3.1.1 \(\int (a+a \sec (c+d x)) \sin ^9(c+d x) \, dx\) [1]

Optimal. Leaf size=152 \[ -\frac {a \cos (c+d x)}{d}+\frac {2 a \cos ^2(c+d x)}{d}+\frac {4 a \cos ^3(c+d x)}{3 d}-\frac {3 a \cos ^4(c+d x)}{2 d}-\frac {6 a \cos ^5(c+d x)}{5 d}+\frac {2 a \cos ^6(c+d x)}{3 d}+\frac {4 a \cos ^7(c+d x)}{7 d}-\frac {a \cos ^8(c+d x)}{8 d}-\frac {a \cos ^9(c+d x)}{9 d}-\frac {a \log (\cos (c+d x))}{d} \]

[Out]

-a*cos(d*x+c)/d+2*a*cos(d*x+c)^2/d+4/3*a*cos(d*x+c)^3/d-3/2*a*cos(d*x+c)^4/d-6/5*a*cos(d*x+c)^5/d+2/3*a*cos(d*
x+c)^6/d+4/7*a*cos(d*x+c)^7/d-1/8*a*cos(d*x+c)^8/d-1/9*a*cos(d*x+c)^9/d-a*ln(cos(d*x+c))/d

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Rubi [A]
time = 0.08, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3957, 2915, 12, 90} \begin {gather*} -\frac {a \cos ^9(c+d x)}{9 d}-\frac {a \cos ^8(c+d x)}{8 d}+\frac {4 a \cos ^7(c+d x)}{7 d}+\frac {2 a \cos ^6(c+d x)}{3 d}-\frac {6 a \cos ^5(c+d x)}{5 d}-\frac {3 a \cos ^4(c+d x)}{2 d}+\frac {4 a \cos ^3(c+d x)}{3 d}+\frac {2 a \cos ^2(c+d x)}{d}-\frac {a \cos (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

Rule 90

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]))

Rule 2915

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/b)*x
)^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 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[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]

Rubi steps

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

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Mathematica [A]
time = 0.14, size = 106, normalized size = 0.70 \begin {gather*} -\frac {a \left (39690 \cos (c+d x)-161280 \cos ^2(c+d x)+120960 \cos ^4(c+d x)-53760 \cos ^6(c+d x)+10080 \cos ^8(c+d x)-8820 \cos (3 (c+d x))+2268 \cos (5 (c+d x))-405 \cos (7 (c+d x))+35 \cos (9 (c+d x))+80640 \log (\cos (c+d x))\right )}{80640 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/80640*(a*(39690*Cos[c + d*x] - 161280*Cos[c + d*x]^2 + 120960*Cos[c + d*x]^4 - 53760*Cos[c + d*x]^6 + 10080
*Cos[c + d*x]^8 - 8820*Cos[3*(c + d*x)] + 2268*Cos[5*(c + d*x)] - 405*Cos[7*(c + d*x)] + 35*Cos[9*(c + d*x)] +
80640*Log[Cos[c + d*x]]))/d

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Maple [A]
time = 0.12, size = 107, normalized size = 0.70

method	result	size
derivativedivides	\(\frac {a \left (-\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{8}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a \left (\frac {128}{35}+\sin ^{8}\left (d x +c \right )+\frac {8 \left (\sin ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\sin ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\sin ^{2}\left (d x +c \right )\right )}{35}\right ) \cos \left (d x +c \right )}{9}}{d}\)	\(107\)
default	\(\frac {a \left (-\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{8}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a \left (\frac {128}{35}+\sin ^{8}\left (d x +c \right )+\frac {8 \left (\sin ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\sin ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\sin ^{2}\left (d x +c \right )\right )}{35}\right ) \cos \left (d x +c \right )}{9}}{d}\)	\(107\)
risch	\(i a x +\frac {2 i a c}{d}+\frac {65 a \,{\mathrm e}^{2 i \left (d x +c \right )}}{256 d}+\frac {65 a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{256 d}-\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}-\frac {63 a \cos \left (d x +c \right )}{128 d}-\frac {a \cos \left (9 d x +9 c \right )}{2304 d}-\frac {a \cos \left (8 d x +8 c \right )}{1024 d}+\frac {9 a \cos \left (7 d x +7 c \right )}{1792 d}+\frac {5 a \cos \left (6 d x +6 c \right )}{384 d}-\frac {9 a \cos \left (5 d x +5 c \right )}{320 d}-\frac {23 a \cos \left (4 d x +4 c \right )}{256 d}+\frac {7 a \cos \left (3 d x +3 c \right )}{64 d}\)	\(180\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))*sin(d*x+c)^9,x,method=_RETURNVERBOSE)

[Out]

1/d*(a*(-1/8*sin(d*x+c)^8-1/6*sin(d*x+c)^6-1/4*sin(d*x+c)^4-1/2*sin(d*x+c)^2-ln(cos(d*x+c)))-1/9*a*(128/35+sin
(d*x+c)^8+8/7*sin(d*x+c)^6+48/35*sin(d*x+c)^4+64/35*sin(d*x+c)^2)*cos(d*x+c))

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Maxima [A]
time = 0.25, size = 113, normalized size = 0.74 \begin {gather*} -\frac {280 \, a \cos \left (d x + c\right )^{9} + 315 \, a \cos \left (d x + c\right )^{8} - 1440 \, a \cos \left (d x + c\right )^{7} - 1680 \, a \cos \left (d x + c\right )^{6} + 3024 \, a \cos \left (d x + c\right )^{5} + 3780 \, a \cos \left (d x + c\right )^{4} - 3360 \, a \cos \left (d x + c\right )^{3} - 5040 \, a \cos \left (d x + c\right )^{2} + 2520 \, a \cos \left (d x + c\right ) + 2520 \, a \log \left (\cos \left (d x + c\right )\right )}{2520 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2520*(280*a*cos(d*x + c)^9 + 315*a*cos(d*x + c)^8 - 1440*a*cos(d*x + c)^7 - 1680*a*cos(d*x + c)^6 + 3024*a*
cos(d*x + c)^5 + 3780*a*cos(d*x + c)^4 - 3360*a*cos(d*x + c)^3 - 5040*a*cos(d*x + c)^2 + 2520*a*cos(d*x + c) +
2520*a*log(cos(d*x + c)))/d

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Fricas [A]
time = 2.62, size = 115, normalized size = 0.76 \begin {gather*} -\frac {280 \, a \cos \left (d x + c\right )^{9} + 315 \, a \cos \left (d x + c\right )^{8} - 1440 \, a \cos \left (d x + c\right )^{7} - 1680 \, a \cos \left (d x + c\right )^{6} + 3024 \, a \cos \left (d x + c\right )^{5} + 3780 \, a \cos \left (d x + c\right )^{4} - 3360 \, a \cos \left (d x + c\right )^{3} - 5040 \, a \cos \left (d x + c\right )^{2} + 2520 \, a \cos \left (d x + c\right ) + 2520 \, a \log \left (-\cos \left (d x + c\right )\right )}{2520 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2520*(280*a*cos(d*x + c)^9 + 315*a*cos(d*x + c)^8 - 1440*a*cos(d*x + c)^7 - 1680*a*cos(d*x + c)^6 + 3024*a*
cos(d*x + c)^5 + 3780*a*cos(d*x + c)^4 - 3360*a*cos(d*x + c)^3 - 5040*a*cos(d*x + c)^2 + 2520*a*cos(d*x + c) +
2520*a*log(-cos(d*x + c)))/d

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4370 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 293 vs. \(2 (138) = 276\).
time = 0.52, size = 293, normalized size = 1.93 \begin {gather*} \frac {2520 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 2520 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {9177 \, a - \frac {87633 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {375732 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {953988 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1594782 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {1336734 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {781956 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {302004 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {69201 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {7129 \, a {\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2520*(2520*a*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)) - 2520*a*log(abs(-(cos(d*x + c) - 1)/(cos(
d*x + c) + 1) - 1)) + (9177*a - 87633*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 375732*a*(cos(d*x + c) - 1)^2/
(cos(d*x + c) + 1)^2 - 953988*a*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 1594782*a*(cos(d*x + c) - 1)^4/(co
s(d*x + c) + 1)^4 - 1336734*a*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 781956*a*(cos(d*x + c) - 1)^6/(cos(d
*x + c) + 1)^6 - 302004*a*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7 + 69201*a*(cos(d*x + c) - 1)^8/(cos(d*x +
c) + 1)^8 - 7129*a*(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

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Mupad [B]
time = 0.13, size = 111, normalized size = 0.73 \begin {gather*} -\frac {a\,\cos \left (c+d\,x\right )-2\,a\,{\cos \left (c+d\,x\right )}^2-\frac {4\,a\,{\cos \left (c+d\,x\right )}^3}{3}+\frac {3\,a\,{\cos \left (c+d\,x\right )}^4}{2}+\frac {6\,a\,{\cos \left (c+d\,x\right )}^5}{5}-\frac {2\,a\,{\cos \left (c+d\,x\right )}^6}{3}-\frac {4\,a\,{\cos \left (c+d\,x\right )}^7}{7}+\frac {a\,{\cos \left (c+d\,x\right )}^8}{8}+\frac {a\,{\cos \left (c+d\,x\right )}^9}{9}+a\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)^9*(a + a/cos(c + d*x)),x)

[Out]

-(a*cos(c + d*x) - 2*a*cos(c + d*x)^2 - (4*a*cos(c + d*x)^3)/3 + (3*a*cos(c + d*x)^4)/2 + (6*a*cos(c + d*x)^5)
/5 - (2*a*cos(c + d*x)^6)/3 - (4*a*cos(c + d*x)^7)/7 + (a*cos(c + d*x)^8)/8 + (a*cos(c + d*x)^9)/9 + a*log(cos
(c + d*x)))/d

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