\(\int \frac {\cos ^4(c+d x)}{(a+b \sin ^3(c+d x))^2} \, dx\) [332]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Int}\left (\frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2},x\right ) \] Output:

Defer(Int)(cos(d*x+c)^4/(a+b*sin(d*x+c)^3)^2,x)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.05 (sec) , antiderivative size = 394, normalized size of antiderivative = 17.13 \[ \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {-i \text {RootSum}\left [-i b+3 i b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 i b \text {$\#$1}^4+i b \text {$\#$1}^6\&,\frac {2 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+4 i a \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+2 a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}+12 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-6 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-4 i a \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-2 a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+2 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{b \text {$\#$1}-4 i a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\&\right ]+\frac {24 \cos (c+d x) (a+b \sin (c+d x))}{4 a+3 b \sin (c+d x)-b \sin (3 (c+d x))}}{18 a b d} \] Input:

Integrate[Cos[c + d*x]^4/(a + b*Sin[c + d*x]^3)^2,x]
 

Output:

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

Rubi [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {3042, 3707}

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.

\(\displaystyle \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (c+d x)^4}{\left (a+b \sin (c+d x)^3\right )^2}dx\)

\(\Big \downarrow \) 3707

\(\displaystyle \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2}dx\)

Input:

Int[Cos[c + d*x]^4/(a + b*Sin[c + d*x]^3)^2,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3707

Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(m_.)*((a_) + (b_.)*((c_.)*sin[(e_.) + 
(f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> Unintegrable[(d*Cos[e + f*x])^m*(a + 
 b*(c*Sin[e + f*x])^n)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x]
 

Maple [N/A] (verified)

Time = 1.99 (sec) , antiderivative size = 241, normalized size of antiderivative = 10.48

method	result	size
derivativedivides	\(\frac {\frac {-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 a}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{3 b}+\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a}+\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3 b}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a}+\frac {2}{3 b}}{a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+3 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b +3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4} b +\textit {\_R}^{3} a +a \textit {\_R} +b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +a \textit {\_R}}\right )}{9 a b}}{d}\)	\(241\)
default	\(\frac {\frac {-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 a}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{3 b}+\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a}+\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3 b}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a}+\frac {2}{3 b}}{a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+3 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b +3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4} b +\textit {\_R}^{3} a +a \textit {\_R} +b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +a \textit {\_R}}\right )}{9 a b}}{d}\)	\(241\)
risch	\(-\frac {2 \,{\mathrm e}^{i \left (d x +c \right )} \left (2 i a \,{\mathrm e}^{3 i \left (d x +c \right )}+b \,{\mathrm e}^{4 i \left (d x +c \right )}+2 i a \,{\mathrm e}^{i \left (d x +c \right )}-b \right )}{3 a b d \left ({\mathrm e}^{6 i \left (d x +c \right )} b -3 b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 b \,{\mathrm e}^{2 i \left (d x +c \right )}-8 i a \,{\mathrm e}^{3 i \left (d x +c \right )}-b \right )}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (531441 a^{10} b^{8} d^{6} \textit {\_Z}^{6}+59049 a^{8} b^{6} d^{4} \textit {\_Z}^{4}+2187 a^{6} b^{4} d^{2} \textit {\_Z}^{2}+a^{6}+15 a^{4} b^{2}+48 b^{4} a^{2}-64 b^{6}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {118098 a^{9} b^{7} d^{5} \textit {\_R}^{5}}{a^{6}-48 b^{4} a^{2}+128 b^{6}}+\left (\frac {6561 i d^{4} b^{5} a^{9}}{a^{6}-48 b^{4} a^{2}+128 b^{6}}+\frac {52488 i d^{4} b^{7} a^{7}}{a^{6}-48 b^{4} a^{2}+128 b^{6}}\right ) \textit {\_R}^{4}+\left (\frac {11664 a^{7} b^{5} d^{3}}{a^{6}-48 b^{4} a^{2}+128 b^{6}}-\frac {11664 d^{3} b^{7} a^{5}}{a^{6}-48 b^{4} a^{2}+128 b^{6}}\right ) \textit {\_R}^{3}+\left (\frac {486 i d^{2} b^{3} a^{7}}{a^{6}-48 b^{4} a^{2}+128 b^{6}}+\frac {3888 i d^{2} b^{5} a^{5}}{a^{6}-48 b^{4} a^{2}+128 b^{6}}\right ) \textit {\_R}^{2}+\left (-\frac {9 d b \,a^{7}}{a^{6}-48 b^{4} a^{2}+128 b^{6}}+\frac {342 a^{5} b^{3} d}{a^{6}-48 b^{4} a^{2}+128 b^{6}}-\frac {576 d \,b^{5} a^{3}}{a^{6}-48 b^{4} a^{2}+128 b^{6}}\right ) \textit {\_R} +\frac {9 i b \,a^{5}}{a^{6}-48 b^{4} a^{2}+128 b^{6}}+\frac {72 i b^{3} a^{3}}{a^{6}-48 b^{4} a^{2}+128 b^{6}}\right )\right )\)	\(584\)

Input:

int(cos(d*x+c)^4/(a+b*sin(d*x+c)^3)^2,x,method=_RETURNVERBOSE)
 

Output:

1/d*(2*(-1/3/a*tan(1/2*d*x+1/2*c)^5+1/3/b*tan(1/2*d*x+1/2*c)^4+4/3/a*tan(1 
/2*d*x+1/2*c)^3+2/3/b*tan(1/2*d*x+1/2*c)^2+1/3/a*tan(1/2*d*x+1/2*c)+1/3/b) 
/(a*tan(1/2*d*x+1/2*c)^6+3*a*tan(1/2*d*x+1/2*c)^4+8*tan(1/2*d*x+1/2*c)^3*b 
+3*tan(1/2*d*x+1/2*c)^2*a+a)+2/9/a/b*sum((_R^4*b+_R^3*a+_R*a+b)/(_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 = 8.86 (sec) , antiderivative size = 9984, normalized size of antiderivative = 434.09 \[ \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(cos(d*x+c)^4/(a+b*sin(d*x+c)^3)^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [N/A]

Not integrable

Time = 140.69 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\int \frac {\cos ^{4}{\left (c + d x \right )}}{\left (a + b \sin ^{3}{\left (c + d x \right )}\right )^{2}}\, dx \] Input:

integrate(cos(d*x+c)**4/(a+b*sin(d*x+c)**3)**2,x)
 

Output:

Integral(cos(c + d*x)**4/(a + b*sin(c + d*x)**3)**2, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(cos(d*x+c)^4/(a+b*sin(d*x+c)^3)^2,x, algorithm="maxima")
 

Output:

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

Giac [N/A]

Not integrable

Time = 3.35 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.13 \[ \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\int { \frac {\cos \left (d x + c\right )^{4}}{{\left (b \sin \left (d x + c\right )^{3} + a\right )}^{2}} \,d x } \] Input:

integrate(cos(d*x+c)^4/(a+b*sin(d*x+c)^3)^2,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

sage0*x
 

Mupad [B] (verification not implemented)

Time = 37.13 (sec) , antiderivative size = 2431, normalized size of antiderivative = 105.70 \[ \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:

int(cos(c + d*x)^4/(a + b*sin(c + d*x)^3)^2,x)
 

Output:

2/(3*d*(a*b + 8*b^2*tan(c/2 + (d*x)/2)^3 + 3*a*b*tan(c/2 + (d*x)/2)^2 + 3* 
a*b*tan(c/2 + (d*x)/2)^4 + a*b*tan(c/2 + (d*x)/2)^6)) + symsum(log((638976 
*a^2*b^4 - 655360*b^6 - 8192*a^6 + 24576*a^4*b^2 - 2949120*root(531441*a^1 
0*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 
 + a^6 - 64*b^6, d, k)*a^3*b^5 + 2138112*root(531441*a^10*b^8*d^6 + 59049* 
a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d 
, k)*a^5*b^3 - 9437184*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187 
*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)*b^8*tan(c/2 + 
 (d*x)/2) - 786432*a*b^5*tan(c/2 + (d*x)/2) + 98304*a^5*b*tan(c/2 + (d*x)/ 
2) - 21233664*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4* 
d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)^2*a^2*b^8 + 18579456*r 
oot(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^ 
4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)^2*a^4*b^6 + 2654208*root(531441*a^10* 
b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + 
 a^6 - 64*b^6, d, k)^2*a^6*b^4 - 167215104*root(531441*a^10*b^8*d^6 + 5904 
9*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, 
 d, k)^3*a^5*b^7 + 113467392*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 
+ 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)^3*a^7*b 
^5 - 107495424*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4 
*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)^4*a^6*b^8 + 107495...
 

Reduce [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 5459, normalized size of antiderivative = 237.35 \[ \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:

int(cos(d*x+c)^4/(a+b*sin(d*x+c)^3)^2,x)
 

Output:

( - 72*cos(c + d*x)*sin(c + d*x)**2*a**2*b**2 - 152*cos(c + d*x)*sin(c + d 
*x)**2*b**4 - 18*cos(c + d*x)*sin(c + d*x)*a**3*b - 36*cos(c + d*x)*sin(c 
+ d*x)*a*b**3 + 72*cos(c + d*x)*a**2*b**2 + 128*cos(c + d*x)*b**4 + 96*int 
(tan((c + d*x)/2)**4/(tan((c + d*x)/2)**12*a**2 + 6*tan((c + d*x)/2)**10*a 
**2 + 16*tan((c + d*x)/2)**9*a*b + 15*tan((c + d*x)/2)**8*a**2 + 48*tan((c 
 + d*x)/2)**7*a*b + 20*tan((c + d*x)/2)**6*a**2 + 64*tan((c + d*x)/2)**6*b 
**2 + 48*tan((c + d*x)/2)**5*a*b + 15*tan((c + d*x)/2)**4*a**2 + 16*tan((c 
 + d*x)/2)**3*a*b + 6*tan((c + d*x)/2)**2*a**2 + a**2),x)*sin(c + d*x)**3* 
a**4*b**2*d + 576*int(tan((c + d*x)/2)**4/(tan((c + d*x)/2)**12*a**2 + 6*t 
an((c + d*x)/2)**10*a**2 + 16*tan((c + d*x)/2)**9*a*b + 15*tan((c + d*x)/2 
)**8*a**2 + 48*tan((c + d*x)/2)**7*a*b + 20*tan((c + d*x)/2)**6*a**2 + 64* 
tan((c + d*x)/2)**6*b**2 + 48*tan((c + d*x)/2)**5*a*b + 15*tan((c + d*x)/2 
)**4*a**2 + 16*tan((c + d*x)/2)**3*a*b + 6*tan((c + d*x)/2)**2*a**2 + a**2 
),x)*sin(c + d*x)**3*a**2*b**4*d - 768*int(tan((c + d*x)/2)**4/(tan((c + d 
*x)/2)**12*a**2 + 6*tan((c + d*x)/2)**10*a**2 + 16*tan((c + d*x)/2)**9*a*b 
 + 15*tan((c + d*x)/2)**8*a**2 + 48*tan((c + d*x)/2)**7*a*b + 20*tan((c + 
d*x)/2)**6*a**2 + 64*tan((c + d*x)/2)**6*b**2 + 48*tan((c + d*x)/2)**5*a*b 
 + 15*tan((c + d*x)/2)**4*a**2 + 16*tan((c + d*x)/2)**3*a*b + 6*tan((c + d 
*x)/2)**2*a**2 + a**2),x)*sin(c + d*x)**3*b**6*d + 96*int(tan((c + d*x)/2) 
**4/(tan((c + d*x)/2)**12*a**2 + 6*tan((c + d*x)/2)**10*a**2 + 16*tan((...