3.4.100 \(\int \frac {\cos ^2(c+d x)}{(a+b \sin ^3(c+d x))^2} \, dx\) [400]

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

[Out]

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

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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 ^2(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]^2/(a + b*Sin[c + d*x]^3)^2,x]

[Out]

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

Rubi steps

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

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Mathematica [A] Result contains complex when optimal does not.
time = 0.18, size = 273, normalized size = 10.50 \begin {gather*} \frac {-i \text {RootSum}\left [-i b+3 i b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 i b \text {$\#$1}^4+i b \text {$\#$1}^6\&,\frac {2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+12 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-6 i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{b \text {$\#$1}-4 i a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\&\right ]+\frac {12 \sin (2 (c+d x))}{4 a+3 b \sin (c+d x)-b \sin (3 (c+d x))}}{18 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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Maple [A] Leaf count of result is larger than twice the leaf count of optimal. \(174\) vs. \(2(25)=50\).
time = 2.01, size = 175, normalized size = 6.73

method	result	size
derivativedivides	\(\frac {\frac {-\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a}}{a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {2 \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}+1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{9 a}}{d}\)	\(175\)
default	\(\frac {\frac {-\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a}}{a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {2 \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}+1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{9 a}}{d}\)	\(175\)
risch	\(-\frac {2 \left ({\mathrm e}^{5 i \left (d x +c \right )}-{\mathrm e}^{i \left (d x +c \right )}\right )}{3 a 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 (\left (531441 a^{12} b^{4} d^{6}-531441 a^{10} b^{6} d^{6}\right ) \textit {\_Z}^{6}+19683 a^{8} b^{4} d^{4} \textit {\_Z}^{4}+729 a^{6} b^{2} d^{2} \textit {\_Z}^{2}+a^{4}-16 a^{2} b^{2}+64 b^{4}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\left (-\frac {59049 d^{5} b^{3} a^{15}}{a^{8}+20 a^{6} b^{2}-304 a^{4} b^{4}+704 a^{2} b^{6}-512 b^{8}}-\frac {59049 d^{5} b^{5} a^{13}}{a^{8}+20 a^{6} b^{2}-304 a^{4} b^{4}+704 a^{2} b^{6}-512 b^{8}}+\frac {590490 d^{5} b^{7} a^{11}}{a^{8}+20 a^{6} b^{2}-304 a^{4} b^{4}+704 a^{2} b^{6}-512 b^{8}}-\frac {472392 d^{5} b^{9} a^{9}}{a^{8}+20 a^{6} b^{2}-304 a^{4} b^{4}+704 a^{2} b^{6}-512 b^{8}}\right ) \textit {\_R}^{5}+\left (\frac {19683 i d^{4} b^{3} a^{13}}{a^{8}+20 a^{6} b^{2}-304 a^{4} b^{4}+704 a^{2} b^{6}-512 b^{8}}-\frac {203391 i d^{4} b^{5} a^{11}}{a^{8}+20 a^{6} b^{2}-304 a^{4} b^{4}+704 a^{2} b^{6}-512 b^{8}}+\frac {393660 i d^{4} b^{7} a^{9}}{a^{8}+20 a^{6} b^{2}-304 a^{4} b^{4}+704 a^{2} b^{6}-512 b^{8}}-\frac {209952 i d^{4} b^{9} a^{7}}{a^{8}+20 a^{6} b^{2}-304 a^{4} b^{4}+704 a^{2} b^{6}-512 b^{8}}\right ) \textit {\_R}^{4}+\left (-\frac {6561 d^{3} b^{3} a^{11}}{a^{8}+20 a^{6} b^{2}-304 a^{4} b^{4}+704 a^{2} b^{6}-512 b^{8}}+\frac {40824 d^{3} b^{5} a^{9}}{a^{8}+20 a^{6} b^{2}-304 a^{4} b^{4}+704 a^{2} b^{6}-512 b^{8}}-\frac {69984 d^{3} b^{7} a^{7}}{a^{8}+20 a^{6} b^{2}-304 a^{4} b^{4}+704 a^{2} b^{6}-512 b^{8}}+\frac {46656 d^{3} b^{9} a^{5}}{a^{8}+20 a^{6} b^{2}-304 a^{4} b^{4}+704 a^{2} b^{6}-512 b^{8}}\right ) \textit {\_R}^{3}+\left (-\frac {81 i d^{2} b \,a^{11}}{a^{8}+20 a^{6} b^{2}-304 a^{4} b^{4}+704 a^{2} b^{6}-512 b^{8}}+\frac {1134 i d^{2} b^{3} a^{9}}{a^{8}+20 a^{6} b^{2}-304 a^{4} b^{4}+704 a^{2} b^{6}-512 b^{8}}-\frac {4536 i d^{2} b^{5} a^{7}}{a^{8}+20 a^{6} b^{2}-304 a^{4} b^{4}+704 a^{2} b^{6}-512 b^{8}}+\frac {5184 i d^{2} b^{7} a^{5}}{a^{8}+20 a^{6} b^{2}-304 a^{4} b^{4}+704 a^{2} b^{6}-512 b^{8}}\right ) \textit {\_R}^{2}+\left (-\frac {63 d b \,a^{9}}{a^{8}+20 a^{6} b^{2}-304 a^{4} b^{4}+704 a^{2} b^{6}-512 b^{8}}-\frac {414 d \,b^{3} a^{7}}{a^{8}+20 a^{6} b^{2}-304 a^{4} b^{4}+704 a^{2} b^{6}-512 b^{8}}+\frac {1656 d \,b^{5} a^{5}}{a^{8}+20 a^{6} b^{2}-304 a^{4} b^{4}+704 a^{2} b^{6}-512 b^{8}}-\frac {1152 d \,b^{7} a^{3}}{a^{8}+20 a^{6} b^{2}-304 a^{4} b^{4}+704 a^{2} b^{6}-512 b^{8}}\right ) \textit {\_R} +\frac {17 i a^{7} b}{a^{8}+20 a^{6} b^{2}-304 a^{4} b^{4}+704 a^{2} b^{6}-512 b^{8}}-\frac {164 i b^{3} a^{5}}{a^{8}+20 a^{6} b^{2}-304 a^{4} b^{4}+704 a^{2} b^{6}-512 b^{8}}+\frac {224 i b^{5} a^{3}}{a^{8}+20 a^{6} b^{2}-304 a^{4} b^{4}+704 a^{2} b^{6}-512 b^{8}}\right )\right )\)	\(1243\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^2/(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/a*tan(1/2*d*x+1/2*c))/(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*sum((_R^4+1)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(ta
n(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)))

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

[Out]

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

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Fricas [A] Result contains complex when optimal does not.
time = 3.28, size = 36403, normalized size = 1400.12 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/324*(3*sqrt(2/3)*sqrt(1/6)*(a^2*d - (a*b*d*cos(d*x + c)^2 - a*b*d)*sin(d*x + c))*sqrt(-((a^4 - a^2*b^2)*((-I
*sqrt(3) + 1)*(3/(a^6*b^2*d^4 - a^4*b^4*d^4) - 1/(a^4*d^2 - a^2*b^2*d^2)^2)/(-1/1062882*(a^4 - 16*a^2*b^2 + 64
*b^4)/(a^12*b^4*d^6 - a^10*b^6*d^6) + 1/118098/((a^6*b^2*d^4 - a^4*b^4*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/53144
1/(a^4*d^2 - a^2*b^2*d^2)^3 + 1/1062882*(a^6 + 28*a^4*b^2 - 80*a^2*b^4 + 64*b^6)/((a^2 - b^2)^2*a^10*b^4*d^6))
^(1/3) - 6561*(I*sqrt(3) + 1)*(-1/1062882*(a^4 - 16*a^2*b^2 + 64*b^4)/(a^12*b^4*d^6 - a^10*b^6*d^6) + 1/118098
/((a^6*b^2*d^4 - a^4*b^4*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/531441/(a^4*d^2 - a^2*b^2*d^2)^3 + 1/1062882*(a^6 +
28*a^4*b^2 - 80*a^2*b^4 + 64*b^6)/((a^2 - b^2)^2*a^10*b^4*d^6))^(1/3) - 162/(a^4*d^2 - a^2*b^2*d^2))*d^2 + 3*
sqrt(1/3)*(a^4 - a^2*b^2)*d^2*sqrt(-((a^8*b^2 - 2*a^6*b^4 + a^4*b^6)*((-I*sqrt(3) + 1)*(3/(a^6*b^2*d^4 - a^4*b
^4*d^4) - 1/(a^4*d^2 - a^2*b^2*d^2)^2)/(-1/1062882*(a^4 - 16*a^2*b^2 + 64*b^4)/(a^12*b^4*d^6 - a^10*b^6*d^6) +
1/118098/ ...

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos ^{2}{\left (c + d x \right )}}{\left (a + b \sin ^{3}{\left (c + d x \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Mupad [A]
time = 15.75, size = 1648, normalized size = 63.38 \begin {gather*} \frac {\sum _{k=1}^6\ln \left (-\frac {-\frac {16384\,a^2}{243}+\frac {131072\,b^2}{243}+\frac {\mathrm {root}\left (531441\,a^{12}\,b^4\,d^6-531441\,a^{10}\,b^6\,d^6+19683\,a^8\,b^4\,d^4+729\,a^6\,b^2\,d^2-16\,a^2\,b^2+a^4+64\,b^4,d,k\right )\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,8192}{27}+\frac {\mathrm {root}\left (531441\,a^{12}\,b^4\,d^6-531441\,a^{10}\,b^6\,d^6+19683\,a^8\,b^4\,d^4+729\,a^6\,b^2\,d^2-16\,a^2\,b^2+a^4+64\,b^4,d,k\right )\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1048576}{27}+\frac {{\mathrm {root}\left (531441\,a^{12}\,b^4\,d^6-531441\,a^{10}\,b^6\,d^6+19683\,a^8\,b^4\,d^4+729\,a^6\,b^2\,d^2-16\,a^2\,b^2+a^4+64\,b^4,d,k\right )}^2\,a^2\,b^4\,262144}{3}-\frac {{\mathrm {root}\left (531441\,a^{12}\,b^4\,d^6-531441\,a^{10}\,b^6\,d^6+19683\,a^8\,b^4\,d^4+729\,a^6\,b^2\,d^2-16\,a^2\,b^2+a^4+64\,b^4,d,k\right )}^2\,a^4\,b^2\,131072}{3}-{\mathrm {root}\left (531441\,a^{12}\,b^4\,d^6-531441\,a^{10}\,b^6\,d^6+19683\,a^8\,b^4\,d^4+729\,a^6\,b^2\,d^2-16\,a^2\,b^2+a^4+64\,b^4,d,k\right )}^3\,a^5\,b^3\,98304+{\mathrm {root}\left (531441\,a^{12}\,b^4\,d^6-531441\,a^{10}\,b^6\,d^6+19683\,a^8\,b^4\,d^4+729\,a^6\,b^2\,d^2-16\,a^2\,b^2+a^4+64\,b^4,d,k\right )}^4\,a^6\,b^4\,442368+{\mathrm {root}\left (531441\,a^{12}\,b^4\,d^6-531441\,a^{10}\,b^6\,d^6+19683\,a^8\,b^4\,d^4+729\,a^6\,b^2\,d^2-16\,a^2\,b^2+a^4+64\,b^4,d,k\right )}^4\,a^8\,b^2\,221184+{\mathrm {root}\left (531441\,a^{12}\,b^4\,d^6-531441\,a^{10}\,b^6\,d^6+19683\,a^8\,b^4\,d^4+729\,a^6\,b^2\,d^2-16\,a^2\,b^2+a^4+64\,b^4,d,k\right )}^5\,a^7\,b^5\,7962624-{\mathrm {root}\left (531441\,a^{12}\,b^4\,d^6-531441\,a^{10}\,b^6\,d^6+19683\,a^8\,b^4\,d^4+729\,a^6\,b^2\,d^2-16\,a^2\,b^2+a^4+64\,b^4,d,k\right )}^5\,a^9\,b^3\,5971968+\frac {\mathrm {root}\left (531441\,a^{12}\,b^4\,d^6-531441\,a^{10}\,b^6\,d^6+19683\,a^8\,b^4\,d^4+729\,a^6\,b^2\,d^2-16\,a^2\,b^2+a^4+64\,b^4,d,k\right )\,a\,b^3\,131072}{27}-\frac {\mathrm {root}\left (531441\,a^{12}\,b^4\,d^6-531441\,a^{10}\,b^6\,d^6+19683\,a^8\,b^4\,d^4+729\,a^6\,b^2\,d^2-16\,a^2\,b^2+a^4+64\,b^4,d,k\right )\,a^3\,b\,65536}{27}-\frac {\mathrm {root}\left (531441\,a^{12}\,b^4\,d^6-531441\,a^{10}\,b^6\,d^6+19683\,a^8\,b^4\,d^4+729\,a^6\,b^2\,d^2-16\,a^2\,b^2+a^4+64\,b^4,d,k\right )\,a^2\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,131072}{9}-\frac {{\mathrm {root}\left (531441\,a^{12}\,b^4\,d^6-531441\,a^{10}\,b^6\,d^6+19683\,a^8\,b^4\,d^4+729\,a^6\,b^2\,d^2-16\,a^2\,b^2+a^4+64\,b^4,d,k\right )}^2\,a^5\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,32768}{3}-\frac {{\mathrm {root}\left (531441\,a^{12}\,b^4\,d^6-531441\,a^{10}\,b^6\,d^6+19683\,a^8\,b^4\,d^4+729\,a^6\,b^2\,d^2-16\,a^2\,b^2+a^4+64\,b^4,d,k\right )}^2\,a^3\,b^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,131072}{3}+{\mathrm {root}\left (531441\,a^{12}\,b^4\,d^6-531441\,a^{10}\,b^6\,d^6+19683\,a^8\,b^4\,d^4+729\,a^6\,b^2\,d^2-16\,a^2\,b^2+a^4+64\,b^4,d,k\right )}^3\,a^6\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,245760+{\mathrm {root}\left (531441\,a^{12}\,b^4\,d^6-531441\,a^{10}\,b^6\,d^6+19683\,a^8\,b^4\,d^4+729\,a^6\,b^2\,d^2-16\,a^2\,b^2+a^4+64\,b^4,d,k\right )}^4\,a^5\,b^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,3538944-{\mathrm {root}\left (531441\,a^{12}\,b^4\,d^6-531441\,a^{10}\,b^6\,d^6+19683\,a^8\,b^4\,d^4+729\,a^6\,b^2\,d^2-16\,a^2\,b^2+a^4+64\,b^4,d,k\right )}^4\,a^7\,b^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,2654208+{\mathrm {root}\left (531441\,a^{12}\,b^4\,d^6-531441\,a^{10}\,b^6\,d^6+19683\,a^8\,b^4\,d^4+729\,a^6\,b^2\,d^2-16\,a^2\,b^2+a^4+64\,b^4,d,k\right )}^5\,a^8\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1990656}{a^3}\right )\,\mathrm {root}\left (531441\,a^{12}\,b^4\,d^6-531441\,a^{10}\,b^6\,d^6+19683\,a^8\,b^4\,d^4+729\,a^6\,b^2\,d^2-16\,a^2\,b^2+a^4+64\,b^4,d,k\right )}{d}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3\,d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^2+8\,b\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}+\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3\,d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^2+8\,b\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log(-((131072*b^2)/243 - (16384*a^2)/243 + (8192*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683
*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)*a^4*tan(c/2 + (d*x)/2))/27 + (1048576*root(5
31441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d,
k)*b^4*tan(c/2 + (d*x)/2))/27 + (262144*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 +
729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)^2*a^2*b^4)/3 - (131072*root(531441*a^12*b^4*d^6 - 531441*a^
10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)^2*a^4*b^2)/3 - 98304*root(
531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d
, k)^3*a^5*b^3 + 442368*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 -
16*a^2*b^2 + a^4 + 64*b^4, d, k)^4*a^6*b^4 + 221184*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^
8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)^4*a^8*b^2 + 7962624*root(531441*a^12*b^4*d^6 -
531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)^5*a^7*b^5 - 59719
68*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 6
4*b^4, d, k)^5*a^9*b^3 + (131072*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*
b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)*a*b^3)/27 - (65536*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 +
19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)*a^3*b)/27 - (131072*root(531441*a^12*b
^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)*a^2*b^2*
tan(c/2 + (d*x)/2))/9 - (32768*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^
2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)^2*a^5*b*tan(c/2 + (d*x)/2))/3 - (131072*root(531441*a^12*b^4*d^6 - 53
1441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)^2*a^3*b^3*tan(c/2 +
(d*x)/2))/3 + 245760*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 1
6*a^2*b^2 + a^4 + 64*b^4, d, k)^3*a^6*b^2*tan(c/2 + (d*x)/2) + 3538944*root(531441*a^12*b^4*d^6 - 531441*a^10*
b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)^4*a^5*b^5*tan(c/2 + (d*x)/2)
- 2654208*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 +
a^4 + 64*b^4, d, k)^4*a^7*b^3*tan(c/2 + (d*x)/2) + 1990656*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19
683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)^5*a^8*b^4*tan(c/2 + (d*x)/2))/a^3)*root(5
31441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d,
k), k, 1, 6)/d - (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)) + (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))

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