3.845 \(\int \frac{\sec ^3(c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=123 \[ \frac{4 \tan ^3(c+d x)}{63 a^3 d}+\frac{4 \tan (c+d x)}{21 a^3 d}-\frac{\sec ^3(c+d x)}{21 d \left (a^3 \sin (c+d x)+a^3\right )}-\frac{\sec ^3(c+d x)}{21 a d (a \sin (c+d x)+a)^2}+\frac{\sec ^3(c+d x)}{9 d (a \sin (c+d x)+a)^3} \]

[Out]

Sec[c + d*x]^3/(9*d*(a + a*Sin[c + d*x])^3) - Sec[c + d*x]^3/(21*a*d*(a + a*Sin[c + d*x])^2) - Sec[c + d*x]^3/
(21*d*(a^3 + a^3*Sin[c + d*x])) + (4*Tan[c + d*x])/(21*a^3*d) + (4*Tan[c + d*x]^3)/(63*a^3*d)

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Rubi [A]  time = 0.160907, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2859, 2672, 3767} \[ \frac{4 \tan ^3(c+d x)}{63 a^3 d}+\frac{4 \tan (c+d x)}{21 a^3 d}-\frac{\sec ^3(c+d x)}{21 d \left (a^3 \sin (c+d x)+a^3\right )}-\frac{\sec ^3(c+d x)}{21 a d (a \sin (c+d x)+a)^2}+\frac{\sec ^3(c+d x)}{9 d (a \sin (c+d x)+a)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sec[c + d*x]^3/(9*d*(a + a*Sin[c + d*x])^3) - Sec[c + d*x]^3/(21*a*d*(a + a*Sin[c + d*x])^2) - Sec[c + d*x]^3/
(21*d*(a^3 + a^3*Sin[c + d*x])) + (4*Tan[c + d*x])/(21*a^3*d) + (4*Tan[c + d*x]^3)/(63*a^3*d)

Rule 2859

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*(2*m +
p + 1)), x] + Dist[(a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^
(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[
m + p], 0]) && NeQ[2*m + p + 1, 0]

Rule 2672

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

Rubi steps

\begin{align*} \int \frac{\sec ^3(c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}+\frac{\int \frac{\sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx}{3 a}\\ &=\frac{\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}-\frac{\sec ^3(c+d x)}{21 a d (a+a \sin (c+d x))^2}+\frac{5 \int \frac{\sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx}{21 a^2}\\ &=\frac{\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}-\frac{\sec ^3(c+d x)}{21 a d (a+a \sin (c+d x))^2}-\frac{\sec ^3(c+d x)}{21 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac{4 \int \sec ^4(c+d x) \, dx}{21 a^3}\\ &=\frac{\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}-\frac{\sec ^3(c+d x)}{21 a d (a+a \sin (c+d x))^2}-\frac{\sec ^3(c+d x)}{21 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac{4 \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{21 a^3 d}\\ &=\frac{\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}-\frac{\sec ^3(c+d x)}{21 a d (a+a \sin (c+d x))^2}-\frac{\sec ^3(c+d x)}{21 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac{4 \tan (c+d x)}{21 a^3 d}+\frac{4 \tan ^3(c+d x)}{63 a^3 d}\\ \end{align*}

Mathematica [A]  time = 0.276557, size = 185, normalized size = 1.5 \[ \frac{9216 \sin (c+d x)+675 \sin (2 (c+d x))+512 \sin (3 (c+d x))+300 \sin (4 (c+d x))-1536 \sin (5 (c+d x))-25 \sin (6 (c+d x))+900 \cos (c+d x)-6912 \cos (2 (c+d x))+50 \cos (3 (c+d x))-3072 \cos (4 (c+d x))-150 \cos (5 (c+d x))+256 \cos (6 (c+d x))+10752}{64512 d (a \sin (c+d x)+a)^3 \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10752 + 900*Cos[c + d*x] - 6912*Cos[2*(c + d*x)] + 50*Cos[3*(c + d*x)] - 3072*Cos[4*(c + d*x)] - 150*Cos[5*(c
+ d*x)] + 256*Cos[6*(c + d*x)] + 9216*Sin[c + d*x] + 675*Sin[2*(c + d*x)] + 512*Sin[3*(c + d*x)] + 300*Sin[4*
(c + d*x)] - 1536*Sin[5*(c + d*x)] - 25*Sin[6*(c + d*x)])/(64512*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Co
s[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(a + a*Sin[c + d*x])^3)

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Maple [A]  time = 0.113, size = 190, normalized size = 1.5 \begin{align*} 4\,{\frac{1}{d{a}^{3}} \left ( -{\frac{1}{96\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{3}}}-{\frac{1}{64\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{2}}}-{\frac{5}{128\,\tan \left ( 1/2\,dx+c/2 \right ) -128}}+2/9\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-9}- \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-8}+{\frac{16}{7\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{7}}}-10/3\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-6}+{\frac{27}{8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{5}}}-{\frac{39}{16\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{4}}}+{\frac{59}{48\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{3}}}-{\frac{13}{32\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{2}}}+{\frac{5}{128\,\tan \left ( 1/2\,dx+c/2 \right ) +128}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/d/a^3*(-1/96/(tan(1/2*d*x+1/2*c)-1)^3-1/64/(tan(1/2*d*x+1/2*c)-1)^2-5/128/(tan(1/2*d*x+1/2*c)-1)+2/9/(tan(1/
2*d*x+1/2*c)+1)^9-1/(tan(1/2*d*x+1/2*c)+1)^8+16/7/(tan(1/2*d*x+1/2*c)+1)^7-10/3/(tan(1/2*d*x+1/2*c)+1)^6+27/8/
(tan(1/2*d*x+1/2*c)+1)^5-39/16/(tan(1/2*d*x+1/2*c)+1)^4+59/48/(tan(1/2*d*x+1/2*c)+1)^3-13/32/(tan(1/2*d*x+1/2*
c)+1)^2+5/128/(tan(1/2*d*x+1/2*c)+1))

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Maxima [B]  time = 1.18718, size = 597, normalized size = 4.85 \begin{align*} \frac{2 \,{\left (\frac{6 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{75 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{128 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{162 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{36 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{42 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{189 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{126 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac{63 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + 1\right )}}{63 \,{\left (a^{3} + \frac{6 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{12 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{2 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{27 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{36 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{36 \, a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{27 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac{2 \, a^{3} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac{12 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac{6 \, a^{3} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac{a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/63*(6*sin(d*x + c)/(cos(d*x + c) + 1) + 75*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 128*sin(d*x + c)^3/(cos(d*x
+ c) + 1)^3 + 162*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 36*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 42*sin(d*x +
c)^6/(cos(d*x + c) + 1)^6 + 189*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 126*sin(d*x + c)^9/(cos(d*x + c) + 1)^9
+ 63*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 1)/((a^3 + 6*a^3*sin(d*x + c)/(cos(d*x + c) + 1) + 12*a^3*sin(d*
x + c)^2/(cos(d*x + c) + 1)^2 + 2*a^3*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 27*a^3*sin(d*x + c)^4/(cos(d*x + c
) + 1)^4 - 36*a^3*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 36*a^3*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 27*a^3*si
n(d*x + c)^8/(cos(d*x + c) + 1)^8 - 2*a^3*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 12*a^3*sin(d*x + c)^10/(cos(d*
x + c) + 1)^10 - 6*a^3*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - a^3*sin(d*x + c)^12/(cos(d*x + c) + 1)^12)*d)

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Fricas [A]  time = 1.68669, size = 328, normalized size = 2.67 \begin{align*} -\frac{8 \, \cos \left (d x + c\right )^{6} - 36 \, \cos \left (d x + c\right )^{4} + 15 \, \cos \left (d x + c\right )^{2} -{\left (24 \, \cos \left (d x + c\right )^{4} - 20 \, \cos \left (d x + c\right )^{2} - 7\right )} \sin \left (d x + c\right ) + 14}{63 \,{\left (3 \, a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3} +{\left (a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/63*(8*cos(d*x + c)^6 - 36*cos(d*x + c)^4 + 15*cos(d*x + c)^2 - (24*cos(d*x + c)^4 - 20*cos(d*x + c)^2 - 7)*
sin(d*x + c) + 14)/(3*a^3*d*cos(d*x + c)^5 - 4*a^3*d*cos(d*x + c)^3 + (a^3*d*cos(d*x + c)^5 - 4*a^3*d*cos(d*x
+ c)^3)*sin(d*x + c))

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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*sin(d*x+c)/(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.31876, size = 232, normalized size = 1.89 \begin{align*} -\frac{\frac{21 \,{\left (15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 24 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 13\right )}}{a^{3}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{3}} - \frac{315 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 756 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 4200 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 11340 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 14994 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 13356 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 6768 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 2196 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 209}{a^{3}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{9}}}{2016 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2016*(21*(15*tan(1/2*d*x + 1/2*c)^2 - 24*tan(1/2*d*x + 1/2*c) + 13)/(a^3*(tan(1/2*d*x + 1/2*c) - 1)^3) - (3
15*tan(1/2*d*x + 1/2*c)^8 - 756*tan(1/2*d*x + 1/2*c)^7 - 4200*tan(1/2*d*x + 1/2*c)^6 - 11340*tan(1/2*d*x + 1/2
*c)^5 - 14994*tan(1/2*d*x + 1/2*c)^4 - 13356*tan(1/2*d*x + 1/2*c)^3 - 6768*tan(1/2*d*x + 1/2*c)^2 - 2196*tan(1
/2*d*x + 1/2*c) - 209)/(a^3*(tan(1/2*d*x + 1/2*c) + 1)^9))/d