3.1.100 \(\int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^8} \, dx\) [100]

Optimal. Leaf size=284 \[ \frac {33 \tanh ^{-1}(\sin (c+d x))}{2048 a^8 d}-\frac {a^2}{80 d (a+a \sin (c+d x))^{10}}-\frac {a}{48 d (a+a \sin (c+d x))^9}-\frac {3}{128 d (a+a \sin (c+d x))^8}-\frac {5}{224 a d (a+a \sin (c+d x))^7}-\frac {5}{256 a^2 d (a+a \sin (c+d x))^6}-\frac {21}{1280 a^3 d (a+a \sin (c+d x))^5}-\frac {3}{256 a^5 d (a+a \sin (c+d x))^3}-\frac {7}{512 d \left (a^2+a^2 \sin (c+d x)\right )^4}+\frac {1}{4096 d \left (a^4-a^4 \sin (c+d x)\right )^2}-\frac {45}{4096 d \left (a^4+a^4 \sin (c+d x)\right )^2}+\frac {11}{4096 d \left (a^8-a^8 \sin (c+d x)\right )}-\frac {55}{4096 d \left (a^8+a^8 \sin (c+d x)\right )} \]

[Out]

33/2048*arctanh(sin(d*x+c))/a^8/d-1/80*a^2/d/(a+a*sin(d*x+c))^10-1/48*a/d/(a+a*sin(d*x+c))^9-3/128/d/(a+a*sin(
d*x+c))^8-5/224/a/d/(a+a*sin(d*x+c))^7-5/256/a^2/d/(a+a*sin(d*x+c))^6-21/1280/a^3/d/(a+a*sin(d*x+c))^5-3/256/a
^5/d/(a+a*sin(d*x+c))^3-7/512/d/(a^2+a^2*sin(d*x+c))^4+1/4096/d/(a^4-a^4*sin(d*x+c))^2-45/4096/d/(a^4+a^4*sin(
d*x+c))^2+11/4096/d/(a^8-a^8*sin(d*x+c))-55/4096/d/(a^8+a^8*sin(d*x+c))

________________________________________________________________________________________

Rubi [A]
time = 0.16, antiderivative size = 284, normalized size of antiderivative = 1.00, 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 = {2746, 46, 212} \begin {gather*} \frac {11}{4096 d \left (a^8-a^8 \sin (c+d x)\right )}-\frac {55}{4096 d \left (a^8 \sin (c+d x)+a^8\right )}+\frac {33 \tanh ^{-1}(\sin (c+d x))}{2048 a^8 d}-\frac {3}{256 a^5 d (a \sin (c+d x)+a)^3}+\frac {1}{4096 d \left (a^4-a^4 \sin (c+d x)\right )^2}-\frac {45}{4096 d \left (a^4 \sin (c+d x)+a^4\right )^2}-\frac {21}{1280 a^3 d (a \sin (c+d x)+a)^5}-\frac {a^2}{80 d (a \sin (c+d x)+a)^{10}}-\frac {7}{512 d \left (a^2 \sin (c+d x)+a^2\right )^4}-\frac {5}{256 a^2 d (a \sin (c+d x)+a)^6}-\frac {a}{48 d (a \sin (c+d x)+a)^9}-\frac {3}{128 d (a \sin (c+d x)+a)^8}-\frac {5}{224 a d (a \sin (c+d x)+a)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^5/(a + a*Sin[c + d*x])^8,x]

[Out]

(33*ArcTanh[Sin[c + d*x]])/(2048*a^8*d) - a^2/(80*d*(a + a*Sin[c + d*x])^10) - a/(48*d*(a + a*Sin[c + d*x])^9)
- 3/(128*d*(a + a*Sin[c + d*x])^8) - 5/(224*a*d*(a + a*Sin[c + d*x])^7) - 5/(256*a^2*d*(a + a*Sin[c + d*x])^6
) - 21/(1280*a^3*d*(a + a*Sin[c + d*x])^5) - 3/(256*a^5*d*(a + a*Sin[c + d*x])^3) - 7/(512*d*(a^2 + a^2*Sin[c
+ d*x])^4) + 1/(4096*d*(a^4 - a^4*Sin[c + d*x])^2) - 45/(4096*d*(a^4 + a^4*Sin[c + d*x])^2) + 11/(4096*d*(a^8
- a^8*Sin[c + d*x])) - 55/(4096*d*(a^8 + a^8*Sin[c + d*x]))

Rule 46

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] && Lt
Q[m + n + 2, 0])

Rule 212

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

Rule 2746

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])

Rubi steps

\begin {align*} \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^8} \, dx &=\frac {a^5 \text {Subst}\left (\int \frac {1}{(a-x)^3 (a+x)^{11}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^5 \text {Subst}\left (\int \left (\frac {1}{2048 a^{11} (a-x)^3}+\frac {11}{4096 a^{12} (a-x)^2}+\frac {1}{8 a^3 (a+x)^{11}}+\frac {3}{16 a^4 (a+x)^{10}}+\frac {3}{16 a^5 (a+x)^9}+\frac {5}{32 a^6 (a+x)^8}+\frac {15}{128 a^7 (a+x)^7}+\frac {21}{256 a^8 (a+x)^6}+\frac {7}{128 a^9 (a+x)^5}+\frac {9}{256 a^{10} (a+x)^4}+\frac {45}{2048 a^{11} (a+x)^3}+\frac {55}{4096 a^{12} (a+x)^2}+\frac {33}{2048 a^{12} \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {a^2}{80 d (a+a \sin (c+d x))^{10}}-\frac {a}{48 d (a+a \sin (c+d x))^9}-\frac {3}{128 d (a+a \sin (c+d x))^8}-\frac {5}{224 a d (a+a \sin (c+d x))^7}-\frac {5}{256 a^2 d (a+a \sin (c+d x))^6}-\frac {21}{1280 a^3 d (a+a \sin (c+d x))^5}-\frac {3}{256 a^5 d (a+a \sin (c+d x))^3}-\frac {7}{512 d \left (a^2+a^2 \sin (c+d x)\right )^4}+\frac {1}{4096 d \left (a^4-a^4 \sin (c+d x)\right )^2}-\frac {45}{4096 d \left (a^4+a^4 \sin (c+d x)\right )^2}+\frac {11}{4096 d \left (a^8-a^8 \sin (c+d x)\right )}-\frac {55}{4096 d \left (a^8+a^8 \sin (c+d x)\right )}+\frac {33 \text {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{2048 a^7 d}\\ &=\frac {33 \tanh ^{-1}(\sin (c+d x))}{2048 a^8 d}-\frac {a^2}{80 d (a+a \sin (c+d x))^{10}}-\frac {a}{48 d (a+a \sin (c+d x))^9}-\frac {3}{128 d (a+a \sin (c+d x))^8}-\frac {5}{224 a d (a+a \sin (c+d x))^7}-\frac {5}{256 a^2 d (a+a \sin (c+d x))^6}-\frac {21}{1280 a^3 d (a+a \sin (c+d x))^5}-\frac {3}{256 a^5 d (a+a \sin (c+d x))^3}-\frac {7}{512 d \left (a^2+a^2 \sin (c+d x)\right )^4}+\frac {1}{4096 d \left (a^4-a^4 \sin (c+d x)\right )^2}-\frac {45}{4096 d \left (a^4+a^4 \sin (c+d x)\right )^2}+\frac {11}{4096 d \left (a^8-a^8 \sin (c+d x)\right )}-\frac {55}{4096 d \left (a^8+a^8 \sin (c+d x)\right )}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.57, size = 195, normalized size = 0.69 \begin {gather*} \frac {\sec ^4(c+d x) \left (-34816+3465 \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 )^4 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^{20}-66953 \sin (c+d x)-72776 \sin ^2(c+d x)+21395 \sin ^3(c+d x)+190080 \sin ^4(c+d x)+255222 \sin ^5(c+d x)+114576 \sin ^6(c+d x)-82698 \sin ^7(c+d x)-147840 \sin ^8(c+d x)-91245 \sin ^9(c+d x)-27720 \sin ^{10}(c+d x)-3465 \sin ^{11}(c+d x)\right )}{215040 a^8 d (1+\sin (c+d x))^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^5/(a + a*Sin[c + d*x])^8,x]

[Out]

(Sec[c + d*x]^4*(-34816 + 3465*ArcTanh[Sin[c + d*x]]*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4*(Cos[(c + d*x)/2]
+ Sin[(c + d*x)/2])^20 - 66953*Sin[c + d*x] - 72776*Sin[c + d*x]^2 + 21395*Sin[c + d*x]^3 + 190080*Sin[c + d*
x]^4 + 255222*Sin[c + d*x]^5 + 114576*Sin[c + d*x]^6 - 82698*Sin[c + d*x]^7 - 147840*Sin[c + d*x]^8 - 91245*Si
n[c + d*x]^9 - 27720*Sin[c + d*x]^10 - 3465*Sin[c + d*x]^11))/(215040*a^8*d*(1 + Sin[c + d*x])^8)

________________________________________________________________________________________

Maple [A]
time = 0.41, size = 175, normalized size = 0.62

method	result	size
derivativedivides	\(\frac {\frac {1}{4096 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {11}{4096 \left (\sin \left (d x +c \right )-1\right )}-\frac {33 \ln \left (\sin \left (d x +c \right )-1\right )}{4096}-\frac {1}{80 \left (1+\sin \left (d x +c \right )\right )^{10}}-\frac {1}{48 \left (1+\sin \left (d x +c \right )\right )^{9}}-\frac {3}{128 \left (1+\sin \left (d x +c \right )\right )^{8}}-\frac {5}{224 \left (1+\sin \left (d x +c \right )\right )^{7}}-\frac {5}{256 \left (1+\sin \left (d x +c \right )\right )^{6}}-\frac {21}{1280 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {7}{512 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {3}{256 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {45}{4096 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {55}{4096 \left (1+\sin \left (d x +c \right )\right )}+\frac {33 \ln \left (1+\sin \left (d x +c \right )\right )}{4096}}{d \,a^{8}}\)	\(175\)
default	\(\frac {\frac {1}{4096 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {11}{4096 \left (\sin \left (d x +c \right )-1\right )}-\frac {33 \ln \left (\sin \left (d x +c \right )-1\right )}{4096}-\frac {1}{80 \left (1+\sin \left (d x +c \right )\right )^{10}}-\frac {1}{48 \left (1+\sin \left (d x +c \right )\right )^{9}}-\frac {3}{128 \left (1+\sin \left (d x +c \right )\right )^{8}}-\frac {5}{224 \left (1+\sin \left (d x +c \right )\right )^{7}}-\frac {5}{256 \left (1+\sin \left (d x +c \right )\right )^{6}}-\frac {21}{1280 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {7}{512 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {3}{256 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {45}{4096 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {55}{4096 \left (1+\sin \left (d x +c \right )\right )}+\frac {33 \ln \left (1+\sin \left (d x +c \right )\right )}{4096}}{d \,a^{8}}\)	\(175\)
risch	\(-\frac {i \left (-1737120 i {\mathrm e}^{4 i \left (d x +c \right )}+3465 \,{\mathrm e}^{23 i \left (d x +c \right )}+55440 i {\mathrm e}^{2 i \left (d x +c \right )}-403095 \,{\mathrm e}^{21 i \left (d x +c \right )}-23276992 i {\mathrm e}^{12 i \left (d x +c \right )}+4798563 \,{\mathrm e}^{19 i \left (d x +c \right )}-37181408 i {\mathrm e}^{10 i \left (d x +c \right )}-6638973 \,{\mathrm e}^{17 i \left (d x +c \right )}-1737120 i {\mathrm e}^{20 i \left (d x +c \right )}-27559862 \,{\mathrm e}^{15 i \left (d x +c \right )}+6559872 i {\mathrm e}^{8 i \left (d x +c \right )}[phone] \,{\mathrm e}^{13 i \left (d x +c \right )}+8290128 i {\mathrm e}^{6 i \left (d x +c \right )}-17314378 \,{\mathrm e}^{11 i \left (d x +c \right )}+6559872 i {\mathrm e}^{16 i \left (d x +c \right )}[phone] \,{\mathrm e}^{9 i \left (d x +c \right )}-37181408 i {\mathrm e}^{14 i \left (d x +c \right )}+6638973 \,{\mathrm e}^{7 i \left (d x +c \right )}+55440 i {\mathrm e}^{22 i \left (d x +c \right )}-4798563 \,{\mathrm e}^{5 i \left (d x +c \right )}+8290128 i {\mathrm e}^{18 i \left (d x +c \right )}+403095 \,{\mathrm e}^{3 i \left (d x +c \right )}-3465 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{107520 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{20} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{4} a^{8} d}-\frac {33 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2048 a^{8} d}+\frac {33 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2048 a^{8} d}\)	\(346\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^5/(a+a*sin(d*x+c))^8,x,method=_RETURNVERBOSE)

[Out]

1/d/a^8*(1/4096/(sin(d*x+c)-1)^2-11/4096/(sin(d*x+c)-1)-33/4096*ln(sin(d*x+c)-1)-1/80/(1+sin(d*x+c))^10-1/48/(
1+sin(d*x+c))^9-3/128/(1+sin(d*x+c))^8-5/224/(1+sin(d*x+c))^7-5/256/(1+sin(d*x+c))^6-21/1280/(1+sin(d*x+c))^5-
7/512/(1+sin(d*x+c))^4-3/256/(1+sin(d*x+c))^3-45/4096/(1+sin(d*x+c))^2-55/4096/(1+sin(d*x+c))+33/4096*ln(1+sin
(d*x+c)))

________________________________________________________________________________________

Maxima [A]
time = 0.31, size = 305, normalized size = 1.07 \begin {gather*} -\frac {\frac {2 \, {\left (3465 \, \sin \left (d x + c\right )^{11} + 27720 \, \sin \left (d x + c\right )^{10} + 91245 \, \sin \left (d x + c\right )^{9} + 147840 \, \sin \left (d x + c\right )^{8} + 82698 \, \sin \left (d x + c\right )^{7} - 114576 \, \sin \left (d x + c\right )^{6} - 255222 \, \sin \left (d x + c\right )^{5} - 190080 \, \sin \left (d x + c\right )^{4} - 21395 \, \sin \left (d x + c\right )^{3} + 72776 \, \sin \left (d x + c\right )^{2} + 66953 \, \sin \left (d x + c\right ) + 34816\right )}}{a^{8} \sin \left (d x + c\right )^{12} + 8 \, a^{8} \sin \left (d x + c\right )^{11} + 26 \, a^{8} \sin \left (d x + c\right )^{10} + 40 \, a^{8} \sin \left (d x + c\right )^{9} + 15 \, a^{8} \sin \left (d x + c\right )^{8} - 48 \, a^{8} \sin \left (d x + c\right )^{7} - 84 \, a^{8} \sin \left (d x + c\right )^{6} - 48 \, a^{8} \sin \left (d x + c\right )^{5} + 15 \, a^{8} \sin \left (d x + c\right )^{4} + 40 \, a^{8} \sin \left (d x + c\right )^{3} + 26 \, a^{8} \sin \left (d x + c\right )^{2} + 8 \, a^{8} \sin \left (d x + c\right ) + a^{8}} - \frac {3465 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{8}} + \frac {3465 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{8}}}{430080 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5/(a+a*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

-1/430080*(2*(3465*sin(d*x + c)^11 + 27720*sin(d*x + c)^10 + 91245*sin(d*x + c)^9 + 147840*sin(d*x + c)^8 + 82
698*sin(d*x + c)^7 - 114576*sin(d*x + c)^6 - 255222*sin(d*x + c)^5 - 190080*sin(d*x + c)^4 - 21395*sin(d*x + c
)^3 + 72776*sin(d*x + c)^2 + 66953*sin(d*x + c) + 34816)/(a^8*sin(d*x + c)^12 + 8*a^8*sin(d*x + c)^11 + 26*a^8
*sin(d*x + c)^10 + 40*a^8*sin(d*x + c)^9 + 15*a^8*sin(d*x + c)^8 - 48*a^8*sin(d*x + c)^7 - 84*a^8*sin(d*x + c)
^6 - 48*a^8*sin(d*x + c)^5 + 15*a^8*sin(d*x + c)^4 + 40*a^8*sin(d*x + c)^3 + 26*a^8*sin(d*x + c)^2 + 8*a^8*sin
(d*x + c) + a^8) - 3465*log(sin(d*x + c) + 1)/a^8 + 3465*log(sin(d*x + c) - 1)/a^8)/d

________________________________________________________________________________________

Fricas [A]
time = 0.42, size = 466, normalized size = 1.64 \begin {gather*} \frac {55440 \, \cos \left (d x + c\right )^{10} - 572880 \, \cos \left (d x + c\right )^{8} + 1507968 \, \cos \left (d x + c\right )^{6} - 1260864 \, \cos \left (d x + c\right )^{4} + 157696 \, \cos \left (d x + c\right )^{2} + 3465 \, {\left (\cos \left (d x + c\right )^{12} - 32 \, \cos \left (d x + c\right )^{10} + 160 \, \cos \left (d x + c\right )^{8} - 256 \, \cos \left (d x + c\right )^{6} + 128 \, \cos \left (d x + c\right )^{4} - 8 \, {\left (\cos \left (d x + c\right )^{10} - 10 \, \cos \left (d x + c\right )^{8} + 24 \, \cos \left (d x + c\right )^{6} - 16 \, \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3465 \, {\left (\cos \left (d x + c\right )^{12} - 32 \, \cos \left (d x + c\right )^{10} + 160 \, \cos \left (d x + c\right )^{8} - 256 \, \cos \left (d x + c\right )^{6} + 128 \, \cos \left (d x + c\right )^{4} - 8 \, {\left (\cos \left (d x + c\right )^{10} - 10 \, \cos \left (d x + c\right )^{8} + 24 \, \cos \left (d x + c\right )^{6} - 16 \, \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (3465 \, \cos \left (d x + c\right )^{10} - 108570 \, \cos \left (d x + c\right )^{8} + 482328 \, \cos \left (d x + c\right )^{6} - 574992 \, \cos \left (d x + c\right )^{4} + 98560 \, \cos \left (d x + c\right )^{2} + 32256\right )} \sin \left (d x + c\right ) + 43008}{430080 \, {\left (a^{8} d \cos \left (d x + c\right )^{12} - 32 \, a^{8} d \cos \left (d x + c\right )^{10} + 160 \, a^{8} d \cos \left (d x + c\right )^{8} - 256 \, a^{8} d \cos \left (d x + c\right )^{6} + 128 \, a^{8} d \cos \left (d x + c\right )^{4} - 8 \, {\left (a^{8} d \cos \left (d x + c\right )^{10} - 10 \, a^{8} d \cos \left (d x + c\right )^{8} + 24 \, a^{8} d \cos \left (d x + c\right )^{6} - 16 \, a^{8} d \cos \left (d x + c\right )^{4}\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)^5/(a+a*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

1/430080*(55440*cos(d*x + c)^10 - 572880*cos(d*x + c)^8 + 1507968*cos(d*x + c)^6 - 1260864*cos(d*x + c)^4 + 15
7696*cos(d*x + c)^2 + 3465*(cos(d*x + c)^12 - 32*cos(d*x + c)^10 + 160*cos(d*x + c)^8 - 256*cos(d*x + c)^6 + 1
28*cos(d*x + c)^4 - 8*(cos(d*x + c)^10 - 10*cos(d*x + c)^8 + 24*cos(d*x + c)^6 - 16*cos(d*x + c)^4)*sin(d*x +
c))*log(sin(d*x + c) + 1) - 3465*(cos(d*x + c)^12 - 32*cos(d*x + c)^10 + 160*cos(d*x + c)^8 - 256*cos(d*x + c)
^6 + 128*cos(d*x + c)^4 - 8*(cos(d*x + c)^10 - 10*cos(d*x + c)^8 + 24*cos(d*x + c)^6 - 16*cos(d*x + c)^4)*sin(
d*x + c))*log(-sin(d*x + c) + 1) + 2*(3465*cos(d*x + c)^10 - 108570*cos(d*x + c)^8 + 482328*cos(d*x + c)^6 - 5
74992*cos(d*x + c)^4 + 98560*cos(d*x + c)^2 + 32256)*sin(d*x + c) + 43008)/(a^8*d*cos(d*x + c)^12 - 32*a^8*d*c
os(d*x + c)^10 + 160*a^8*d*cos(d*x + c)^8 - 256*a^8*d*cos(d*x + c)^6 + 128*a^8*d*cos(d*x + c)^4 - 8*(a^8*d*cos
(d*x + c)^10 - 10*a^8*d*cos(d*x + c)^8 + 24*a^8*d*cos(d*x + c)^6 - 16*a^8*d*cos(d*x + c)^4)*sin(d*x + c))

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 4.88, size = 186, normalized size = 0.65 \begin {gather*} \frac {\frac {27720 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{8}} - \frac {27720 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{8}} + \frac {420 \, {\left (99 \, \sin \left (d x + c\right )^{2} - 220 \, \sin \left (d x + c\right ) + 123\right )}}{a^{8} {\left (\sin \left (d x + c\right ) - 1\right )}^{2}} - \frac {81191 \, \sin \left (d x + c\right )^{10} + 858110 \, \sin \left (d x + c\right )^{9} + 4107195 \, \sin \left (d x + c\right )^{8} + 11748840 \, \sin \left (d x + c\right )^{7} + 22318590 \, \sin \left (d x + c\right )^{6} + 29583540 \, \sin \left (d x + c\right )^{5} + 27983550 \, \sin \left (d x + c\right )^{4} + 19002600 \, \sin \left (d x + c\right )^{3} + 9206235 \, \sin \left (d x + c\right )^{2} + 3108990 \, \sin \left (d x + c\right ) + 648327}{a^{8} {\left (\sin \left (d x + c\right ) + 1\right )}^{10}}}{3440640 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5/(a+a*sin(d*x+c))^8,x, algorithm="giac")

[Out]

1/3440640*(27720*log(abs(sin(d*x + c) + 1))/a^8 - 27720*log(abs(sin(d*x + c) - 1))/a^8 + 420*(99*sin(d*x + c)^
2 - 220*sin(d*x + c) + 123)/(a^8*(sin(d*x + c) - 1)^2) - (81191*sin(d*x + c)^10 + 858110*sin(d*x + c)^9 + 4107
195*sin(d*x + c)^8 + 11748840*sin(d*x + c)^7 + 22318590*sin(d*x + c)^6 + 29583540*sin(d*x + c)^5 + 27983550*si
n(d*x + c)^4 + 19002600*sin(d*x + c)^3 + 9206235*sin(d*x + c)^2 + 3108990*sin(d*x + c) + 648327)/(a^8*(sin(d*x
+ c) + 1)^10))/d

________________________________________________________________________________________

Mupad [B]
time = 0.80, size = 290, normalized size = 1.02 \begin {gather*} \frac {33\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )}{2048\,a^8\,d}-\frac {\frac {33\,{\sin \left (c+d\,x\right )}^{11}}{2048}+\frac {33\,{\sin \left (c+d\,x\right )}^{10}}{256}+\frac {869\,{\sin \left (c+d\,x\right )}^9}{2048}+\frac {11\,{\sin \left (c+d\,x\right )}^8}{16}+\frac {1969\,{\sin \left (c+d\,x\right )}^7}{5120}-\frac {341\,{\sin \left (c+d\,x\right )}^6}{640}-\frac {42537\,{\sin \left (c+d\,x\right )}^5}{35840}-\frac {99\,{\sin \left (c+d\,x\right )}^4}{112}-\frac {4279\,{\sin \left (c+d\,x\right )}^3}{43008}+\frac {9097\,{\sin \left (c+d\,x\right )}^2}{26880}+\frac {66953\,\sin \left (c+d\,x\right )}{215040}+\frac {17}{105}}{d\,\left (a^8\,{\sin \left (c+d\,x\right )}^{12}+8\,a^8\,{\sin \left (c+d\,x\right )}^{11}+26\,a^8\,{\sin \left (c+d\,x\right )}^{10}+40\,a^8\,{\sin \left (c+d\,x\right )}^9+15\,a^8\,{\sin \left (c+d\,x\right )}^8-48\,a^8\,{\sin \left (c+d\,x\right )}^7-84\,a^8\,{\sin \left (c+d\,x\right )}^6-48\,a^8\,{\sin \left (c+d\,x\right )}^5+15\,a^8\,{\sin \left (c+d\,x\right )}^4+40\,a^8\,{\sin \left (c+d\,x\right )}^3+26\,a^8\,{\sin \left (c+d\,x\right )}^2+8\,a^8\,\sin \left (c+d\,x\right )+a^8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(c + d*x)^5*(a + a*sin(c + d*x))^8),x)

[Out]

(33*atanh(sin(c + d*x)))/(2048*a^8*d) - ((66953*sin(c + d*x))/215040 + (9097*sin(c + d*x)^2)/26880 - (4279*sin
(c + d*x)^3)/43008 - (99*sin(c + d*x)^4)/112 - (42537*sin(c + d*x)^5)/35840 - (341*sin(c + d*x)^6)/640 + (1969
*sin(c + d*x)^7)/5120 + (11*sin(c + d*x)^8)/16 + (869*sin(c + d*x)^9)/2048 + (33*sin(c + d*x)^10)/256 + (33*si
n(c + d*x)^11)/2048 + 17/105)/(d*(8*a^8*sin(c + d*x) + a^8 + 26*a^8*sin(c + d*x)^2 + 40*a^8*sin(c + d*x)^3 + 1
5*a^8*sin(c + d*x)^4 - 48*a^8*sin(c + d*x)^5 - 84*a^8*sin(c + d*x)^6 - 48*a^8*sin(c + d*x)^7 + 15*a^8*sin(c +
d*x)^8 + 40*a^8*sin(c + d*x)^9 + 26*a^8*sin(c + d*x)^10 + 8*a^8*sin(c + d*x)^11 + a^8*sin(c + d*x)^12))

________________________________________________________________________________________