\(\int \frac {1}{(a+b \sin ^3(c+d x))^2} \, dx\) [401]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int \frac {1}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Int}\left (\frac {1}{\left (a+b \sin ^3(c+d x)\right )^2},x\right ) \]

[Out]

Unintegrable(1/(a+b*sin(d*x+c)^3)^2,x)

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\int \frac {1}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[In]

Int[(a + b*Sin[c + d*x]^3)^(-2),x]

[Out]

Defer[Int][(a + b*Sin[c + d*x]^3)^(-2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.61 (sec) , antiderivative size = 502, normalized size of antiderivative = 35.86 \[ \int \frac {1}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {\frac {i \text {RootSum}\left [-i b+3 i b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 i b \text {$\#$1}^4+i b \text {$\#$1}^6\&,\frac {2 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+4 i a b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+2 a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}-24 a^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+12 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+12 i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-6 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-4 i a b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-2 a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+2 b^2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{b \text {$\#$1}-4 i a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\&\right ]}{a^2-b^2}-\frac {12 b \cos (c+d x) (-3 a+a \cos (2 (c+d x))+2 b \sin (c+d x))}{(a-b) (a+b) (4 a+3 b \sin (c+d x)-b \sin (3 (c+d x)))}}{18 a d} \]

[In]

Integrate[(a + b*Sin[c + d*x]^3)^(-2),x]

[Out]

((I*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2*b^2*ArcTan[Sin[c + d*x]/(Cos[c +
d*x] - #1)] - I*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (4*I)*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1
+ 2*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 - 24*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 12*b^2*
ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (12*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (6*I)*b^2*
Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (4*I)*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - 2*a*b*Log[1
- 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 2*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - I*b^2*Log[1 - 2*Cos[
c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ])/(a^2 - b^2) - (12*b*Cos[c + d*x]*(-3*
a + a*Cos[2*(c + d*x)] + 2*b*Sin[c + d*x]))/((a - b)*(a + b)*(4*a + 3*b*Sin[c + d*x] - b*Sin[3*(c + d*x)])))/(
18*a*d)

Maple [N/A] (verified)

Time = 2.88 (sec) , antiderivative size = 349, normalized size of antiderivative = 24.93

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 b^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a \left (a^{2}-b^{2}\right )}-\frac {2 b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-b^{2}\right )}+\frac {8 b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a \left (a^{2}-b^{2}\right )}+\frac {8 b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-b^{2}\right )}-\frac {2 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a \left (a^{2}-b^{2}\right )}+\frac {2 b}{3 a^{2}-3 b^{2}}}{a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\left (3 a^{2}-2 b^{2}\right ) \textit {\_R}^{4}-2 a b \,\textit {\_R}^{3}+6 a^{2} \textit {\_R}^{2}-2 a \textit {\_R} b +3 a^{2}-2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}}{9 a \left (a^{2}-b^{2}\right )}}{d}\)	\(349\)
default	\(\frac {\frac {\frac {2 b^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a \left (a^{2}-b^{2}\right )}-\frac {2 b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-b^{2}\right )}+\frac {8 b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a \left (a^{2}-b^{2}\right )}+\frac {8 b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-b^{2}\right )}-\frac {2 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a \left (a^{2}-b^{2}\right )}+\frac {2 b}{3 a^{2}-3 b^{2}}}{a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\left (3 a^{2}-2 b^{2}\right ) \textit {\_R}^{4}-2 a b \,\textit {\_R}^{3}+6 a^{2} \textit {\_R}^{2}-2 a \textit {\_R} b +3 a^{2}-2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}}{9 a \left (a^{2}-b^{2}\right )}}{d}\)	\(349\)
risch	\(\text {Expression too large to display}\)	\(2453\)

[In]

int(1/(a+b*sin(d*x+c)^3)^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(2*(1/3*b^2/a/(a^2-b^2)*tan(1/2*d*x+1/2*c)^5-1/3/(a^2-b^2)*b*tan(1/2*d*x+1/2*c)^4+4/3*b^2/a/(a^2-b^2)*tan(
1/2*d*x+1/2*c)^3+4/3/(a^2-b^2)*b*tan(1/2*d*x+1/2*c)^2-1/3*b^2/a/(a^2-b^2)*tan(1/2*d*x+1/2*c)+1/3/(a^2-b^2)*b)/
(a*tan(1/2*d*x+1/2*c)^6+3*tan(1/2*d*x+1/2*c)^4*a+8*tan(1/2*d*x+1/2*c)^3*b+3*tan(1/2*d*x+1/2*c)^2*a+a)+1/9/a/(a
^2-b^2)*sum(((3*a^2-2*b^2)*_R^4-2*a*b*_R^3+6*a^2*_R^2-2*a*_R*b+3*a^2-2*b^2)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln
(tan(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a)))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 8.47 (sec) , antiderivative size = 70185, normalized size of antiderivative = 5013.21 \[ \int \frac {1}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate(1/(a+b*sin(d*x+c)^3)^2,x, algorithm="fricas")

[Out]

Too large to include

Sympy [N/A]

Not integrable

Time = 99.86 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\int \frac {1}{\left (a + b \sin ^{3}{\left (c + d x \right )}\right )^{2}}\, dx \]

[In]

integrate(1/(a+b*sin(d*x+c)**3)**2,x)

[Out]

Integral((a + b*sin(c + d*x)**3)**(-2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(1/(a+b*sin(d*x+c)^3)^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [N/A]

Not integrable

Time = 4.34 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.21 \[ \int \frac {1}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\int { \frac {1}{{\left (b \sin \left (d x + c\right )^{3} + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(a+b*sin(d*x+c)^3)^2,x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 17.02 (sec) , antiderivative size = 1567, normalized size of antiderivative = 111.93 \[ \int \frac {1}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Too large to display} \]

[In]

int(1/(a + b*sin(c + d*x)^3)^2,x)

[Out]

symsum(log(- (8192*(80*b^6 - 270*a^2*b^4))/(243*(a^7 + a^3*b^4 - 2*a^5*b^2)) - root(1594323*a^14*b^2*d^6 - 159
4323*a^12*b^4*d^6 + 531441*a^10*b^6*d^6 - 531441*a^16*d^6 - 59049*a^10*b^2*d^4 + 59049*a^8*b^4*d^4 - 177147*a^
12*d^4 + 8019*a^6*b^2*d^2 - 19683*a^8*d^2 + 432*a^2*b^2 - 64*b^4 - 729*a^4, d, k)*((8192*(144*a*b^7 + 648*a^3*
b^5 - 2187*a^5*b^3))/(243*(a^7 + a^3*b^4 - 2*a^5*b^2)) - root(1594323*a^14*b^2*d^6 - 1594323*a^12*b^4*d^6 + 53
1441*a^10*b^6*d^6 - 531441*a^16*d^6 - 59049*a^10*b^2*d^4 + 59049*a^8*b^4*d^4 - 177147*a^12*d^4 + 8019*a^6*b^2*
d^2 - 19683*a^8*d^2 + 432*a^2*b^2 - 64*b^4 - 729*a^4, d, k)*(root(1594323*a^14*b^2*d^6 - 1594323*a^12*b^4*d^6
+ 531441*a^10*b^6*d^6 - 531441*a^16*d^6 - 59049*a^10*b^2*d^4 + 59049*a^8*b^4*d^4 - 177147*a^12*d^4 + 8019*a^6*
b^2*d^2 - 19683*a^8*d^2 + 432*a^2*b^2 - 64*b^4 - 729*a^4, d, k)*((8192*(26973*a^7*b^5 - 20412*a^5*b^7 + 39366*
a^9*b^3))/(243*(a^7 + a^3*b^4 - 2*a^5*b^2)) - root(1594323*a^14*b^2*d^6 - 1594323*a^12*b^4*d^6 + 531441*a^10*b
^6*d^6 - 531441*a^16*d^6 - 59049*a^10*b^2*d^4 + 59049*a^8*b^4*d^4 - 177147*a^12*d^4 + 8019*a^6*b^2*d^2 - 19683
*a^8*d^2 + 432*a^2*b^2 - 64*b^4 - 729*a^4, d, k)*(root(1594323*a^14*b^2*d^6 - 1594323*a^12*b^4*d^6 + 531441*a^
10*b^6*d^6 - 531441*a^16*d^6 - 59049*a^10*b^2*d^4 + 59049*a^8*b^4*d^4 - 177147*a^12*d^4 + 8019*a^6*b^2*d^2 - 1
9683*a^8*d^2 + 432*a^2*b^2 - 64*b^4 - 729*a^4, d, k)*((8192*(236196*a^7*b^9 - 649539*a^9*b^7 + 590490*a^11*b^5
- 177147*a^13*b^3))/(243*(a^7 + a^3*b^4 - 2*a^5*b^2)) + (8192*tan(c/2 + (d*x)/2)*(6561*a^8*b^8 - 13122*a^10*b
^6 + 6561*a^12*b^4))/(27*(a^7 + a^3*b^4 - 2*a^5*b^2))) + (8192*(13122*a^6*b^8 - 85293*a^8*b^6 + 72171*a^10*b^4
))/(243*(a^7 + a^3*b^4 - 2*a^5*b^2)) + (8192*tan(c/2 + (d*x)/2)*(11664*a^5*b^9 - 40824*a^7*b^7 + 37908*a^9*b^5
- 8748*a^11*b^3))/(27*(a^7 + a^3*b^4 - 2*a^5*b^2))) + (8192*tan(c/2 + (d*x)/2)*(3078*a^6*b^6 - 8181*a^8*b^4))
/(27*(a^7 + a^3*b^4 - 2*a^5*b^2))) - (8192*(2592*a^2*b^8 - 11340*a^4*b^6 + 11664*a^6*b^4))/(243*(a^7 + a^3*b^4
- 2*a^5*b^2)) + (8192*tan(c/2 + (d*x)/2)*(1260*a^5*b^5 - 720*a^3*b^7 + 1944*a^7*b^3))/(27*(a^7 + a^3*b^4 - 2*
a^5*b^2))) + (8192*tan(c/2 + (d*x)/2)*(128*b^8 - 688*a^2*b^6 + 1053*a^4*b^4))/(27*(a^7 + a^3*b^4 - 2*a^5*b^2))
) - (8192*tan(c/2 + (d*x)/2)*(32*a*b^5 - 108*a^3*b^3))/(27*(a^7 + a^3*b^4 - 2*a^5*b^2)))*root(1594323*a^14*b^2
*d^6 - 1594323*a^12*b^4*d^6 + 531441*a^10*b^6*d^6 - 531441*a^16*d^6 - 59049*a^10*b^2*d^4 + 59049*a^8*b^4*d^4 -
177147*a^12*d^4 + 8019*a^6*b^2*d^2 - 19683*a^8*d^2 + 432*a^2*b^2 - 64*b^4 - 729*a^4, d, k), k, 1, 6)/d + ((2*
b)/(3*(a^2 - b^2)) + (8*b*tan(c/2 + (d*x)/2)^2)/(3*(a^2 - b^2)) - (2*b*tan(c/2 + (d*x)/2)^4)/(3*(a^2 - b^2)) -
(2*b^2*tan(c/2 + (d*x)/2))/(3*a*(a^2 - b^2)) + (8*b^2*tan(c/2 + (d*x)/2)^3)/(3*a*(a^2 - b^2)) + (2*b^2*tan(c/
2 + (d*x)/2)^5)/(3*a*(a^2 - b^2)))/(d*(a + 3*a*tan(c/2 + (d*x)/2)^2 + 3*a*tan(c/2 + (d*x)/2)^4 + a*tan(c/2 + (
d*x)/2)^6 + 8*b*tan(c/2 + (d*x)/2)^3))