3.409 \(\int \sec ^4(c+d x) (a+b \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=84 \[ \frac{2 a \left (a^2-b^2\right ) \tan (c+d x)}{3 d}+\frac{2 b \left (a^2-b^2\right ) \sec (c+d x)}{3 d}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{3 d} \]

[Out]

(2*b*(a^2 - b^2)*Sec[c + d*x])/(3*d) + (Sec[c + d*x]^3*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(3*d) + (2
*a*(a^2 - b^2)*Tan[c + d*x])/(3*d)

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Rubi [A]  time = 0.088474, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2691, 12, 2669, 3767, 8} \[ \frac{2 a \left (a^2-b^2\right ) \tan (c+d x)}{3 d}+\frac{2 b \left (a^2-b^2\right ) \sec (c+d x)}{3 d}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^4*(a + b*Sin[c + d*x])^3,x]

[Out]

(2*b*(a^2 - b^2)*Sec[c + d*x])/(3*d) + (Sec[c + d*x]^3*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(3*d) + (2
*a*(a^2 - b^2)*Tan[c + d*x])/(3*d)

Rule 2691

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2669

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b*(g*Cos[
e + f*x])^(p + 1))/(f*g*(p + 1)), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x]
&& (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \sec ^4(c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac{\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{3 d}-\frac{1}{3} \int \left (-2 a^2+2 b^2\right ) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx\\ &=\frac{\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{3 d}+\frac{1}{3} \left (2 \left (a^2-b^2\right )\right ) \int \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx\\ &=\frac{2 b \left (a^2-b^2\right ) \sec (c+d x)}{3 d}+\frac{\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{3 d}+\frac{1}{3} \left (2 a \left (a^2-b^2\right )\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac{2 b \left (a^2-b^2\right ) \sec (c+d x)}{3 d}+\frac{\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{3 d}-\frac{\left (2 a \left (a^2-b^2\right )\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=\frac{2 b \left (a^2-b^2\right ) \sec (c+d x)}{3 d}+\frac{\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{3 d}+\frac{2 a \left (a^2-b^2\right ) \tan (c+d x)}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.412579, size = 136, normalized size = 1.62 \[ \frac{\sec ^3(c+d x) \left (\left (15 b^3-9 a^2 b\right ) \cos (c+d x)-3 a^2 b \cos (3 (c+d x))+24 a^2 b+12 a^3 \sin (c+d x)+4 a^3 \sin (3 (c+d x))+18 a b^2 \sin (c+d x)-6 a b^2 \sin (3 (c+d x))-12 b^3 \cos (2 (c+d x))+5 b^3 \cos (3 (c+d x))-4 b^3\right )}{24 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^4*(a + b*Sin[c + d*x])^3,x]

[Out]

(Sec[c + d*x]^3*(24*a^2*b - 4*b^3 + (-9*a^2*b + 15*b^3)*Cos[c + d*x] - 12*b^3*Cos[2*(c + d*x)] - 3*a^2*b*Cos[3
*(c + d*x)] + 5*b^3*Cos[3*(c + d*x)] + 12*a^3*Sin[c + d*x] + 18*a*b^2*Sin[c + d*x] + 4*a^3*Sin[3*(c + d*x)] -
6*a*b^2*Sin[3*(c + d*x)]))/(24*d)

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Maple [A]  time = 0.068, size = 122, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ( -{a}^{3} \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \tan \left ( dx+c \right ) +{\frac{{a}^{2}b}{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{a{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{b}^{3} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{3\,\cos \left ( dx+c \right ) }}-{\frac{ \left ( 2+ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \right ) \cos \left ( dx+c \right ) }{3}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^4*(a+b*sin(d*x+c))^3,x)

[Out]

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

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Maxima [A]  time = 0.954564, size = 108, normalized size = 1.29 \begin{align*} \frac{3 \, a b^{2} \tan \left (d x + c\right )^{3} +{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{3} - \frac{{\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} b^{3}}{\cos \left (d x + c\right )^{3}} + \frac{3 \, a^{2} b}{\cos \left (d x + c\right )^{3}}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.42204, size = 176, normalized size = 2.1 \begin{align*} -\frac{3 \, b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b - b^{3} -{\left (a^{3} + 3 \, a b^{2} +{\left (2 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{3 \, d \cos \left (d x + c\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

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

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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)**4*(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.11149, size = 173, normalized size = 2.06 \begin{align*} -\frac{2 \,{\left (3 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 9 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 2 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 12 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 3 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, a^{2} b - 2 \, b^{3}\right )}}{3 \,{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-2/3*(3*a^3*tan(1/2*d*x + 1/2*c)^5 + 9*a^2*b*tan(1/2*d*x + 1/2*c)^4 - 2*a^3*tan(1/2*d*x + 1/2*c)^3 + 12*a*b^2*
tan(1/2*d*x + 1/2*c)^3 + 6*b^3*tan(1/2*d*x + 1/2*c)^2 + 3*a^3*tan(1/2*d*x + 1/2*c) + 3*a^2*b - 2*b^3)/((tan(1/
2*d*x + 1/2*c)^2 - 1)^3*d)