1/2
*d*x + 1/2*c)) - 1/4*pi*m + pi*floor(d*x/pi + c/pi + 1/2))*tan(1/2*d*x + 1/2*c)^3 - cos(2*pi*m*floor(-1/8*sgn(
4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2
*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) + 2) + 5/8) + 1/4*pi*m*sgn(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 +
8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) +
2) - 1/4*pi*m)*e^(m*log(sqrt(2)*sqrt(abs(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d
*x + 1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) + 2)*tan(d*x + c)^2*tan(1/2
*d*x + 1/2*c)^2 + abs(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(
d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) + 2)*tan(d*x + c)^2 + abs(4*tan(d*x + c)^2*tan(
1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*t
an(1/2*d*x + 1/2*c) + 2)*tan(1/2*d*x + 1/2*c)^2 + abs(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)
^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) + 2))*abs(a)/(t
an(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + tan(d*x + c)^2 + tan(1/2*d*x + 1/2*c)^2 + 1)) - m*log(4*abs(tan(1/2*d*x
+ 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)) - 2*log(4*abs(tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)))*ta
n(-1/2*pi + 1/4*pi*m*sgn(2*a*tan(1/2*d*x + 1/2*c)^4 + 4*a*tan(1/2*d*x + 1/2*c)^3 - 4*a*tan(1/2*d*x + 1/2*c) -
2*a)*sgn(4*a*tan(1/2*d*x + 1/2*c)^3 + 8*a*tan(1/2*d*x + 1/2*c)^2 + 4*a*tan(1/2*d*x + 1/2*c)) - 1/4*pi*m*sgn(ta
n(1/2*d*x + 1/2*c)^2 - 1)*sgn(tan(1/2*d*x + 1/2*c)) + 1/4*pi*m*sgn(4*a*tan(1/2*d*x + 1/2*c)^3 + 8*a*tan(1/2*d*
x + 1/2*c)^2 + 4*a*tan(1/2*d*x + 1/2*c)) - 1/2*pi*sgn(tan(1/2*d*x + 1/2*c)^2 - 1)*sgn(tan(1/2*d*x + 1/2*c)) -
1/4*pi*m + pi*floor(d*x/pi + c/pi + 1/2))^2*tan(1/2*d*x + 1/2*c) - cos(2*pi*m*floor(-1/8*sgn(4*tan(d*x + c)^2*
tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 +
8*tan(1/2*d*x + 1/2*c) + 2) + 5/8) + 1/4*pi*m*sgn(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)^2*
tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) + 2) - 1/4*pi*m)*e
^(m*log(sqrt(2)*sqrt(abs(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*t
an(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) + 2)*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 +
abs(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c)^2 + 2*ta
n(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) + 2)*tan(d*x + c)^2 + abs(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^
2 + 8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*
c) + 2)*tan(1/2*d*x + 1/2*c)^2 + abs(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d*x +
1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) + 2))*abs(a)/(tan(d*x + c)^2*tan
(1/2*d*x + 1/2*c)^2 + tan(d*x + c)^2 + tan(1/2*d*x + 1/2*c)^2 + 1)) - m*log(4*abs(tan(1/2*d*x + 1/2*c))/(tan(1
/2*d*x + 1/2*c)^2 + 1)) - 2*log(4*abs(tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)))*tan(1/2*d*x + 1/2*c
)^3 + 2*e^(m*log(sqrt(2)*sqrt(abs(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d*x + 1/2
*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) + 2)*tan(d*x + c)^2*tan(1/2*d*x + 1
/2*c)^2 + abs(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c)
^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) + 2)*tan(d*x + c)^2 + abs(4*tan(d*x + c)^2*tan(1/2*d*x
+ 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d
*x + 1/2*c) + 2)*tan(1/2*d*x + 1/2*c)^2 + abs(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1
/2*d*x + 1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) + 2))*abs(a)/(tan(d*x +
c)^2*tan(1/2*d*x + 1/2*c)^2 + tan(d*x + c)^2 + tan(1/2*d*x + 1/2*c)^2 + 1)) - m*log(4*abs(tan(1/2*d*x + 1/2*c
))/(tan(1/2*d*x + 1/2*c)^2 + 1)) - 2*log(4*abs(tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)))*sin(2*pi*m
*floor(-1/8*sgn(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x +
c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) + 2) + 5/8) + 1/4*pi*m*sgn(4*tan(d*x + c)^2*tan(1/2*d
*x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/
2*d*x + 1/2*c) + 2) - 1/4*pi*m)*tan(-1/2*pi + 1/4*pi*m*sgn(2*a*tan(1/2*d*x + 1/2*c)^4 + 4*a*tan(1/2*d*x + 1/2*
c)^3 - 4*a*tan(1/2*d*x + 1/2*c) - 2*a)*sgn(4*a*tan(1/2*d*x + 1/2*c)^3 + 8*a*tan(1/2*d*x + 1/2*c)^2 + 4*a*tan(1
/2*d*x + 1/2*c)) - 1/4*pi*m*sgn(tan(1/2*d*x + 1/2*c)^2 - 1)*sgn(tan(1/2*d*x + 1/2*c)) + 1/4*pi*m*sgn(4*a*tan(1
/2*d*x + 1/2*c)^3 + 8*a*tan(1/2*d*x + 1/2*c)^2 + 4*a*tan(1/2*d*x + 1/2*c)) - 1/2*pi*sgn(tan(1/2*d*x + 1/2*c)^2
- 1)*sgn(tan(1/2*d*x + 1/2*c)) - 1/4*pi*m + pi*floor(d*x/pi + c/pi + 1/2))*tan(1/2*d*x + 1/2*c) + cos(2*pi*m*
floor(-1/8*sgn(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c
)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) + 2) + 5/8) + 1/4*pi*m*sgn(4*tan(d*x + c)^2*tan(1/2*d*
x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2
*d*x + 1/2*c) + 2) - 1/4*pi*m)*e^(m*log(sqrt(2)*sqrt(abs(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x +
c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) + 2)*tan(d*x
+ c)^2*tan(1/2*d*x + 1/2*c)^2 + abs(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d*x +
1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) + 2)*tan(d*x + c)^2 + abs(4*tan(
d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*d*x +
1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c) + 2)*tan(1/2*d*x + 1/2*c)^2 + abs(4*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 +
8*tan(d*x + c)^2*tan(1/2*d*x + 1/2*c) + 4*tan(d*x + c)^2 + 2*tan(1/2*d*x + 1/2*c)^2 + 8*tan(1/2*d*x + 1/2*c)
+ 2))*abs(a)/(tan(d*x + c)^2*tan(1/2*d*x + 1/2*c)^2 + tan(d*x + c)^2 + tan(1/2*d*x + 1/2*c)^2 + 1)) - m*log(4*
abs(tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)) - 2*log(4*abs(tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2
*c)^2 + 1)))*tan(1/2*d*x + 1/2*c))/(d*tan(-1/2*pi + 1/4*pi*m*sgn(2*a*tan(1/2*d*x + 1/2*c)^4 + 4*a*tan(1/2*d*x
+ 1/2*c)^3 - 4*a*tan(1/2*d*x + 1/2*c) - 2*a)*sgn(4*a*tan(1/2*d*x + 1/2*c)^3 + 8*a*tan(1/2*d*x + 1/2*c)^2 + 4*a
*tan(1/2*d*x + 1/2*c)) - 1/4*pi*m*sgn(tan(1/2*d*x + 1/2*c)^2 - 1)*sgn(tan(1/2*d*x + 1/2*c)) + 1/4*pi*m*sgn(4*a
*tan(1/2*d*x + 1/2*c)^3 + 8*a*tan(1/2*d*x + 1/2*c)^2 + 4*a*tan(1/2*d*x + 1/2*c)) - 1/2*pi*sgn(tan(1/2*d*x + 1/
2*c)^2 - 1)*sgn(tan(1/2*d*x + 1/2*c)) - 1/4*pi*m + pi*floor(d*x/pi + c/pi + 1/2))^2*tan(1/2*d*x + 1/2*c)^4 + 2
*d*tan(-1/2*pi + 1/4*pi*m*sgn(2*a*tan(1/2*d*x + 1/2*c)^4 + 4*a*tan(1/2*d*x + 1/2*c)^3 - 4*a*tan(1/2*d*x + 1/2*
c) - 2*a)*sgn(4*a*tan(1/2*d*x + 1/2*c)^3 + 8*a*tan(1/2*d*x + 1/2*c)^2 + 4*a*tan(1/2*d*x + 1/2*c)) - 1/4*pi*m*s
gn(tan(1/2*d*x + 1/2*c)^2 - 1)*sgn(tan(1/2*d*x + 1/2*c)) + 1/4*pi*m*sgn(4*a*tan(1/2*d*x + 1/2*c)^3 + 8*a*tan(1
/2*d*x + 1/2*c)^2 + 4*a*tan(1/2*d*x + 1/2*c)) - 1/2*pi*sgn(tan(1/2*d*x + 1/2*c)^2 - 1)*sgn(tan(1/2*d*x + 1/2*c
)) - 1/4*pi*m + pi*floor(d*x/pi + c/pi + 1/2))^2*tan(1/2*d*x + 1/2*c)^2 + d*tan(1/2*d*x + 1/2*c)^4 + d*tan(-1/
2*pi + 1/4*pi*m*sgn(2*a*tan(1/2*d*x + 1/2*c)^4 + 4*a*tan(1/2*d*x + 1/2*c)^3 - 4*a*tan(1/2*d*x + 1/2*c) - 2*a)*
sgn(4*a*tan(1/2*d*x + 1/2*c)^3 + 8*a*tan(1/2*d*x + 1/2*c)^2 + 4*a*tan(1/2*d*x + 1/2*c)) - 1/4*pi*m*sgn(tan(1/2
*d*x + 1/2*c)^2 - 1)*sgn(tan(1/2*d*x + 1/2*c)) + 1/4*pi*m*sgn(4*a*tan(1/2*d*x + 1/2*c)^3 + 8*a*tan(1/2*d*x + 1
/2*c)^2 + 4*a*tan(1/2*d*x + 1/2*c)) - 1/2*pi*sgn(tan(1/2*d*x + 1/2*c)^2 - 1)*sgn(tan(1/2*d*x + 1/2*c)) - 1/4*p
i*m + pi*floor(d*x/pi + c/pi + 1/2))^2 + 2*d*tan(1/2*d*x + 1/2*c)^2 + d)
Mupad [B] (verification not implemented)
Time = 14.39 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.09
\[
\int \sin ^{-2-m}(c+d x) (a+a \sin (c+d x))^m (1+m-m \sin (c+d x)) \, dx=-\frac {\sin \left (2\,c+2\,d\,x\right )\,{\left (a\,\left (\sin \left (c+d\,x\right )+1\right )\right )}^m}{2\,d\,{\sin \left (c+d\,x\right )}^{m+2}}
\]
[In]
int(((a + a*sin(c + d*x))^m*(m - m*sin(c + d*x) + 1))/sin(c + d*x)^(m + 2),x)
[Out]
-(sin(2*c + 2*d*x)*(a*(sin(c + d*x) + 1))^m)/(2*d*sin(c + d*x)^(m + 2))