3.4.99 \(\int \frac {\cos ^4(c+d x)}{(a+b \sin ^3(c+d x))^2} \, dx\) [399]

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx &=\int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A] Result contains complex when optimal does not.
time = 0.28, size = 394, normalized size = 15.15 \begin {gather*} \frac {-i \text {RootSum}\left [-i b+3 i b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 i b \text {$\#$1}^4+i b \text {$\#$1}^6\&,\frac {2 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+4 i a \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+2 a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}+12 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-6 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-4 i a \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-2 a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+2 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{b \text {$\#$1}-4 i a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$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} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^4/(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*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*S
in[c + d*x] - b*Sin[3*(c + d*x)]))/(18*a*b*d)

________________________________________________________________________________________

Maple [A] Leaf count of result is larger than twice the leaf count of optimal. \(240\) vs. \(2(25)=50\).
time = 1.64, size = 241, normalized size = 9.27

method	result	size
derivativedivides	\(\frac {\frac {-\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a}+\frac {2 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 b}+\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a}+\frac {4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 b}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a}+\frac {2}{3 b}}{a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {2 \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4} b +\textit {\_R}^{3} a +\textit {\_R} a +b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{9 a b}}{d}\)	\(241\)
default	\(\frac {\frac {-\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a}+\frac {2 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 b}+\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a}+\frac {4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 b}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a}+\frac {2}{3 b}}{a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {2 \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4} b +\textit {\_R}^{3} a +\textit {\_R} a +b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{9 a b}}{d}\)	\(241\)
risch	\(-\frac {2 \left (2 i a \,{\mathrm e}^{4 i \left (d x +c \right )}+b \,{\mathrm e}^{5 i \left (d x +c \right )}+2 i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 a b d \left (b \,{\mathrm e}^{6 i \left (d x +c \right )}-3 b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 b \,{\mathrm e}^{2 i \left (d x +c \right )}-8 i a \,{\mathrm e}^{3 i \left (d x +c \right )}-b \right )}+\left (\munderset {\textit {\_R} =\RootOf \left (531441 a^{10} b^{8} d^{6} \textit {\_Z}^{6}+59049 a^{8} b^{6} d^{4} \textit {\_Z}^{4}+2187 a^{6} b^{4} d^{2} \textit {\_Z}^{2}+a^{6}+15 a^{4} b^{2}+48 a^{2} b^{4}-64 b^{6}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {118098 a^{9} b^{7} d^{5} \textit {\_R}^{5}}{a^{6}-48 a^{2} b^{4}+128 b^{6}}+\left (\frac {6561 i d^{4} b^{5} a^{9}}{a^{6}-48 a^{2} b^{4}+128 b^{6}}+\frac {52488 i d^{4} b^{7} a^{7}}{a^{6}-48 a^{2} b^{4}+128 b^{6}}\right ) \textit {\_R}^{4}+\left (\frac {11664 a^{7} b^{5} d^{3}}{a^{6}-48 a^{2} b^{4}+128 b^{6}}-\frac {11664 d^{3} b^{7} a^{5}}{a^{6}-48 a^{2} b^{4}+128 b^{6}}\right ) \textit {\_R}^{3}+\left (\frac {486 i d^{2} b^{3} a^{7}}{a^{6}-48 a^{2} b^{4}+128 b^{6}}+\frac {3888 i d^{2} b^{5} a^{5}}{a^{6}-48 a^{2} b^{4}+128 b^{6}}\right ) \textit {\_R}^{2}+\left (-\frac {9 d b \,a^{7}}{a^{6}-48 a^{2} b^{4}+128 b^{6}}+\frac {342 a^{5} b^{3} d}{a^{6}-48 a^{2} b^{4}+128 b^{6}}-\frac {576 d \,b^{5} a^{3}}{a^{6}-48 a^{2} b^{4}+128 b^{6}}\right ) \textit {\_R} +\frac {9 i a^{5} b}{a^{6}-48 a^{2} b^{4}+128 b^{6}}+\frac {72 i b^{3} a^{3}}{a^{6}-48 a^{2} b^{4}+128 b^{6}}\right )\right )\)	\(572\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*b*tan(1/2*d*x+1/2*c)
^3+3*a*tan(1/2*d*x+1/2*c)^2+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)))

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [A]
time = 15.99, size = 2431, normalized size = 93.50 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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
^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^3*b^5 + 2
138112*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*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 4
8*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 + 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^5*b^7 + 1134673
92*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 + 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^8*b^6 - 1934917632*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)^5*a^7*b^9 + 1451188224*root(5314
41*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)^5*a^9*b
^7 + 688128*a^3*b^3*tan(c/2 + (d*x)/2) - 1179648*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*b^7 + 12976128*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^2*b^6*tan(c/2 + (d*x)/2) - 6266880*roo
t(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
^4*b^4*tan(c/2 + (d*x)/2) + 737280*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^6*b^2*tan(c/2 + (d*x)/2) - 53084160*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^3*b^7*tan(c/2 + (d*x)/2) + 50
429952*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^5*b^5*tan(c/2 + (d*x)/2) + 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^7*b^3*tan(c/2 + (d*x)/2) - 59719680*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^6*b^6*tan(c/2
+ (d*x)/2) + 5971968*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^8*b^4*tan(c/2 + (d*x)/2) - 859963392*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^5*b^9*tan(c/2 + (d*x)/2) + 859963392*roo
t(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^7*b^7*tan(c/2 + (d*x)/2) - 483729408*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)^5*a^8*b^8*tan(c/2 + (d*x)/2))/(a^3*b^4))*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), k, 1, 6)/d + (8*tan(c/2
+ (d*x)/2)^3)/(3*d*(3*a^2*tan(c/2 + (d*x)/2)^2 + 3*a^2*tan(c/2 + (d*x)/2)^4 + a^2*tan(c/2 + (d*x)/2)^6 + a^2 +
8*a*b*tan(c/2 + (d*x)/2)^3)) - (2*tan(c/2 + (d*x)/2)^5)/(3*d*(3*a^2*tan(c/2 + (d*x)/2)^2 + 3*a^2*tan(c/2 + (d
*x)/2)^4 + a^2*tan(c/2 + (d*x)/2)^6 + a^2 + 8*a*b*tan(c/2 + (d*x)/2)^3)) + (4*tan(c/2 + (d*x)/2)^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)) + (2*tan(c/2 + (d*x)/2)^4)/(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*t
an(c/2 + (d*x)/2)^4 + a*b*tan(c/2 + (d*x)/2)^6)) + (2*tan(c/2 + (d*x)/2))/(3*d*(3*a^2*tan(c/2 + (d*x)/2)^2 + 3
*a^2*tan(c/2 + (d*x)/2)^4 + a^2*tan(c/2 + (d*x)/2)^6 + a^2 + 8*a*b*tan(c/2 + (d*x)/2)^3))

________________________________________________________________________________________