3.4.0 ∫ aB+bB tan(c+dx) _ (a+b tan(c+dx))3∕2 dx [366]

Integrand size = 28, antiderivative size = 406 \[ \int \frac {a B+b B \tan (c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\frac {b B \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}} d}-\frac {b B \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}} d}-\frac {b B \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {b B \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d} \]

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))/d*2^(1/2

)/(a^2+b^2)^(1/2)/(a-(a^2+b^2)^(1/2))^(1/2)-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))/d*2^(1/2)/(a^2+b^2)^(1/2)/(a-(a^2+b^2)^(1/2))^(1/2)-1/4*b*B*ln(a+(a^2+b^2)

^(1/2)-2^(1/2)*(a+(a^2+b^2)^(1/2))^(1/2)*(a+b*tan(d*x+c))^(1/2)+b*tan(d*x+c))/d*2^(1/2)/(a^2+b^2)^(1/2)/(a+(a^

2+b^2)^(1/2))^(1/2)+1/4*b*B*ln(a+(a^2+b^2)^(1/2)+2^(1/2)*(a+(a^2+b^2)^(1/2))^(1/2)*(a+b*tan(d*x+c))^(1/2)+b*ta

n(d*x+c))/d*2^(1/2)/(a^2+b^2)^(1/2)/(a+(a^2+b^2)^(1/2))^(1/2)

Rubi [A] (veriﬁed)

Time = 0.59 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {21, 3566, 722, 1108, 648, 632, 212, 642} \[ \int \frac {a B+b B \tan (c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\frac {b B \text {arctanh}\left (\frac {\sqrt {\sqrt {a^2+b^2}+a}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} d \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}}}-\frac {b B \text {arctanh}\left (\frac {\sqrt {\sqrt {a^2+b^2}+a}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} d \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}}}-\frac {b B \log \left (-\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \tan (c+d x)}+\sqrt {a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt {2} d \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}+\frac {b B \log \left (\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \tan (c+d x)}+\sqrt {a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt {2} d \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}} \]

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

(b*B*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[

2]*Sqrt[a^2 + b^2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) - (b*B*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqrt[a + b

*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) - (b*B*Log[a

+ Sqrt[a^2 + b^2] + b*Tan[c + d*x] - Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*

Sqrt[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) + (b*B*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] + Sqrt[2]*Sqrt[a

+ Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b^2]]*d)

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(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 +

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] && (GtQ[a, 0] || LtQ[b, 0])

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

, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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]

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +

b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &

& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && !NiceSqrtQ[b^2 - 4*a*c]

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(c*d^2 + a*e^2 - 2*c

*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]

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]}

, Dist[1/(2*c*q*r), Int[(r - x)/(q - r*x + x^2), x], x] + Dist[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]

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[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]

Rubi steps \begin{align*} \text {integral}& = B \int \frac {1}{\sqrt {a+b \tan (c+d x)}} \, dx \\ & = \frac {(b B) \text {Subst}\left (\int \frac {1}{\sqrt {a+x} \left (b^2+x^2\right )} \, dx,x,b \tan (c+d x)\right )}{d} \\ & = \frac {(2 b B) \text {Subst}\left (\int \frac {1}{a^2+b^2-2 a x^2+x^4} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{d} \\ & = \frac {(b B) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}-x}{\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+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 {Subst}\left (\int \frac {\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+x}{\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+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 {Subst}\left (\int \frac {1}{\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {a^2+b^2} d}+\frac {(b B) \text {Subst}\left (\int \frac {1}{\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {a^2+b^2} d}-\frac {(b B) \text {Subst}\left (\int \frac {-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 x}{\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {(b B) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 x}{\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d} \\ & = -\frac {b B \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {b B \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {(b B) \text {Subst}\left (\int \frac {1}{2 \left (a-\sqrt {a^2+b^2}\right )-x^2} \, dx,x,-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 \sqrt {a+b \tan (c+d x)}\right )}{\sqrt {a^2+b^2} d}-\frac {(b B) \text {Subst}\left (\int \frac {1}{2 \left (a-\sqrt {a^2+b^2}\right )-x^2} \, dx,x,\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 \sqrt {a+b \tan (c+d x)}\right )}{\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}-\frac {b B \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {b B \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d} \\ \end{align*}

Mathematica [C] (veriﬁed)

Time = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.22 \[ \int \frac {a B+b B \tan (c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=-\frac {i B \left (\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b}}\right )}{d} \]

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

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

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(1574\) vs. \(2(327)=654\).

Time = 0.12 (sec) , antiderivative size = 1575, normalized size of antiderivative = 3.88


method	result	size

derivativedivides	\(\text {Expression too large to display}\)	\(1575\)
default	\(\text {Expression too large to display}\)	\(1575\)
parts	\(\text {Expression too large to display}\)	\(3686\)

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

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*ta

n(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^2-1/4/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)+1/4/d/b/(a^2+b^2)^(3/2)*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^3

+1/4/d*b/(a^2+b^2)^(3/2)*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-3/d*b/(a^2+b^2)^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^

(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(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^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(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^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*ta

n(d*x+c))^(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)*arct

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

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 805 vs. \(2 (329) = 658\).

Time = 0.26 (sec) , antiderivative size = 805, normalized size of antiderivative = 1.98 \[ \int \frac {a B+b B \tan (c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\frac {1}{2} \, \sqrt {-\frac {{\left (a^{2} + b^{2}\right )} \sqrt {-\frac {B^{4} b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} d^{2} + B^{2} a}{{\left (a^{2} + b^{2}\right )} d^{2}}} \log \left (\sqrt {b \tan \left (d x + c\right ) + a} B^{3} b + {\left (B^{2} b^{2} d + {\left (a^{3} + a b^{2}\right )} \sqrt {-\frac {B^{4} b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} d^{3}\right )} \sqrt {-\frac {{\left (a^{2} + b^{2}\right )} \sqrt {-\frac {B^{4} b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} d^{2} + B^{2} a}{{\left (a^{2} + b^{2}\right )} d^{2}}}\right ) - \frac {1}{2} \, \sqrt {-\frac {{\left (a^{2} + b^{2}\right )} \sqrt {-\frac {B^{4} b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} d^{2} + B^{2} a}{{\left (a^{2} + b^{2}\right )} d^{2}}} \log \left (\sqrt {b \tan \left (d x + c\right ) + a} B^{3} b - {\left (B^{2} b^{2} d + {\left (a^{3} + a b^{2}\right )} \sqrt {-\frac {B^{4} b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} d^{3}\right )} \sqrt {-\frac {{\left (a^{2} + b^{2}\right )} \sqrt {-\frac {B^{4} b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} d^{2} + B^{2} a}{{\left (a^{2} + b^{2}\right )} d^{2}}}\right ) + \frac {1}{2} \, \sqrt {\frac {{\left (a^{2} + b^{2}\right )} \sqrt {-\frac {B^{4} b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} d^{2} - B^{2} a}{{\left (a^{2} + b^{2}\right )} d^{2}}} \log \left (\sqrt {b \tan \left (d x + c\right ) + a} B^{3} b + {\left (B^{2} b^{2} d - {\left (a^{3} + a b^{2}\right )} \sqrt {-\frac {B^{4} b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} d^{3}\right )} \sqrt {\frac {{\left (a^{2} + b^{2}\right )} \sqrt {-\frac {B^{4} b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} d^{2} - B^{2} a}{{\left (a^{2} + b^{2}\right )} d^{2}}}\right ) - \frac {1}{2} \, \sqrt {\frac {{\left (a^{2} + b^{2}\right )} \sqrt {-\frac {B^{4} b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} d^{2} - B^{2} a}{{\left (a^{2} + b^{2}\right )} d^{2}}} \log \left (\sqrt {b \tan \left (d x + c\right ) + a} B^{3} b - {\left (B^{2} b^{2} d - {\left (a^{3} + a b^{2}\right )} \sqrt {-\frac {B^{4} b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} d^{3}\right )} \sqrt {\frac {{\left (a^{2} + b^{2}\right )} \sqrt {-\frac {B^{4} b^{2}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}}} d^{2} - B^{2} a}{{\left (a^{2} + b^{2}\right )} d^{2}}}\right ) \]

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

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)*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

Sympy [F]

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

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

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

Maxima [F(-2)]

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

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

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a

ssume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for mor

Giac [F(-1)]

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

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

Mupad [B] (veriﬁcation not implemented)

Time = 12.47 (sec) , antiderivative size = 6453, normalized size of antiderivative = 15.89 \[ \int \frac {a B+b B \tan (c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \]

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

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 + 1

6*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 - 144*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*d^4 + 16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*d^4))^(1/2)*((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)

+ (-((96*B^4*a^6*b^4*d^4 - 16*B^4*a^4*b^6*d^4 - 144*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*d^4 + 16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*d^4))^(1/2)*(64*B*a^2*b^11*d^4 - (-((96*B^4*a^6*b^4*

d^4 - 16*B^4*a^4*b^6*d^4 - 144*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*d^4 + 16*b

^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*d^4))^(1/2)*(a + b*tan(c + d*x))^(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) + 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)) + 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)*(-

((96*B^4*a^6*b^4*d^4 - 16*B^4*a^4*b^6*d^4 - 144*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*d^4 + 16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*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 - 144*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*d^4 + 16*b

^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*d^4))^(1/2)*((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) + (((96*B^4*a^6*b^4*d^4 - 16*B^4*a^4*b^6*d^4 - 144*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*d^4 + 16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*

d^4))^(1/2)*(64*B*a^2*b^11*d^4 - (((96*B^4*a^6*b^4*d^4 - 16*B^4*a^4*b^6*d^4 - 144*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*d^4 + 16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*d^4))^(1/2)*(a + b*tan

(c + d*x))^(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) + 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)) + 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)*(((96*B^4*a^6*b^4*d^4 - 16*B^4*a^4*b^6*d^4 - 144*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*d^4 + 16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*d^4))^

(1/2) + 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 + 64

0*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(- ((a + b*tan(c + d*x))^(1/2)*(16*B^2*b^12*d^3 + 32*B^2*a^2*b^10*d^3 - 32*B^2*a^

6*b^6*d^3 - 16*B^2*a^8*b^4*d^3) + (-(((8*B^2*a^3*b^2*d^2 - 24*B^2*a*b^4*d^2)^2/4 - B^4*b^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^3*b^2*d^2 + 12*B^2*a*b^4*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)*(32*B*b^13*d^4 + (a + b*tan(c + d*x))^(1/2)*(-(((8*B^2*a^3*b^2*d^2 -

24*B^2*a*b^4*d^2)^2/4 - B^4*b^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^

3*b^2*d^2 + 12*B^2*a*b^4*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) + 96*B*a^2*b^11*d^4

+ 64*B*a^4*b^9*d^4 - 64*B*a^6*b^7*d^4 - 96*B*a^8*b^5*d^4 - 32*B*a^10*b^3*d^4))*(-(((8*B^2*a^3*b^2*d^2 - 24*B^

2*a*b^4*d^2)^2/4 - B^4*b^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^3*b^2*

d^2 + 12*B^2*a*b^4*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) - 24*B^3*a^3*b^9*d^2 -

24*B^3*a^5*b^7*d^2 - 8*B^3*a^7*b^5*d^2 - 8*B^3*a*b^11*d^2)*(-(((8*B^2*a^3*b^2*d^2 - 24*B^2*a*b^4*d^2)^2/4 - B

^4*b^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^3*b^2*d^2 + 12*B^2*a*b^4*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(- ((a + b*tan(c + d*x))^(1/2)*(16*B^

2*b^12*d^3 + 32*B^2*a^2*b^10*d^3 - 32*B^2*a^6*b^6*d^3 - 16*B^2*a^8*b^4*d^3) + ((((8*B^2*a^3*b^2*d^2 - 24*B^2*a

*b^4*d^2)^2/4 - B^4*b^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^3*b^2*d^2

- 12*B^2*a*b^4*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)*(32*B*b^13*d^4 + (a + b*t

an(c + d*x))^(1/2)*((((8*B^2*a^3*b^2*d^2 - 24*B^2*a*b^4*d^2)^2/4 - B^4*b^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^3*b^2*d^2 - 12*B^2*a*b^4*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) + 96*B*a^2*b^11*d^4 + 64*B*a^4*b^9*d^4 - 64*B*a^6*b^7*d^4 - 96*B*a^8*b^5*d^4 - 32*B*a^1

0*b^3*d^4))*((((8*B^2*a^3*b^2*d^2 - 24*B^2*a*b^4*d^2)^2/4 - B^4*b^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^3*b^2*d^2 - 12*B^2*a*b^4*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) - 24*B^3*a^3*b^9*d^2 - 24*B^3*a^5*b^7*d^2 - 8*B^3*a^7*b^5*d^2 - 8*B^3*a*b^11*d^2)*((((8*B^2

*a^3*b^2*d^2 - 24*B^2*a*b^4*d^2)^2/4 - B^4*b^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^3*b^2*d^2 - 12*B^2*a*b^4*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((((96*B^4*a^2*b^8*d^4 - 16*B^4*b^10*d^4 - 144*B^4*a^4*b^6*d^4)^(1/2) + 4*B^2*a^3*b^2*d^2 - 12*B^2*a*b^4*d^

2)/(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)*((a + b*tan(c + d*x))^(1/2)*(16*B^2*b^12

*d^3 + 32*B^2*a^2*b^10*d^3 - 32*B^2*a^6*b^6*d^3 - 16*B^2*a^8*b^4*d^3) + (((96*B^4*a^2*b^8*d^4 - 16*B^4*b^10*d^

4 - 144*B^4*a^4*b^6*d^4)^(1/2) + 4*B^2*a^3*b^2*d^2 - 12*B^2*a*b^4*d^2)/(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)*((((96*B^4*a^2*b^8*d^4 - 16*B^4*b^10*d^4 - 144*B^4*a^4*b^6*d^4)^(1/2) + 4*B^2*a^3*

b^2*d^2 - 12*B^2*a*b^4*d^2)/(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)*(a + b*tan(c +

d*x))^(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) - 32*B*b^13*d^4 - 96*B*a^2*b^11*d^4 - 64*B*a^4*b^9*d^4 + 64*B*a^6*b^7*d^4 + 96*B*a^8*b^5*d^4 + 32*B*a

^10*b^3*d^4)) - 24*B^3*a^3*b^9*d^2 - 24*B^3*a^5*b^7*d^2 - 8*B^3*a^7*b^5*d^2 - 8*B^3*a*b^11*d^2)*(((96*B^4*a^2*

b^8*d^4 - 16*B^4*b^10*d^4 - 144*B^4*a^4*b^6*d^4)^(1/2) + 4*B^2*a^3*b^2*d^2 - 12*B^2*a*b^4*d^2)/(16*a^6*d^4 + 1

6*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*d^4))^(1/2) - log((-((96*B^4*a^2*b^8*d^4 - 16*B^4*b^10*d^4 - 144*B^4*a

^4*b^6*d^4)^(1/2) - 4*B^2*a^3*b^2*d^2 + 12*B^2*a*b^4*d^2)/(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)*((a + b*tan(c + d*x))^(1/2)*(16*B^2*b^12*d^3 + 32*B^2*a^2*b^10*d^3 - 32*B^2*a^6*b^6*d^3 - 16*B^

2*a^8*b^4*d^3) + (-((96*B^4*a^2*b^8*d^4 - 16*B^4*b^10*d^4 - 144*B^4*a^4*b^6*d^4)^(1/2) - 4*B^2*a^3*b^2*d^2 + 1

2*B^2*a*b^4*d^2)/(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)*((-((96*B^4*a^2*b^8*d^4 -

16*B^4*b^10*d^4 - 144*B^4*a^4*b^6*d^4)^(1/2) - 4*B^2*a^3*b^2*d^2 + 12*B^2*a*b^4*d^2)/(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)*(a + b*tan(c + d*x))^(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) - 32*B*b^13*d^4 - 96*B*a^2*b^11*d^4 - 64*B*a

^4*b^9*d^4 + 64*B*a^6*b^7*d^4 + 96*B*a^8*b^5*d^4 + 32*B*a^10*b^3*d^4)) - 24*B^3*a^3*b^9*d^2 - 24*B^3*a^5*b^7*d

^2 - 8*B^3*a^7*b^5*d^2 - 8*B^3*a*b^11*d^2)*(-((96*B^4*a^2*b^8*d^4 - 16*B^4*b^10*d^4 - 144*B^4*a^4*b^6*d^4)^(1/

2) - 4*B^2*a^3*b^2*d^2 + 12*B^2*a*b^4*d^2)/(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)

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

Optimal result