\(\int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {9}{2}}(c+d x)} \, dx\) [622]

Optimal result

Integrand size = 25, antiderivative size = 266 \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {9}{2}}(c+d x)} \, dx=-\frac {(i a-b)^{3/2} \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {(i a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 a \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {16 b \sqrt {a+b \tan (c+d x)}}{35 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2-3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {4 b \left (70 a^2+3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{105 a^2 d \sqrt {\tan (c+d x)}} \] Output:

-(I*a-b)^(3/2)*arctan((I*a-b)^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(1/2 
))/d-(I*a+b)^(3/2)*arctanh((I*a+b)^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c)) 
^(1/2))/d-2/7*a*(a+b*tan(d*x+c))^(1/2)/d/tan(d*x+c)^(7/2)-16/35*b*(a+b*tan 
(d*x+c))^(1/2)/d/tan(d*x+c)^(5/2)+2/105*(35*a^2-3*b^2)*(a+b*tan(d*x+c))^(1 
/2)/a/d/tan(d*x+c)^(3/2)+4/105*b*(70*a^2+3*b^2)*(a+b*tan(d*x+c))^(1/2)/a^2 
/d/tan(d*x+c)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.98 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {9}{2}}(c+d x)} \, dx=\frac {105 \sqrt [4]{-1} \sqrt {-a+i b} (i a+b) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )-105 (-1)^{3/4} (a+i b)^{3/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+\frac {2 \sqrt {a+b \tan (c+d x)} \left (-15 a^3-24 a^2 b \tan (c+d x)+a \left (35 a^2-3 b^2\right ) \tan ^2(c+d x)+2 b \left (70 a^2+3 b^2\right ) \tan ^3(c+d x)\right )}{a^2 \tan ^{\frac {7}{2}}(c+d x)}}{105 d} \] Input:

Integrate[(a + b*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(9/2),x]
 

Output:

(105*(-1)^(1/4)*Sqrt[-a + I*b]*(I*a + b)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b] 
*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - 105*(-1)^(3/4)*(a + I*b)^ 
(3/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[ 
c + d*x]]] + (2*Sqrt[a + b*Tan[c + d*x]]*(-15*a^3 - 24*a^2*b*Tan[c + d*x] 
+ a*(35*a^2 - 3*b^2)*Tan[c + d*x]^2 + 2*b*(70*a^2 + 3*b^2)*Tan[c + d*x]^3) 
)/(a^2*Tan[c + d*x]^(7/2)))/(105*d)
 

Rubi [A] (verified)

Time = 1.93 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.15, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.760, Rules used = {3042, 4050, 27, 3042, 4132, 27, 3042, 4132, 27, 3042, 4132, 27, 3042, 4099, 3042, 4098, 104, 216, 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.

\(\displaystyle \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {9}{2}}(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan (c+d x)^{9/2}}dx\)

\(\Big \downarrow \) 4050

\(\displaystyle -\frac {2}{7} \int -\frac {-6 a b \tan ^2(c+d x)-7 \left (a^2-b^2\right ) \tan (c+d x)+8 a b}{2 \tan ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx-\frac {2 a \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \int \frac {-6 a b \tan ^2(c+d x)-7 \left (a^2-b^2\right ) \tan (c+d x)+8 a b}{\tan ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx-\frac {2 a \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \int \frac {-6 a b \tan (c+d x)^2-7 \left (a^2-b^2\right ) \tan (c+d x)+8 a b}{\tan (c+d x)^{7/2} \sqrt {a+b \tan (c+d x)}}dx-\frac {2 a \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 4132

\(\displaystyle \frac {1}{7} \left (-\frac {2 \int \frac {70 b \tan (c+d x) a^2+32 b^2 \tan ^2(c+d x) a+\left (35 a^2-3 b^2\right ) a}{2 \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx}{5 a}-\frac {16 b \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )-\frac {2 a \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (-\frac {\int \frac {70 b \tan (c+d x) a^2+32 b^2 \tan ^2(c+d x) a+\left (35 a^2-3 b^2\right ) a}{\tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx}{5 a}-\frac {16 b \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )-\frac {2 a \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (-\frac {\int \frac {70 b \tan (c+d x) a^2+32 b^2 \tan (c+d x)^2 a+\left (35 a^2-3 b^2\right ) a}{\tan (c+d x)^{5/2} \sqrt {a+b \tan (c+d x)}}dx}{5 a}-\frac {16 b \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )-\frac {2 a \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 4132

\(\displaystyle \frac {1}{7} \left (-\frac {-\frac {2 \int -\frac {-105 \left (a^2-b^2\right ) \tan (c+d x) a^2-2 b \left (35 a^2-3 b^2\right ) \tan ^2(c+d x) a+2 b \left (70 a^2+3 b^2\right ) a}{2 \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx}{3 a}-\frac {2 \left (35 a^2-3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {16 b \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )-\frac {2 a \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (-\frac {\frac {\int \frac {-105 \left (a^2-b^2\right ) \tan (c+d x) a^2-2 b \left (35 a^2-3 b^2\right ) \tan ^2(c+d x) a+2 b \left (70 a^2+3 b^2\right ) a}{\tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}dx}{3 a}-\frac {2 \left (35 a^2-3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {16 b \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )-\frac {2 a \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (-\frac {\frac {\int \frac {-105 \left (a^2-b^2\right ) \tan (c+d x) a^2-2 b \left (35 a^2-3 b^2\right ) \tan (c+d x)^2 a+2 b \left (70 a^2+3 b^2\right ) a}{\tan (c+d x)^{3/2} \sqrt {a+b \tan (c+d x)}}dx}{3 a}-\frac {2 \left (35 a^2-3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {16 b \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )-\frac {2 a \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 4132

\(\displaystyle \frac {1}{7} \left (-\frac {\frac {-\frac {2 \int \frac {105 \left (2 b \tan (c+d x) a^4+\left (a^2-b^2\right ) a^3\right )}{2 \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx}{a}-\frac {4 b \left (70 a^2+3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {2 \left (35 a^2-3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {16 b \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )-\frac {2 a \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (-\frac {\frac {-\frac {105 \int \frac {2 b \tan (c+d x) a^4+\left (a^2-b^2\right ) a^3}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx}{a}-\frac {4 b \left (70 a^2+3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {2 \left (35 a^2-3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {16 b \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )-\frac {2 a \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (-\frac {\frac {-\frac {105 \int \frac {2 b \tan (c+d x) a^4+\left (a^2-b^2\right ) a^3}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx}{a}-\frac {4 b \left (70 a^2+3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}}{3 a}-\frac {2 \left (35 a^2-3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}}{5 a}-\frac {16 b \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )-\frac {2 a \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 4099

\(\displaystyle -\frac {2 a \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {1}{7} \left (-\frac {16 b \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {-\frac {2 \left (35 a^2-3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {-\frac {4 b \left (70 a^2+3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}-\frac {105 \left (\frac {1}{2} a^3 (a+i b)^2 \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} a^3 (a-i b)^2 \int \frac {i \tan (c+d x)+1}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx\right )}{a}}{3 a}}{5 a}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 a \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {1}{7} \left (-\frac {16 b \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {-\frac {2 \left (35 a^2-3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {-\frac {4 b \left (70 a^2+3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}-\frac {105 \left (\frac {1}{2} a^3 (a+i b)^2 \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} a^3 (a-i b)^2 \int \frac {i \tan (c+d x)+1}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx\right )}{a}}{3 a}}{5 a}\right )\)

\(\Big \downarrow \) 4098

\(\displaystyle -\frac {2 a \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {1}{7} \left (-\frac {16 b \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {-\frac {2 \left (35 a^2-3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {-\frac {4 b \left (70 a^2+3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}-\frac {105 \left (\frac {a^3 (a-i b)^2 \int \frac {1}{(1-i \tan (c+d x)) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{2 d}+\frac {a^3 (a+i b)^2 \int \frac {1}{(i \tan (c+d x)+1) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{2 d}\right )}{a}}{3 a}}{5 a}\right )\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {2 a \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {1}{7} \left (-\frac {16 b \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {-\frac {2 \left (35 a^2-3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {-\frac {4 b \left (70 a^2+3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}-\frac {105 \left (\frac {a^3 (a+i b)^2 \int \frac {1}{\frac {(i a-b) \tan (c+d x)}{a+b \tan (c+d x)}+1}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}}{d}+\frac {a^3 (a-i b)^2 \int \frac {1}{1-\frac {(i a+b) \tan (c+d x)}{a+b \tan (c+d x)}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}}{d}\right )}{a}}{3 a}}{5 a}\right )\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {2 a \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {1}{7} \left (-\frac {16 b \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {-\frac {2 \left (35 a^2-3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {-\frac {4 b \left (70 a^2+3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}-\frac {105 \left (\frac {a^3 (a-i b)^2 \int \frac {1}{1-\frac {(i a+b) \tan (c+d x)}{a+b \tan (c+d x)}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}}{d}+\frac {a^3 (a+i b)^2 \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {-b+i a}}\right )}{a}}{3 a}}{5 a}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {2 a \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {1}{7} \left (-\frac {16 b \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {-\frac {2 \left (35 a^2-3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {-\frac {4 b \left (70 a^2+3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}-\frac {105 \left (\frac {a^3 (a+i b)^2 \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {-b+i a}}+\frac {a^3 (a-i b)^2 \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {b+i a}}\right )}{a}}{3 a}}{5 a}\right )\)

Input:

Int[(a + b*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(9/2),x]
 

Output:

(-2*a*Sqrt[a + b*Tan[c + d*x]])/(7*d*Tan[c + d*x]^(7/2)) + ((-16*b*Sqrt[a 
+ b*Tan[c + d*x]])/(5*d*Tan[c + d*x]^(5/2)) - ((-2*(35*a^2 - 3*b^2)*Sqrt[a 
 + b*Tan[c + d*x]])/(3*d*Tan[c + d*x]^(3/2)) + ((-105*((a^3*(a + I*b)^2*Ar 
cTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I 
*a - b]*d) + (a^3*(a - I*b)^2*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/S 
qrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a + b]*d)))/a - (4*b*(70*a^2 + 3*b^2)*Sq 
rt[a + b*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]]))/(3*a))/(5*a))/7
 

Defintions of rubi rules used

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 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 216

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

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 3042

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

rule 4050

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

rule 4098

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[A^2/f   Subst[Int[(a + b*x)^m*((c + d*x)^n/(A - B*x)), x], x, Tan[e + f* 
x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && 
 NeQ[a^2 + b^2, 0] && EqQ[A^2 + B^2, 0]
 

rule 4099

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[(A + I*B)/2   Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 - I*T 
an[e + f*x]), x], x] + Simp[(A - I*B)/2   Int[(a + b*Tan[e + f*x])^m*(c + d 
*Tan[e + f*x])^n*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A 
, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[A^2 + B^2, 
0]
 

rule 4132

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) 
 + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + 
 f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + 
b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2))   Int[(a + b*Tan[e + 
f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* 
(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d 
)*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta 
n[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[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && 
!(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
 

Maple [B] (warning: unable to verify)

result has leaf size over 500,000. Avoiding possible recursion issues.

Time = 0.72 (sec) , antiderivative size = 1347641, normalized size of antiderivative = 5066.32

\[\text {output too large to display}\]

Input:

int((a+b*tan(d*x+c))^(3/2)/tan(d*x+c)^(9/2),x)
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3585 vs. \(2 (218) = 436\).

Time = 0.45 (sec) , antiderivative size = 3585, normalized size of antiderivative = 13.48 \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {9}{2}}(c+d x)} \, dx=\text {Too large to display} \] Input:

integrate((a+b*tan(d*x+c))^(3/2)/tan(d*x+c)^(9/2),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:

integrate((a+b*tan(d*x+c))**(3/2)/tan(d*x+c)**(9/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\tan \left (d x + c\right )^{\frac {9}{2}}} \,d x } \] Input:

integrate((a+b*tan(d*x+c))^(3/2)/tan(d*x+c)^(9/2),x, algorithm="maxima")
 

Output:

integrate((b*tan(d*x + c) + a)^(3/2)/tan(d*x + c)^(9/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {9}{2}}(c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+b*tan(d*x+c))^(3/2)/tan(d*x+c)^(9/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {9}{2}}(c+d x)} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}}{{\mathrm {tan}\left (c+d\,x\right )}^{9/2}} \,d x \] Input:

int((a + b*tan(c + d*x))^(3/2)/tan(c + d*x)^(9/2),x)
 

Output:

int((a + b*tan(c + d*x))^(3/2)/tan(c + d*x)^(9/2), x)
 

Reduce [F]

\[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {9}{2}}(c+d x)} \, dx=\frac {-140 \sqrt {\tan \left (d x +c \right )}\, \sqrt {a +\tan \left (d x +c \right ) b}\, \tan \left (d x +c \right )^{3} a^{2} b +12 \sqrt {\tan \left (d x +c \right )}\, \sqrt {a +\tan \left (d x +c \right ) b}\, \tan \left (d x +c \right )^{3} b^{3}+70 \sqrt {\tan \left (d x +c \right )}\, \sqrt {a +\tan \left (d x +c \right ) b}\, \tan \left (d x +c \right )^{2} a^{3}-6 \sqrt {\tan \left (d x +c \right )}\, \sqrt {a +\tan \left (d x +c \right ) b}\, \tan \left (d x +c \right )^{2} a \,b^{2}-48 \sqrt {\tan \left (d x +c \right )}\, \sqrt {a +\tan \left (d x +c \right ) b}\, \tan \left (d x +c \right ) a^{2} b -30 \sqrt {\tan \left (d x +c \right )}\, \sqrt {a +\tan \left (d x +c \right ) b}\, a^{3}-210 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \sqrt {a +\tan \left (d x +c \right ) b}}{\tan \left (d x +c \right )^{3} b +\tan \left (d x +c \right )^{2} a}d x \right ) \tan \left (d x +c \right )^{4} a^{3} b d +105 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \sqrt {a +\tan \left (d x +c \right ) b}}{\tan \left (d x +c \right )^{2} b +\tan \left (d x +c \right ) a}d x \right ) \tan \left (d x +c \right )^{4} a^{4} d -105 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \sqrt {a +\tan \left (d x +c \right ) b}}{\tan \left (d x +c \right )^{2} b +\tan \left (d x +c \right ) a}d x \right ) \tan \left (d x +c \right )^{4} a^{2} b^{2} d}{105 \tan \left (d x +c \right )^{4} a^{2} d} \] Input:

int((a+b*tan(d*x+c))^(3/2)/tan(d*x+c)^(9/2),x)
 

Output:

( - 140*sqrt(tan(c + d*x))*sqrt(tan(c + d*x)*b + a)*tan(c + d*x)**3*a**2*b 
 + 12*sqrt(tan(c + d*x))*sqrt(tan(c + d*x)*b + a)*tan(c + d*x)**3*b**3 + 7 
0*sqrt(tan(c + d*x))*sqrt(tan(c + d*x)*b + a)*tan(c + d*x)**2*a**3 - 6*sqr 
t(tan(c + d*x))*sqrt(tan(c + d*x)*b + a)*tan(c + d*x)**2*a*b**2 - 48*sqrt( 
tan(c + d*x))*sqrt(tan(c + d*x)*b + a)*tan(c + d*x)*a**2*b - 30*sqrt(tan(c 
 + d*x))*sqrt(tan(c + d*x)*b + a)*a**3 - 210*int((sqrt(tan(c + d*x))*sqrt( 
tan(c + d*x)*b + a))/(tan(c + d*x)**3*b + tan(c + d*x)**2*a),x)*tan(c + d* 
x)**4*a**3*b*d + 105*int((sqrt(tan(c + d*x))*sqrt(tan(c + d*x)*b + a))/(ta 
n(c + d*x)**2*b + tan(c + d*x)*a),x)*tan(c + d*x)**4*a**4*d - 105*int((sqr 
t(tan(c + d*x))*sqrt(tan(c + d*x)*b + a))/(tan(c + d*x)**2*b + tan(c + d*x 
)*a),x)*tan(c + d*x)**4*a**2*b**2*d)/(105*tan(c + d*x)**4*a**2*d)