3.38 \(\int (a+a \sec (c+d x))^3 \sin ^9(c+d x) \, dx\)

Optimal. Leaf size=203 \[ -\frac{a^3 \cos ^9(c+d x)}{9 d}-\frac{3 a^3 \cos ^8(c+d x)}{8 d}+\frac{a^3 \cos ^7(c+d x)}{7 d}+\frac{11 a^3 \cos ^6(c+d x)}{6 d}+\frac{6 a^3 \cos ^5(c+d x)}{5 d}-\frac{7 a^3 \cos ^4(c+d x)}{2 d}-\frac{14 a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a^3 \cos ^2(c+d x)}{d}+\frac{11 a^3 \cos (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{a^3 \log (\cos (c+d x))}{d} \]

[Out]

(11*a^3*Cos[c + d*x])/d + (3*a^3*Cos[c + d*x]^2)/d - (14*a^3*Cos[c + d*x]^3)/(3*d) - (7*a^3*Cos[c + d*x]^4)/(2
*d) + (6*a^3*Cos[c + d*x]^5)/(5*d) + (11*a^3*Cos[c + d*x]^6)/(6*d) + (a^3*Cos[c + d*x]^7)/(7*d) - (3*a^3*Cos[c
+ d*x]^8)/(8*d) - (a^3*Cos[c + d*x]^9)/(9*d) + (a^3*Log[Cos[c + d*x]])/d + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[
c + d*x]^2)/(2*d)

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Rubi [A]  time = 0.195633, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3872, 2836, 12, 88} \[ -\frac{a^3 \cos ^9(c+d x)}{9 d}-\frac{3 a^3 \cos ^8(c+d x)}{8 d}+\frac{a^3 \cos ^7(c+d x)}{7 d}+\frac{11 a^3 \cos ^6(c+d x)}{6 d}+\frac{6 a^3 \cos ^5(c+d x)}{5 d}-\frac{7 a^3 \cos ^4(c+d x)}{2 d}-\frac{14 a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a^3 \cos ^2(c+d x)}{d}+\frac{11 a^3 \cos (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{a^3 \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[c + d*x])^3*Sin[c + d*x]^9,x]

[Out]

(11*a^3*Cos[c + d*x])/d + (3*a^3*Cos[c + d*x]^2)/d - (14*a^3*Cos[c + d*x]^3)/(3*d) - (7*a^3*Cos[c + d*x]^4)/(2
*d) + (6*a^3*Cos[c + d*x]^5)/(5*d) + (11*a^3*Cos[c + d*x]^6)/(6*d) + (a^3*Cos[c + d*x]^7)/(7*d) - (3*a^3*Cos[c
+ d*x]^8)/(8*d) - (a^3*Cos[c + d*x]^9)/(9*d) + (a^3*Log[Cos[c + d*x]])/d + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[
c + d*x]^2)/(2*d)

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rule 2836

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

Rule 12

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

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int (a+a \sec (c+d x))^3 \sin ^9(c+d x) \, dx &=-\int (-a-a \cos (c+d x))^3 \sin ^6(c+d x) \tan ^3(c+d x) \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{a^3 (-a-x)^4 (-a+x)^7}{x^3} \, dx,x,-a \cos (c+d x)\right )}{a^9 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(-a-x)^4 (-a+x)^7}{x^3} \, dx,x,-a \cos (c+d x)\right )}{a^6 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-11 a^8-\frac{a^{11}}{x^3}+\frac{3 a^{10}}{x^2}+\frac{a^9}{x}+6 a^7 x+14 a^6 x^2-14 a^5 x^3-6 a^4 x^4+11 a^3 x^5-a^2 x^6-3 a x^7+x^8\right ) \, dx,x,-a \cos (c+d x)\right )}{a^6 d}\\ &=\frac{11 a^3 \cos (c+d x)}{d}+\frac{3 a^3 \cos ^2(c+d x)}{d}-\frac{14 a^3 \cos ^3(c+d x)}{3 d}-\frac{7 a^3 \cos ^4(c+d x)}{2 d}+\frac{6 a^3 \cos ^5(c+d x)}{5 d}+\frac{11 a^3 \cos ^6(c+d x)}{6 d}+\frac{a^3 \cos ^7(c+d x)}{7 d}-\frac{3 a^3 \cos ^8(c+d x)}{8 d}-\frac{a^3 \cos ^9(c+d x)}{9 d}+\frac{a^3 \log (\cos (c+d x))}{d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 1.69703, size = 148, normalized size = 0.73 \[ \frac{a^3 \sec ^2(c+d x) (11624760 \cos (c+d x)+2188872 \cos (3 (c+d x))+41160 \cos (4 (c+d x))-204156 \cos (5 (c+d x))-35805 \cos (6 (c+d x))+22972 \cos (7 (c+d x))+9030 \cos (8 (c+d x))-820 \cos (9 (c+d x))-945 \cos (10 (c+d x))-140 \cos (11 (c+d x))+645120 \log (\cos (c+d x))+210 \cos (2 (c+d x)) (3072 \log (\cos (c+d x))-413)+471450)}{1290240 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sec[c + d*x])^3*Sin[c + d*x]^9,x]

[Out]

(a^3*(471450 + 11624760*Cos[c + d*x] + 2188872*Cos[3*(c + d*x)] + 41160*Cos[4*(c + d*x)] - 204156*Cos[5*(c + d
*x)] - 35805*Cos[6*(c + d*x)] + 22972*Cos[7*(c + d*x)] + 9030*Cos[8*(c + d*x)] - 820*Cos[9*(c + d*x)] - 945*Co
s[10*(c + d*x)] - 140*Cos[11*(c + d*x)] + 645120*Log[Cos[c + d*x]] + 210*Cos[2*(c + d*x)]*(-413 + 3072*Log[Cos
[c + d*x]]))*Sec[c + d*x]^2)/(1290240*d)

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Maple [A]  time = 0.05, size = 230, normalized size = 1.1 \begin{align*}{\frac{3328\,{a}^{3}\cos \left ( dx+c \right ) }{315\,d}}+{\frac{26\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{8}\cos \left ( dx+c \right ) }{9\,d}}+{\frac{208\,{a}^{3}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{63\,d}}+{\frac{416\,{a}^{3}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{105\,d}}+{\frac{1664\,{a}^{3}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{315\,d}}+{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{8\,d}}+{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{6\,d}}+{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{{a}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+3\,{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{d\cos \left ( dx+c \right ) }}+{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{10}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))^3*sin(d*x+c)^9,x)

[Out]

3328/315*a^3*cos(d*x+c)/d+26/9/d*a^3*sin(d*x+c)^8*cos(d*x+c)+208/63/d*a^3*cos(d*x+c)*sin(d*x+c)^6+416/105/d*a^
3*cos(d*x+c)*sin(d*x+c)^4+1664/315/d*a^3*cos(d*x+c)*sin(d*x+c)^2+1/8/d*a^3*sin(d*x+c)^8+1/6/d*a^3*sin(d*x+c)^6
+1/4/d*a^3*sin(d*x+c)^4+1/2/d*a^3*sin(d*x+c)^2+a^3*ln(cos(d*x+c))/d+3/d*a^3*sin(d*x+c)^10/cos(d*x+c)+1/2/d*a^3
*sin(d*x+c)^10/cos(d*x+c)^2

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Maxima [A]  time = 0.997478, size = 213, normalized size = 1.05 \begin{align*} -\frac{280 \, a^{3} \cos \left (d x + c\right )^{9} + 945 \, a^{3} \cos \left (d x + c\right )^{8} - 360 \, a^{3} \cos \left (d x + c\right )^{7} - 4620 \, a^{3} \cos \left (d x + c\right )^{6} - 3024 \, a^{3} \cos \left (d x + c\right )^{5} + 8820 \, a^{3} \cos \left (d x + c\right )^{4} + 11760 \, a^{3} \cos \left (d x + c\right )^{3} - 7560 \, a^{3} \cos \left (d x + c\right )^{2} - 27720 \, a^{3} \cos \left (d x + c\right ) - 2520 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) - \frac{1260 \,{\left (6 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )}}{\cos \left (d x + c\right )^{2}}}{2520 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^3*sin(d*x+c)^9,x, algorithm="maxima")

[Out]

-1/2520*(280*a^3*cos(d*x + c)^9 + 945*a^3*cos(d*x + c)^8 - 360*a^3*cos(d*x + c)^7 - 4620*a^3*cos(d*x + c)^6 -
3024*a^3*cos(d*x + c)^5 + 8820*a^3*cos(d*x + c)^4 + 11760*a^3*cos(d*x + c)^3 - 7560*a^3*cos(d*x + c)^2 - 27720
*a^3*cos(d*x + c) - 2520*a^3*log(cos(d*x + c)) - 1260*(6*a^3*cos(d*x + c) + a^3)/cos(d*x + c)^2)/d

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Fricas [A]  time = 2.21011, size = 539, normalized size = 2.66 \begin{align*} -\frac{35840 \, a^{3} \cos \left (d x + c\right )^{11} + 120960 \, a^{3} \cos \left (d x + c\right )^{10} - 46080 \, a^{3} \cos \left (d x + c\right )^{9} - 591360 \, a^{3} \cos \left (d x + c\right )^{8} - 387072 \, a^{3} \cos \left (d x + c\right )^{7} + 1128960 \, a^{3} \cos \left (d x + c\right )^{6} + 1505280 \, a^{3} \cos \left (d x + c\right )^{5} - 967680 \, a^{3} \cos \left (d x + c\right )^{4} - 3548160 \, a^{3} \cos \left (d x + c\right )^{3} - 322560 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) + 212205 \, a^{3} \cos \left (d x + c\right )^{2} - 967680 \, a^{3} \cos \left (d x + c\right ) - 161280 \, a^{3}}{322560 \, d \cos \left (d x + c\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^3*sin(d*x+c)^9,x, algorithm="fricas")

[Out]

-1/322560*(35840*a^3*cos(d*x + c)^11 + 120960*a^3*cos(d*x + c)^10 - 46080*a^3*cos(d*x + c)^9 - 591360*a^3*cos(
d*x + c)^8 - 387072*a^3*cos(d*x + c)^7 + 1128960*a^3*cos(d*x + c)^6 + 1505280*a^3*cos(d*x + c)^5 - 967680*a^3*
cos(d*x + c)^4 - 3548160*a^3*cos(d*x + c)^3 - 322560*a^3*cos(d*x + c)^2*log(-cos(d*x + c)) + 212205*a^3*cos(d*
x + c)^2 - 967680*a^3*cos(d*x + c) - 161280*a^3)/(d*cos(d*x + c)^2)

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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((a+a*sec(d*x+c))**3*sin(d*x+c)**9,x)

[Out]

Timed out

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Giac [B]  time = 1.42928, size = 535, normalized size = 2.64 \begin{align*} -\frac{2520 \, a^{3} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 2520 \, a^{3} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) - \frac{1260 \,{\left (9 \, a^{3} + \frac{2 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{3 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}} + \frac{45257 \, a^{3} - \frac{392193 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{1467972 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{3001908 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{3232782 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{2359854 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{1190196 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{397764 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{79281 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac{7129 \, a^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{9}}}{2520 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^3*sin(d*x+c)^9,x, algorithm="giac")

[Out]

-1/2520*(2520*a^3*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)) - 2520*a^3*log(abs(-(cos(d*x + c) - 1)/
(cos(d*x + c) + 1) - 1)) - 1260*(9*a^3 + 2*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 3*a^3*(cos(d*x + c) - 1
)^2/(cos(d*x + c) + 1)^2)/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)^2 + (45257*a^3 - 392193*a^3*(cos(d*x + c
) - 1)/(cos(d*x + c) + 1) + 1467972*a^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 3001908*a^3*(cos(d*x + c)
- 1)^3/(cos(d*x + c) + 1)^3 + 3232782*a^3*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 2359854*a^3*(cos(d*x + c
) - 1)^5/(cos(d*x + c) + 1)^5 + 1190196*a^3*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 - 397764*a^3*(cos(d*x +
c) - 1)^7/(cos(d*x + c) + 1)^7 + 79281*a^3*(cos(d*x + c) - 1)^8/(cos(d*x + c) + 1)^8 - 7129*a^3*(cos(d*x + c)
- 1)^9/(cos(d*x + c) + 1)^9)/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^9)/d