\(\int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx\) [585]

Optimal result

Integrand size = 21, antiderivative size = 310 \[ \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {5 b^4 \left (6 a^2-b^2\right ) \text {arctanh}\left (\frac {b-a \tan (c+d x)}{\sqrt {a^2+b^2} \sqrt {\sec ^2(c+d x)}}\right ) \sec (c+d x)}{2 \left (a^2+b^2\right )^{9/2} d \sqrt {\sec ^2(c+d x)}}+\frac {b \left (4 a^4+24 a^2 b^2-15 b^4\right ) \sec (c+d x)}{6 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^2}+\frac {\cos ^3(c+d x) (b+a \tan (c+d x))}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a b \left (4 a^4+28 a^2 b^2-81 b^4\right ) \sec (c+d x)}{6 \left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}-\frac {\cos (c+d x) \left (b \left (2 a^2-5 b^2\right )-a \left (2 a^2+9 b^2\right ) \tan (c+d x)\right )}{3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2} \] Output:

-5/2*b^4*(6*a^2-b^2)*arctanh((b-a*tan(d*x+c))/(a^2+b^2)^(1/2)/(sec(d*x+c)^ 
2)^(1/2))*sec(d*x+c)/(a^2+b^2)^(9/2)/d/(sec(d*x+c)^2)^(1/2)+1/6*b*(4*a^4+2 
4*a^2*b^2-15*b^4)*sec(d*x+c)/(a^2+b^2)^3/d/(a+b*tan(d*x+c))^2+1/3*cos(d*x+ 
c)^3*(b+a*tan(d*x+c))/(a^2+b^2)/d/(a+b*tan(d*x+c))^2+1/6*a*b*(4*a^4+28*a^2 
*b^2-81*b^4)*sec(d*x+c)/(a^2+b^2)^4/d/(a+b*tan(d*x+c))-1/3*cos(d*x+c)*(b*( 
2*a^2-5*b^2)-a*(2*a^2+9*b^2)*tan(d*x+c))/(a^2+b^2)^2/d/(a+b*tan(d*x+c))^2
 

Mathematica [A] (verified)

Time = 1.74 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.20 \[ \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left (\frac {9 b \left (a^4+14 a^2 b^2-3 b^4\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{\left (a^2+b^2\right )^4}+\frac {6 b^6 \tan (c+d x)}{a \left (a^2+b^2\right )^3}+\frac {9 a \left (a^4+6 a^2 b^2-11 b^4\right ) (a \cos (c+d x)+b \sin (c+d x))^2 \tan (c+d x)}{\left (a^2+b^2\right )^4}-\frac {6 b^5 \left (12 a^2+b^2\right ) (a+b \tan (c+d x))}{a \left (a^2+b^2\right )^4}-\frac {60 b^4 \left (-6 a^2+b^2\right ) \text {arctanh}\left (\frac {-b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right ) \cos (c+d x) (a+b \tan (c+d x))^2}{\left (a^2+b^2\right )^{9/2}}-\frac {b \left (-3 a^2+b^2\right ) \cos (c+d x) \cos (3 (c+d x)) (a+b \tan (c+d x))^2}{\left (a^2+b^2\right )^3}+\frac {a \left (a^2-3 b^2\right ) \cos (c+d x) \sin (3 (c+d x)) (a+b \tan (c+d x))^2}{\left (a^2+b^2\right )^3}\right )}{12 d (a+b \tan (c+d x))^3} \] Input:

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

Output:

(Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])*((9*b*(a^4 + 14*a^2*b^2 
- 3*b^4)*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(a^2 + b^2)^4 + (6*b^6*Tan[c 
 + d*x])/(a*(a^2 + b^2)^3) + (9*a*(a^4 + 6*a^2*b^2 - 11*b^4)*(a*Cos[c + d* 
x] + b*Sin[c + d*x])^2*Tan[c + d*x])/(a^2 + b^2)^4 - (6*b^5*(12*a^2 + b^2) 
*(a + b*Tan[c + d*x]))/(a*(a^2 + b^2)^4) - (60*b^4*(-6*a^2 + b^2)*ArcTanh[ 
(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]]*Cos[c + d*x]*(a + b*Tan[c + d*x 
])^2)/(a^2 + b^2)^(9/2) - (b*(-3*a^2 + b^2)*Cos[c + d*x]*Cos[3*(c + d*x)]* 
(a + b*Tan[c + d*x])^2)/(a^2 + b^2)^3 + (a*(a^2 - 3*b^2)*Cos[c + d*x]*Sin[ 
3*(c + d*x)]*(a + b*Tan[c + d*x])^2)/(a^2 + b^2)^3))/(12*d*(a + b*Tan[c + 
d*x])^3)
 

Rubi [A] (warning: unable to verify)

Time = 0.64 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.16, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3042, 3992, 496, 25, 27, 686, 25, 25, 27, 688, 25, 27, 679, 488, 219}

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.

\(\displaystyle \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sec (c+d x)^3 (a+b \tan (c+d x))^3}dx\)

\(\Big \downarrow \) 3992

\(\displaystyle \frac {\sec (c+d x) \int \frac {1}{(a+b \tan (c+d x))^3 \left (\tan ^2(c+d x)+1\right )^{5/2}}d(b \tan (c+d x))}{b d \sqrt {\sec ^2(c+d x)}}\)

\(\Big \downarrow \) 496

\(\displaystyle \frac {\sec (c+d x) \left (\frac {a b \tan (c+d x)+b^2}{3 \left (a^2+b^2\right ) \left (\tan ^2(c+d x)+1\right )^{3/2} (a+b \tan (c+d x))^2}-\frac {b^2 \int -\frac {\left (\frac {2 a^2}{b^2}+5\right ) b^2+4 a \tan (c+d x) b}{b^2 (a+b \tan (c+d x))^3 \left (\tan ^2(c+d x)+1\right )^{3/2}}d(b \tan (c+d x))}{3 \left (a^2+b^2\right )}\right )}{b d \sqrt {\sec ^2(c+d x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sec (c+d x) \left (\frac {b^2 \int \frac {2 a^2+4 b \tan (c+d x) a+5 b^2}{b^2 (a+b \tan (c+d x))^3 \left (\tan ^2(c+d x)+1\right )^{3/2}}d(b \tan (c+d x))}{3 \left (a^2+b^2\right )}+\frac {a b \tan (c+d x)+b^2}{3 \left (a^2+b^2\right ) \left (\tan ^2(c+d x)+1\right )^{3/2} (a+b \tan (c+d x))^2}\right )}{b d \sqrt {\sec ^2(c+d x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sec (c+d x) \left (\frac {\int \frac {2 a^2+4 b \tan (c+d x) a+5 b^2}{(a+b \tan (c+d x))^3 \left (\tan ^2(c+d x)+1\right )^{3/2}}d(b \tan (c+d x))}{3 \left (a^2+b^2\right )}+\frac {a b \tan (c+d x)+b^2}{3 \left (a^2+b^2\right ) \left (\tan ^2(c+d x)+1\right )^{3/2} (a+b \tan (c+d x))^2}\right )}{b d \sqrt {\sec ^2(c+d x)}}\)

\(\Big \downarrow \) 686

\(\displaystyle \frac {\sec (c+d x) \left (\frac {\frac {a b \left (2 a^2+9 b^2\right ) \tan (c+d x)+b^4 \left (5-\frac {2 a^2}{b^2}\right )}{\left (a^2+b^2\right ) \sqrt {\tan ^2(c+d x)+1} (a+b \tan (c+d x))^2}-\frac {b^4 \int -\frac {3 \left (5-\frac {2 a^2}{b^2}\right ) b^4+2 a \left (2 a^2+9 b^2\right ) \tan (c+d x) b}{b^4 (a+b \tan (c+d x))^3 \sqrt {\tan ^2(c+d x)+1}}d(b \tan (c+d x))}{a^2+b^2}}{3 \left (a^2+b^2\right )}+\frac {a b \tan (c+d x)+b^2}{3 \left (a^2+b^2\right ) \left (\tan ^2(c+d x)+1\right )^{3/2} (a+b \tan (c+d x))^2}\right )}{b d \sqrt {\sec ^2(c+d x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sec (c+d x) \left (\frac {\frac {b^4 \int -\frac {3 b^2 \left (2 a^2-5 b^2\right )-2 a b \left (2 a^2+9 b^2\right ) \tan (c+d x)}{b^4 (a+b \tan (c+d x))^3 \sqrt {\tan ^2(c+d x)+1}}d(b \tan (c+d x))}{a^2+b^2}+\frac {a b \left (2 a^2+9 b^2\right ) \tan (c+d x)+b^4 \left (5-\frac {2 a^2}{b^2}\right )}{\left (a^2+b^2\right ) \sqrt {\tan ^2(c+d x)+1} (a+b \tan (c+d x))^2}}{3 \left (a^2+b^2\right )}+\frac {a b \tan (c+d x)+b^2}{3 \left (a^2+b^2\right ) \left (\tan ^2(c+d x)+1\right )^{3/2} (a+b \tan (c+d x))^2}\right )}{b d \sqrt {\sec ^2(c+d x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sec (c+d x) \left (\frac {\frac {a b \left (2 a^2+9 b^2\right ) \tan (c+d x)+b^4 \left (5-\frac {2 a^2}{b^2}\right )}{\left (a^2+b^2\right ) \sqrt {\tan ^2(c+d x)+1} (a+b \tan (c+d x))^2}-\frac {b^4 \int \frac {3 b^2 \left (2 a^2-5 b^2\right )-2 a b \left (2 a^2+9 b^2\right ) \tan (c+d x)}{b^4 (a+b \tan (c+d x))^3 \sqrt {\tan ^2(c+d x)+1}}d(b \tan (c+d x))}{a^2+b^2}}{3 \left (a^2+b^2\right )}+\frac {a b \tan (c+d x)+b^2}{3 \left (a^2+b^2\right ) \left (\tan ^2(c+d x)+1\right )^{3/2} (a+b \tan (c+d x))^2}\right )}{b d \sqrt {\sec ^2(c+d x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sec (c+d x) \left (\frac {\frac {a b \left (2 a^2+9 b^2\right ) \tan (c+d x)+b^4 \left (5-\frac {2 a^2}{b^2}\right )}{\left (a^2+b^2\right ) \sqrt {\tan ^2(c+d x)+1} (a+b \tan (c+d x))^2}-\frac {\int \frac {3 b^2 \left (2 a^2-5 b^2\right )-2 a b \left (2 a^2+9 b^2\right ) \tan (c+d x)}{(a+b \tan (c+d x))^3 \sqrt {\tan ^2(c+d x)+1}}d(b \tan (c+d x))}{a^2+b^2}}{3 \left (a^2+b^2\right )}+\frac {a b \tan (c+d x)+b^2}{3 \left (a^2+b^2\right ) \left (\tan ^2(c+d x)+1\right )^{3/2} (a+b \tan (c+d x))^2}\right )}{b d \sqrt {\sec ^2(c+d x)}}\)

\(\Big \downarrow \) 688

\(\displaystyle \frac {\sec (c+d x) \left (\frac {\frac {a b \left (2 a^2+9 b^2\right ) \tan (c+d x)+b^4 \left (5-\frac {2 a^2}{b^2}\right )}{\left (a^2+b^2\right ) \sqrt {\tan ^2(c+d x)+1} (a+b \tan (c+d x))^2}-\frac {-\frac {b^2 \int -\frac {2 a b^2 \left (2 a^2-33 b^2\right )-b \left (4 a^4+24 b^2 a^2-15 b^4\right ) \tan (c+d x)}{b^2 (a+b \tan (c+d x))^2 \sqrt {\tan ^2(c+d x)+1}}d(b \tan (c+d x))}{2 \left (a^2+b^2\right )}-\frac {b^2 \left (4 a^4+24 a^2 b^2-15 b^4\right ) \sqrt {\tan ^2(c+d x)+1}}{2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a^2+b^2}}{3 \left (a^2+b^2\right )}+\frac {a b \tan (c+d x)+b^2}{3 \left (a^2+b^2\right ) \left (\tan ^2(c+d x)+1\right )^{3/2} (a+b \tan (c+d x))^2}\right )}{b d \sqrt {\sec ^2(c+d x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sec (c+d x) \left (\frac {\frac {a b \left (2 a^2+9 b^2\right ) \tan (c+d x)+b^4 \left (5-\frac {2 a^2}{b^2}\right )}{\left (a^2+b^2\right ) \sqrt {\tan ^2(c+d x)+1} (a+b \tan (c+d x))^2}-\frac {\frac {b^2 \int \frac {2 a b^2 \left (2 a^2-33 b^2\right )-b \left (4 a^4+24 b^2 a^2-15 b^4\right ) \tan (c+d x)}{b^2 (a+b \tan (c+d x))^2 \sqrt {\tan ^2(c+d x)+1}}d(b \tan (c+d x))}{2 \left (a^2+b^2\right )}-\frac {b^2 \left (4 a^4+24 a^2 b^2-15 b^4\right ) \sqrt {\tan ^2(c+d x)+1}}{2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a^2+b^2}}{3 \left (a^2+b^2\right )}+\frac {a b \tan (c+d x)+b^2}{3 \left (a^2+b^2\right ) \left (\tan ^2(c+d x)+1\right )^{3/2} (a+b \tan (c+d x))^2}\right )}{b d \sqrt {\sec ^2(c+d x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sec (c+d x) \left (\frac {\frac {a b \left (2 a^2+9 b^2\right ) \tan (c+d x)+b^4 \left (5-\frac {2 a^2}{b^2}\right )}{\left (a^2+b^2\right ) \sqrt {\tan ^2(c+d x)+1} (a+b \tan (c+d x))^2}-\frac {\frac {\int \frac {2 a b^2 \left (2 a^2-33 b^2\right )-b \left (4 a^4+24 b^2 a^2-15 b^4\right ) \tan (c+d x)}{(a+b \tan (c+d x))^2 \sqrt {\tan ^2(c+d x)+1}}d(b \tan (c+d x))}{2 \left (a^2+b^2\right )}-\frac {b^2 \left (4 a^4+24 a^2 b^2-15 b^4\right ) \sqrt {\tan ^2(c+d x)+1}}{2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a^2+b^2}}{3 \left (a^2+b^2\right )}+\frac {a b \tan (c+d x)+b^2}{3 \left (a^2+b^2\right ) \left (\tan ^2(c+d x)+1\right )^{3/2} (a+b \tan (c+d x))^2}\right )}{b d \sqrt {\sec ^2(c+d x)}}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {\sec (c+d x) \left (\frac {\frac {a b \left (2 a^2+9 b^2\right ) \tan (c+d x)+b^4 \left (5-\frac {2 a^2}{b^2}\right )}{\left (a^2+b^2\right ) \sqrt {\tan ^2(c+d x)+1} (a+b \tan (c+d x))^2}-\frac {\frac {-\frac {15 b^4 \left (6 a^2-b^2\right ) \int \frac {1}{(a+b \tan (c+d x)) \sqrt {\tan ^2(c+d x)+1}}d(b \tan (c+d x))}{a^2+b^2}-\frac {a b^2 \left (4 a^4+28 a^2 b^2-81 b^4\right ) \sqrt {\tan ^2(c+d x)+1}}{\left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 \left (a^2+b^2\right )}-\frac {b^2 \left (4 a^4+24 a^2 b^2-15 b^4\right ) \sqrt {\tan ^2(c+d x)+1}}{2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a^2+b^2}}{3 \left (a^2+b^2\right )}+\frac {a b \tan (c+d x)+b^2}{3 \left (a^2+b^2\right ) \left (\tan ^2(c+d x)+1\right )^{3/2} (a+b \tan (c+d x))^2}\right )}{b d \sqrt {\sec ^2(c+d x)}}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {\sec (c+d x) \left (\frac {\frac {a b \left (2 a^2+9 b^2\right ) \tan (c+d x)+b^4 \left (5-\frac {2 a^2}{b^2}\right )}{\left (a^2+b^2\right ) \sqrt {\tan ^2(c+d x)+1} (a+b \tan (c+d x))^2}-\frac {\frac {\frac {15 b^4 \left (6 a^2-b^2\right ) \int \frac {1}{\frac {a^2}{b^2}-b^2 \tan ^2(c+d x)+1}d\frac {1-\frac {a \tan (c+d x)}{b}}{\sqrt {\tan ^2(c+d x)+1}}}{a^2+b^2}-\frac {a b^2 \left (4 a^4+28 a^2 b^2-81 b^4\right ) \sqrt {\tan ^2(c+d x)+1}}{\left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 \left (a^2+b^2\right )}-\frac {b^2 \left (4 a^4+24 a^2 b^2-15 b^4\right ) \sqrt {\tan ^2(c+d x)+1}}{2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a^2+b^2}}{3 \left (a^2+b^2\right )}+\frac {a b \tan (c+d x)+b^2}{3 \left (a^2+b^2\right ) \left (\tan ^2(c+d x)+1\right )^{3/2} (a+b \tan (c+d x))^2}\right )}{b d \sqrt {\sec ^2(c+d x)}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sec (c+d x) \left (\frac {a b \tan (c+d x)+b^2}{3 \left (a^2+b^2\right ) \left (\tan ^2(c+d x)+1\right )^{3/2} (a+b \tan (c+d x))^2}+\frac {\frac {a b \left (2 a^2+9 b^2\right ) \tan (c+d x)+b^4 \left (5-\frac {2 a^2}{b^2}\right )}{\left (a^2+b^2\right ) \sqrt {\tan ^2(c+d x)+1} (a+b \tan (c+d x))^2}-\frac {\frac {\frac {15 b^5 \left (6 a^2-b^2\right ) \text {arctanh}\left (\frac {b^2 \tan (c+d x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {a b^2 \left (4 a^4+28 a^2 b^2-81 b^4\right ) \sqrt {\tan ^2(c+d x)+1}}{\left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 \left (a^2+b^2\right )}-\frac {b^2 \left (4 a^4+24 a^2 b^2-15 b^4\right ) \sqrt {\tan ^2(c+d x)+1}}{2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}}{a^2+b^2}}{3 \left (a^2+b^2\right )}\right )}{b d \sqrt {\sec ^2(c+d x)}}\)

Input:

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

Output:

(Sec[c + d*x]*((b^2 + a*b*Tan[c + d*x])/(3*(a^2 + b^2)*(a + b*Tan[c + d*x] 
)^2*(1 + Tan[c + d*x]^2)^(3/2)) + (((5 - (2*a^2)/b^2)*b^4 + a*b*(2*a^2 + 9 
*b^2)*Tan[c + d*x])/((a^2 + b^2)*(a + b*Tan[c + d*x])^2*Sqrt[1 + Tan[c + d 
*x]^2]) - (-1/2*(b^2*(4*a^4 + 24*a^2*b^2 - 15*b^4)*Sqrt[1 + Tan[c + d*x]^2 
])/((a^2 + b^2)*(a + b*Tan[c + d*x])^2) + ((15*b^5*(6*a^2 - b^2)*ArcTanh[( 
b^2*Tan[c + d*x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2) - (a*b^2*(4*a^4 + 28 
*a^2*b^2 - 81*b^4)*Sqrt[1 + Tan[c + d*x]^2])/((a^2 + b^2)*(a + b*Tan[c + d 
*x])))/(2*(a^2 + b^2)))/(a^2 + b^2))/(3*(a^2 + b^2))))/(b*d*Sqrt[Sec[c + d 
*x]^2])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 488

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, x]
 

rule 496

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-(a*d + b*c*x))*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1)*(b*c^2 
 + a*d^2))), x] + Simp[1/(2*a*(p + 1)*(b*c^2 + a*d^2))   Int[(c + d*x)^n*(a 
 + b*x^2)^(p + 1)*Simp[b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3) + b*c*d*(n + 2 
*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && IntQuad 
raticQ[a, 0, b, c, d, n, p, x]
 

rule 679

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 
)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) 
 Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, 
 p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 686

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + 
a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 
1/(2*a*c*(p + 1)*(c*d^2 + a*e^2))   Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim 
p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f 
+ a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ 
[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 688

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( 
(m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2))   Int[(d + 
 e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m 
 + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] 
&& (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 3042

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

rule 3992

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n 
_), x_Symbol] :> Simp[Sec[e + f*x]/(b*f*Sqrt[Sec[e + f*x]^2])   Subst[Int[( 
a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b 
, e, f, n}, x] && NeQ[a^2 + b^2, 0] && IntegerQ[(m - 1)/2]
 

Maple [A] (verified)

Time = 24.19 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.47

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (\left (-a^{5}-4 a^{3} b^{2}+9 a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-3 a^{4} b -12 a^{2} b^{3}+3 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-\frac {2}{3} a^{5}-\frac {32}{3} a^{3} b^{2}+14 a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-20 a^{2} b^{3}+4 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-a^{5}-4 a^{3} b^{2}+9 a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a^{4} b -\frac {32 a^{2} b^{3}}{3}+\frac {7 b^{5}}{3}\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-\frac {2 b^{4} \left (\frac {-\frac {b^{2} \left (13 a^{2}+2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a}-\frac {b \left (12 a^{4}-23 b^{2} a^{2}-2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a^{2}}+\frac {b^{2} \left (35 a^{2}+2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}+6 a^{2} b +\frac {b^{3}}{2}}{\left (a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )^{2}}-\frac {5 \left (6 a^{2}-b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right ) \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}}{d}\)	\(457\)
default	\(\frac {-\frac {2 \left (\left (-a^{5}-4 a^{3} b^{2}+9 a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-3 a^{4} b -12 a^{2} b^{3}+3 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-\frac {2}{3} a^{5}-\frac {32}{3} a^{3} b^{2}+14 a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-20 a^{2} b^{3}+4 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-a^{5}-4 a^{3} b^{2}+9 a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a^{4} b -\frac {32 a^{2} b^{3}}{3}+\frac {7 b^{5}}{3}\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-\frac {2 b^{4} \left (\frac {-\frac {b^{2} \left (13 a^{2}+2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a}-\frac {b \left (12 a^{4}-23 b^{2} a^{2}-2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a^{2}}+\frac {b^{2} \left (35 a^{2}+2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}+6 a^{2} b +\frac {b^{3}}{2}}{\left (a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )^{2}}-\frac {5 \left (6 a^{2}-b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right ) \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}}{d}\)	\(457\)
risch	\(-\frac {i {\mathrm e}^{3 i \left (d x +c \right )}}{24 \left (-3 i a^{2} b +i b^{3}+a^{3}-3 a \,b^{2}\right ) d}-\frac {9 \,{\mathrm e}^{i \left (d x +c \right )} b}{8 \left (-4 i a^{3} b +4 i a \,b^{3}+a^{4}-6 b^{2} a^{2}+b^{4}\right ) d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} a}{8 \left (-4 i a^{3} b +4 i a \,b^{3}+a^{4}-6 b^{2} a^{2}+b^{4}\right ) d}-\frac {9 \,{\mathrm e}^{-i \left (d x +c \right )} b}{8 \left (i b +a \right )^{2} \left (2 i a b +a^{2}-b^{2}\right ) d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} a}{8 \left (i b +a \right )^{2} \left (2 i a b +a^{2}-b^{2}\right ) d}+\frac {i {\mathrm e}^{-3 i \left (d x +c \right )}}{24 \left (2 i a b +a^{2}-b^{2}\right ) \left (i b +a \right ) d}+\frac {b^{5} {\mathrm e}^{i \left (d x +c \right )} \left (-11 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}+12 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+11 i a b +12 a^{2}+b^{2}\right )}{\left (-i a +b \right )^{4} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{2} d \left (i a +b \right )^{4}}+\frac {15 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{9}+4 i a^{7} b^{2}+6 i a^{5} b^{4}+4 i a^{3} b^{6}+i a \,b^{8}-a^{8} b -4 a^{6} b^{3}-6 a^{4} b^{5}-4 a^{2} b^{7}-b^{9}}{\left (a^{2}+b^{2}\right )^{\frac {9}{2}}}\right ) a^{2}}{\left (a^{2}+b^{2}\right )^{\frac {9}{2}} d}-\frac {5 b^{6} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{9}+4 i a^{7} b^{2}+6 i a^{5} b^{4}+4 i a^{3} b^{6}+i a \,b^{8}-a^{8} b -4 a^{6} b^{3}-6 a^{4} b^{5}-4 a^{2} b^{7}-b^{9}}{\left (a^{2}+b^{2}\right )^{\frac {9}{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {9}{2}} d}-\frac {15 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{9}+4 i a^{7} b^{2}+6 i a^{5} b^{4}+4 i a^{3} b^{6}+i a \,b^{8}-a^{8} b -4 a^{6} b^{3}-6 a^{4} b^{5}-4 a^{2} b^{7}-b^{9}}{\left (a^{2}+b^{2}\right )^{\frac {9}{2}}}\right ) a^{2}}{\left (a^{2}+b^{2}\right )^{\frac {9}{2}} d}+\frac {5 b^{6} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{9}+4 i a^{7} b^{2}+6 i a^{5} b^{4}+4 i a^{3} b^{6}+i a \,b^{8}-a^{8} b -4 a^{6} b^{3}-6 a^{4} b^{5}-4 a^{2} b^{7}-b^{9}}{\left (a^{2}+b^{2}\right )^{\frac {9}{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {9}{2}} d}\)	\(856\)

Input:

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

Output:

1/d*(-2/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/(a^2+b^2)*((-a^5-4*a^3*b^2+9*a*b^4)* 
tan(1/2*d*x+1/2*c)^5+(-3*a^4*b-12*a^2*b^3+3*b^5)*tan(1/2*d*x+1/2*c)^4+(-2/ 
3*a^5-32/3*a^3*b^2+14*a*b^4)*tan(1/2*d*x+1/2*c)^3+(-20*a^2*b^3+4*b^5)*tan( 
1/2*d*x+1/2*c)^2+(-a^5-4*a^3*b^2+9*a*b^4)*tan(1/2*d*x+1/2*c)-a^4*b-32/3*a^ 
2*b^3+7/3*b^5)/(1+tan(1/2*d*x+1/2*c)^2)^3-2*b^4/(a^2+b^2)/(a^6+3*a^4*b^2+3 
*a^2*b^4+b^6)*((-1/2*b^2*(13*a^2+2*b^2)/a*tan(1/2*d*x+1/2*c)^3-1/2*b*(12*a 
^4-23*a^2*b^2-2*b^4)/a^2*tan(1/2*d*x+1/2*c)^2+1/2*b^2*(35*a^2+2*b^2)/a*tan 
(1/2*d*x+1/2*c)+6*a^2*b+1/2*b^3)/(a*tan(1/2*d*x+1/2*c)^2-2*b*tan(1/2*d*x+1 
/2*c)-a)^2-5/2*(6*a^2-b^2)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tan(1/2*d*x+1/ 
2*c)-2*b)/(a^2+b^2)^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 619 vs. \(2 (294) = 588\).

Time = 0.14 (sec) , antiderivative size = 619, normalized size of antiderivative = 2.00 \[ \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {4 \, {\left (a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} \cos \left (d x + c\right )^{5} - 4 \, {\left (2 \, a^{8} b + a^{6} b^{3} - 9 \, a^{4} b^{5} - 13 \, a^{2} b^{7} - 5 \, b^{9}\right )} \cos \left (d x + c\right )^{3} - 15 \, {\left (6 \, a^{2} b^{6} - b^{8} + {\left (6 \, a^{4} b^{4} - 7 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{3} b^{5} - a b^{7}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) + 2 \, {\left (8 \, a^{8} b + 64 \, a^{6} b^{3} - 16 \, a^{4} b^{5} - 87 \, a^{2} b^{7} - 15 \, b^{9}\right )} \cos \left (d x + c\right ) + 2 \, {\left (4 \, a^{7} b^{2} + 32 \, a^{5} b^{4} - 53 \, a^{3} b^{6} - 81 \, a b^{8} + 2 \, {\left (a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 4 \, a^{3} b^{6} + a b^{8}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (2 \, a^{9} + 15 \, a^{7} b^{2} + 33 \, a^{5} b^{4} + 29 \, a^{3} b^{6} + 9 \, a b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{12 \, {\left ({\left (a^{12} + 4 \, a^{10} b^{2} + 5 \, a^{8} b^{4} - 5 \, a^{4} b^{8} - 4 \, a^{2} b^{10} - b^{12}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{11} b + 5 \, a^{9} b^{3} + 10 \, a^{7} b^{5} + 10 \, a^{5} b^{7} + 5 \, a^{3} b^{9} + a b^{11}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{10} b^{2} + 5 \, a^{8} b^{4} + 10 \, a^{6} b^{6} + 10 \, a^{4} b^{8} + 5 \, a^{2} b^{10} + b^{12}\right )} d\right )}} \] Input:

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

Output:

1/12*(4*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cos(d*x + c)^5 - 
 4*(2*a^8*b + a^6*b^3 - 9*a^4*b^5 - 13*a^2*b^7 - 5*b^9)*cos(d*x + c)^3 - 1 
5*(6*a^2*b^6 - b^8 + (6*a^4*b^4 - 7*a^2*b^6 + b^8)*cos(d*x + c)^2 + 2*(6*a 
^3*b^5 - a*b^7)*cos(d*x + c)*sin(d*x + c))*sqrt(a^2 + b^2)*log((2*a*b*cos( 
d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 - 2*a^2 - b^2 - 2*sqrt( 
a^2 + b^2)*(b*cos(d*x + c) - a*sin(d*x + c)))/(2*a*b*cos(d*x + c)*sin(d*x 
+ c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2)) + 2*(8*a^8*b + 64*a^6*b^3 - 16*a 
^4*b^5 - 87*a^2*b^7 - 15*b^9)*cos(d*x + c) + 2*(4*a^7*b^2 + 32*a^5*b^4 - 5 
3*a^3*b^6 - 81*a*b^8 + 2*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8) 
*cos(d*x + c)^4 + 2*(2*a^9 + 15*a^7*b^2 + 33*a^5*b^4 + 29*a^3*b^6 + 9*a*b^ 
8)*cos(d*x + c)^2)*sin(d*x + c))/((a^12 + 4*a^10*b^2 + 5*a^8*b^4 - 5*a^4*b 
^8 - 4*a^2*b^10 - b^12)*d*cos(d*x + c)^2 + 2*(a^11*b + 5*a^9*b^3 + 10*a^7* 
b^5 + 10*a^5*b^7 + 5*a^3*b^9 + a*b^11)*d*cos(d*x + c)*sin(d*x + c) + (a^10 
*b^2 + 5*a^8*b^4 + 10*a^6*b^6 + 10*a^4*b^8 + 5*a^2*b^10 + b^12)*d)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1229 vs. \(2 (294) = 588\).

Time = 0.20 (sec) , antiderivative size = 1229, normalized size of antiderivative = 3.96 \[ \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/6*(15*(6*a^2*b^4 - b^6)*log((b - a*sin(d*x + c)/(cos(d*x + c) + 1) + sq 
rt(a^2 + b^2))/(b - a*sin(d*x + c)/(cos(d*x + c) + 1) - sqrt(a^2 + b^2)))/ 
((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*sqrt(a^2 + b^2)) - 2*(6*a 
^8*b + 64*a^6*b^3 - 50*a^4*b^5 - 3*a^2*b^7 + (6*a^9 + 48*a^7*b^2 + 202*a^5 
*b^4 - 161*a^3*b^6 - 6*a*b^8)*sin(d*x + c)/(cos(d*x + c) + 1) + 2*(6*a^8*b 
 + 56*a^6*b^3 - 14*a^4*b^5 - 67*a^2*b^7 - 3*b^9)*sin(d*x + c)^2/(cos(d*x + 
 c) + 1)^2 - 4*(2*a^9 - 4*a^7*b^2 - 86*a^5*b^4 + 133*a^3*b^6 + 3*a*b^8)*si 
n(d*x + c)^3/(cos(d*x + c) + 1)^3 + 2*(8*a^8*b + 28*a^6*b^3 + 188*a^4*b^5 
- 156*a^2*b^7 - 9*b^9)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 2*(2*a^9 + 4* 
a^7*b^2 + 62*a^5*b^4 - 255*a^3*b^6)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 
2*(14*a^8*b + 56*a^6*b^3 - 246*a^4*b^5 + 141*a^2*b^7 + 9*b^9)*sin(d*x + c) 
^6/(cos(d*x + c) + 1)^6 - 4*(2*a^9 + 8*a^7*b^2 + 42*a^5*b^4 + 33*a^3*b^6 - 
 3*a*b^8)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 3*(2*a^8*b + 8*a^6*b^3 - 7 
8*a^4*b^5 + 23*a^2*b^7 + 2*b^9)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 3*(2 
*a^9 + 8*a^7*b^2 - 18*a^5*b^4 + 13*a^3*b^6 + 2*a*b^8)*sin(d*x + c)^9/(cos( 
d*x + c) + 1)^9)/(a^12 + 4*a^10*b^2 + 6*a^8*b^4 + 4*a^6*b^6 + a^4*b^8 + 4* 
(a^11*b + 4*a^9*b^3 + 6*a^7*b^5 + 4*a^5*b^7 + a^3*b^9)*sin(d*x + c)/(cos(d 
*x + c) + 1) + (a^12 + 8*a^10*b^2 + 22*a^8*b^4 + 28*a^6*b^6 + 17*a^4*b^8 + 
 4*a^2*b^10)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 8*(a^11*b + 4*a^9*b^3 + 
 6*a^7*b^5 + 4*a^5*b^7 + a^3*b^9)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 -...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 640 vs. \(2 (294) = 588\).

Time = 0.41 (sec) , antiderivative size = 640, normalized size of antiderivative = 2.06 \[ \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/6*(15*(6*a^2*b^4 - b^6)*log(abs(2*a*tan(1/2*d*x + 1/2*c) - 2*b - 2*sqrt 
(a^2 + b^2))/abs(2*a*tan(1/2*d*x + 1/2*c) - 2*b + 2*sqrt(a^2 + b^2)))/((a^ 
8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*sqrt(a^2 + b^2)) - 6*(13*a^3* 
b^6*tan(1/2*d*x + 1/2*c)^3 + 2*a*b^8*tan(1/2*d*x + 1/2*c)^3 + 12*a^4*b^5*t 
an(1/2*d*x + 1/2*c)^2 - 23*a^2*b^7*tan(1/2*d*x + 1/2*c)^2 - 2*b^9*tan(1/2* 
d*x + 1/2*c)^2 - 35*a^3*b^6*tan(1/2*d*x + 1/2*c) - 2*a*b^8*tan(1/2*d*x + 1 
/2*c) - 12*a^4*b^5 - a^2*b^7)/((a^10 + 4*a^8*b^2 + 6*a^6*b^4 + 4*a^4*b^6 + 
 a^2*b^8)*(a*tan(1/2*d*x + 1/2*c)^2 - 2*b*tan(1/2*d*x + 1/2*c) - a)^2) - 4 
*(3*a^5*tan(1/2*d*x + 1/2*c)^5 + 12*a^3*b^2*tan(1/2*d*x + 1/2*c)^5 - 27*a* 
b^4*tan(1/2*d*x + 1/2*c)^5 + 9*a^4*b*tan(1/2*d*x + 1/2*c)^4 + 36*a^2*b^3*t 
an(1/2*d*x + 1/2*c)^4 - 9*b^5*tan(1/2*d*x + 1/2*c)^4 + 2*a^5*tan(1/2*d*x + 
 1/2*c)^3 + 32*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 - 42*a*b^4*tan(1/2*d*x + 1/2 
*c)^3 + 60*a^2*b^3*tan(1/2*d*x + 1/2*c)^2 - 12*b^5*tan(1/2*d*x + 1/2*c)^2 
+ 3*a^5*tan(1/2*d*x + 1/2*c) + 12*a^3*b^2*tan(1/2*d*x + 1/2*c) - 27*a*b^4* 
tan(1/2*d*x + 1/2*c) + 3*a^4*b + 32*a^2*b^3 - 7*b^5)/((a^8 + 4*a^6*b^2 + 6 
*a^4*b^4 + 4*a^2*b^6 + b^8)*(tan(1/2*d*x + 1/2*c)^2 + 1)^3))/d
 

Mupad [B] (verification not implemented)

Time = 6.24 (sec) , antiderivative size = 1128, normalized size of antiderivative = 3.64 \[ \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \] Input:

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

Output:

((2*tan(c/2 + (d*x)/2)^5*(2*a^7 - 255*a*b^6 + 62*a^3*b^4 + 4*a^5*b^2))/(3* 
(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) + (6*a^6*b - 3*b^7 - 50*a 
^2*b^5 + 64*a^4*b^3)/(3*(a^2 + b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + 
 (2*tan(c/2 + (d*x)/2)^2*(6*a^6*b - 3*b^7 - 64*a^2*b^5 + 50*a^4*b^3))/(3*a 
^2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c/2 + (d*x)/2)^9*(2*a^8 + 2 
*b^8 + 13*a^2*b^6 - 18*a^4*b^4 + 8*a^6*b^2))/(a*(a^8 + b^8 + 4*a^2*b^6 + 6 
*a^4*b^4 + 4*a^6*b^2)) - (4*tan(c/2 + (d*x)/2)^7*(2*a^6 - 3*b^6 + 36*a^2*b 
^4 + 6*a^4*b^2))/(3*a*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (tan(c/2 + (d 
*x)/2)^8*(2*a^8*b + 2*b^9 + 23*a^2*b^7 - 78*a^4*b^5 + 8*a^6*b^3))/(a^2*(a^ 
8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) + (2*tan(c/2 + (d*x)/2)^4*(8 
*a^8*b - 9*b^9 - 156*a^2*b^7 + 188*a^4*b^5 + 28*a^6*b^3))/(3*a^2*(a^2 + b^ 
2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (2*tan(c/2 + (d*x)/2)^6*(14*a^8* 
b + 9*b^9 + 141*a^2*b^7 - 246*a^4*b^5 + 56*a^6*b^3))/(3*a^2*(a^2 + b^2)*(a 
^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c/2 + (d*x)/2)*(6*a^8 - 6*b^8 - 
161*a^2*b^6 + 202*a^4*b^4 + 48*a^6*b^2))/(3*a*(a^2 + b^2)*(a^6 + b^6 + 3*a 
^2*b^4 + 3*a^4*b^2)) - (4*tan(c/2 + (d*x)/2)^3*(2*a^8 + 3*b^8 + 133*a^2*b^ 
6 - 86*a^4*b^4 - 4*a^6*b^2))/(3*a*(a^2 + b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a 
^4*b^2)))/(d*(a^2*tan(c/2 + (d*x)/2)^10 - tan(c/2 + (d*x)/2)^6*(2*a^2 - 12 
*b^2) - tan(c/2 + (d*x)/2)^4*(2*a^2 - 12*b^2) + a^2 + tan(c/2 + (d*x)/2)^2 
*(a^2 + 4*b^2) + tan(c/2 + (d*x)/2)^8*(a^2 + 4*b^2) + 8*a*b*tan(c/2 + (...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1324, normalized size of antiderivative = 4.27 \[ \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 720*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b* 
*2))*cos(c + d*x)*sin(c + d*x)*a**3*b**6*i + 120*sqrt(a**2 + b**2)*atan((t 
an((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*cos(c + d*x)*sin(c + d*x)*a* 
b**8*i + 360*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 
 + b**2))*sin(c + d*x)**2*a**4*b**5*i - 420*sqrt(a**2 + b**2)*atan((tan((c 
 + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*sin(c + d*x)**2*a**2*b**7*i + 60* 
sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*sin 
(c + d*x)**2*b**9*i - 360*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b 
*i)/sqrt(a**2 + b**2))*a**4*b**5*i + 60*sqrt(a**2 + b**2)*atan((tan((c + d 
*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*a**2*b**7*i + 4*cos(c + d*x)*sin(c + 
d*x)**4*a**8*b**2 + 16*cos(c + d*x)*sin(c + d*x)**4*a**6*b**4 + 24*cos(c + 
 d*x)*sin(c + d*x)**4*a**4*b**6 + 16*cos(c + d*x)*sin(c + d*x)**4*a**2*b** 
8 + 4*cos(c + d*x)*sin(c + d*x)**4*b**10 - 28*cos(c + d*x)*sin(c + d*x)**2 
*a**6*b**4 - 84*cos(c + d*x)*sin(c + d*x)**2*a**4*b**6 - 84*cos(c + d*x)*s 
in(c + d*x)**2*a**2*b**8 - 28*cos(c + d*x)*sin(c + d*x)**2*b**10 + 12*cos( 
c + d*x)*sin(c + d*x)*a**9*b + 84*cos(c + d*x)*sin(c + d*x)*a**7*b**3 + 22 
0*cos(c + d*x)*sin(c + d*x)*a**5*b**5 + 26*cos(c + d*x)*sin(c + d*x)*a**3* 
b**7 - 122*cos(c + d*x)*sin(c + d*x)*a*b**9 + 12*cos(c + d*x)*a**8*b**2 + 
140*cos(c + d*x)*a**6*b**4 + 28*cos(c + d*x)*a**4*b**6 - 106*cos(c + d*x)* 
a**2*b**8 - 6*cos(c + d*x)*b**10 + 4*sin(c + d*x)**5*a**9*b + 16*sin(c ...