3.980 \(\int \sec ^2(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx\)

Optimal. Leaf size=55 \[ \frac {a^2 (A+2 B) \cos (c+d x)}{d}-\left (a^2 x (A+2 B)\right )+\frac {(A+B) \sec (c+d x) (a \sin (c+d x)+a)^2}{d} \]

[Out]

-a^2*(A+2*B)*x+a^2*(A+2*B)*cos(d*x+c)/d+(A+B)*sec(d*x+c)*(a+a*sin(d*x+c))^2/d

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2855, 2638} \[ \frac {a^2 (A+2 B) \cos (c+d x)}{d}+a^2 x (-(A+2 B))+\frac {(A+B) \sec (c+d x) (a \sin (c+d x)+a)^2}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]

[Out]

-(a^2*(A + 2*B)*x) + (a^2*(A + 2*B)*Cos[c + d*x])/d + ((A + B)*Sec[c + d*x]*(a + a*Sin[c + d*x])^2)/d

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2855

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

Rubi steps

\begin {align*} \int \sec ^2(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx &=\frac {(A+B) \sec (c+d x) (a+a \sin (c+d x))^2}{d}-(a (A+2 B)) \int (a+a \sin (c+d x)) \, dx\\ &=-a^2 (A+2 B) x+\frac {(A+B) \sec (c+d x) (a+a \sin (c+d x))^2}{d}-\left (a^2 (A+2 B)\right ) \int \sin (c+d x) \, dx\\ &=-a^2 (A+2 B) x+\frac {a^2 (A+2 B) \cos (c+d x)}{d}+\frac {(A+B) \sec (c+d x) (a+a \sin (c+d x))^2}{d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.24, size = 91, normalized size = 1.65 \[ \frac {a^2 \sec (c+d x) \left (4 (A+2 B) \sin ^{-1}\left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right ) \sqrt {\cos ^2(c+d x)}+4 A \sin (c+d x)+4 A+4 B \sin (c+d x)+B \cos (2 (c+d x))+5 B\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]),x]

[Out]

(a^2*Sec[c + d*x]*(4*A + 5*B + 4*(A + 2*B)*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[Cos[c + d*x]^2] + B*Cos
[2*(c + d*x)] + 4*A*Sin[c + d*x] + 4*B*Sin[c + d*x]))/(2*d)

________________________________________________________________________________________

fricas [B]  time = 0.68, size = 128, normalized size = 2.33 \[ -\frac {{\left (A + 2 \, B\right )} a^{2} d x - B a^{2} \cos \left (d x + c\right )^{2} - 2 \, {\left (A + B\right )} a^{2} + {\left ({\left (A + 2 \, B\right )} a^{2} d x - {\left (2 \, A + 3 \, B\right )} a^{2}\right )} \cos \left (d x + c\right ) - {\left ({\left (A + 2 \, B\right )} a^{2} d x - B a^{2} \cos \left (d x + c\right ) + 2 \, {\left (A + B\right )} a^{2}\right )} \sin \left (d x + c\right )}{d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2*(a+a*sin(d*x+c))^2*(A+B*sin(d*x+c)),x, algorithm="fricas")

[Out]

-((A + 2*B)*a^2*d*x - B*a^2*cos(d*x + c)^2 - 2*(A + B)*a^2 + ((A + 2*B)*a^2*d*x - (2*A + 3*B)*a^2)*cos(d*x + c
) - ((A + 2*B)*a^2*d*x - B*a^2*cos(d*x + c) + 2*(A + B)*a^2)*sin(d*x + c))/(d*cos(d*x + c) - d*sin(d*x + c) +
d)

________________________________________________________________________________________

giac [B]  time = 0.20, size = 125, normalized size = 2.27 \[ -\frac {{\left (A a^{2} + 2 \, B a^{2}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (2 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, A a^{2} + 3 \, B a^{2}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2*(a+a*sin(d*x+c))^2*(A+B*sin(d*x+c)),x, algorithm="giac")

[Out]

-((A*a^2 + 2*B*a^2)*(d*x + c) + 2*(2*A*a^2*tan(1/2*d*x + 1/2*c)^2 + 2*B*a^2*tan(1/2*d*x + 1/2*c)^2 - B*a^2*tan
(1/2*d*x + 1/2*c) + 2*A*a^2 + 3*B*a^2)/(tan(1/2*d*x + 1/2*c)^3 - tan(1/2*d*x + 1/2*c)^2 + tan(1/2*d*x + 1/2*c)
- 1))/d

________________________________________________________________________________________

maple [B]  time = 0.61, size = 123, normalized size = 2.24 \[ \frac {a^{2} A \left (\tan \left (d x +c \right )-d x -c \right )+B \,a^{2} \left (\frac {\sin ^{4}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )\right )+\frac {2 a^{2} A}{\cos \left (d x +c \right )}+2 B \,a^{2} \left (\tan \left (d x +c \right )-d x -c \right )+a^{2} A \tan \left (d x +c \right )+\frac {B \,a^{2}}{\cos \left (d x +c \right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^2*(a+a*sin(d*x+c))^2*(A+B*sin(d*x+c)),x)

[Out]

1/d*(a^2*A*(tan(d*x+c)-d*x-c)+B*a^2*(sin(d*x+c)^4/cos(d*x+c)+(2+sin(d*x+c)^2)*cos(d*x+c))+2*a^2*A/cos(d*x+c)+2
*B*a^2*(tan(d*x+c)-d*x-c)+a^2*A*tan(d*x+c)+B*a^2/cos(d*x+c))

________________________________________________________________________________________

maxima [A]  time = 0.50, size = 104, normalized size = 1.89 \[ -\frac {{\left (d x + c - \tan \left (d x + c\right )\right )} A a^{2} + 2 \, {\left (d x + c - \tan \left (d x + c\right )\right )} B a^{2} - B a^{2} {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} - A a^{2} \tan \left (d x + c\right ) - \frac {2 \, A a^{2}}{\cos \left (d x + c\right )} - \frac {B a^{2}}{\cos \left (d x + c\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2*(a+a*sin(d*x+c))^2*(A+B*sin(d*x+c)),x, algorithm="maxima")

[Out]

-((d*x + c - tan(d*x + c))*A*a^2 + 2*(d*x + c - tan(d*x + c))*B*a^2 - B*a^2*(1/cos(d*x + c) + cos(d*x + c)) -
A*a^2*tan(d*x + c) - 2*A*a^2/cos(d*x + c) - B*a^2/cos(d*x + c))/d

________________________________________________________________________________________

mupad [B]  time = 9.30, size = 110, normalized size = 2.00 \[ -\frac {4\,A\,a^2+6\,B\,a^2+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (4\,A\,a^2+4\,B\,a^2\right )-2\,B\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}-A\,a^2\,x-2\,B\,a^2\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*sin(c + d*x))*(a + a*sin(c + d*x))^2)/cos(c + d*x)^2,x)

[Out]

- (4*A*a^2 + 6*B*a^2 + tan(c/2 + (d*x)/2)^2*(4*A*a^2 + 4*B*a^2) - 2*B*a^2*tan(c/2 + (d*x)/2))/(d*(tan(c/2 + (d
*x)/2) - tan(c/2 + (d*x)/2)^2 + tan(c/2 + (d*x)/2)^3 - 1)) - A*a^2*x - 2*B*a^2*x

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ a^{2} \left (\int A \sec ^{2}{\left (c + d x \right )}\, dx + \int 2 A \sin {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int A \sin ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int B \sin {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 2 B \sin ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int B \sin ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**2*(a+a*sin(d*x+c))**2*(A+B*sin(d*x+c)),x)

[Out]

a**2*(Integral(A*sec(c + d*x)**2, x) + Integral(2*A*sin(c + d*x)*sec(c + d*x)**2, x) + Integral(A*sin(c + d*x)
**2*sec(c + d*x)**2, x) + Integral(B*sin(c + d*x)*sec(c + d*x)**2, x) + Integral(2*B*sin(c + d*x)**2*sec(c + d
*x)**2, x) + Integral(B*sin(c + d*x)**3*sec(c + d*x)**2, x))

________________________________________________________________________________________