\(\int \frac {(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^{7/2}} \, dx\) [1292]

Optimal result

Integrand size = 29, antiderivative size = 391 \[ \int \frac {(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^{7/2}} \, dx=-\frac {i (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a-i b)^{7/2} f}+\frac {i (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{7/2} f}-\frac {2 (b c-a d) \sqrt {c+d \tan (e+f x)}}{5 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}-\frac {4 \left (5 a b c-2 a^2 d+3 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{15 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}+\frac {2 \left (50 a^3 b c d-70 a b^3 c d-8 a^4 d^2-a^2 b^2 \left (45 c^2-49 d^2\right )+3 b^4 \left (5 c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{15 \left (a^2+b^2\right )^3 (b c-a d) f \sqrt {a+b \tan (e+f x)}} \] Output:

-I*(c-I*d)^(3/2)*arctanh((c-I*d)^(1/2)*(a+b*tan(f*x+e))^(1/2)/(a-I*b)^(1/2 
)/(c+d*tan(f*x+e))^(1/2))/(a-I*b)^(7/2)/f+I*(c+I*d)^(3/2)*arctanh((c+I*d)^ 
(1/2)*(a+b*tan(f*x+e))^(1/2)/(a+I*b)^(1/2)/(c+d*tan(f*x+e))^(1/2))/(a+I*b) 
^(7/2)/f-2/5*(-a*d+b*c)*(c+d*tan(f*x+e))^(1/2)/(a^2+b^2)/f/(a+b*tan(f*x+e) 
)^(5/2)-4/15*(-2*a^2*d+5*a*b*c+3*b^2*d)*(c+d*tan(f*x+e))^(1/2)/(a^2+b^2)^2 
/f/(a+b*tan(f*x+e))^(3/2)+2/15*(50*a^3*b*c*d-70*a*b^3*c*d-8*a^4*d^2-a^2*b^ 
2*(45*c^2-49*d^2)+3*b^4*(5*c^2-d^2))*(c+d*tan(f*x+e))^(1/2)/(a^2+b^2)^3/(- 
a*d+b*c)/f/(a+b*tan(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 6.04 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.15 \[ \int \frac {(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^{7/2}} \, dx=\frac {-\frac {3 b (c+d \tan (e+f x))^{5/2}}{(i a+b) (b c-a d) (a+b \tan (e+f x))^{5/2}}+\frac {3 i b (c+d \tan (e+f x))^{5/2}}{(a+i b) (-b c+a d) (a+b \tan (e+f x))^{5/2}}+\frac {5 \left (-\frac {3 (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{5/2}}+\frac {\sqrt {c+d \tan (e+f x)} (4 a c+i b c+3 i a d+(3 b c+a d+4 i b d) \tan (e+f x))}{(a+i b)^2 (a+b \tan (e+f x))^{3/2}}\right )}{i a-b}+\frac {5 i \left (\frac {(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^{3/2}}+3 (c-i d) \left (\frac {\sqrt {-c+i d} \text {arctanh}\left (\frac {\sqrt {-c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {-a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(-a+i b)^{3/2}}+\frac {\sqrt {c+d \tan (e+f x)}}{(a-i b) \sqrt {a+b \tan (e+f x)}}\right )\right )}{(a-i b)^2}}{15 f} \] Input:

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

Output:

((-3*b*(c + d*Tan[e + f*x])^(5/2))/((I*a + b)*(b*c - a*d)*(a + b*Tan[e + f 
*x])^(5/2)) + ((3*I)*b*(c + d*Tan[e + f*x])^(5/2))/((a + I*b)*(-(b*c) + a* 
d)*(a + b*Tan[e + f*x])^(5/2)) + (5*((-3*(c + I*d)^(3/2)*ArcTanh[(Sqrt[c + 
 I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])]) 
/(a + I*b)^(5/2) + (Sqrt[c + d*Tan[e + f*x]]*(4*a*c + I*b*c + (3*I)*a*d + 
(3*b*c + a*d + (4*I)*b*d)*Tan[e + f*x]))/((a + I*b)^2*(a + b*Tan[e + f*x]) 
^(3/2))))/(I*a - b) + ((5*I)*((c + d*Tan[e + f*x])^(3/2)/(a + b*Tan[e + f* 
x])^(3/2) + 3*(c - I*d)*((Sqrt[-c + I*d]*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + 
b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(-a + I*b)^(3 
/2) + Sqrt[c + d*Tan[e + f*x]]/((a - I*b)*Sqrt[a + b*Tan[e + f*x]]))))/(a 
- I*b)^2)/(15*f)
 

Rubi [A] (verified)

Time = 2.94 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.22, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {3042, 4050, 27, 3042, 4132, 27, 3042, 4132, 27, 3042, 4099, 3042, 4098, 104, 221}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4050

\(\displaystyle -\frac {2 \int -\frac {-4 d (b c-a d) \tan ^2(e+f x)+5 \left (2 a c d-b \left (c^2-d^2\right )\right ) \tan (e+f x)+6 b c d+a \left (5 c^2-d^2\right )}{2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}dx}{5 \left (a^2+b^2\right )}-\frac {2 (b c-a d) \sqrt {c+d \tan (e+f x)}}{5 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {5 a c^2+6 b d c-a d^2-4 d (b c-a d) \tan ^2(e+f x)+5 \left (2 a c d-b \left (c^2-d^2\right )\right ) \tan (e+f x)}{(a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}dx}{5 \left (a^2+b^2\right )}-\frac {2 (b c-a d) \sqrt {c+d \tan (e+f x)}}{5 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {5 a c^2+6 b d c-a d^2-4 d (b c-a d) \tan (e+f x)^2+5 \left (2 a c d-b \left (c^2-d^2\right )\right ) \tan (e+f x)}{(a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}dx}{5 \left (a^2+b^2\right )}-\frac {2 (b c-a d) \sqrt {c+d \tan (e+f x)}}{5 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\)

\(\Big \downarrow \) 4132

\(\displaystyle \frac {-\frac {2 \int -\frac {-30 (a c+b d) \tan (e+f x) (b c-a d)^2-4 d \left (-2 d a^2+5 b c a+3 b^2 d\right ) \tan ^2(e+f x) (b c-a d)+\left (\left (15 c^2-7 d^2\right ) a^2+40 b c d a-3 b^2 \left (5 c^2-d^2\right )\right ) (b c-a d)}{2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}dx}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {4 \left (-2 a^2 d+5 a b c+3 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{3 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}}{5 \left (a^2+b^2\right )}-\frac {2 (b c-a d) \sqrt {c+d \tan (e+f x)}}{5 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {-30 (a c+b d) \tan (e+f x) (b c-a d)^2-4 d \left (-2 d a^2+5 b c a+3 b^2 d\right ) \tan ^2(e+f x) (b c-a d)+\left (\left (15 c^2-7 d^2\right ) a^2+40 b c d a-3 b^2 \left (5 c^2-d^2\right )\right ) (b c-a d)}{(a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}dx}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {4 \left (-2 a^2 d+5 a b c+3 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{3 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}}{5 \left (a^2+b^2\right )}-\frac {2 (b c-a d) \sqrt {c+d \tan (e+f x)}}{5 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {-30 (a c+b d) \tan (e+f x) (b c-a d)^2-4 d \left (-2 d a^2+5 b c a+3 b^2 d\right ) \tan (e+f x)^2 (b c-a d)+\left (\left (15 c^2-7 d^2\right ) a^2+40 b c d a-3 b^2 \left (5 c^2-d^2\right )\right ) (b c-a d)}{(a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}dx}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {4 \left (-2 a^2 d+5 a b c+3 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{3 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}}{5 \left (a^2+b^2\right )}-\frac {2 (b c-a d) \sqrt {c+d \tan (e+f x)}}{5 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\)

\(\Big \downarrow \) 4132

\(\displaystyle \frac {\frac {\frac {2 \left (-8 a^4 d^2+50 a^3 b c d-a^2 b^2 \left (45 c^2-49 d^2\right )-70 a b^3 c d+3 b^4 \left (5 c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}-\frac {2 \int -\frac {15 \left (\left (\left (c^2-d^2\right ) a^3+6 b c d a^2-3 b^2 \left (c^2-d^2\right ) a-2 b^3 c d\right ) (b c-a d)^2+\left (2 c d a^3-3 b \left (c^2-d^2\right ) a^2-6 b^2 c d a+b^3 \left (c^2-d^2\right )\right ) \tan (e+f x) (b c-a d)^2\right )}{2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {4 \left (-2 a^2 d+5 a b c+3 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{3 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}}{5 \left (a^2+b^2\right )}-\frac {2 (b c-a d) \sqrt {c+d \tan (e+f x)}}{5 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {15 \int \frac {\left (\left (c^2-d^2\right ) a^3+6 b c d a^2-3 b^2 \left (c^2-d^2\right ) a-2 b^3 c d\right ) (b c-a d)^2+\left (2 c d a^3-3 b \left (c^2-d^2\right ) a^2-6 b^2 c d a+b^3 \left (c^2-d^2\right )\right ) \tan (e+f x) (b c-a d)^2}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}+\frac {2 \left (-8 a^4 d^2+50 a^3 b c d-a^2 b^2 \left (45 c^2-49 d^2\right )-70 a b^3 c d+3 b^4 \left (5 c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {4 \left (-2 a^2 d+5 a b c+3 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{3 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}}{5 \left (a^2+b^2\right )}-\frac {2 (b c-a d) \sqrt {c+d \tan (e+f x)}}{5 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {15 \int \frac {\left (\left (c^2-d^2\right ) a^3+6 b c d a^2-3 b^2 \left (c^2-d^2\right ) a-2 b^3 c d\right ) (b c-a d)^2+\left (2 c d a^3-3 b \left (c^2-d^2\right ) a^2-6 b^2 c d a+b^3 \left (c^2-d^2\right )\right ) \tan (e+f x) (b c-a d)^2}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}+\frac {2 \left (-8 a^4 d^2+50 a^3 b c d-a^2 b^2 \left (45 c^2-49 d^2\right )-70 a b^3 c d+3 b^4 \left (5 c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {4 \left (-2 a^2 d+5 a b c+3 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{3 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}}{5 \left (a^2+b^2\right )}-\frac {2 (b c-a d) \sqrt {c+d \tan (e+f x)}}{5 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\)

\(\Big \downarrow \) 4099

\(\displaystyle -\frac {2 (b c-a d) \sqrt {c+d \tan (e+f x)}}{5 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}+\frac {-\frac {4 \left (-2 a^2 d+5 a b c+3 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{3 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac {\frac {2 \left (-8 a^4 d^2+50 a^3 b c d-a^2 b^2 \left (45 c^2-49 d^2\right )-70 a b^3 c d+3 b^4 \left (5 c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}+\frac {15 \left (\frac {1}{2} (a-i b)^3 (c+i d)^2 (b c-a d)^2 \int \frac {1-i \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} (a+i b)^3 (c-i d)^2 (b c-a d)^2 \int \frac {i \tan (e+f x)+1}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 \left (a^2+b^2\right ) (b c-a d)}}{5 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 (b c-a d) \sqrt {c+d \tan (e+f x)}}{5 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}+\frac {-\frac {4 \left (-2 a^2 d+5 a b c+3 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{3 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac {\frac {2 \left (-8 a^4 d^2+50 a^3 b c d-a^2 b^2 \left (45 c^2-49 d^2\right )-70 a b^3 c d+3 b^4 \left (5 c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}+\frac {15 \left (\frac {1}{2} (a-i b)^3 (c+i d)^2 (b c-a d)^2 \int \frac {1-i \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} (a+i b)^3 (c-i d)^2 (b c-a d)^2 \int \frac {i \tan (e+f x)+1}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 \left (a^2+b^2\right ) (b c-a d)}}{5 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 4098

\(\displaystyle -\frac {2 (b c-a d) \sqrt {c+d \tan (e+f x)}}{5 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}+\frac {-\frac {4 \left (-2 a^2 d+5 a b c+3 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{3 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac {\frac {2 \left (-8 a^4 d^2+50 a^3 b c d-a^2 b^2 \left (45 c^2-49 d^2\right )-70 a b^3 c d+3 b^4 \left (5 c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}+\frac {15 \left (\frac {(a-i b)^3 (c+i d)^2 (b c-a d)^2 \int \frac {1}{(i \tan (e+f x)+1) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{2 f}+\frac {(a+i b)^3 (c-i d)^2 (b c-a d)^2 \int \frac {1}{(1-i \tan (e+f x)) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{2 f}\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 \left (a^2+b^2\right ) (b c-a d)}}{5 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {2 (b c-a d) \sqrt {c+d \tan (e+f x)}}{5 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}+\frac {-\frac {4 \left (-2 a^2 d+5 a b c+3 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{3 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac {\frac {2 \left (-8 a^4 d^2+50 a^3 b c d-a^2 b^2 \left (45 c^2-49 d^2\right )-70 a b^3 c d+3 b^4 \left (5 c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}+\frac {15 \left (\frac {(a-i b)^3 (c+i d)^2 (b c-a d)^2 \int \frac {1}{-i a+b+\frac {(i c-d) (a+b \tan (e+f x))}{c+d \tan (e+f x)}}d\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}}{f}+\frac {(a+i b)^3 (c-i d)^2 (b c-a d)^2 \int \frac {1}{i a+b-\frac {(i c+d) (a+b \tan (e+f x))}{c+d \tan (e+f x)}}d\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}}{f}\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 \left (a^2+b^2\right ) (b c-a d)}}{5 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {2 (b c-a d) \sqrt {c+d \tan (e+f x)}}{5 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}+\frac {-\frac {4 \left (-2 a^2 d+5 a b c+3 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{3 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac {\frac {2 \left (-8 a^4 d^2+50 a^3 b c d-a^2 b^2 \left (45 c^2-49 d^2\right )-70 a b^3 c d+3 b^4 \left (5 c^2-d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}+\frac {15 \left (\frac {i (a-i b)^3 (c+i d)^{3/2} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a+i b}}-\frac {i (a+i b)^3 (c-i d)^{3/2} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a-i b}}\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 \left (a^2+b^2\right ) (b c-a d)}}{5 \left (a^2+b^2\right )}\)

Input:

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

Output:

(-2*(b*c - a*d)*Sqrt[c + d*Tan[e + f*x]])/(5*(a^2 + b^2)*f*(a + b*Tan[e + 
f*x])^(5/2)) + ((-4*(5*a*b*c - 2*a^2*d + 3*b^2*d)*Sqrt[c + d*Tan[e + f*x]] 
)/(3*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^(3/2)) + ((15*(((-I)*(a + I*b)^3*( 
c - I*d)^(3/2)*(b*c - a*d)^2*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x 
]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a - I*b]*f) + (I*(a - 
 I*b)^3*(c + I*d)^(3/2)*(b*c - a*d)^2*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Ta 
n[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*f)) 
)/((a^2 + b^2)*(b*c - a*d)) + (2*(50*a^3*b*c*d - 70*a*b^3*c*d - 8*a^4*d^2 
- a^2*b^2*(45*c^2 - 49*d^2) + 3*b^4*(5*c^2 - d^2))*Sqrt[c + d*Tan[e + f*x] 
])/((a^2 + b^2)*f*Sqrt[a + b*Tan[e + f*x]]))/(3*(a^2 + b^2)*(b*c - a*d)))/ 
(5*(a^2 + b^2))
 

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 221

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

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 [F(-1)]

Timed out.

\[\int \frac {\left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{\left (a +b \tan \left (f x +e \right )\right )^{\frac {7}{2}}}d x\]

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 28332 vs. \(2 (329) = 658\).

Time = 30.96 (sec) , antiderivative size = 28332, normalized size of antiderivative = 72.46 \[ \int \frac {(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((c+d*tan(f*x+e))^(3/2)/(a+b*tan(f*x+e))^(7/2),x, algorithm="fric 
as")
 

Output:

Too large to include
 

Sympy [F]

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

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

Output:

Integral((c + d*tan(e + f*x))**(3/2)/(a + b*tan(e + f*x))**(7/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^{7/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c+d*tan(f*x+e))^(3/2)/(a+b*tan(f*x+e))^(7/2),x, algorithm="maxi 
ma")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(((2*b*d+2*a*c)^2>0)', see `assum 
e?` for mo
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^{7/2}} \, dx=\text {too large to display} \] Input:

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

Output:

(16*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)**2*b**2 
*d**4 + 40*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)* 
a*b*d**4 - 8*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x 
)*b**2*c*d**3 + 30*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*a**2* 
d**4 - 20*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*a*b*c*d**3 + 6 
*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*b**2*c**2*d**2 + 30*int 
((sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x))/(tan(e + 
 f*x)**5*b**4*d + 4*tan(e + f*x)**4*a*b**3*d + tan(e + f*x)**4*b**4*c + 6* 
tan(e + f*x)**3*a**2*b**2*d + 4*tan(e + f*x)**3*a*b**3*c + 4*tan(e + f*x)* 
*2*a**3*b*d + 6*tan(e + f*x)**2*a**2*b**2*c + tan(e + f*x)*a**4*d + 4*tan( 
e + f*x)*a**3*b*c + a**4*c),x)*tan(e + f*x)**3*a**3*b**3*c*d**4*f - 90*int 
((sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x))/(tan(e + 
 f*x)**5*b**4*d + 4*tan(e + f*x)**4*a*b**3*d + tan(e + f*x)**4*b**4*c + 6* 
tan(e + f*x)**3*a**2*b**2*d + 4*tan(e + f*x)**3*a*b**3*c + 4*tan(e + f*x)* 
*2*a**3*b*d + 6*tan(e + f*x)**2*a**2*b**2*c + tan(e + f*x)*a**4*d + 4*tan( 
e + f*x)*a**3*b*c + a**4*c),x)*tan(e + f*x)**3*a**2*b**4*c**2*d**3*f + 90* 
int((sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x))/(tan( 
e + f*x)**5*b**4*d + 4*tan(e + f*x)**4*a*b**3*d + tan(e + f*x)**4*b**4*c + 
 6*tan(e + f*x)**3*a**2*b**2*d + 4*tan(e + f*x)**3*a*b**3*c + 4*tan(e + f* 
x)**2*a**3*b*d + 6*tan(e + f*x)**2*a**2*b**2*c + tan(e + f*x)*a**4*d + ...