3.98 \(\int \frac{\sec ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx\)

Optimal. Leaf size=238 \[ \frac{1}{1024 d \left (a^8-a^8 \sin (c+d x)\right )}-\frac{9}{1024 d \left (a^8 \sin (c+d x)+a^8\right )}-\frac{1}{128 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac{3}{256 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac{1}{48 a^2 d (a \sin (c+d x)+a)^6}-\frac{1}{64 a^3 d (a \sin (c+d x)+a)^5}-\frac{7}{768 a^5 d (a \sin (c+d x)+a)^3}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{512 a^8 d}-\frac{a}{36 d (a \sin (c+d x)+a)^9}-\frac{1}{32 d (a \sin (c+d x)+a)^8}-\frac{3}{112 a d (a \sin (c+d x)+a)^7} \]

[Out]

(5*ArcTanh[Sin[c + d*x]])/(512*a^8*d) - a/(36*d*(a + a*Sin[c + d*x])^9) - 1/(32*d*(a + a*Sin[c + d*x])^8) - 3/
(112*a*d*(a + a*Sin[c + d*x])^7) - 1/(48*a^2*d*(a + a*Sin[c + d*x])^6) - 1/(64*a^3*d*(a + a*Sin[c + d*x])^5) -
7/(768*a^5*d*(a + a*Sin[c + d*x])^3) - 3/(256*d*(a^2 + a^2*Sin[c + d*x])^4) - 1/(128*d*(a^4 + a^4*Sin[c + d*x
])^2) + 1/(1024*d*(a^8 - a^8*Sin[c + d*x])) - 9/(1024*d*(a^8 + a^8*Sin[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.171267, antiderivative size = 238, 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} \[ \frac{1}{1024 d \left (a^8-a^8 \sin (c+d x)\right )}-\frac{9}{1024 d \left (a^8 \sin (c+d x)+a^8\right )}-\frac{1}{128 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac{3}{256 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac{1}{48 a^2 d (a \sin (c+d x)+a)^6}-\frac{1}{64 a^3 d (a \sin (c+d x)+a)^5}-\frac{7}{768 a^5 d (a \sin (c+d x)+a)^3}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{512 a^8 d}-\frac{a}{36 d (a \sin (c+d x)+a)^9}-\frac{1}{32 d (a \sin (c+d x)+a)^8}-\frac{3}{112 a d (a \sin (c+d x)+a)^7} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^3/(a + a*Sin[c + d*x])^8,x]

[Out]

(5*ArcTanh[Sin[c + d*x]])/(512*a^8*d) - a/(36*d*(a + a*Sin[c + d*x])^9) - 1/(32*d*(a + a*Sin[c + d*x])^8) - 3/
(112*a*d*(a + a*Sin[c + d*x])^7) - 1/(48*a^2*d*(a + a*Sin[c + d*x])^6) - 1/(64*a^3*d*(a + a*Sin[c + d*x])^5) -
7/(768*a^5*d*(a + a*Sin[c + d*x])^3) - 3/(256*d*(a^2 + a^2*Sin[c + d*x])^4) - 1/(128*d*(a^4 + a^4*Sin[c + d*x
])^2) + 1/(1024*d*(a^8 - a^8*Sin[c + d*x])) - 9/(1024*d*(a^8 + a^8*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 ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{(a-x)^2 (a+x)^{10}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \left (\frac{1}{1024 a^{10} (a-x)^2}+\frac{1}{4 a^2 (a+x)^{10}}+\frac{1}{4 a^3 (a+x)^9}+\frac{3}{16 a^4 (a+x)^8}+\frac{1}{8 a^5 (a+x)^7}+\frac{5}{64 a^6 (a+x)^6}+\frac{3}{64 a^7 (a+x)^5}+\frac{7}{256 a^8 (a+x)^4}+\frac{1}{64 a^9 (a+x)^3}+\frac{9}{1024 a^{10} (a+x)^2}+\frac{5}{512 a^{10} \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{a}{36 d (a+a \sin (c+d x))^9}-\frac{1}{32 d (a+a \sin (c+d x))^8}-\frac{3}{112 a d (a+a \sin (c+d x))^7}-\frac{1}{48 a^2 d (a+a \sin (c+d x))^6}-\frac{1}{64 a^3 d (a+a \sin (c+d x))^5}-\frac{7}{768 a^5 d (a+a \sin (c+d x))^3}-\frac{3}{256 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac{1}{128 d \left (a^4+a^4 \sin (c+d x)\right )^2}+\frac{1}{1024 d \left (a^8-a^8 \sin (c+d x)\right )}-\frac{9}{1024 d \left (a^8+a^8 \sin (c+d x)\right )}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{512 a^7 d}\\ &=\frac{5 \tanh ^{-1}(\sin (c+d x))}{512 a^8 d}-\frac{a}{36 d (a+a \sin (c+d x))^9}-\frac{1}{32 d (a+a \sin (c+d x))^8}-\frac{3}{112 a d (a+a \sin (c+d x))^7}-\frac{1}{48 a^2 d (a+a \sin (c+d x))^6}-\frac{1}{64 a^3 d (a+a \sin (c+d x))^5}-\frac{7}{768 a^5 d (a+a \sin (c+d x))^3}-\frac{3}{256 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac{1}{128 d \left (a^4+a^4 \sin (c+d x)\right )^2}+\frac{1}{1024 d \left (a^8-a^8 \sin (c+d x)\right )}-\frac{9}{1024 d \left (a^8+a^8 \sin (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 1.76929, size = 175, normalized size = 0.74 \[ -\frac{\sec ^2(c+d x) \left (-315 \sin ^9(c+d x)-2520 \sin ^8(c+d x)-8610 \sin ^7(c+d x)-15960 \sin ^6(c+d x)-16128 \sin ^5(c+d x)-5544 \sin ^4(c+d x)+7074 \sin ^3(c+d x)+11736 \sin ^2(c+d x)+9019 \sin (c+d x)-315 \tanh ^{-1}(\sin (c+d x)) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^{18}+5120\right )}{32256 a^8 d (\sin (c+d x)+1)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^3/(a + a*Sin[c + d*x])^8,x]

[Out]

-(Sec[c + d*x]^2*(5120 - 315*ArcTanh[Sin[c + d*x]]*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(Cos[(c + d*x)/2] +
Sin[(c + d*x)/2])^18 + 9019*Sin[c + d*x] + 11736*Sin[c + d*x]^2 + 7074*Sin[c + d*x]^3 - 5544*Sin[c + d*x]^4 -
16128*Sin[c + d*x]^5 - 15960*Sin[c + d*x]^6 - 8610*Sin[c + d*x]^7 - 2520*Sin[c + d*x]^8 - 315*Sin[c + d*x]^9)
)/(32256*a^8*d*(1 + Sin[c + d*x])^8)

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Maple [A]  time = 0.135, size = 216, normalized size = 0.9 \begin{align*} -{\frac{1}{1024\,d{a}^{8} \left ( \sin \left ( dx+c \right ) -1 \right ) }}-{\frac{5\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{1024\,d{a}^{8}}}-{\frac{1}{36\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{1}{32\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{3}{112\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{1}{48\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{1}{64\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{3}{256\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{7}{768\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{1}{128\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{9}{1024\,d{a}^{8} \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{5\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{1024\,d{a}^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^3/(a+a*sin(d*x+c))^8,x)

[Out]

-1/1024/d/a^8/(sin(d*x+c)-1)-5/1024/d/a^8*ln(sin(d*x+c)-1)-1/36/d/a^8/(1+sin(d*x+c))^9-1/32/d/a^8/(1+sin(d*x+c
))^8-3/112/d/a^8/(1+sin(d*x+c))^7-1/48/d/a^8/(1+sin(d*x+c))^6-1/64/d/a^8/(1+sin(d*x+c))^5-3/256/d/a^8/(1+sin(d
*x+c))^4-7/768/d/a^8/(1+sin(d*x+c))^3-1/128/d/a^8/(1+sin(d*x+c))^2-9/1024/d/a^8/(1+sin(d*x+c))+5/1024/d/a^8*ln
(1+sin(d*x+c))

________________________________________________________________________________________

Maxima [A]  time = 0.980043, size = 335, normalized size = 1.41 \begin{align*} -\frac{\frac{2 \,{\left (315 \, \sin \left (d x + c\right )^{9} + 2520 \, \sin \left (d x + c\right )^{8} + 8610 \, \sin \left (d x + c\right )^{7} + 15960 \, \sin \left (d x + c\right )^{6} + 16128 \, \sin \left (d x + c\right )^{5} + 5544 \, \sin \left (d x + c\right )^{4} - 7074 \, \sin \left (d x + c\right )^{3} - 11736 \, \sin \left (d x + c\right )^{2} - 9019 \, \sin \left (d x + c\right ) - 5120\right )}}{a^{8} \sin \left (d x + c\right )^{10} + 8 \, a^{8} \sin \left (d x + c\right )^{9} + 27 \, a^{8} \sin \left (d x + c\right )^{8} + 48 \, a^{8} \sin \left (d x + c\right )^{7} + 42 \, a^{8} \sin \left (d x + c\right )^{6} - 42 \, a^{8} \sin \left (d x + c\right )^{4} - 48 \, a^{8} \sin \left (d x + c\right )^{3} - 27 \, a^{8} \sin \left (d x + c\right )^{2} - 8 \, a^{8} \sin \left (d x + c\right ) - a^{8}} - \frac{315 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{8}} + \frac{315 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{8}}}{64512 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3/(a+a*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

-1/64512*(2*(315*sin(d*x + c)^9 + 2520*sin(d*x + c)^8 + 8610*sin(d*x + c)^7 + 15960*sin(d*x + c)^6 + 16128*sin
(d*x + c)^5 + 5544*sin(d*x + c)^4 - 7074*sin(d*x + c)^3 - 11736*sin(d*x + c)^2 - 9019*sin(d*x + c) - 5120)/(a^
8*sin(d*x + c)^10 + 8*a^8*sin(d*x + c)^9 + 27*a^8*sin(d*x + c)^8 + 48*a^8*sin(d*x + c)^7 + 42*a^8*sin(d*x + c)
^6 - 42*a^8*sin(d*x + c)^4 - 48*a^8*sin(d*x + c)^3 - 27*a^8*sin(d*x + c)^2 - 8*a^8*sin(d*x + c) - a^8) - 315*l
og(sin(d*x + c) + 1)/a^8 + 315*log(sin(d*x + c) - 1)/a^8)/d

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Fricas [B]  time = 2.16685, size = 1241, normalized size = 5.21 \begin{align*} \frac{5040 \, \cos \left (d x + c\right )^{8} - 52080 \, \cos \left (d x + c\right )^{6} + 137088 \, \cos \left (d x + c\right )^{4} - 114624 \, \cos \left (d x + c\right )^{2} + 315 \,{\left (\cos \left (d x + c\right )^{10} - 32 \, \cos \left (d x + c\right )^{8} + 160 \, \cos \left (d x + c\right )^{6} - 256 \, \cos \left (d x + c\right )^{4} + 128 \, \cos \left (d x + c\right )^{2} - 8 \,{\left (\cos \left (d x + c\right )^{8} - 10 \, \cos \left (d x + c\right )^{6} + 24 \, \cos \left (d x + c\right )^{4} - 16 \, \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 315 \,{\left (\cos \left (d x + c\right )^{10} - 32 \, \cos \left (d x + c\right )^{8} + 160 \, \cos \left (d x + c\right )^{6} - 256 \, \cos \left (d x + c\right )^{4} + 128 \, \cos \left (d x + c\right )^{2} - 8 \,{\left (\cos \left (d x + c\right )^{8} - 10 \, \cos \left (d x + c\right )^{6} + 24 \, \cos \left (d x + c\right )^{4} - 16 \, \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (315 \, \cos \left (d x + c\right )^{8} - 9870 \, \cos \left (d x + c\right )^{6} + 43848 \, \cos \left (d x + c\right )^{4} - 52272 \, \cos \left (d x + c\right )^{2} + 8960\right )} \sin \left (d x + c\right ) + 14336}{64512 \,{\left (a^{8} d \cos \left (d x + c\right )^{10} - 32 \, a^{8} d \cos \left (d x + c\right )^{8} + 160 \, a^{8} d \cos \left (d x + c\right )^{6} - 256 \, a^{8} d \cos \left (d x + c\right )^{4} + 128 \, a^{8} d \cos \left (d x + c\right )^{2} - 8 \,{\left (a^{8} d \cos \left (d x + c\right )^{8} - 10 \, a^{8} d \cos \left (d x + c\right )^{6} + 24 \, a^{8} d \cos \left (d x + c\right )^{4} - 16 \, a^{8} d \cos \left (d x + c\right )^{2}\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)^3/(a+a*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

1/64512*(5040*cos(d*x + c)^8 - 52080*cos(d*x + c)^6 + 137088*cos(d*x + c)^4 - 114624*cos(d*x + c)^2 + 315*(cos
(d*x + c)^10 - 32*cos(d*x + c)^8 + 160*cos(d*x + c)^6 - 256*cos(d*x + c)^4 + 128*cos(d*x + c)^2 - 8*(cos(d*x +
c)^8 - 10*cos(d*x + c)^6 + 24*cos(d*x + c)^4 - 16*cos(d*x + c)^2)*sin(d*x + c))*log(sin(d*x + c) + 1) - 315*(
cos(d*x + c)^10 - 32*cos(d*x + c)^8 + 160*cos(d*x + c)^6 - 256*cos(d*x + c)^4 + 128*cos(d*x + c)^2 - 8*(cos(d*
x + c)^8 - 10*cos(d*x + c)^6 + 24*cos(d*x + c)^4 - 16*cos(d*x + c)^2)*sin(d*x + c))*log(-sin(d*x + c) + 1) + 2
*(315*cos(d*x + c)^8 - 9870*cos(d*x + c)^6 + 43848*cos(d*x + c)^4 - 52272*cos(d*x + c)^2 + 8960)*sin(d*x + c)
+ 14336)/(a^8*d*cos(d*x + c)^10 - 32*a^8*d*cos(d*x + c)^8 + 160*a^8*d*cos(d*x + c)^6 - 256*a^8*d*cos(d*x + c)^
4 + 128*a^8*d*cos(d*x + c)^2 - 8*(a^8*d*cos(d*x + c)^8 - 10*a^8*d*cos(d*x + c)^6 + 24*a^8*d*cos(d*x + c)^4 - 1
6*a^8*d*cos(d*x + c)^2)*sin(d*x + c))

________________________________________________________________________________________

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)**3/(a+a*sin(d*x+c))**8,x)

[Out]

Timed out

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Giac [A]  time = 1.23705, size = 224, normalized size = 0.94 \begin{align*} \frac{\frac{2520 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{8}} - \frac{2520 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{8}} + \frac{504 \,{\left (5 \, \sin \left (d x + c\right ) - 6\right )}}{a^{8}{\left (\sin \left (d x + c\right ) - 1\right )}} - \frac{7129 \, \sin \left (d x + c\right )^{9} + 68697 \, \sin \left (d x + c\right )^{8} + 296964 \, \sin \left (d x + c\right )^{7} + 758772 \, \sin \left (d x + c\right )^{6} + 1271214 \, \sin \left (d x + c\right )^{5} + 1465758 \, \sin \left (d x + c\right )^{4} + 1191540 \, \sin \left (d x + c\right )^{3} + 693828 \, \sin \left (d x + c\right )^{2} + 295425 \, \sin \left (d x + c\right ) + 89553}{a^{8}{\left (\sin \left (d x + c\right ) + 1\right )}^{9}}}{516096 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3/(a+a*sin(d*x+c))^8,x, algorithm="giac")

[Out]

1/516096*(2520*log(abs(sin(d*x + c) + 1))/a^8 - 2520*log(abs(sin(d*x + c) - 1))/a^8 + 504*(5*sin(d*x + c) - 6)
/(a^8*(sin(d*x + c) - 1)) - (7129*sin(d*x + c)^9 + 68697*sin(d*x + c)^8 + 296964*sin(d*x + c)^7 + 758772*sin(d
*x + c)^6 + 1271214*sin(d*x + c)^5 + 1465758*sin(d*x + c)^4 + 1191540*sin(d*x + c)^3 + 693828*sin(d*x + c)^2 +
295425*sin(d*x + c) + 89553)/(a^8*(sin(d*x + c) + 1)^9))/d