\(\int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx\) [814]

Optimal result

Integrand size = 23, antiderivative size = 216 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {2 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{5 d}-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))}{5 d} \] Output:

-1/2*(a+b)*(a^2-4*a*b+b^2)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/d-1 
/2*(a+b)*(a^2-4*a*b+b^2)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/d-1/2* 
(a-b)*(a^2+4*a*b+b^2)*arctanh(2^(1/2)*cot(d*x+c)^(1/2)/(1+cot(d*x+c)))*2^( 
1/2)/d+2*a*(a^2-3*b^2)*cot(d*x+c)^(1/2)/d-8/5*a^2*b*cot(d*x+c)^(3/2)/d-2/5 
*a^2*cot(d*x+c)^(3/2)*(b+a*cot(d*x+c))/d
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 1.41 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.04 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {2 a^2 b \cot ^{\frac {3}{2}}(c+d x)+\frac {2}{5} a^3 \cot ^{\frac {5}{2}}(c+d x)+\frac {2}{3} b \left (-3 a^2+b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )-\frac {1}{4} a \left (a^2-3 b^2\right ) \left (2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+8 \sqrt {\cot (c+d x)}+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{d} \] Input:

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

Output:

-((2*a^2*b*Cot[c + d*x]^(3/2) + (2*a^3*Cot[c + d*x]^(5/2))/5 + (2*b*(-3*a^ 
2 + b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2 
])/3 - (a*(a^2 - 3*b^2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] 
- 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqrt[Cot[c + d*x]] 
+ Sqrt[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]] + Cot[c + d*x]]))/4)/d)
 

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.14, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.870, Rules used = {3042, 4156, 3042, 4049, 27, 3042, 4113, 3042, 4011, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4156

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4049

\(\displaystyle -\frac {2}{5} \int \frac {1}{2} \sqrt {\cot (c+d x)} \left (-12 a^2 b \cot ^2(c+d x)+5 a \left (a^2-3 b^2\right ) \cot (c+d x)+b \left (3 a^2-5 b^2\right )\right )dx-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{5} \int \sqrt {\cot (c+d x)} \left (-12 a^2 b \cot ^2(c+d x)+5 a \left (a^2-3 b^2\right ) \cot (c+d x)+b \left (3 a^2-5 b^2\right )\right )dx-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{5} \int \sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (-12 a^2 b \tan \left (c+d x+\frac {\pi }{2}\right )^2-5 a \left (a^2-3 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )+b \left (3 a^2-5 b^2\right )\right )dx-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\)

\(\Big \downarrow \) 4113

\(\displaystyle \frac {1}{5} \left (-\int \sqrt {\cot (c+d x)} \left (5 b \left (3 a^2-b^2\right )+5 a \left (a^2-3 b^2\right ) \cot (c+d x)\right )dx-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{5} \left (-\int \sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (5 b \left (3 a^2-b^2\right )-5 a \left (a^2-3 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\)

\(\Big \downarrow \) 4011

\(\displaystyle \frac {1}{5} \left (-\int \frac {5 b \left (3 a^2-b^2\right ) \cot (c+d x)-5 a \left (a^2-3 b^2\right )}{\sqrt {\cot (c+d x)}}dx+\frac {10 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{5} \left (-\int \frac {-5 a \left (a^2-3 b^2\right )-5 b \left (3 a^2-b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\)

\(\Big \downarrow \) 4017

\(\displaystyle \frac {1}{5} \left (-\frac {2 \int \frac {5 \left (a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {10 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \left (-\frac {10 \int \frac {a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {10 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\)

\(\Big \downarrow \) 1482

\(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}+\frac {10 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}+\frac {10 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}+\frac {10 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {10 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {10 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {10 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {10 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {1}{5} \left (-\frac {10 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}+\frac {10 a \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {8 a^2 b \cot ^{\frac {3}{2}}(c+d x)}{d}\right )-\frac {2 a^2 \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)}{5 d}\)

Input:

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

Output:

(-2*a^2*Cot[c + d*x]^(3/2)*(b + a*Cot[c + d*x]))/(5*d) + ((10*a*(a^2 - 3*b 
^2)*Sqrt[Cot[c + d*x]])/d - (8*a^2*b*Cot[c + d*x]^(3/2))/d - (10*(((a + b) 
*(a^2 - 4*a*b + b^2)*(-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + 
ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]))/2 + ((a - b)*(a^2 + 4*a*b 
 + b^2)*(-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]] + Cot[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 4011

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int 
[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] 
, x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 
 0] && GtQ[m, 0]
 

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 4049

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c 
+ d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Simp[1/(d*(m + n - 1)) 
 Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n 
- 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[ 
e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2, x], x], 
x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2 
, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || I 
ntegerQ[m]) &&  !(IGtQ[n, 2] && ( !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 4156

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(572\) vs. \(2(188)=376\).

Time = 0.58 (sec) , antiderivative size = 573, normalized size of antiderivative = 2.65

method	result	size
derivativedivides	\(-\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {7}{2}} \tan \left (d x +c \right ) \left (15 \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {2}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) a^{2} b -5 \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {2}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) b^{3}-10 \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{3}+30 \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{2} b +30 \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a \,b^{2}-10 \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) b^{3}-10 \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{3}+30 \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{2} b +30 \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a \,b^{2}-10 \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) b^{3}-5 \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right ) a^{3}+15 \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right ) a \,b^{2}-40 \tan \left (d x +c \right )^{2} a^{3}+120 \tan \left (d x +c \right )^{2} a \,b^{2}+40 \tan \left (d x +c \right ) a^{2} b +8 a^{3}\right )}{20 d}\)	\(573\)
default	\(-\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {7}{2}} \tan \left (d x +c \right ) \left (15 \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {2}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) a^{2} b -5 \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {2}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) b^{3}-10 \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{3}+30 \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{2} b +30 \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a \,b^{2}-10 \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) b^{3}-10 \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{3}+30 \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{2} b +30 \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a \,b^{2}-10 \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) b^{3}-5 \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right ) a^{3}+15 \tan \left (d x +c \right )^{\frac {5}{2}} \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right ) a \,b^{2}-40 \tan \left (d x +c \right )^{2} a^{3}+120 \tan \left (d x +c \right )^{2} a \,b^{2}+40 \tan \left (d x +c \right ) a^{2} b +8 a^{3}\right )}{20 d}\)	\(573\)

Input:

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

Output:

-1/20/d*(1/tan(d*x+c))^(7/2)*tan(d*x+c)*(15*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^2*b-5*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^3-10*tan(d*x+c)^(5/2)* 
2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^3+30*tan(d*x+c)^(5/2)*2^(1/2) 
*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b+30*tan(d*x+c)^(5/2)*2^(1/2)*arct 
an(1+2^(1/2)*tan(d*x+c)^(1/2))*a*b^2-10*tan(d*x+c)^(5/2)*2^(1/2)*arctan(1+ 
2^(1/2)*tan(d*x+c)^(1/2))*b^3-10*tan(d*x+c)^(5/2)*2^(1/2)*arctan(-1+2^(1/2 
)*tan(d*x+c)^(1/2))*a^3+30*tan(d*x+c)^(5/2)*2^(1/2)*arctan(-1+2^(1/2)*tan( 
d*x+c)^(1/2))*a^2*b+30*tan(d*x+c)^(5/2)*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+ 
c)^(1/2))*a*b^2-10*tan(d*x+c)^(5/2)*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^( 
1/2))*b^3-5*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^3+15*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*b^2-40*tan(d*x+c)^2*a^3+120*tan(d*x+c)^2*a*b^2+40*tan(d 
*x+c)*a^2*b+8*a^3)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 973 vs. \(2 (188) = 376\).

Time = 0.10 (sec) , antiderivative size = 973, normalized size of antiderivative = 4.50 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx =\text {Too large to display} \] Input:

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

Output:

1/10*(10*sqrt(1/2)*d*sqrt((a^6 - 6*a^5*b + 3*a^4*b^2 + 20*a^3*b^3 + 3*a^2* 
b^4 - 6*a*b^5 + b^6)/d^2)*arctan((2*sqrt(1/2)*(a^3 + 3*a^2*b - 3*a*b^2 - b 
^3)*d*sqrt((a^6 - 6*a^5*b + 3*a^4*b^2 + 20*a^3*b^3 + 3*a^2*b^4 - 6*a*b^5 + 
 b^6)/d^2)*sqrt(tan(d*x + c)) + d^2*sqrt((a^6 + 6*a^5*b + 3*a^4*b^2 - 20*a 
^3*b^3 + 3*a^2*b^4 + 6*a*b^5 + b^6)/d^2)*sqrt((a^6 - 6*a^5*b + 3*a^4*b^2 + 
 20*a^3*b^3 + 3*a^2*b^4 - 6*a*b^5 + b^6)/d^2))/(a^6 - 15*a^4*b^2 + 15*a^2* 
b^4 - b^6))*tan(d*x + c)^2 + 10*sqrt(1/2)*d*sqrt((a^6 - 6*a^5*b + 3*a^4*b^ 
2 + 20*a^3*b^3 + 3*a^2*b^4 - 6*a*b^5 + b^6)/d^2)*arctan((2*sqrt(1/2)*(a^3 
+ 3*a^2*b - 3*a*b^2 - b^3)*d*sqrt((a^6 - 6*a^5*b + 3*a^4*b^2 + 20*a^3*b^3 
+ 3*a^2*b^4 - 6*a*b^5 + b^6)/d^2)*sqrt(tan(d*x + c)) - d^2*sqrt((a^6 + 6*a 
^5*b + 3*a^4*b^2 - 20*a^3*b^3 + 3*a^2*b^4 + 6*a*b^5 + b^6)/d^2)*sqrt((a^6 
- 6*a^5*b + 3*a^4*b^2 + 20*a^3*b^3 + 3*a^2*b^4 - 6*a*b^5 + b^6)/d^2))/(a^6 
 - 15*a^4*b^2 + 15*a^2*b^4 - b^6))*tan(d*x + c)^2 + 5*sqrt(1/2)*d*sqrt((a^ 
6 + 6*a^5*b + 3*a^4*b^2 - 20*a^3*b^3 + 3*a^2*b^4 + 6*a*b^5 + b^6)/d^2)*log 
(-a^3 - 3*a^2*b + 3*a*b^2 + b^3 + 2*sqrt(1/2)*d*sqrt((a^6 + 6*a^5*b + 3*a^ 
4*b^2 - 20*a^3*b^3 + 3*a^2*b^4 + 6*a*b^5 + b^6)/d^2)*sqrt(tan(d*x + c)) - 
(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*tan(d*x + c))*tan(d*x + c)^2 - 5*sqrt(1/2) 
*d*sqrt((a^6 + 6*a^5*b + 3*a^4*b^2 - 20*a^3*b^3 + 3*a^2*b^4 + 6*a*b^5 + b^ 
6)/d^2)*log(-a^3 - 3*a^2*b + 3*a*b^2 + b^3 - 2*sqrt(1/2)*d*sqrt((a^6 + 6*a 
^5*b + 3*a^4*b^2 - 20*a^3*b^3 + 3*a^2*b^4 + 6*a*b^5 + b^6)/d^2)*sqrt(ta...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.11 \[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {10 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 10 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 5 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - 5 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \frac {40 \, a^{2} b}{\tan \left (d x + c\right )^{\frac {3}{2}}} + \frac {8 \, a^{3}}{\tan \left (d x + c\right )^{\frac {5}{2}}} - \frac {40 \, {\left (a^{3} - 3 \, a b^{2}\right )}}{\sqrt {\tan \left (d x + c\right )}}}{20 \, d} \] Input:

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

Output:

-1/20*(10*sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*arctan(1/2*sqrt(2)*(sqrt 
(2) + 2/sqrt(tan(d*x + c)))) + 10*sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)* 
arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) + 5*sqrt(2)*(a^3 + 3 
*a^2*b - 3*a*b^2 - b^3)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 
1) - 5*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*log(-sqrt(2)/sqrt(tan(d*x + 
 c)) + 1/tan(d*x + c) + 1) + 40*a^2*b/tan(d*x + c)^(3/2) + 8*a^3/tan(d*x + 
 c)^(5/2) - 40*(a^3 - 3*a*b^2)/sqrt(tan(d*x + c)))/d
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \cot ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {-2 \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2} a^{3}+10 \sqrt {\cot \left (d x +c \right )}\, a^{3}+5 \left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )}d x \right ) a^{3} d +5 \left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{3} \tan \left (d x +c \right )^{3}d x \right ) b^{3} d +15 \left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{3} \tan \left (d x +c \right )^{2}d x \right ) a \,b^{2} d +15 \left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{3} \tan \left (d x +c \right )d x \right ) a^{2} b d}{5 d} \] Input:

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

Output:

( - 2*sqrt(cot(c + d*x))*cot(c + d*x)**2*a**3 + 10*sqrt(cot(c + d*x))*a**3 
 + 5*int(sqrt(cot(c + d*x))/cot(c + d*x),x)*a**3*d + 5*int(sqrt(cot(c + d* 
x))*cot(c + d*x)**3*tan(c + d*x)**3,x)*b**3*d + 15*int(sqrt(cot(c + d*x))* 
cot(c + d*x)**3*tan(c + d*x)**2,x)*a*b**2*d + 15*int(sqrt(cot(c + d*x))*co 
t(c + d*x)**3*tan(c + d*x),x)*a**2*b*d)/(5*d)