3.767 \(\int \sec (c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx\)

Optimal. Leaf size=67 \[ \frac{6 a^3 \cos (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{9 a^3 x}{2}+\frac{\sec (c+d x) (a \sin (c+d x)+a)^3}{d} \]

[Out]

(-9*a^3*x)/2 + (6*a^3*Cos[c + d*x])/d + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (Sec[c + d*x]*(a + a*Sin[c +
d*x])^3)/d

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Rubi [A]  time = 0.063891, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2855, 2644} \[ \frac{6 a^3 \cos (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{9 a^3 x}{2}+\frac{\sec (c+d x) (a \sin (c+d x)+a)^3}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]*(a + a*Sin[c + d*x])^3*Tan[c + d*x],x]

[Out]

(-9*a^3*x)/2 + (6*a^3*Cos[c + d*x])/d + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (Sec[c + d*x]*(a + a*Sin[c +
d*x])^3)/d

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]

Rule 2644

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^2, x_Symbol] :> Simp[((2*a^2 + b^2)*x)/2, x] + (-Simp[(2*a*b*Cos[c
+ d*x])/d, x] - Simp[(b^2*Cos[c + d*x]*Sin[c + d*x])/(2*d), x]) /; FreeQ[{a, b, c, d}, x]

Rubi steps

\begin{align*} \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx &=\frac{\sec (c+d x) (a+a \sin (c+d x))^3}{d}-(3 a) \int (a+a \sin (c+d x))^2 \, dx\\ &=-\frac{9 a^3 x}{2}+\frac{6 a^3 \cos (c+d x)}{d}+\frac{3 a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac{\sec (c+d x) (a+a \sin (c+d x))^3}{d}\\ \end{align*}

Mathematica [B]  time = 0.500428, size = 145, normalized size = 2.16 \[ -\frac{a^3 (\sin (c+d x)+1)^3 \left (\cos \left (\frac{1}{2} (c+d x)\right ) (18 (c+d x)-\sin (2 (c+d x))-12 \cos (c+d x))+\sin \left (\frac{1}{2} (c+d x)\right ) (-2 (9 c+9 d x+16)+\sin (2 (c+d x))+12 \cos (c+d x))\right )}{4 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])^3*Tan[c + d*x],x]

[Out]

-(a^3*(1 + Sin[c + d*x])^3*(Cos[(c + d*x)/2]*(18*(c + d*x) - 12*Cos[c + d*x] - Sin[2*(c + d*x)]) + Sin[(c + d*
x)/2]*(-2*(16 + 9*c + 9*d*x) + 12*Cos[c + d*x] + Sin[2*(c + d*x)])))/(4*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]
)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)

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Maple [B]  time = 0.053, size = 130, normalized size = 1.9 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{\cos \left ( dx+c \right ) }}+ \left ( \left ( \sin \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\sin \left ( dx+c \right ) }{2}} \right ) \cos \left ( dx+c \right ) -{\frac{3\,dx}{2}}-{\frac{3\,c}{2}} \right ) +3\,{a}^{3} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{\cos \left ( dx+c \right ) }}+ \left ( 2+ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \right ) \cos \left ( dx+c \right ) \right ) +3\,{a}^{3} \left ( \tan \left ( dx+c \right ) -dx-c \right ) +{\frac{{a}^{3}}{\cos \left ( dx+c \right ) }} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^2*sin(d*x+c)*(a+a*sin(d*x+c))^3,x)

[Out]

1/d*(a^3*(sin(d*x+c)^5/cos(d*x+c)+(sin(d*x+c)^3+3/2*sin(d*x+c))*cos(d*x+c)-3/2*d*x-3/2*c)+3*a^3*(sin(d*x+c)^4/
cos(d*x+c)+(2+sin(d*x+c)^2)*cos(d*x+c))+3*a^3*(tan(d*x+c)-d*x-c)+a^3/cos(d*x+c))

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Maxima [A]  time = 1.5146, size = 131, normalized size = 1.96 \begin{align*} -\frac{{\left (3 \, d x + 3 \, c - \frac{\tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 2 \, \tan \left (d x + c\right )\right )} a^{3} + 6 \,{\left (d x + c - \tan \left (d x + c\right )\right )} a^{3} - 6 \, a^{3}{\left (\frac{1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} - \frac{2 \, a^{3}}{\cos \left (d x + c\right )}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2*sin(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.08699, size = 297, normalized size = 4.43 \begin{align*} \frac{a^{3} \cos \left (d x + c\right )^{3} - 9 \, a^{3} d x + 6 \, a^{3} \cos \left (d x + c\right )^{2} + 8 \, a^{3} -{\left (9 \, a^{3} d x - 13 \, a^{3}\right )} \cos \left (d x + c\right ) +{\left (9 \, a^{3} d x + a^{3} \cos \left (d x + c\right )^{2} - 5 \, a^{3} \cos \left (d x + c\right ) + 8 \, a^{3}\right )} \sin \left (d x + c\right )}{2 \,{\left (d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2*sin(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/2*(a^3*cos(d*x + c)^3 - 9*a^3*d*x + 6*a^3*cos(d*x + c)^2 + 8*a^3 - (9*a^3*d*x - 13*a^3)*cos(d*x + c) + (9*a^
3*d*x + a^3*cos(d*x + c)^2 - 5*a^3*cos(d*x + c) + 8*a^3)*sin(d*x + c))/(d*cos(d*x + c) - d*sin(d*x + c) + d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**2*sin(d*x+c)*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.26537, size = 138, normalized size = 2.06 \begin{align*} -\frac{9 \,{\left (d x + c\right )} a^{3} + \frac{16 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1} + \frac{2 \,{\left (a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 6 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 6 \, a^{3}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2*sin(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/2*(9*(d*x + c)*a^3 + 16*a^3/(tan(1/2*d*x + 1/2*c) - 1) + 2*(a^3*tan(1/2*d*x + 1/2*c)^3 - 6*a^3*tan(1/2*d*x
+ 1/2*c)^2 - a^3*tan(1/2*d*x + 1/2*c) - 6*a^3)/(tan(1/2*d*x + 1/2*c)^2 + 1)^2)/d