\(\int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx\) [588]

Optimal result

Integrand size = 33, antiderivative size = 304 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {2 a \left (5 a^2 A-14 A b^2-15 a b B\right ) \sqrt {\cot (c+d x)}}{5 d}-\frac {2 a^2 (9 A b+5 a B) \cot ^{\frac {3}{2}}(c+d x)}{15 d}-\frac {2 a A \sqrt {\cot (c+d x)} (b+a \cot (c+d x))^2}{5 d} \] Output:

-1/2*(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*arctan(-1+2^(1/2)*c 
ot(d*x+c)^(1/2))*2^(1/2)/d-1/2*(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3* 
(A+B))*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/d-1/2*(3*a^2*b*(A-B)-b^3 
*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*arctanh(2^(1/2)*cot(d*x+c)^(1/2)/(1+cot(d* 
x+c)))*2^(1/2)/d+2/5*a*(5*A*a^2-14*A*b^2-15*B*a*b)*cot(d*x+c)^(1/2)/d-2/15 
*a^2*(9*A*b+5*B*a)*cot(d*x+c)^(3/2)/d-2/5*a*A*cot(d*x+c)^(1/2)*(b+a*cot(d* 
x+c))^2/d
 

Mathematica [A] (verified)

Time = 1.54 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.94 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {2 \sqrt {\cot (c+d x)} \left (-\frac {\left (a^3 (A-B)+3 a b^2 (-A+B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \left (\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )\right )}{2 \sqrt {2}}+\frac {\left (3 a^2 b (A-B)+b^3 (-A+B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \left (\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )}{4 \sqrt {2}}-\frac {a^3 A}{5 \tan ^{\frac {5}{2}}(c+d x)}-\frac {a^2 (3 A b+a B)}{3 \tan ^{\frac {3}{2}}(c+d x)}+\frac {a \left (a^2 A-3 A b^2-3 a b B\right )}{\sqrt {\tan (c+d x)}}\right ) \sqrt {\tan (c+d x)}}{d} \] Input:

Integrate[Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x 
]
 

Output:

(2*Sqrt[Cot[c + d*x]]*(-1/2*((a^3*(A - B) + 3*a*b^2*(-A + B) - 3*a^2*b*(A 
+ B) + b^3*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + S 
qrt[2]*Sqrt[Tan[c + d*x]]]))/Sqrt[2] + ((3*a^2*b*(A - B) + b^3*(-A + B) + 
a^3*(A + B) - 3*a*b^2*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c 
 + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]))/(4*Sqrt[2] 
) - (a^3*A)/(5*Tan[c + d*x]^(5/2)) - (a^2*(3*A*b + a*B))/(3*Tan[c + d*x]^( 
3/2)) + (a*(a^2*A - 3*A*b^2 - 3*a*b*B))/Sqrt[Tan[c + d*x]])*Sqrt[Tan[c + d 
*x]])/d
 

Rubi [A] (verified)

Time = 1.36 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.03, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3042, 4064, 3042, 4090, 27, 3042, 4120, 27, 3042, 4113, 3042, 4017, 27, 1482, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cot (c+d x)^{7/2} (a+b \tan (c+d x))^3 (A+B \tan (c+d x))dx\)

\(\Big \downarrow \) 4064

\(\displaystyle \int \frac {(a \cot (c+d x)+b)^3 (A \cot (c+d x)+B)}{\sqrt {\cot (c+d x)}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (B-A \tan \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx\)

\(\Big \downarrow \) 4090

\(\displaystyle -\frac {2}{5} \int \frac {(b+a \cot (c+d x)) \left (-a (9 A b+5 a B) \cot ^2(c+d x)+5 \left (A a^2-2 b B a-A b^2\right ) \cot (c+d x)+b (a A-5 b B)\right )}{2 \sqrt {\cot (c+d x)}}dx-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{5} \int \frac {(b+a \cot (c+d x)) \left (-a (9 A b+5 a B) \cot ^2(c+d x)+5 \left (A a^2-2 b B a-A b^2\right ) \cot (c+d x)+b (a A-5 b B)\right )}{\sqrt {\cot (c+d x)}}dx-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{5} \int \frac {\left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-a (9 A b+5 a B) \tan \left (c+d x+\frac {\pi }{2}\right )^2-5 \left (A a^2-2 b B a-A b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )+b (a A-5 b B)\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\)

\(\Big \downarrow \) 4120

\(\displaystyle \frac {1}{5} \left (-\frac {2}{3} \int \frac {3 \left ((a A-5 b B) b^2+a \left (5 A a^2-15 b B a-14 A b^2\right ) \cot ^2(c+d x)+5 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \cot (c+d x)\right )}{2 \sqrt {\cot (c+d x)}}dx-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \left (-\int \frac {(a A-5 b B) b^2+a \left (5 A a^2-15 b B a-14 A b^2\right ) \cot ^2(c+d x)+5 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{5} \left (-\int \frac {(a A-5 b B) b^2+a \left (5 A a^2-15 b B a-14 A b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2-5 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\)

\(\Big \downarrow \) 4113

\(\displaystyle \frac {1}{5} \left (-\int \frac {5 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \cot (c+d x)-5 \left (A a^3-3 b B a^2-3 A b^2 a+b^3 B\right )}{\sqrt {\cot (c+d x)}}dx+\frac {2 a \left (5 a^2 A-15 a b B-14 A b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{5} \left (-\int \frac {-5 \left (A a^3-3 b B a^2-3 A b^2 a+b^3 B\right )-5 \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a \left (5 a^2 A-15 a b B-14 A b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\)

\(\Big \downarrow \) 4017

\(\displaystyle \frac {1}{5} \left (-\frac {2 \int \frac {5 \left (A a^3-3 b B a^2-3 A b^2 a+b^3 B-\left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {2 a \left (5 a^2 A-15 a b B-14 A b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \left (-\frac {10 \int \frac {A a^3-3 b B a^2-3 A b^2 a+b^3 B-\left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {2 a \left (5 a^2 A-15 a b B-14 A b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\)

\(\Big \downarrow \) 1482

\(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} \left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}+\frac {2 a \left (5 a^2 A-15 a b B-14 A b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} \left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}+\frac {2 a \left (5 a^2 A-15 a b B-14 A b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} \left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a \left (5 a^2 A-15 a b B-14 A b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} \left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a \left (5 a^2 A-15 a b B-14 A b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} \left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a \left (5 a^2 A-15 a b B-14 A b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} \left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a \left (5 a^2 A-15 a b B-14 A b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} \left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )+\frac {1}{2} \left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a \left (5 a^2 A-15 a b B-14 A b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {1}{5} \left (\frac {2 a \left (5 a^2 A-15 a b B-14 A b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {2 a^2 (5 a B+9 A b) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {10 \left (\frac {1}{2} \left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}\right )-\frac {2 a A \sqrt {\cot (c+d x)} (a \cot (c+d x)+b)^2}{5 d}\)

Input:

Int[Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]
 

Output:

(-2*a*A*Sqrt[Cot[c + d*x]]*(b + a*Cot[c + d*x])^2)/(5*d) + ((2*a*(5*a^2*A 
- 14*A*b^2 - 15*a*b*B)*Sqrt[Cot[c + d*x]])/d - (2*a^2*(9*A*b + 5*a*B)*Cot[ 
c + d*x]^(3/2))/(3*d) - (10*(((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A 
+ B) + b^3*(A + B))*(-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + A 
rcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]))/2 + ((3*a^2*b*(A - B) - b^ 
3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*(-1/2*Log[1 - Sqrt[2]*Sqrt[Cot[ 
c + d*x]] + Cot[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + C 
ot[c + d*x]]/(2*Sqrt[2])))/2))/d)/5
 

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 217

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

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 1482

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
a*c, 2]}, Simp[(d*q + a*e)/(2*a*c)   Int[(q + c*x^2)/(a + c*x^4), x], x] + 
Simp[(d*q - a*e)/(2*a*c)   Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a 
, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- 
a)*c]
 

rule 3042

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

rule 4017

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ 
)]], x_Symbol] :> Simp[2/f   Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq 
rt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] & 
& NeQ[c^2 + d^2, 0]
 

rule 4064

Int[(cot[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_.) + (b_.)*tan[(e_.) + (f_.)*( 
x_)])^(m_.)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp 
[g^(m + n)   Int[(g*Cot[e + f*x])^(p - m - n)*(b + a*Cot[e + f*x])^m*(d + c 
*Cot[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&  !Integer 
Q[p] && IntegerQ[m] && IntegerQ[n]
 

rule 4090

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + 
 (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si 
mp[b*B*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + 
 n))), x] + Simp[1/(d*(m + n))   Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Ta 
n[e + f*x])^n*Simp[a^2*A*d*(m + n) - b*B*(b*c*(m - 1) + a*d*(n + 1)) + d*(m 
 + n)*(2*a*A*b + B*(a^2 - b^2))*Tan[e + f*x] - (b*B*(b*c - a*d)*(m - 1) - b 
*(A*b + a*B)*d*(m + n))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, 
 f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2 
, 0] && GtQ[m, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&  !(IGtQ[n, 1] 
&& ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
 

rule 4113

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) 
+ (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + 
 b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Si 
mp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && 
NeQ[A*b^2 - a*b*B + a^2*C, 0] &&  !LeQ[m, -1]
 

rule 4120

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.) 
*(x_)])^(n_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f 
_.)*(x_)]^2), x_Symbol] :> Simp[b*C*Tan[e + f*x]*((c + d*Tan[e + f*x])^(n + 
 1)/(d*f*(n + 2))), x] - Simp[1/(d*(n + 2))   Int[(c + d*Tan[e + f*x])^n*Si 
mp[b*c*C - a*A*d*(n + 2) - (A*b + a*B - b*C)*d*(n + 2)*Tan[e + f*x] - (a*C* 
d*(n + 2) - b*(c*C - B*d*(n + 2)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, 
b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && 
  !LtQ[n, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1133\) vs. \(2(274)=548\).

Time = 1.12 (sec) , antiderivative size = 1134, normalized size of antiderivative = 3.73

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1134\)
default	\(\text {Expression too large to display}\)	\(1134\)

Input:

int(cot(d*x+c)^(7/2)*(a+b*tan(d*x+c))^3*(A+B*tan(d*x+c)),x,method=_RETURNV 
ERBOSE)
 

Output:

-1/60/d*(1/tan(d*x+c))^(7/2)*tan(d*x+c)*(-30*A*tan(d*x+c)^(5/2)*2^(1/2)*ar 
ctan(1+2^(1/2)*tan(d*x+c)^(1/2))*b^3-30*A*tan(d*x+c)^(5/2)*2^(1/2)*arctan( 
-1+2^(1/2)*tan(d*x+c)^(1/2))*a^3-30*A*tan(d*x+c)^(5/2)*2^(1/2)*arctan(-1+2 
^(1/2)*tan(d*x+c)^(1/2))*b^3-15*A*tan(d*x+c)^(5/2)*2^(1/2)*ln(-(2^(1/2)*ta 
n(d*x+c)^(1/2)-tan(d*x+c)-1)/(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1))*a^3- 
15*B*tan(d*x+c)^(5/2)*2^(1/2)*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/ 
(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1))*b^3+15*B*tan(d*x+c)^(5/2)*2^(1/2) 
*ln(-(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1)/(2^(1/2)*tan(d*x+c)^(1/2)-tan 
(d*x+c)-1))*a^3+30*B*tan(d*x+c)^(5/2)*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^ 
(1/2))*a^3+40*B*tan(d*x+c)*a^3-30*B*tan(d*x+c)^(5/2)*2^(1/2)*arctan(1+2^(1 
/2)*tan(d*x+c)^(1/2))*b^3+30*B*tan(d*x+c)^(5/2)*2^(1/2)*arctan(-1+2^(1/2)* 
tan(d*x+c)^(1/2))*a^3-30*B*tan(d*x+c)^(5/2)*2^(1/2)*arctan(-1+2^(1/2)*tan( 
d*x+c)^(1/2))*b^3-120*A*tan(d*x+c)^2*a^3-15*A*tan(d*x+c)^(5/2)*2^(1/2)*ln( 
-(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1)/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x 
+c)-1))*b^3-30*A*tan(d*x+c)^(5/2)*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2 
))*a^3+45*B*tan(d*x+c)^(5/2)*2^(1/2)*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x 
+c)-1)/(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1))*a^2*b-90*B*tan(d*x+c)^(5/2 
)*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a*b^2+90*B*tan(d*x+c)^(5/2)* 
2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b-45*B*tan(d*x+c)^(5/2)*2^ 
(1/2)*ln(-(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1)/(2^(1/2)*tan(d*x+c)^(...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2099 vs. \(2 (274) = 548\).

Time = 0.13 (sec) , antiderivative size = 2099, normalized size of antiderivative = 6.90 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\text {Too large to display} \] Input:

integrate(cot(d*x+c)^(7/2)*(a+b*tan(d*x+c))^3*(A+B*tan(d*x+c)),x, algorith 
m="fricas")
 

Output:

1/30*(30*sqrt(1/2)*d*sqrt(((A^2 - 2*A*B + B^2)*a^6 - 6*(A^2 - B^2)*a^5*b + 
 3*(A^2 + 10*A*B + B^2)*a^4*b^2 + 20*(A^2 - B^2)*a^3*b^3 + 3*(A^2 - 10*A*B 
 + B^2)*a^2*b^4 - 6*(A^2 - B^2)*a*b^5 + (A^2 + 2*A*B + B^2)*b^6)/d^2)*arct 
an(-(2*sqrt(1/2)*((A + B)*a^3 + 3*(A - B)*a^2*b - 3*(A + B)*a*b^2 - (A - B 
)*b^3)*d*sqrt(((A^2 - 2*A*B + B^2)*a^6 - 6*(A^2 - B^2)*a^5*b + 3*(A^2 + 10 
*A*B + B^2)*a^4*b^2 + 20*(A^2 - B^2)*a^3*b^3 + 3*(A^2 - 10*A*B + B^2)*a^2* 
b^4 - 6*(A^2 - B^2)*a*b^5 + (A^2 + 2*A*B + B^2)*b^6)/d^2)*sqrt(tan(d*x + c 
)) + d^2*sqrt(((A^2 + 2*A*B + B^2)*a^6 + 6*(A^2 - B^2)*a^5*b + 3*(A^2 - 10 
*A*B + B^2)*a^4*b^2 - 20*(A^2 - B^2)*a^3*b^3 + 3*(A^2 + 10*A*B + B^2)*a^2* 
b^4 + 6*(A^2 - B^2)*a*b^5 + (A^2 - 2*A*B + B^2)*b^6)/d^2)*sqrt(((A^2 - 2*A 
*B + B^2)*a^6 - 6*(A^2 - B^2)*a^5*b + 3*(A^2 + 10*A*B + B^2)*a^4*b^2 + 20* 
(A^2 - B^2)*a^3*b^3 + 3*(A^2 - 10*A*B + B^2)*a^2*b^4 - 6*(A^2 - B^2)*a*b^5 
 + (A^2 + 2*A*B + B^2)*b^6)/d^2))/(12*A*B*a^5*b - 40*A*B*a^3*b^3 + 12*A*B* 
a*b^5 - (A^2 - B^2)*a^6 + 15*(A^2 - B^2)*a^4*b^2 - 15*(A^2 - B^2)*a^2*b^4 
+ (A^2 - B^2)*b^6))*tan(d*x + c)^2 + 30*sqrt(1/2)*d*sqrt(((A^2 - 2*A*B + B 
^2)*a^6 - 6*(A^2 - B^2)*a^5*b + 3*(A^2 + 10*A*B + B^2)*a^4*b^2 + 20*(A^2 - 
 B^2)*a^3*b^3 + 3*(A^2 - 10*A*B + B^2)*a^2*b^4 - 6*(A^2 - B^2)*a*b^5 + (A^ 
2 + 2*A*B + B^2)*b^6)/d^2)*arctan(-(2*sqrt(1/2)*((A + B)*a^3 + 3*(A - B)*a 
^2*b - 3*(A + B)*a*b^2 - (A - B)*b^3)*d*sqrt(((A^2 - 2*A*B + B^2)*a^6 - 6* 
(A^2 - B^2)*a^5*b + 3*(A^2 + 10*A*B + B^2)*a^4*b^2 + 20*(A^2 - B^2)*a^3...
 

Sympy [F(-1)]

Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)**(7/2)*(a+b*tan(d*x+c))**3*(A+B*tan(d*x+c)),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.09 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=-\frac {30 \, \sqrt {2} {\left ({\left (A - B\right )} a^{3} - 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} + {\left (A + B\right )} b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 30 \, \sqrt {2} {\left ({\left (A - B\right )} a^{3} - 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} + {\left (A + B\right )} b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 15 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} + 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} - {\left (A - B\right )} b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - 15 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} + 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} - {\left (A - B\right )} b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \frac {24 \, A a^{3}}{\tan \left (d x + c\right )^{\frac {5}{2}}} - \frac {120 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2}\right )}}{\sqrt {\tan \left (d x + c\right )}} + \frac {40 \, {\left (B a^{3} + 3 \, A a^{2} b\right )}}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{60 \, d} \] Input:

integrate(cot(d*x+c)^(7/2)*(a+b*tan(d*x+c))^3*(A+B*tan(d*x+c)),x, algorith 
m="maxima")
 

Output:

-1/60*(30*sqrt(2)*((A - B)*a^3 - 3*(A + B)*a^2*b - 3*(A - B)*a*b^2 + (A + 
B)*b^3)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 30*sqrt(2)* 
((A - B)*a^3 - 3*(A + B)*a^2*b - 3*(A - B)*a*b^2 + (A + B)*b^3)*arctan(-1/ 
2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) + 15*sqrt(2)*((A + B)*a^3 + 3* 
(A - B)*a^2*b - 3*(A + B)*a*b^2 - (A - B)*b^3)*log(sqrt(2)/sqrt(tan(d*x + 
c)) + 1/tan(d*x + c) + 1) - 15*sqrt(2)*((A + B)*a^3 + 3*(A - B)*a^2*b - 3* 
(A + B)*a*b^2 - (A - B)*b^3)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + 
 c) + 1) + 24*A*a^3/tan(d*x + c)^(5/2) - 120*(A*a^3 - 3*B*a^2*b - 3*A*a*b^ 
2)/sqrt(tan(d*x + c)) + 40*(B*a^3 + 3*A*a^2*b)/tan(d*x + c)^(3/2))/d
 

Giac [F]

\[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3} \cot \left (d x + c\right )^{\frac {7}{2}} \,d x } \] Input:

integrate(cot(d*x+c)^(7/2)*(a+b*tan(d*x+c))^3*(A+B*tan(d*x+c)),x, algorith 
m="giac")
 

Output:

integrate((B*tan(d*x + c) + A)*(b*tan(d*x + c) + a)^3*cot(d*x + c)^(7/2), 
x)
 

Mupad [F(-1)]

Timed out. \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{7/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3 \,d x \] Input:

int(cot(c + d*x)^(7/2)*(A + B*tan(c + d*x))*(a + b*tan(c + d*x))^3,x)
 

Output:

int(cot(c + d*x)^(7/2)*(A + B*tan(c + d*x))*(a + b*tan(c + d*x))^3, x)
 

Reduce [F]

\[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\int \cot \left (d x +c \right )^{\frac {7}{2}} \left (a +\tan \left (d x +c \right ) b \right )^{3} \left (A +B \tan \left (d x +c \right )\right )d x \] Input:

int(cot(d*x+c)^(7/2)*(a+b*tan(d*x+c))^3*(A+B*tan(d*x+c)),x)
 

Output:

int(cot(d*x+c)^(7/2)*(a+b*tan(d*x+c))^3*(A+B*tan(d*x+c)),x)