Integrand size = 10, antiderivative size = 141 \[ \int \frac {1}{a+a \sin ^8(x)} \, dx=\frac {\arctan \left (\sqrt {1+\sqrt [4]{-1}} \tan (x)\right )}{4 \sqrt {1+\sqrt [4]{-1}} a}+\frac {\arctan \left (\sqrt {1-(-1)^{3/4}} \tan (x)\right )}{4 \sqrt {1-(-1)^{3/4}} a}+\frac {\text {arctanh}\left (\sqrt {-1+\sqrt [4]{-1}} \tan (x)\right )}{4 \sqrt {-1+\sqrt [4]{-1}} a}+\frac {\text {arctanh}\left (\sqrt {-1-(-1)^{3/4}} \tan (x)\right )}{4 \sqrt {-1-(-1)^{3/4}} a} \] Output:
1/4*arctan((1+(-1)^(1/4))^(1/2)*tan(x))/(1+(-1)^(1/4))^(1/2)/a+1/4*arctan( (1-(-1)^(3/4))^(1/2)*tan(x))/(1-(-1)^(3/4))^(1/2)/a+1/4*arctanh((-1+(-1)^( 1/4))^(1/2)*tan(x))/(-1+(-1)^(1/4))^(1/2)/a+1/4*arctanh((-1-(-1)^(3/4))^(1 /2)*tan(x))/(-1-(-1)^(3/4))^(1/2)/a
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 5.12 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.02 \[ \int \frac {1}{a+a \sin ^8(x)} \, dx=\frac {8 \text {RootSum}\left [1-8 \text {$\#$1}+28 \text {$\#$1}^2-56 \text {$\#$1}^3+326 \text {$\#$1}^4-56 \text {$\#$1}^5+28 \text {$\#$1}^6-8 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {2 \arctan \left (\frac {\sin (2 x)}{\cos (2 x)-\text {$\#$1}}\right ) \text {$\#$1}^3-i \log \left (1-2 \cos (2 x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3}{-1+7 \text {$\#$1}-21 \text {$\#$1}^2+163 \text {$\#$1}^3-35 \text {$\#$1}^4+21 \text {$\#$1}^5-7 \text {$\#$1}^6+\text {$\#$1}^7}\&\right ]}{a} \] Input:
Integrate[(a + a*Sin[x]^8)^(-1),x]
Output:
(8*RootSum[1 - 8*#1 + 28*#1^2 - 56*#1^3 + 326*#1^4 - 56*#1^5 + 28*#1^6 - 8 *#1^7 + #1^8 & , (2*ArcTan[Sin[2*x]/(Cos[2*x] - #1)]*#1^3 - I*Log[1 - 2*Co s[2*x]*#1 + #1^2]*#1^3)/(-1 + 7*#1 - 21*#1^2 + 163*#1^3 - 35*#1^4 + 21*#1^ 5 - 7*#1^6 + #1^7) & ])/a
Time = 0.40 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3690, 3042, 3660, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{a \sin ^8(x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{a \sin (x)^8+a}dx\) |
\(\Big \downarrow \) 3690 |
\(\displaystyle \frac {\int \frac {1}{1-\sqrt [4]{-1} \sin ^2(x)}dx}{4 a}+\frac {\int \frac {1}{\sqrt [4]{-1} \sin ^2(x)+1}dx}{4 a}+\frac {\int \frac {1}{1-(-1)^{3/4} \sin ^2(x)}dx}{4 a}+\frac {\int \frac {1}{(-1)^{3/4} \sin ^2(x)+1}dx}{4 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {1}{1-\sqrt [4]{-1} \sin (x)^2}dx}{4 a}+\frac {\int \frac {1}{\sqrt [4]{-1} \sin (x)^2+1}dx}{4 a}+\frac {\int \frac {1}{1-(-1)^{3/4} \sin (x)^2}dx}{4 a}+\frac {\int \frac {1}{(-1)^{3/4} \sin (x)^2+1}dx}{4 a}\) |
\(\Big \downarrow \) 3660 |
\(\displaystyle \frac {\int \frac {1}{\left (1-\sqrt [4]{-1}\right ) \tan ^2(x)+1}d\tan (x)}{4 a}+\frac {\int \frac {1}{\left (1+\sqrt [4]{-1}\right ) \tan ^2(x)+1}d\tan (x)}{4 a}+\frac {\int \frac {1}{\left (1-(-1)^{3/4}\right ) \tan ^2(x)+1}d\tan (x)}{4 a}+\frac {\int \frac {1}{\left (1+(-1)^{3/4}\right ) \tan ^2(x)+1}d\tan (x)}{4 a}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\arctan \left (\sqrt {1-\sqrt [4]{-1}} \tan (x)\right )}{4 \sqrt {1-\sqrt [4]{-1}} a}+\frac {\arctan \left (\sqrt {1+\sqrt [4]{-1}} \tan (x)\right )}{4 \sqrt {1+\sqrt [4]{-1}} a}+\frac {\arctan \left (\sqrt {1-(-1)^{3/4}} \tan (x)\right )}{4 \sqrt {1-(-1)^{3/4}} a}+\frac {\arctan \left (\sqrt {1+(-1)^{3/4}} \tan (x)\right )}{4 \sqrt {1+(-1)^{3/4}} a}\) |
Input:
Int[(a + a*Sin[x]^8)^(-1),x]
Output:
ArcTan[Sqrt[1 - (-1)^(1/4)]*Tan[x]]/(4*Sqrt[1 - (-1)^(1/4)]*a) + ArcTan[Sq rt[1 + (-1)^(1/4)]*Tan[x]]/(4*Sqrt[1 + (-1)^(1/4)]*a) + ArcTan[Sqrt[1 - (- 1)^(3/4)]*Tan[x]]/(4*Sqrt[1 - (-1)^(3/4)]*a) + ArcTan[Sqrt[1 + (-1)^(3/4)] *Tan[x]]/(4*Sqrt[1 + (-1)^(3/4)]*a)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[1/(a + (a + b)*ff^2*x^ 2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(-1), x_Symbol] :> Module[{ k}, Simp[2/(a*n) Sum[Int[1/(1 - Sin[e + f*x]^2/((-1)^(4*(k/n))*Rt[-a/b, n /2])), x], {k, 1, n/2}], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[n/2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.36 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.52
method | result | size |
default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{8}+4 \textit {\_Z}^{6}+6 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )}{\sum }\frac {\left (\textit {\_R}^{6}+3 \textit {\_R}^{4}+3 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (x \right )-\textit {\_R} \right )}{2 \textit {\_R}^{7}+3 \textit {\_R}^{5}+3 \textit {\_R}^{3}+\textit {\_R}}}{8 a}\) | \(74\) |
risch | \(\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8192 a^{4} \textit {\_Z}^{4}+\left (-128 i a^{2}+128 a^{2}\right ) \textit {\_Z}^{2}+1-i\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (-1024 i a^{3}-1024 a^{3}\right ) \textit {\_R}^{3}+\left (-128 i a^{2}+128 a^{2}\right ) \textit {\_R}^{2}+\left (16 i a -16 a \right ) \textit {\_R} -1-2 i\right )\right )+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8192 a^{4} \textit {\_Z}^{4}+\left (128 i a^{2}+128 a^{2}\right ) \textit {\_Z}^{2}+1+i\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (-1024 i a^{3}+1024 a^{3}\right ) \textit {\_R}^{3}+\left (128 i a^{2}+128 a^{2}\right ) \textit {\_R}^{2}+\left (16 i a +16 a \right ) \textit {\_R} -1+2 i\right )\right )\) | \(174\) |
Input:
int(1/(a+a*sin(x)^8),x,method=_RETURNVERBOSE)
Output:
1/8/a*sum((_R^6+3*_R^4+3*_R^2+1)/(2*_R^7+3*_R^5+3*_R^3+_R)*ln(tan(x)-_R),_ R=RootOf(2*_Z^8+4*_Z^6+6*_Z^4+4*_Z^2+1))
Leaf count of result is larger than twice the leaf count of optimal. 1977 vs. \(2 (101) = 202\).
Time = 0.39 (sec) , antiderivative size = 1977, normalized size of antiderivative = 14.02 \[ \int \frac {1}{a+a \sin ^8(x)} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+a*sin(x)^8),x, algorithm="fricas")
Output:
-1/16*sqrt(1/2)*sqrt(-(a^2*sqrt(-(4*sqrt(1/2)*a^4*sqrt(a^(-8)) + 3)/a^4) + 1)/a^2)*log(1/2*sqrt(1/2)*(sqrt(1/2)*a^5*sqrt(a^(-8))*cos(x)*sin(x) - a*c os(x)*sin(x) + (sqrt(1/2)*a^7*sqrt(a^(-8))*cos(x)*sin(x) - a^3*cos(x)*sin( x))*sqrt(-(4*sqrt(1/2)*a^4*sqrt(a^(-8)) + 3)/a^4))*sqrt(-(a^2*sqrt(-(4*sqr t(1/2)*a^4*sqrt(a^(-8)) + 3)/a^4) + 1)/a^2) + 1/4*sqrt(1/2)*(2*a^4*cos(x)^ 2 - a^4)*sqrt(a^(-8)) - 1/4*cos(x)^2 - 1/4*(2*a^2*cos(x)^2 - a^2 - sqrt(1/ 2)*(2*a^6*cos(x)^2 - a^6)*sqrt(a^(-8)))*sqrt(-(4*sqrt(1/2)*a^4*sqrt(a^(-8) ) + 3)/a^4) + 1/4) + 1/16*sqrt(1/2)*sqrt(-(a^2*sqrt(-(4*sqrt(1/2)*a^4*sqrt (a^(-8)) + 3)/a^4) + 1)/a^2)*log(-1/2*sqrt(1/2)*(sqrt(1/2)*a^5*sqrt(a^(-8) )*cos(x)*sin(x) - a*cos(x)*sin(x) + (sqrt(1/2)*a^7*sqrt(a^(-8))*cos(x)*sin (x) - a^3*cos(x)*sin(x))*sqrt(-(4*sqrt(1/2)*a^4*sqrt(a^(-8)) + 3)/a^4))*sq rt(-(a^2*sqrt(-(4*sqrt(1/2)*a^4*sqrt(a^(-8)) + 3)/a^4) + 1)/a^2) + 1/4*sqr t(1/2)*(2*a^4*cos(x)^2 - a^4)*sqrt(a^(-8)) - 1/4*cos(x)^2 - 1/4*(2*a^2*cos (x)^2 - a^2 - sqrt(1/2)*(2*a^6*cos(x)^2 - a^6)*sqrt(a^(-8)))*sqrt(-(4*sqrt (1/2)*a^4*sqrt(a^(-8)) + 3)/a^4) + 1/4) + 1/16*sqrt(1/2)*sqrt((a^2*sqrt(-( 4*sqrt(1/2)*a^4*sqrt(a^(-8)) + 3)/a^4) - 1)/a^2)*log(1/2*sqrt(1/2)*(sqrt(1 /2)*a^5*sqrt(a^(-8))*cos(x)*sin(x) - a*cos(x)*sin(x) - (sqrt(1/2)*a^7*sqrt (a^(-8))*cos(x)*sin(x) - a^3*cos(x)*sin(x))*sqrt(-(4*sqrt(1/2)*a^4*sqrt(a^ (-8)) + 3)/a^4))*sqrt((a^2*sqrt(-(4*sqrt(1/2)*a^4*sqrt(a^(-8)) + 3)/a^4) - 1)/a^2) - 1/4*sqrt(1/2)*(2*a^4*cos(x)^2 - a^4)*sqrt(a^(-8)) + 1/4*cos(...
\[ \int \frac {1}{a+a \sin ^8(x)} \, dx=\frac {\int \frac {1}{\sin ^{8}{\left (x \right )} + 1}\, dx}{a} \] Input:
integrate(1/(a+a*sin(x)**8),x)
Output:
Integral(1/(sin(x)**8 + 1), x)/a
\[ \int \frac {1}{a+a \sin ^8(x)} \, dx=\int { \frac {1}{a \sin \left (x\right )^{8} + a} \,d x } \] Input:
integrate(1/(a+a*sin(x)^8),x, algorithm="maxima")
Output:
integrate(1/(a*sin(x)^8 + a), x)
Exception generated. \[ \int \frac {1}{a+a \sin ^8(x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(a+a*sin(x)^8),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to find common minimal polyn omial Error: Bad Argument Value
Time = 37.65 (sec) , antiderivative size = 1169, normalized size of antiderivative = 8.29 \[ \int \frac {1}{a+a \sin ^8(x)} \, dx=\text {Too large to display} \] Input:
int(1/(a + a*sin(x)^8),x)
Output:
atan((tan(x)*(- (- 2*2^(1/2) - 3)^(1/2)/(128*a^2) - 1/(128*a^2))^(1/2)*32i )/((16*(- 2*2^(1/2) - 3)^(1/2))/a - (12*2^(1/2)*(- 2*2^(1/2) - 3)^(1/2))/a ) - (2^(1/2)*tan(x)*(- (- 2*2^(1/2) - 3)^(1/2)/(128*a^2) - 1/(128*a^2))^(1 /2)*32i)/((16*(- 2*2^(1/2) - 3)^(1/2))/a - (12*2^(1/2)*(- 2*2^(1/2) - 3)^( 1/2))/a) + (tan(x)*(- (- 2*2^(1/2) - 3)^(1/2)/(128*a^2) - 1/(128*a^2))^(1/ 2)*(- 2*2^(1/2) - 3)^(1/2)*224i)/((16*(- 2*2^(1/2) - 3)^(1/2))/a - (12*2^( 1/2)*(- 2*2^(1/2) - 3)^(1/2))/a) - (2^(1/2)*tan(x)*(- (- 2*2^(1/2) - 3)^(1 /2)/(128*a^2) - 1/(128*a^2))^(1/2)*(- 2*2^(1/2) - 3)^(1/2)*160i)/((16*(- 2 *2^(1/2) - 3)^(1/2))/a - (12*2^(1/2)*(- 2*2^(1/2) - 3)^(1/2))/a))*(-((- 2* 2^(1/2) - 3)^(1/2) + 1)/(128*a^2))^(1/2)*2i - atan((tan(x)*((- 2*2^(1/2) - 3)^(1/2)/(128*a^2) - 1/(128*a^2))^(1/2)*32i)/((16*(- 2*2^(1/2) - 3)^(1/2) )/a - (12*2^(1/2)*(- 2*2^(1/2) - 3)^(1/2))/a) - (2^(1/2)*tan(x)*((- 2*2^(1 /2) - 3)^(1/2)/(128*a^2) - 1/(128*a^2))^(1/2)*32i)/((16*(- 2*2^(1/2) - 3)^ (1/2))/a - (12*2^(1/2)*(- 2*2^(1/2) - 3)^(1/2))/a) - (tan(x)*((- 2*2^(1/2) - 3)^(1/2)/(128*a^2) - 1/(128*a^2))^(1/2)*(- 2*2^(1/2) - 3)^(1/2)*224i)/( (16*(- 2*2^(1/2) - 3)^(1/2))/a - (12*2^(1/2)*(- 2*2^(1/2) - 3)^(1/2))/a) + (2^(1/2)*tan(x)*((- 2*2^(1/2) - 3)^(1/2)/(128*a^2) - 1/(128*a^2))^(1/2)*( - 2*2^(1/2) - 3)^(1/2)*160i)/((16*(- 2*2^(1/2) - 3)^(1/2))/a - (12*2^(1/2) *(- 2*2^(1/2) - 3)^(1/2))/a))*(((- 2*2^(1/2) - 3)^(1/2) - 1)/(128*a^2))^(1 /2)*2i + atan((tan(x)*(- (2*2^(1/2) - 3)^(1/2)/(128*a^2) - 1/(128*a^2))...
\[ \int \frac {1}{a+a \sin ^8(x)} \, dx=\frac {\int \frac {1}{\sin \left (x \right )^{8}+1}d x}{a} \] Input:
int(1/(a+a*sin(x)^8),x)
Output:
int(1/(sin(x)**8 + 1),x)/a