\(\int \frac {a B+b B \tan (c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx\) [366]

Optimal result

Integrand size = 28, antiderivative size = 303 \[ \int \frac {a B+b B \tan (c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\frac {b B \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}{a+\sqrt {a^2+b^2}+b \tan (c+d x)}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {b B \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}} d}-\frac {b B \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}} d} \] Output:

1/2*b*B*arctanh(2^(1/2)*(a+(a^2+b^2)^(1/2))^(1/2)*(a+b*tan(d*x+c))^(1/2)/( 
a+(a^2+b^2)^(1/2)+b*tan(d*x+c)))*2^(1/2)/(a^2+b^2)^(1/2)/(a+(a^2+b^2)^(1/2 
))^(1/2)/d+1/2*b*B*arctanh(((a+(a^2+b^2)^(1/2))^(1/2)-2^(1/2)*(a+b*tan(d*x 
+c))^(1/2))/(a-(a^2+b^2)^(1/2))^(1/2))*2^(1/2)/(a^2+b^2)^(1/2)/(a-(a^2+b^2 
)^(1/2))^(1/2)/d-1/2*b*B*arctanh(((a+(a^2+b^2)^(1/2))^(1/2)+2^(1/2)*(a+b*t 
an(d*x+c))^(1/2))/(a-(a^2+b^2)^(1/2))^(1/2))*2^(1/2)/(a^2+b^2)^(1/2)/(a-(a 
^2+b^2)^(1/2))^(1/2)/d
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.02 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.29 \[ \int \frac {a B+b B \tan (c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=-\frac {i B \left (\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b}}\right )}{d} \] Input:

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

Output:

((-I)*B*(ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/Sqrt[a - I*b] - A 
rcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/Sqrt[a + I*b]))/d
 

Rubi [A] (warning: unable to verify)

Time = 0.64 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.41, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {2011, 3042, 3966, 484, 1407, 1142, 25, 27, 1083, 219, 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.

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

\(\Big \downarrow \) 2011

\(\displaystyle B \int \frac {1}{\sqrt {a+b \tan (c+d x)}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle B \int \frac {1}{\sqrt {a+b \tan (c+d x)}}dx\)

\(\Big \downarrow \) 3966

\(\displaystyle \frac {b B \int \frac {1}{\sqrt {a+b \tan (c+d x)} \left (\tan ^2(c+d x) b^2+b^2\right )}d(b \tan (c+d x))}{d}\)

\(\Big \downarrow \) 484

\(\displaystyle \frac {2 b B \int \frac {1}{b^4 \tan ^4(c+d x)-2 a b^2 \tan ^2(c+d x)+a^2+b^2}d\sqrt {a+b \tan (c+d x)}}{d}\)

\(\Big \downarrow \) 1407

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

\(\Big \downarrow \) 1142

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1083

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {2 b B \left (\frac {\frac {\int \frac {\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}-\frac {\sqrt {\sqrt {a^2+b^2}+a} \text {arctanh}\left (\frac {2 \sqrt {a+b \tan (c+d x)}-\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {a-\sqrt {a^2+b^2}}}}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}+\frac {\frac {\int \frac {\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{b^2 \tan ^2(c+d x)+\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}}d\sqrt {a+b \tan (c+d x)}}{\sqrt {2}}-\frac {\sqrt {\sqrt {a^2+b^2}+a} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}+2 \sqrt {a+b \tan (c+d x)}}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {a-\sqrt {a^2+b^2}}}}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}\right )}{d}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {2 b B \left (\frac {-\frac {\sqrt {\sqrt {a^2+b^2}+a} \text {arctanh}\left (\frac {2 \sqrt {a+b \tan (c+d x)}-\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {a-\sqrt {a^2+b^2}}}-\frac {1}{2} \log \left (-\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \tan (c+d x)}+\sqrt {a^2+b^2}+b^2 \tan ^2(c+d x)\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}+\frac {\frac {1}{2} \log \left (\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \tan (c+d x)}+\sqrt {a^2+b^2}+b^2 \tan ^2(c+d x)\right )-\frac {\sqrt {\sqrt {a^2+b^2}+a} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a}+2 \sqrt {a+b \tan (c+d x)}}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {a-\sqrt {a^2+b^2}}}}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}\right )}{d}\)

Input:

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

Output:

(2*b*B*((-((Sqrt[a + Sqrt[a^2 + b^2]]*ArcTanh[(-(Sqrt[2]*Sqrt[a + Sqrt[a^2 
 + b^2]]) + 2*Sqrt[a + b*Tan[c + d*x]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]] 
)])/Sqrt[a - Sqrt[a^2 + b^2]]) - Log[Sqrt[a^2 + b^2] + b^2*Tan[c + d*x]^2 
- Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]]/2)/(2*Sqrt[2 
]*Sqrt[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b^2]]) + (-((Sqrt[a + Sqrt[a^2 + b^2 
]]*ArcTanh[(Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]] + 2*Sqrt[a + b*Tan[c + d*x]] 
)/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]])])/Sqrt[a - Sqrt[a^2 + b^2]]) + Log[S 
qrt[a^2 + b^2] + b^2*Tan[c + d*x]^2 + Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sq 
rt[a + b*Tan[c + d*x]]]/2)/(2*Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + 
b^2]])))/d
 

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 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 484

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^2)), x_Symbol] :> Simp[2* 
d   Subst[Int[1/(b*c^2 + a*d^2 - 2*b*c*x^2 + b*x^4), x], x, Sqrt[c + d*x]], 
 x] /; FreeQ[{a, b, c, d}, x]
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{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 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 1407

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/ 
c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r)   Int[(r - x)/(q - r* 
x + x^2), x], x] + Simp[1/(2*c*q*r)   Int[(r + x)/(q + r*x + x^2), x], x]]] 
 /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]
 

rule 2011

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> 
 Simp[(b/d)^m   Int[u*(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c + d*x 
, a + b*x])
 

rule 3042

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

rule 3966

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d   Su 
bst[Int[(a + x)^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c 
, d, n}, x] && NeQ[a^2 + b^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1574\) vs. \(2(247)=494\).

Time = 0.12 (sec) , antiderivative size = 1575, normalized size of antiderivative = 5.20

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1575\)
default	\(\text {Expression too large to display}\)	\(1575\)
parts	\(\text {Expression too large to display}\)	\(3681\)

Input:

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

Output:

1/4/d/b/(a^2+b^2)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1 
/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^2+1/4/d* 
b/(a^2+b^2)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2* 
a)^(1/2)+(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-1/4/d/b/(a^2+b^2 
)^(3/2)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^( 
1/2)+(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3-1/4/d*b/(a^2+b^2 
)^(3/2)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^( 
1/2)+(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a+3/d*b/(a^2+b^2)^(3 
/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2 
+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B*a^2+2/d*b^3/(a^2+ 
b^2)^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+ 
(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B-1/d/b/(a^2 
+b^2)^(1/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2) 
+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B*a^2-1/d*b 
/(a^2+b^2)^(1/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^ 
(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B+1/d/ 
b/(a^2+b^2)^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c)) 
^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B*a^4 
-1/4/d/b/(a^2+b^2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2) 
-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^2-1/...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 805 vs. \(2 (249) = 498\).

Time = 0.10 (sec) , antiderivative size = 805, normalized size of antiderivative = 2.66 \[ \int \frac {a B+b B \tan (c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

1/2*sqrt(-((a^2 + b^2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + 
B^2*a)/((a^2 + b^2)*d^2))*log(sqrt(b*tan(d*x + c) + a)*B^3*b + (B^2*b^2*d 
+ (a^3 + a*b^2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^3)*sqrt(-(( 
a^2 + b^2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + B^2*a)/((a^2 
 + b^2)*d^2))) - 1/2*sqrt(-((a^2 + b^2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*b^2 + 
b^4)*d^4))*d^2 + B^2*a)/((a^2 + b^2)*d^2))*log(sqrt(b*tan(d*x + c) + a)*B^ 
3*b - (B^2*b^2*d + (a^3 + a*b^2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^ 
4))*d^3)*sqrt(-((a^2 + b^2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d 
^2 + B^2*a)/((a^2 + b^2)*d^2))) + 1/2*sqrt(((a^2 + b^2)*sqrt(-B^4*b^2/((a^ 
4 + 2*a^2*b^2 + b^4)*d^4))*d^2 - B^2*a)/((a^2 + b^2)*d^2))*log(sqrt(b*tan( 
d*x + c) + a)*B^3*b + (B^2*b^2*d - (a^3 + a*b^2)*sqrt(-B^4*b^2/((a^4 + 2*a 
^2*b^2 + b^4)*d^4))*d^3)*sqrt(((a^2 + b^2)*sqrt(-B^4*b^2/((a^4 + 2*a^2*b^2 
 + b^4)*d^4))*d^2 - B^2*a)/((a^2 + b^2)*d^2))) - 1/2*sqrt(((a^2 + b^2)*sqr 
t(-B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 - B^2*a)/((a^2 + b^2)*d^2))* 
log(sqrt(b*tan(d*x + c) + a)*B^3*b - (B^2*b^2*d - (a^3 + a*b^2)*sqrt(-B^4* 
b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^3)*sqrt(((a^2 + b^2)*sqrt(-B^4*b^2/(( 
a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 - B^2*a)/((a^2 + b^2)*d^2)))
 

Sympy [F]

\[ \int \frac {a B+b B \tan (c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=B \int \frac {1}{\sqrt {a + b \tan {\left (c + d x \right )}}}\, dx \] Input:

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

Output:

B*Integral(1/sqrt(a + b*tan(c + d*x)), x)
 

Maxima [F(-2)]

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

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

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(b-a>0)', see `assume?` for more 
details)Is
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 8.01 (sec) , antiderivative size = 6453, normalized size of antiderivative = 21.30 \[ \int \frac {a B+b B \tan (c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

log(((a + b*tan(c + d*x))^(1/2)*(16*B^2*a^2*b^10*d^3 + 32*B^2*a^4*b^8*d^3 
- 32*B^2*a^8*b^4*d^3 - 16*B^2*a^10*b^2*d^3) - ((((8*B^2*a^5*d^2 - 24*B^2*a 
^3*b^2*d^2)^2/4 - B^4*a^4*(16*a^6*d^4 + 16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a 
^4*b^2*d^4))^(1/2) - 4*B^2*a^5*d^2 + 12*B^2*a^3*b^2*d^2)/(16*(a^6*d^4 + b^ 
6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2)*((a + b*tan(c + d*x))^(1/2) 
*((((8*B^2*a^5*d^2 - 24*B^2*a^3*b^2*d^2)^2/4 - B^4*a^4*(16*a^6*d^4 + 16*b^ 
6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*d^4))^(1/2) - 4*B^2*a^5*d^2 + 12*B^2*a 
^3*b^2*d^2)/(16*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2 
)*(64*a*b^12*d^5 + 320*a^3*b^10*d^5 + 640*a^5*b^8*d^5 + 640*a^7*b^6*d^5 + 
320*a^9*b^4*d^5 + 64*a^11*b^2*d^5) + 64*B*a^2*b^11*d^4 + 256*B*a^4*b^9*d^4 
 + 384*B*a^6*b^7*d^4 + 256*B*a^8*b^5*d^4 + 64*B*a^10*b^3*d^4))*((((8*B^2*a 
^5*d^2 - 24*B^2*a^3*b^2*d^2)^2/4 - B^4*a^4*(16*a^6*d^4 + 16*b^6*d^4 + 48*a 
^2*b^4*d^4 + 48*a^4*b^2*d^4))^(1/2) - 4*B^2*a^5*d^2 + 12*B^2*a^3*b^2*d^2)/ 
(16*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2) + 8*B^3*a^ 
3*b^9*d^2 + 24*B^3*a^5*b^7*d^2 + 24*B^3*a^7*b^5*d^2 + 8*B^3*a^9*b^3*d^2)*( 
(((8*B^2*a^5*d^2 - 24*B^2*a^3*b^2*d^2)^2/4 - B^4*a^4*(16*a^6*d^4 + 16*b^6* 
d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*d^4))^(1/2) - 4*B^2*a^5*d^2 + 12*B^2*a^3 
*b^2*d^2)/(16*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2) 
- log(8*B^3*a^3*b^9*d^2 - (-((96*B^4*a^6*b^4*d^4 - 16*B^4*a^4*b^6*d^4 - 14 
4*B^4*a^8*b^2*d^4)^(1/2) + 4*B^2*a^5*d^2 - 12*B^2*a^3*b^2*d^2)/(16*a^6*...
 

Reduce [F]

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

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

Output:

(2*sqrt(tan(c + d*x)*b + a) - int((sqrt(tan(c + d*x)*b + a)*tan(c + d*x)** 
2)/(tan(c + d*x)*b + a),x)*b*d)/d