3.907 \(\int \frac{\sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=210 \[ -\frac{a^4}{160 d (a \sin (c+d x)+a)^5}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}-\frac{5 a^3}{256 d (a \sin (c+d x)+a)^4}+\frac{a^2}{64 d (a-a \sin (c+d x))^3}-\frac{5 a^2}{128 d (a \sin (c+d x)+a)^3}+\frac{21 a}{512 d (a-a \sin (c+d x))^2}-\frac{35 a}{512 d (a \sin (c+d x)+a)^2}+\frac{7}{64 d (a-a \sin (c+d x))}-\frac{35}{256 d (a \sin (c+d x)+a)}+\frac{63 \tanh ^{-1}(\sin (c+d x))}{256 a d} \]

[Out]

(63*ArcTanh[Sin[c + d*x]])/(256*a*d) + a^3/(256*d*(a - a*Sin[c + d*x])^4) + a^2/(64*d*(a - a*Sin[c + d*x])^3)
+ (21*a)/(512*d*(a - a*Sin[c + d*x])^2) + 7/(64*d*(a - a*Sin[c + d*x])) - a^4/(160*d*(a + a*Sin[c + d*x])^5) -
(5*a^3)/(256*d*(a + a*Sin[c + d*x])^4) - (5*a^2)/(128*d*(a + a*Sin[c + d*x])^3) - (35*a)/(512*d*(a + a*Sin[c
+ d*x])^2) - 35/(256*d*(a + a*Sin[c + d*x]))

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Rubi [A]  time = 0.167037, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2667, 44, 206} \[ -\frac{a^4}{160 d (a \sin (c+d x)+a)^5}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}-\frac{5 a^3}{256 d (a \sin (c+d x)+a)^4}+\frac{a^2}{64 d (a-a \sin (c+d x))^3}-\frac{5 a^2}{128 d (a \sin (c+d x)+a)^3}+\frac{21 a}{512 d (a-a \sin (c+d x))^2}-\frac{35 a}{512 d (a \sin (c+d x)+a)^2}+\frac{7}{64 d (a-a \sin (c+d x))}-\frac{35}{256 d (a \sin (c+d x)+a)}+\frac{63 \tanh ^{-1}(\sin (c+d x))}{256 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(63*ArcTanh[Sin[c + d*x]])/(256*a*d) + a^3/(256*d*(a - a*Sin[c + d*x])^4) + a^2/(64*d*(a - a*Sin[c + d*x])^3)
+ (21*a)/(512*d*(a - a*Sin[c + d*x])^2) + 7/(64*d*(a - a*Sin[c + d*x])) - a^4/(160*d*(a + a*Sin[c + d*x])^5) -
(5*a^3)/(256*d*(a + a*Sin[c + d*x])^4) - (5*a^2)/(128*d*(a + a*Sin[c + d*x])^3) - (35*a)/(512*d*(a + a*Sin[c
+ d*x])^2) - 35/(256*d*(a + a*Sin[c + d*x]))

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sec ^9(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{a^9 \operatorname{Subst}\left (\int \frac{1}{(a-x)^5 (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^9 \operatorname{Subst}\left (\int \left (\frac{1}{64 a^6 (a-x)^5}+\frac{3}{64 a^7 (a-x)^4}+\frac{21}{256 a^8 (a-x)^3}+\frac{7}{64 a^9 (a-x)^2}+\frac{1}{32 a^5 (a+x)^6}+\frac{5}{64 a^6 (a+x)^5}+\frac{15}{128 a^7 (a+x)^4}+\frac{35}{256 a^8 (a+x)^3}+\frac{35}{256 a^9 (a+x)^2}+\frac{63}{256 a^9 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^3}{256 d (a-a \sin (c+d x))^4}+\frac{a^2}{64 d (a-a \sin (c+d x))^3}+\frac{21 a}{512 d (a-a \sin (c+d x))^2}+\frac{7}{64 d (a-a \sin (c+d x))}-\frac{a^4}{160 d (a+a \sin (c+d x))^5}-\frac{5 a^3}{256 d (a+a \sin (c+d x))^4}-\frac{5 a^2}{128 d (a+a \sin (c+d x))^3}-\frac{35 a}{512 d (a+a \sin (c+d x))^2}-\frac{35}{256 d (a+a \sin (c+d x))}+\frac{63 \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{256 d}\\ &=\frac{63 \tanh ^{-1}(\sin (c+d x))}{256 a d}+\frac{a^3}{256 d (a-a \sin (c+d x))^4}+\frac{a^2}{64 d (a-a \sin (c+d x))^3}+\frac{21 a}{512 d (a-a \sin (c+d x))^2}+\frac{7}{64 d (a-a \sin (c+d x))}-\frac{a^4}{160 d (a+a \sin (c+d x))^5}-\frac{5 a^3}{256 d (a+a \sin (c+d x))^4}-\frac{5 a^2}{128 d (a+a \sin (c+d x))^3}-\frac{35 a}{512 d (a+a \sin (c+d x))^2}-\frac{35}{256 d (a+a \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 1.4314, size = 165, normalized size = 0.79 \[ \frac{\sec ^8(c+d x) \left (-315 \sin ^8(c+d x)-315 \sin ^7(c+d x)+1155 \sin ^6(c+d x)+1155 \sin ^5(c+d x)-1533 \sin ^4(c+d x)-1533 \sin ^3(c+d x)+837 \sin ^2(c+d x)+837 \sin (c+d x)+315 \tanh ^{-1}(\sin (c+d x)) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^8 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^{10}-128\right )}{1280 a d (\sin (c+d x)+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[c + d*x]^8*(-128 + 315*ArcTanh[Sin[c + d*x]]*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^8*(Cos[(c + d*x)/2] +
Sin[(c + d*x)/2])^10 + 837*Sin[c + d*x] + 837*Sin[c + d*x]^2 - 1533*Sin[c + d*x]^3 - 1533*Sin[c + d*x]^4 + 115
5*Sin[c + d*x]^5 + 1155*Sin[c + d*x]^6 - 315*Sin[c + d*x]^7 - 315*Sin[c + d*x]^8))/(1280*a*d*(1 + Sin[c + d*x]
))

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Maple [A]  time = 0.073, size = 198, normalized size = 0.9 \begin{align*}{\frac{1}{256\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{4}}}-{\frac{1}{64\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{3}}}+{\frac{21}{512\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}-{\frac{7}{64\,da \left ( \sin \left ( dx+c \right ) -1 \right ) }}-{\frac{63\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{512\,da}}-{\frac{1}{160\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{5}{256\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{5}{128\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{35}{512\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{35}{256\,da \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{63\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{512\,da}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/256/d/a/(sin(d*x+c)-1)^4-1/64/d/a/(sin(d*x+c)-1)^3+21/512/d/a/(sin(d*x+c)-1)^2-7/64/a/d/(sin(d*x+c)-1)-63/51
2/a/d*ln(sin(d*x+c)-1)-1/160/d/a/(1+sin(d*x+c))^5-5/256/d/a/(1+sin(d*x+c))^4-5/128/d/a/(1+sin(d*x+c))^3-35/512
/a/d/(1+sin(d*x+c))^2-35/256/a/d/(1+sin(d*x+c))+63/512*ln(1+sin(d*x+c))/a/d

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Maxima [A]  time = 1.04454, size = 289, normalized size = 1.38 \begin{align*} -\frac{\frac{2 \,{\left (315 \, \sin \left (d x + c\right )^{8} + 315 \, \sin \left (d x + c\right )^{7} - 1155 \, \sin \left (d x + c\right )^{6} - 1155 \, \sin \left (d x + c\right )^{5} + 1533 \, \sin \left (d x + c\right )^{4} + 1533 \, \sin \left (d x + c\right )^{3} - 837 \, \sin \left (d x + c\right )^{2} - 837 \, \sin \left (d x + c\right ) + 128\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} - 4 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} + 6 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} - 4 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + a} - \frac{315 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac{315 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{2560 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2560*(2*(315*sin(d*x + c)^8 + 315*sin(d*x + c)^7 - 1155*sin(d*x + c)^6 - 1155*sin(d*x + c)^5 + 1533*sin(d*x
+ c)^4 + 1533*sin(d*x + c)^3 - 837*sin(d*x + c)^2 - 837*sin(d*x + c) + 128)/(a*sin(d*x + c)^9 + a*sin(d*x + c
)^8 - 4*a*sin(d*x + c)^7 - 4*a*sin(d*x + c)^6 + 6*a*sin(d*x + c)^5 + 6*a*sin(d*x + c)^4 - 4*a*sin(d*x + c)^3 -
4*a*sin(d*x + c)^2 + a*sin(d*x + c) + a) - 315*log(sin(d*x + c) + 1)/a + 315*log(sin(d*x + c) - 1)/a)/d

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Fricas [A]  time = 2.10542, size = 520, normalized size = 2.48 \begin{align*} -\frac{630 \, \cos \left (d x + c\right )^{8} - 210 \, \cos \left (d x + c\right )^{6} - 84 \, \cos \left (d x + c\right )^{4} - 48 \, \cos \left (d x + c\right )^{2} - 315 \,{\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 315 \,{\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 6 \,{\left (105 \, \cos \left (d x + c\right )^{6} + 70 \, \cos \left (d x + c\right )^{4} + 56 \, \cos \left (d x + c\right )^{2} + 48\right )} \sin \left (d x + c\right ) - 32}{2560 \,{\left (a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2560*(630*cos(d*x + c)^8 - 210*cos(d*x + c)^6 - 84*cos(d*x + c)^4 - 48*cos(d*x + c)^2 - 315*(cos(d*x + c)^8
*sin(d*x + c) + cos(d*x + c)^8)*log(sin(d*x + c) + 1) + 315*(cos(d*x + c)^8*sin(d*x + c) + cos(d*x + c)^8)*log
(-sin(d*x + c) + 1) - 6*(105*cos(d*x + c)^6 + 70*cos(d*x + c)^4 + 56*cos(d*x + c)^2 + 48)*sin(d*x + c) - 32)/(
a*d*cos(d*x + c)^8*sin(d*x + c) + a*d*cos(d*x + c)^8)

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

[Out]

Timed out

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Giac [A]  time = 1.33553, size = 211, normalized size = 1. \begin{align*} \frac{\frac{1260 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac{1260 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac{5 \,{\left (525 \, \sin \left (d x + c\right )^{4} - 2324 \, \sin \left (d x + c\right )^{3} + 3906 \, \sin \left (d x + c\right )^{2} - 2972 \, \sin \left (d x + c\right ) + 873\right )}}{a{\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac{2877 \, \sin \left (d x + c\right )^{5} + 15785 \, \sin \left (d x + c\right )^{4} + 35070 \, \sin \left (d x + c\right )^{3} + 39670 \, \sin \left (d x + c\right )^{2} + 23085 \, \sin \left (d x + c\right ) + 5641}{a{\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{10240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10240*(1260*log(abs(sin(d*x + c) + 1))/a - 1260*log(abs(sin(d*x + c) - 1))/a + 5*(525*sin(d*x + c)^4 - 2324*
sin(d*x + c)^3 + 3906*sin(d*x + c)^2 - 2972*sin(d*x + c) + 873)/(a*(sin(d*x + c) - 1)^4) - (2877*sin(d*x + c)^
5 + 15785*sin(d*x + c)^4 + 35070*sin(d*x + c)^3 + 39670*sin(d*x + c)^2 + 23085*sin(d*x + c) + 5641)/(a*(sin(d*
x + c) + 1)^5))/d