3.1.48 \(\int \sec ^3(c+d x) (a+a \sin (c+d x))^8 \, dx\) [48]

Optimal. Leaf size=121 \[ \frac {192 a^8 \log (1-\sin (c+d x))}{d}+\frac {129 a^8 \sin (c+d x)}{d}+\frac {36 a^8 \sin ^2(c+d x)}{d}+\frac {10 a^8 \sin ^3(c+d x)}{d}+\frac {2 a^8 \sin ^4(c+d x)}{d}+\frac {a^8 \sin ^5(c+d x)}{5 d}+\frac {64 a^9}{d (a-a \sin (c+d x))} \]

[Out]

192*a^8*ln(1-sin(d*x+c))/d+129*a^8*sin(d*x+c)/d+36*a^8*sin(d*x+c)^2/d+10*a^8*sin(d*x+c)^3/d+2*a^8*sin(d*x+c)^4
/d+1/5*a^8*sin(d*x+c)^5/d+64*a^9/d/(a-a*sin(d*x+c))

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Rubi [A]
time = 0.06, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2746, 45} \begin {gather*} \frac {64 a^9}{d (a-a \sin (c+d x))}+\frac {a^8 \sin ^5(c+d x)}{5 d}+\frac {2 a^8 \sin ^4(c+d x)}{d}+\frac {10 a^8 \sin ^3(c+d x)}{d}+\frac {36 a^8 \sin ^2(c+d x)}{d}+\frac {129 a^8 \sin (c+d x)}{d}+\frac {192 a^8 \log (1-\sin (c+d x))}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(192*a^8*Log[1 - Sin[c + d*x]])/d + (129*a^8*Sin[c + d*x])/d + (36*a^8*Sin[c + d*x]^2)/d + (10*a^8*Sin[c + d*x
]^3)/d + (2*a^8*Sin[c + d*x]^4)/d + (a^8*Sin[c + d*x]^5)/(5*d) + (64*a^9)/(d*(a - a*Sin[c + d*x]))

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 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 \sec ^3(c+d x) (a+a \sin (c+d x))^8 \, dx &=\frac {a^3 \text {Subst}\left (\int \frac {(a+x)^6}{(a-x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^3 \text {Subst}\left (\int \left (129 a^4+\frac {64 a^6}{(a-x)^2}-\frac {192 a^5}{a-x}+72 a^3 x+30 a^2 x^2+8 a x^3+x^4\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {192 a^8 \log (1-\sin (c+d x))}{d}+\frac {129 a^8 \sin (c+d x)}{d}+\frac {36 a^8 \sin ^2(c+d x)}{d}+\frac {10 a^8 \sin ^3(c+d x)}{d}+\frac {2 a^8 \sin ^4(c+d x)}{d}+\frac {a^8 \sin ^5(c+d x)}{5 d}+\frac {64 a^9}{d (a-a \sin (c+d x))}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 111, normalized size = 0.92 \begin {gather*} \frac {a^8 \sec ^2(c+d x) (1-\sin (c+d x)) (1+\sin (c+d x)) \left (192 \log (1-\sin (c+d x))+\frac {64}{1-\sin (c+d x)}+129 \sin (c+d x)+36 \sin ^2(c+d x)+10 \sin ^3(c+d x)+2 \sin ^4(c+d x)+\frac {1}{5} \sin ^5(c+d x)\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^8*Sec[c + d*x]^2*(1 - Sin[c + d*x])*(1 + Sin[c + d*x])*(192*Log[1 - Sin[c + d*x]] + 64/(1 - Sin[c + d*x]) +
129*Sin[c + d*x] + 36*Sin[c + d*x]^2 + 10*Sin[c + d*x]^3 + 2*Sin[c + d*x]^4 + Sin[c + d*x]^5/5))/d

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(441\) vs. \(2(119)=238\).
time = 0.21, size = 442, normalized size = 3.65

method	result	size
risch	\(-192 i a^{8} x -\frac {1093 i a^{8} {\mathrm e}^{i \left (d x +c \right )}}{16 d}+\frac {1093 i a^{8} {\mathrm e}^{-i \left (d x +c \right )}}{16 d}-\frac {384 i a^{8} c}{d}-\frac {128 i a^{8} {\mathrm e}^{i \left (d x +c \right )}}{\left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{2} d}+\frac {384 a^{8} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {a^{8} \sin \left (5 d x +5 c \right )}{80 d}+\frac {a^{8} \cos \left (4 d x +4 c \right )}{4 d}-\frac {41 a^{8} \sin \left (3 d x +3 c \right )}{16 d}-\frac {19 a^{8} \cos \left (2 d x +2 c \right )}{d}\)	\(176\)
derivativedivides	\(\frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{7}\left (d x +c \right )\right )}{2}+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{10}+\frac {7 \left (\sin ^{3}\left (d x +c \right )\right )}{6}+\frac {7 \sin \left (d x +c \right )}{2}-\frac {7 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+8 a^{8} \left (\frac {\sin ^{8}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{2}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {3 \left (\sin ^{2}\left (d x +c \right )\right )}{2}+3 \ln \left (\cos \left (d x +c \right )\right )\right )+28 a^{8} \left (\frac {\sin ^{7}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{2}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{6}+\frac {5 \sin \left (d x +c \right )}{2}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+56 a^{8} \left (\frac {\sin ^{6}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{2}+\sin ^{2}\left (d x +c \right )+2 \ln \left (\cos \left (d x +c \right )\right )\right )+70 a^{8} \left (\frac {\sin ^{5}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+56 a^{8} \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )+28 a^{8} \left (\frac {\sin ^{3}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {4 a^{8}}{\cos \left (d x +c \right )^{2}}+a^{8} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\)	\(442\)
default	\(\frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{7}\left (d x +c \right )\right )}{2}+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{10}+\frac {7 \left (\sin ^{3}\left (d x +c \right )\right )}{6}+\frac {7 \sin \left (d x +c \right )}{2}-\frac {7 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+8 a^{8} \left (\frac {\sin ^{8}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{2}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {3 \left (\sin ^{2}\left (d x +c \right )\right )}{2}+3 \ln \left (\cos \left (d x +c \right )\right )\right )+28 a^{8} \left (\frac {\sin ^{7}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{2}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{6}+\frac {5 \sin \left (d x +c \right )}{2}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+56 a^{8} \left (\frac {\sin ^{6}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{2}+\sin ^{2}\left (d x +c \right )+2 \ln \left (\cos \left (d x +c \right )\right )\right )+70 a^{8} \left (\frac {\sin ^{5}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+56 a^{8} \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )+28 a^{8} \left (\frac {\sin ^{3}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {4 a^{8}}{\cos \left (d x +c \right )^{2}}+a^{8} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\)	\(442\)

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,method=_RETURNVERBOSE)

[Out]

1/d*(a^8*(1/2*sin(d*x+c)^9/cos(d*x+c)^2+1/2*sin(d*x+c)^7+7/10*sin(d*x+c)^5+7/6*sin(d*x+c)^3+7/2*sin(d*x+c)-7/2
*ln(sec(d*x+c)+tan(d*x+c)))+8*a^8*(1/2*sin(d*x+c)^8/cos(d*x+c)^2+1/2*sin(d*x+c)^6+3/4*sin(d*x+c)^4+3/2*sin(d*x
+c)^2+3*ln(cos(d*x+c)))+28*a^8*(1/2*sin(d*x+c)^7/cos(d*x+c)^2+1/2*sin(d*x+c)^5+5/6*sin(d*x+c)^3+5/2*sin(d*x+c)
-5/2*ln(sec(d*x+c)+tan(d*x+c)))+56*a^8*(1/2*sin(d*x+c)^6/cos(d*x+c)^2+1/2*sin(d*x+c)^4+sin(d*x+c)^2+2*ln(cos(d
*x+c)))+70*a^8*(1/2*sin(d*x+c)^5/cos(d*x+c)^2+1/2*sin(d*x+c)^3+3/2*sin(d*x+c)-3/2*ln(sec(d*x+c)+tan(d*x+c)))+5
6*a^8*(1/2*tan(d*x+c)^2+ln(cos(d*x+c)))+28*a^8*(1/2*sin(d*x+c)^3/cos(d*x+c)^2+1/2*sin(d*x+c)-1/2*ln(sec(d*x+c)
+tan(d*x+c)))+4*a^8/cos(d*x+c)^2+a^8*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c))))

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Maxima [A]
time = 0.29, size = 97, normalized size = 0.80 \begin {gather*} \frac {a^{8} \sin \left (d x + c\right )^{5} + 10 \, a^{8} \sin \left (d x + c\right )^{4} + 50 \, a^{8} \sin \left (d x + c\right )^{3} + 180 \, a^{8} \sin \left (d x + c\right )^{2} + 960 \, a^{8} \log \left (\sin \left (d x + c\right ) - 1\right ) + 645 \, a^{8} \sin \left (d x + c\right ) - \frac {320 \, a^{8}}{\sin \left (d x + c\right ) - 1}}{5 \, d} \end {gather*}

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]

1/5*(a^8*sin(d*x + c)^5 + 10*a^8*sin(d*x + c)^4 + 50*a^8*sin(d*x + c)^3 + 180*a^8*sin(d*x + c)^2 + 960*a^8*log
(sin(d*x + c) - 1) + 645*a^8*sin(d*x + c) - 320*a^8/(sin(d*x + c) - 1))/d

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Fricas [A]
time = 0.38, size = 130, normalized size = 1.07 \begin {gather*} -\frac {4 \, a^{8} \cos \left (d x + c\right )^{6} - 172 \, a^{8} \cos \left (d x + c\right )^{4} + 2192 \, a^{8} \cos \left (d x + c\right )^{2} - 1119 \, a^{8} - 3840 \, {\left (a^{8} \sin \left (d x + c\right ) - a^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - {\left (36 \, a^{8} \cos \left (d x + c\right )^{4} - 592 \, a^{8} \cos \left (d x + c\right )^{2} - 2399 \, a^{8}\right )} \sin \left (d x + c\right )}{20 \, {\left (d \sin \left (d x + c\right ) - d\right )}} \end {gather*}

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]

-1/20*(4*a^8*cos(d*x + c)^6 - 172*a^8*cos(d*x + c)^4 + 2192*a^8*cos(d*x + c)^2 - 1119*a^8 - 3840*(a^8*sin(d*x
+ c) - a^8)*log(-sin(d*x + c) + 1) - (36*a^8*cos(d*x + c)^4 - 592*a^8*cos(d*x + c)^2 - 2399*a^8)*sin(d*x + c))
/(d*sin(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(sec(d*x+c)**3*(a+a*sin(d*x+c))**8,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6190 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (120) = 240\).
time = 5.32, size = 275, normalized size = 2.27 \begin {gather*} -\frac {2 \, {\left (480 \, a^{8} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 960 \, a^{8} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {160 \, {\left (9 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 20 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, a^{8}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{2}} - \frac {1096 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 645 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 5840 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 2780 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12120 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 4286 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12120 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2780 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5840 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 645 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1096 \, a^{8}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}\right )}}{5 \, d} \end {gather*}

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]

-2/5*(480*a^8*log(tan(1/2*d*x + 1/2*c)^2 + 1) - 960*a^8*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 160*(9*a^8*tan(1/
2*d*x + 1/2*c)^2 - 20*a^8*tan(1/2*d*x + 1/2*c) + 9*a^8)/(tan(1/2*d*x + 1/2*c) - 1)^2 - (1096*a^8*tan(1/2*d*x +
1/2*c)^10 + 645*a^8*tan(1/2*d*x + 1/2*c)^9 + 5840*a^8*tan(1/2*d*x + 1/2*c)^8 + 2780*a^8*tan(1/2*d*x + 1/2*c)^
7 + 12120*a^8*tan(1/2*d*x + 1/2*c)^6 + 4286*a^8*tan(1/2*d*x + 1/2*c)^5 + 12120*a^8*tan(1/2*d*x + 1/2*c)^4 + 27
80*a^8*tan(1/2*d*x + 1/2*c)^3 + 5840*a^8*tan(1/2*d*x + 1/2*c)^2 + 645*a^8*tan(1/2*d*x + 1/2*c) + 1096*a^8)/(ta
n(1/2*d*x + 1/2*c)^2 + 1)^5)/d

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Mupad [B]
time = 4.62, size = 97, normalized size = 0.80 \begin {gather*} \frac {192\,a^8\,\ln \left (\sin \left (c+d\,x\right )-1\right )-\frac {64\,a^8}{\sin \left (c+d\,x\right )-1}+129\,a^8\,\sin \left (c+d\,x\right )+36\,a^8\,{\sin \left (c+d\,x\right )}^2+10\,a^8\,{\sin \left (c+d\,x\right )}^3+2\,a^8\,{\sin \left (c+d\,x\right )}^4+\frac {a^8\,{\sin \left (c+d\,x\right )}^5}{5}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(192*a^8*log(sin(c + d*x) - 1) - (64*a^8)/(sin(c + d*x) - 1) + 129*a^8*sin(c + d*x) + 36*a^8*sin(c + d*x)^2 +
10*a^8*sin(c + d*x)^3 + 2*a^8*sin(c + d*x)^4 + (a^8*sin(c + d*x)^5)/5)/d

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