\(\int \frac {(a+b \tan (e+f x))^3}{(c+d \tan (e+f x))^{5/2}} \, dx\) [1262]

Optimal result

Integrand size = 27, antiderivative size = 219 \[ \int \frac {(a+b \tan (e+f x))^3}{(c+d \tan (e+f x))^{5/2}} \, dx=\frac {(i a+b)^3 \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{5/2} f}-\frac {(i a-b)^3 \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(c+i d)^{5/2} f}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {4 (b c-a d)^2 \left (3 a c d+b \left (c^2+4 d^2\right )\right )}{3 d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}} \] Output:

(I*a+b)^3*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(c-I*d)^(5/2)/f-(I 
*a-b)^3*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/(c+I*d)^(5/2)/f-2/3* 
(-a*d+b*c)^2*(a+b*tan(f*x+e))/d/(c^2+d^2)/f/(c+d*tan(f*x+e))^(3/2)-4/3*(-a 
*d+b*c)^2*(3*a*c*d+b*(c^2+4*d^2))/d^2/(c^2+d^2)^2/f/(c+d*tan(f*x+e))^(1/2)
 

Mathematica [C] (verified)

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

Time = 1.06 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.30 \[ \int \frac {(a+b \tan (e+f x))^3}{(c+d \tan (e+f x))^{5/2}} \, dx=-\frac {4 b^3 c \left (c^2+d^2\right )-d \left (-3 a^2 b c+b^3 c+a^3 d-3 a b^2 d\right ) \left (i (c+i d) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {c+d \tan (e+f x)}{c-i d}\right )-(i c+d) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {c+d \tan (e+f x)}{c+i d}\right )\right )+6 b^2 (c-i d) (c+i d) d (a+b \tan (e+f x))-3 b \left (3 a^2-b^2\right ) d \left (i (c+i d) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {c+d \tan (e+f x)}{c-i d}\right )-(i c+d) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {c+d \tan (e+f x)}{c+i d}\right )\right ) (c+d \tan (e+f x))}{3 d^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}} \] Input:

Integrate[(a + b*Tan[e + f*x])^3/(c + d*Tan[e + f*x])^(5/2),x]
 

Output:

-1/3*(4*b^3*c*(c^2 + d^2) - d*(-3*a^2*b*c + b^3*c + a^3*d - 3*a*b^2*d)*(I* 
(c + I*d)*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c - I*d)] 
 - (I*c + d)*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c + I* 
d)]) + 6*b^2*(c - I*d)*(c + I*d)*d*(a + b*Tan[e + f*x]) - 3*b*(3*a^2 - b^2 
)*d*(I*(c + I*d)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c - 
 I*d)] - (I*c + d)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c 
 + I*d)])*(c + d*Tan[e + f*x]))/(d^2*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3 
/2))
 

Rubi [A] (warning: unable to verify)

Time = 1.38 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.13, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {3042, 4048, 27, 3042, 4111, 27, 3042, 4022, 3042, 4020, 25, 73, 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.

\(\displaystyle \int \frac {(a+b \tan (e+f x))^3}{(c+d \tan (e+f x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+b \tan (e+f x))^3}{(c+d \tan (e+f x))^{5/2}}dx\)

\(\Big \downarrow \) 4048

\(\displaystyle \frac {2 \int \frac {3 c d a^3+8 b d^2 a^2-7 b^2 c d a+2 b^3 c^2+b \left (\left (2 c^2+3 d^2\right ) b^2+2 a c d b-a^2 d^2\right ) \tan ^2(e+f x)+3 d \left (-d a^3+3 b c a^2+3 b^2 d a-b^3 c\right ) \tan (e+f x)}{2 (c+d \tan (e+f x))^{3/2}}dx}{3 d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {3 c d a^3+8 b d^2 a^2-7 b^2 c d a+2 b^3 c^2+b \left (\left (2 c^2+3 d^2\right ) b^2+2 a c d b-a^2 d^2\right ) \tan ^2(e+f x)+3 d \left (-d a^3+3 b c a^2+3 b^2 d a-b^3 c\right ) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{3 d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {3 c d a^3+8 b d^2 a^2-7 b^2 c d a+2 b^3 c^2+b \left (\left (2 c^2+3 d^2\right ) b^2+2 a c d b-a^2 d^2\right ) \tan (e+f x)^2+3 d \left (-d a^3+3 b c a^2+3 b^2 d a-b^3 c\right ) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}}dx}{3 d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 4111

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4022

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4020

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

\(\displaystyle -\frac {2 (b c-a d)^2 (a+b \tan (e+f x))}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}+\frac {-\frac {4 (b c-a d)^2 \left (3 a c d+b \left (c^2+4 d^2\right )\right )}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {3 \left (\frac {d (a-i b)^3 (c+i d)^2 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}+\frac {d (a+i b)^3 (c-i d)^2 \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}\right )}{c^2+d^2}}{3 d \left (c^2+d^2\right )}\)

Input:

Int[(a + b*Tan[e + f*x])^3/(c + d*Tan[e + f*x])^(5/2),x]
 

Output:

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

Defintions of rubi rules used

rule 25

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

rule 27

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

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, 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 4020

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

rule 4022

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

rule 4048

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m 
 - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Simp[1 
/(d*(n + 1)*(c^2 + d^2))   Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + 
f*x])^(n + 1)*Simp[a^2*d*(b*d*(m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c 
*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d) 
*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*( 
n + 1)))*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] && GtQ[m, 2] && LtQ 
[n, -1] && IntegerQ[2*m]
 

rule 4111

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(8960\) vs. \(2(193)=386\).

Time = 0.54 (sec) , antiderivative size = 8961, normalized size of antiderivative = 40.92

method	result	size
parts	\(\text {Expression too large to display}\)	\(8961\)
derivativedivides	\(\text {Expression too large to display}\)	\(26274\)
default	\(\text {Expression too large to display}\)	\(26274\)

Input:

int((a+b*tan(f*x+e))^3/(c+d*tan(f*x+e))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 14680 vs. \(2 (187) = 374\).

Time = 25.92 (sec) , antiderivative size = 14680, normalized size of antiderivative = 67.03 \[ \int \frac {(a+b \tan (e+f x))^3}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((a+b*tan(f*x+e))^3/(c+d*tan(f*x+e))^(5/2),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {(a+b \tan (e+f x))^3}{(c+d \tan (e+f x))^{5/2}} \, dx=\int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{3}}{\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((a+b*tan(f*x+e))**3/(c+d*tan(f*x+e))**(5/2),x)
 

Output:

Integral((a + b*tan(e + f*x))**3/(c + d*tan(e + f*x))**(5/2), x)
                                                                                    
                                                                                    
 

Maxima [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (e+f x))^3}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((a+b*tan(f*x+e))^3/(c+d*tan(f*x+e))^(5/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(a+b \tan (e+f x))^3}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+b*tan(f*x+e))^3/(c+d*tan(f*x+e))^(5/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%%{1,[6,14,5]%%%}+%%%{6,[6,12,5]%%%}+%%%{15,[6,10,5]%%%}+ 
%%%{20,[6
 

Mupad [B] (verification not implemented)

Time = 21.87 (sec) , antiderivative size = 34142, normalized size of antiderivative = 155.90 \[ \int \frac {(a+b \tan (e+f x))^3}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Too large to display} \] Input:

int((a + b*tan(e + f*x))^3/(c + d*tan(e + f*x))^(5/2),x)
 

Output:

- atan(((((((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f^2 + 48*a^5*b*d 
^5*f^2 + 40*a^6*c*d^4*f^2 - 40*b^6*c*d^4*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a 
^4*b^2*c^5*f^2 - 160*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c^3*d^2 
*f^2 - 1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*f^2 + 1200*a^4*b^2* 
c^3*d^2*f^2 + 240*a*b^5*c^4*d*f^2 + 240*a^5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^ 
3*f^2 + 600*a^2*b^4*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^4*b^2*c*d^4* 
f^2 - 480*a^5*b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8 
*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4)*(a^12 + b^12 + 
6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) - 4 
*a^6*c^5*f^2 + 4*b^6*c^5*f^2 - 24*a*b^5*d^5*f^2 - 24*a^5*b*d^5*f^2 - 20*a^ 
6*c*d^4*f^2 + 20*b^6*c*d^4*f^2 - 60*a^2*b^4*c^5*f^2 + 60*a^4*b^2*c^5*f^2 + 
 80*a^3*b^3*d^5*f^2 + 40*a^6*c^3*d^2*f^2 - 40*b^6*c^3*d^2*f^2 + 600*a^2*b^ 
4*c^3*d^2*f^2 - 800*a^3*b^3*c^2*d^3*f^2 - 600*a^4*b^2*c^3*d^2*f^2 - 120*a* 
b^5*c^4*d*f^2 - 120*a^5*b*c^4*d*f^2 + 240*a*b^5*c^2*d^3*f^2 - 300*a^2*b^4* 
c*d^4*f^2 + 400*a^3*b^3*c^4*d*f^2 + 300*a^4*b^2*c*d^4*f^2 + 240*a^5*b*c^2* 
d^3*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^ 
6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*((c + d*tan(e + f*x))^(1/2)*((((8*a^6*c 
^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^ 
4*f^2 - 40*b^6*c*d^4*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 160 
*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c^3*d^2*f^2 - 1200*a^2*b...
 

Reduce [F]

\[ \int \frac {(a+b \tan (e+f x))^3}{(c+d \tan (e+f x))^{5/2}} \, dx =\text {Too large to display} \] Input:

int((a+b*tan(f*x+e))^3/(c+d*tan(f*x+e))^(5/2),x)
 

Output:

( - 2*sqrt(tan(e + f*x)*d + c)*a**3 + 3*int((sqrt(tan(e + f*x)*d + c)*tan( 
e + f*x)**3)/(tan(e + f*x)**3*d**3 + 3*tan(e + f*x)**2*c*d**2 + 3*tan(e + 
f*x)*c**2*d + c**3),x)*tan(e + f*x)**2*b**3*d**3*f + 6*int((sqrt(tan(e + f 
*x)*d + c)*tan(e + f*x)**3)/(tan(e + f*x)**3*d**3 + 3*tan(e + f*x)**2*c*d* 
*2 + 3*tan(e + f*x)*c**2*d + c**3),x)*tan(e + f*x)*b**3*c*d**2*f + 3*int(( 
sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**3)/(tan(e + f*x)**3*d**3 + 3*tan(e 
+ f*x)**2*c*d**2 + 3*tan(e + f*x)*c**2*d + c**3),x)*b**3*c**2*d*f - 3*int( 
(sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**2)/(tan(e + f*x)**3*d**3 + 3*tan(e 
 + f*x)**2*c*d**2 + 3*tan(e + f*x)*c**2*d + c**3),x)*tan(e + f*x)**2*a**3* 
d**3*f + 9*int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**2)/(tan(e + f*x)**3 
*d**3 + 3*tan(e + f*x)**2*c*d**2 + 3*tan(e + f*x)*c**2*d + c**3),x)*tan(e 
+ f*x)**2*a*b**2*d**3*f - 6*int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**2) 
/(tan(e + f*x)**3*d**3 + 3*tan(e + f*x)**2*c*d**2 + 3*tan(e + f*x)*c**2*d 
+ c**3),x)*tan(e + f*x)*a**3*c*d**2*f + 18*int((sqrt(tan(e + f*x)*d + c)*t 
an(e + f*x)**2)/(tan(e + f*x)**3*d**3 + 3*tan(e + f*x)**2*c*d**2 + 3*tan(e 
 + f*x)*c**2*d + c**3),x)*tan(e + f*x)*a*b**2*c*d**2*f - 3*int((sqrt(tan(e 
 + f*x)*d + c)*tan(e + f*x)**2)/(tan(e + f*x)**3*d**3 + 3*tan(e + f*x)**2* 
c*d**2 + 3*tan(e + f*x)*c**2*d + c**3),x)*a**3*c**2*d*f + 9*int((sqrt(tan( 
e + f*x)*d + c)*tan(e + f*x)**2)/(tan(e + f*x)**3*d**3 + 3*tan(e + f*x)**2 
*c*d**2 + 3*tan(e + f*x)*c**2*d + c**3),x)*a*b**2*c**2*d*f + 9*int((sqr...