3.1.97 \(\int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^8} \, dx\) [97]

Optimal. Leaf size=245 \[ -\frac {\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac {3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac {24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac {168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac {112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac {16 \sec (c+d x)}{2431 a^2 d \left (a^2+a^2 \sin (c+d x)\right )^3}-\frac {64 \sec (c+d x)}{12155 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac {64 \sec (c+d x)}{12155 d \left (a^8+a^8 \sin (c+d x)\right )}+\frac {128 \tan (c+d x)}{12155 a^8 d} \]

[Out]

-1/17*sec(d*x+c)/d/(a+a*sin(d*x+c))^8-3/85*sec(d*x+c)/a/d/(a+a*sin(d*x+c))^7-24/1105*sec(d*x+c)/a^2/d/(a+a*sin
(d*x+c))^6-168/12155*sec(d*x+c)/a^3/d/(a+a*sin(d*x+c))^5-112/12155*sec(d*x+c)/d/(a^2+a^2*sin(d*x+c))^4-16/2431
*sec(d*x+c)/a^2/d/(a^2+a^2*sin(d*x+c))^3-64/12155*sec(d*x+c)/d/(a^4+a^4*sin(d*x+c))^2-64/12155*sec(d*x+c)/d/(a
^8+a^8*sin(d*x+c))+128/12155*tan(d*x+c)/a^8/d

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Rubi [A]
time = 0.29, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2751, 3852, 8} \begin {gather*} \frac {128 \tan (c+d x)}{12155 a^8 d}-\frac {64 \sec (c+d x)}{12155 d \left (a^8 \sin (c+d x)+a^8\right )}-\frac {64 \sec (c+d x)}{12155 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac {168 \sec (c+d x)}{12155 a^3 d (a \sin (c+d x)+a)^5}-\frac {16 \sec (c+d x)}{2431 a^2 d \left (a^2 \sin (c+d x)+a^2\right )^3}-\frac {112 \sec (c+d x)}{12155 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac {24 \sec (c+d x)}{1105 a^2 d (a \sin (c+d x)+a)^6}-\frac {3 \sec (c+d x)}{85 a d (a \sin (c+d x)+a)^7}-\frac {\sec (c+d x)}{17 d (a \sin (c+d x)+a)^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^2/(a + a*Sin[c + d*x])^8,x]

[Out]

-1/17*Sec[c + d*x]/(d*(a + a*Sin[c + d*x])^8) - (3*Sec[c + d*x])/(85*a*d*(a + a*Sin[c + d*x])^7) - (24*Sec[c +
d*x])/(1105*a^2*d*(a + a*Sin[c + d*x])^6) - (168*Sec[c + d*x])/(12155*a^3*d*(a + a*Sin[c + d*x])^5) - (112*Se
c[c + d*x])/(12155*d*(a^2 + a^2*Sin[c + d*x])^4) - (16*Sec[c + d*x])/(2431*a^2*d*(a^2 + a^2*Sin[c + d*x])^3) -
(64*Sec[c + d*x])/(12155*d*(a^4 + a^4*Sin[c + d*x])^2) - (64*Sec[c + d*x])/(12155*d*(a^8 + a^8*Sin[c + d*x]))
+ (128*Tan[c + d*x])/(12155*a^8*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2751

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[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 3852

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 ^2(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=-\frac {\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}+\frac {9 \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^7} \, dx}{17 a}\\ &=-\frac {\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac {3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}+\frac {24 \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^6} \, dx}{85 a^2}\\ &=-\frac {\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac {3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac {24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}+\frac {168 \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^5} \, dx}{1105 a^3}\\ &=-\frac {\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac {3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac {24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac {168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}+\frac {1008 \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx}{12155 a^4}\\ &=-\frac {\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac {3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac {24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac {168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac {112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}+\frac {112 \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx}{2431 a^5}\\ &=-\frac {\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac {3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac {24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac {168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac {16 \sec (c+d x)}{2431 a^5 d (a+a \sin (c+d x))^3}-\frac {112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}+\frac {64 \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx}{2431 a^6}\\ &=-\frac {\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac {3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac {24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac {168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac {16 \sec (c+d x)}{2431 a^5 d (a+a \sin (c+d x))^3}-\frac {112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac {64 \sec (c+d x)}{12155 d \left (a^4+a^4 \sin (c+d x)\right )^2}+\frac {192 \int \frac {\sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx}{12155 a^7}\\ &=-\frac {\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac {3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac {24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac {168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac {16 \sec (c+d x)}{2431 a^5 d (a+a \sin (c+d x))^3}-\frac {112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac {64 \sec (c+d x)}{12155 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac {64 \sec (c+d x)}{12155 d \left (a^8+a^8 \sin (c+d x)\right )}+\frac {128 \int \sec ^2(c+d x) \, dx}{12155 a^8}\\ &=-\frac {\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac {3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac {24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac {168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac {16 \sec (c+d x)}{2431 a^5 d (a+a \sin (c+d x))^3}-\frac {112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac {64 \sec (c+d x)}{12155 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac {64 \sec (c+d x)}{12155 d \left (a^8+a^8 \sin (c+d x)\right )}-\frac {128 \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{12155 a^8 d}\\ &=-\frac {\sec (c+d x)}{17 d (a+a \sin (c+d x))^8}-\frac {3 \sec (c+d x)}{85 a d (a+a \sin (c+d x))^7}-\frac {24 \sec (c+d x)}{1105 a^2 d (a+a \sin (c+d x))^6}-\frac {168 \sec (c+d x)}{12155 a^3 d (a+a \sin (c+d x))^5}-\frac {16 \sec (c+d x)}{2431 a^5 d (a+a \sin (c+d x))^3}-\frac {112 \sec (c+d x)}{12155 d \left (a^2+a^2 \sin (c+d x)\right )^4}-\frac {64 \sec (c+d x)}{12155 d \left (a^4+a^4 \sin (c+d x)\right )^2}-\frac {64 \sec (c+d x)}{12155 d \left (a^8+a^8 \sin (c+d x)\right )}+\frac {128 \tan (c+d x)}{12155 a^8 d}\\ \end {align*}

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Mathematica [A]
time = 0.22, size = 113, normalized size = 0.46 \begin {gather*} \frac {\sec (c+d x) (-7072 \cos (2 (c+d x))+3808 \cos (4 (c+d x))-544 \cos (6 (c+d x))+16 \cos (8 (c+d x))+4862 \sin (c+d x)-6188 \sin (3 (c+d x))+1700 \sin (5 (c+d x))-119 \sin (7 (c+d x))+\sin (9 (c+d x)))}{24310 a^8 d (1+\sin (c+d x))^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^2/(a + a*Sin[c + d*x])^8,x]

[Out]

(Sec[c + d*x]*(-7072*Cos[2*(c + d*x)] + 3808*Cos[4*(c + d*x)] - 544*Cos[6*(c + d*x)] + 16*Cos[8*(c + d*x)] + 4
862*Sin[c + d*x] - 6188*Sin[3*(c + d*x)] + 1700*Sin[5*(c + d*x)] - 119*Sin[7*(c + d*x)] + Sin[9*(c + d*x)]))/(
24310*a^8*d*(1 + Sin[c + d*x])^8)

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Maple [A]
time = 0.25, size = 280, normalized size = 1.14

method	result	size
risch	\(\frac {256 i \left (7072 i {\mathrm e}^{7 i \left (d x +c \right )}+4862 \,{\mathrm e}^{8 i \left (d x +c \right )}-3808 i {\mathrm e}^{5 i \left (d x +c \right )}-6188 \,{\mathrm e}^{6 i \left (d x +c \right )}+544 i {\mathrm e}^{3 i \left (d x +c \right )}+1700 \,{\mathrm e}^{4 i \left (d x +c \right )}-16 i {\mathrm e}^{i \left (d x +c \right )}-119 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{12155 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{17} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{8} d}\)	\(132\)
derivativedivides	\(\frac {-\frac {256}{17 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{17}}+\frac {128}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{16}}-\frac {2752}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{15}}+\frac {1568}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{14}}-\frac {42800}{13 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{13}}+\frac {5384}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{12}}-\frac {77908}{11 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{11}}+\frac {38218}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{10}}-\frac {6847}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {10241}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {12799}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {13313}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {57083}{80 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {7937}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {4351}{64 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1793}{128 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {511}{256 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{256 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{a^{8} d}\)	\(280\)
default	\(\frac {-\frac {256}{17 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{17}}+\frac {128}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{16}}-\frac {2752}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{15}}+\frac {1568}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{14}}-\frac {42800}{13 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{13}}+\frac {5384}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{12}}-\frac {77908}{11 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{11}}+\frac {38218}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{10}}-\frac {6847}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {10241}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {12799}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {13313}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {57083}{80 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {7937}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {4351}{64 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1793}{128 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {511}{256 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{256 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{a^{8} d}\)	\(280\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^2/(a+a*sin(d*x+c))^8,x,method=_RETURNVERBOSE)

[Out]

2/d/a^8*(-128/17/(tan(1/2*d*x+1/2*c)+1)^17+64/(tan(1/2*d*x+1/2*c)+1)^16-1376/5/(tan(1/2*d*x+1/2*c)+1)^15+784/(
tan(1/2*d*x+1/2*c)+1)^14-21400/13/(tan(1/2*d*x+1/2*c)+1)^13+2692/(tan(1/2*d*x+1/2*c)+1)^12-38954/11/(tan(1/2*d
*x+1/2*c)+1)^11+19109/5/(tan(1/2*d*x+1/2*c)+1)^10-6847/2/(tan(1/2*d*x+1/2*c)+1)^9+10241/4/(tan(1/2*d*x+1/2*c)+
1)^8-12799/8/(tan(1/2*d*x+1/2*c)+1)^7+13313/16/(tan(1/2*d*x+1/2*c)+1)^6-57083/160/(tan(1/2*d*x+1/2*c)+1)^5+793
7/64/(tan(1/2*d*x+1/2*c)+1)^4-4351/128/(tan(1/2*d*x+1/2*c)+1)^3+1793/256/(tan(1/2*d*x+1/2*c)+1)^2-511/512/(tan
(1/2*d*x+1/2*c)+1)-1/512/(tan(1/2*d*x+1/2*c)-1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 740 vs. \(2 (227) = 454\).
time = 0.35, size = 740, normalized size = 3.02 \begin {gather*} -\frac {2 \, {\left (\frac {18181 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {128384 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {545224 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1667360 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {3612364 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5742464 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {6271096 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {3928496 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {850850 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {5289856 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {7137416 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {5989984 \, \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac {3607604 \, \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {1555840 \, \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} - \frac {486200 \, \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}} - \frac {97240 \, \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}} - \frac {12155 \, \sin \left (d x + c\right )^{17}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{17}} + 1896\right )}}{12155 \, {\left (a^{8} + \frac {16 \, a^{8} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {119 \, a^{8} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {544 \, a^{8} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1700 \, a^{8} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {3808 \, a^{8} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {6188 \, a^{8} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {7072 \, a^{8} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {4862 \, a^{8} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {4862 \, a^{8} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {7072 \, a^{8} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {6188 \, a^{8} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac {3808 \, a^{8} \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {1700 \, a^{8} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}} - \frac {544 \, a^{8} \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}} - \frac {119 \, a^{8} \sin \left (d x + c\right )^{16}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{16}} - \frac {16 \, a^{8} \sin \left (d x + c\right )^{17}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{17}} - \frac {a^{8} \sin \left (d x + c\right )^{18}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{18}}\right )} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2/(a+a*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

-2/12155*(18181*sin(d*x + c)/(cos(d*x + c) + 1) + 128384*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 545224*sin(d*x
+ c)^3/(cos(d*x + c) + 1)^3 + 1667360*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 3612364*sin(d*x + c)^5/(cos(d*x +
c) + 1)^5 + 5742464*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 6271096*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 392849
6*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 850850*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 5289856*sin(d*x + c)^10/(
cos(d*x + c) + 1)^10 - 7137416*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 5989984*sin(d*x + c)^12/(cos(d*x + c) +
1)^12 - 3607604*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 - 1555840*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 - 48620
0*sin(d*x + c)^15/(cos(d*x + c) + 1)^15 - 97240*sin(d*x + c)^16/(cos(d*x + c) + 1)^16 - 12155*sin(d*x + c)^17/
(cos(d*x + c) + 1)^17 + 1896)/((a^8 + 16*a^8*sin(d*x + c)/(cos(d*x + c) + 1) + 119*a^8*sin(d*x + c)^2/(cos(d*x
+ c) + 1)^2 + 544*a^8*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 1700*a^8*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 38
08*a^8*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 6188*a^8*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 7072*a^8*sin(d*x +
c)^7/(cos(d*x + c) + 1)^7 + 4862*a^8*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 4862*a^8*sin(d*x + c)^10/(cos(d*x
+ c) + 1)^10 - 7072*a^8*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 - 6188*a^8*sin(d*x + c)^12/(cos(d*x + c) + 1)^12
- 3808*a^8*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 - 1700*a^8*sin(d*x + c)^14/(cos(d*x + c) + 1)^14 - 544*a^8*s
in(d*x + c)^15/(cos(d*x + c) + 1)^15 - 119*a^8*sin(d*x + c)^16/(cos(d*x + c) + 1)^16 - 16*a^8*sin(d*x + c)^17/
(cos(d*x + c) + 1)^17 - a^8*sin(d*x + c)^18/(cos(d*x + c) + 1)^18)*d)

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Fricas [A]
time = 0.36, size = 225, normalized size = 0.92 \begin {gather*} \frac {1024 \, \cos \left (d x + c\right )^{8} - 10752 \, \cos \left (d x + c\right )^{6} + 29568 \, \cos \left (d x + c\right )^{4} - 27456 \, \cos \left (d x + c\right )^{2} + {\left (128 \, \cos \left (d x + c\right )^{8} - 4032 \, \cos \left (d x + c\right )^{6} + 18480 \, \cos \left (d x + c\right )^{4} - 24024 \, \cos \left (d x + c\right )^{2} + 6435\right )} \sin \left (d x + c\right ) + 5720}{12155 \, {\left (a^{8} d \cos \left (d x + c\right )^{9} - 32 \, a^{8} d \cos \left (d x + c\right )^{7} + 160 \, a^{8} d \cos \left (d x + c\right )^{5} - 256 \, a^{8} d \cos \left (d x + c\right )^{3} + 128 \, a^{8} d \cos \left (d x + c\right ) - 8 \, {\left (a^{8} d \cos \left (d x + c\right )^{7} - 10 \, a^{8} d \cos \left (d x + c\right )^{5} + 24 \, a^{8} d \cos \left (d x + c\right )^{3} - 16 \, a^{8} d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2/(a+a*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

1/12155*(1024*cos(d*x + c)^8 - 10752*cos(d*x + c)^6 + 29568*cos(d*x + c)^4 - 27456*cos(d*x + c)^2 + (128*cos(d
*x + c)^8 - 4032*cos(d*x + c)^6 + 18480*cos(d*x + c)^4 - 24024*cos(d*x + c)^2 + 6435)*sin(d*x + c) + 5720)/(a^
8*d*cos(d*x + c)^9 - 32*a^8*d*cos(d*x + c)^7 + 160*a^8*d*cos(d*x + c)^5 - 256*a^8*d*cos(d*x + c)^3 + 128*a^8*d
*cos(d*x + c) - 8*(a^8*d*cos(d*x + c)^7 - 10*a^8*d*cos(d*x + c)^5 + 24*a^8*d*cos(d*x + c)^3 - 16*a^8*d*cos(d*x
+ c))*sin(d*x + c))

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**2/(a+a*sin(d*x+c))**8,x)

[Out]

Timed out

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Giac [A]
time = 7.84, size = 249, normalized size = 1.02 \begin {gather*} -\frac {\frac {12155}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}} + \frac {6211205 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{16} + 55791450 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 303072770 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} + 1091397450 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 2909561798 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 5901218466 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 9405145178 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 11877161010 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 12017308160 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 9710430158 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 6263238566 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3172666718 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1247921210 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 365303990 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 77883902 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 10498214 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 982907}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{17}}}{3111680 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2/(a+a*sin(d*x+c))^8,x, algorithm="giac")

[Out]

-1/3111680*(12155/(a^8*(tan(1/2*d*x + 1/2*c) - 1)) + (6211205*tan(1/2*d*x + 1/2*c)^16 + 55791450*tan(1/2*d*x +
1/2*c)^15 + 303072770*tan(1/2*d*x + 1/2*c)^14 + 1091397450*tan(1/2*d*x + 1/2*c)^13 + 2909561798*tan(1/2*d*x +
1/2*c)^12 + 5901218466*tan(1/2*d*x + 1/2*c)^11 + 9405145178*tan(1/2*d*x + 1/2*c)^10 + 11877161010*tan(1/2*d*x
+ 1/2*c)^9 + 12017308160*tan(1/2*d*x + 1/2*c)^8 + 9710430158*tan(1/2*d*x + 1/2*c)^7 + 6263238566*tan(1/2*d*x
+ 1/2*c)^6 + 3172666718*tan(1/2*d*x + 1/2*c)^5 + 1247921210*tan(1/2*d*x + 1/2*c)^4 + 365303990*tan(1/2*d*x + 1
/2*c)^3 + 77883902*tan(1/2*d*x + 1/2*c)^2 + 10498214*tan(1/2*d*x + 1/2*c) + 982907)/(a^8*(tan(1/2*d*x + 1/2*c)
+ 1)^17))/d

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Mupad [B]
time = 8.04, size = 233, normalized size = 0.95 \begin {gather*} \frac {\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {519571\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{16}-\frac {576147\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{16}+\frac {213707\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{16}-\frac {183243\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{16}-\frac {18207\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{16}+\frac {13855\,\cos \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{16}+\frac {493\,\cos \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )}{32}-\frac {237\,\cos \left (\frac {17\,c}{2}+\frac {17\,d\,x}{2}\right )}{32}+\frac {56425\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}-\frac {51563\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{2}-\frac {53191\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{2}+\frac {47003\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{2}+\frac {9403\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{2}-\frac {7703\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{2}-\frac {355\,\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{2}+118\,\sin \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )+\frac {\sin \left (\frac {17\,c}{2}+\frac {17\,d\,x}{2}\right )}{2}\right )}{3111680\,a^8\,d\,{\cos \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\right )}^{17}\,\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d\,x}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(c + d*x)^2*(a + a*sin(c + d*x))^8),x)

[Out]

(cos(c/2 + (d*x)/2)*((519571*cos((5*c)/2 + (5*d*x)/2))/16 - (576147*cos((3*c)/2 + (3*d*x)/2))/16 + (213707*cos
((7*c)/2 + (7*d*x)/2))/16 - (183243*cos((9*c)/2 + (9*d*x)/2))/16 - (18207*cos((11*c)/2 + (11*d*x)/2))/16 + (13
855*cos((13*c)/2 + (13*d*x)/2))/16 + (493*cos((15*c)/2 + (15*d*x)/2))/32 - (237*cos((17*c)/2 + (17*d*x)/2))/32
+ (56425*sin(c/2 + (d*x)/2))/2 - (51563*sin((3*c)/2 + (3*d*x)/2))/2 - (53191*sin((5*c)/2 + (5*d*x)/2))/2 + (4
7003*sin((7*c)/2 + (7*d*x)/2))/2 + (9403*sin((9*c)/2 + (9*d*x)/2))/2 - (7703*sin((11*c)/2 + (11*d*x)/2))/2 - (
355*sin((13*c)/2 + (13*d*x)/2))/2 + 118*sin((15*c)/2 + (15*d*x)/2) + sin((17*c)/2 + (17*d*x)/2)/2))/(3111680*a
^8*d*cos(c/2 - pi/4 + (d*x)/2)^17*cos(c/2 + pi/4 + (d*x)/2))

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