3.4.0 ∫ aB+bB tan(c+dx) _ (a+b tan(c+dx))5∕2 dx [368]

Integrand size = 28, antiderivative size = 123 \[ \int \frac {a B+b B \tan (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=-\frac {i B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{3/2} d}+\frac {i B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{3/2} d}-\frac {2 b B}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}} \]

-I*B*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/(a-I*b)^(3/2)/d+I*B*arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*b)^

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

Rubi [A] (veriﬁed)

Time = 0.32 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {21, 3564, 3620, 3618, 65, 214} \[ \int \frac {a B+b B \tan (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=-\frac {2 b B}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}-\frac {i B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d (a-i b)^{3/2}}+\frac {i B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d (a+i b)^{3/2}} \]

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

((-I)*B*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(3/2)*d) + (I*B*ArcTanh[Sqrt[a + b*Tan[c +

d*x]]/Sqrt[a + I*b]])/((a + I*b)^(3/2)*d) - (2*b*B)/((a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])

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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub

st[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] &

& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,

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

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n + 1)/(d*(n + 1)*

(a^2 + b^2))), x] + Dist[1/(a^2 + b^2), Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ

[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[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]

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c

+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(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]

Rubi steps \begin{align*} \text {integral}& = B \int \frac {1}{(a+b \tan (c+d x))^{3/2}} \, dx \\ & = -\frac {2 b B}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {B \int \frac {a-b \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{a^2+b^2} \\ & = -\frac {2 b B}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {B \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (a-i b)}+\frac {B \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (a+i b)} \\ & = -\frac {2 b B}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {B \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 (i a-b) d}-\frac {B \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 (i a+b) d} \\ & = -\frac {2 b B}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}-\frac {B \text {Subst}\left (\int \frac {1}{-1-\frac {i a}{b}+\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{(a-i b) b d}-\frac {B \text {Subst}\left (\int \frac {1}{-1+\frac {i a}{b}-\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{(a+i b) b d} \\ & = -\frac {i B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{3/2} d}+\frac {i B \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{3/2} d}-\frac {2 b B}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}} \\ \end{align*}

Mathematica [C] (veriﬁed)

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

Time = 0.14 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.86 \[ \int \frac {a B+b B \tan (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\frac {B \left (i (a+i b) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {a+b \tan (c+d x)}{a-i b}\right )+(-i a-b) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {a+b \tan (c+d x)}{a+i b}\right )\right )}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}} \]

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

(B*(I*(a + I*b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a - I*b)] + ((-I)*a - b)*Hypergeometric2

F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a + I*b)]))/((a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(1954\) vs. \(2(103)=206\).

Time = 0.09 (sec) , antiderivative size = 1955, normalized size of antiderivative = 15.89


method	result	size

derivativedivides	\(\text {Expression too large to display}\)	\(1955\)
default	\(\text {Expression too large to display}\)	\(1955\)
parts	\(\text {Expression too large to display}\)	\(4475\)

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

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

(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3+1/4/d*B*b/(a^2+b^2)^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))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a-1/4/d*B/b/(a^2+b^2)^(5/2)*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))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^4+1/4/d*

B*b^3/(a^2+b^2)^(5/2)*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))*

(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+4/d*B*b/(a^2+b^2)^(5/2)/(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))*a^3+3/d*B*b^3/(a^2+b^2)^(5/2)/(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))*a-1/d*B/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))*a^3-1/d*B*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))*a-1/d*B*b/(a^2+b^2)^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))*a^2+1/d*B/b/(a^2+b^2)^(5/2)/(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))*a^5-1/d*B*b^3/(a^2+b^2)^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))-1/4/d*B/b/(

a^2+b^2)^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))*(2*(a^2+b^2

)^(1/2)+2*a)^(1/2)*a^3-1/4/d*B*b/(a^2+b^2)^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))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a+1/4/d*B/b/(a^2+b^2)^(5/2)*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))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^4-1/4/d*B*b^3/(a^2+

b^2)^(5/2)*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))*(2*(a^2+b^2

)^(1/2)+2*a)^(1/2)-4/d*B*b/(a^2+b^2)^(5/2)/(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))*a^3-3/d*B*b^3/(a^2+b^2)^(5/2)/(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))*a+1/d*B

/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))*a^3+1/d*B*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))*a+1/d*B*b/(a^2+b^2)^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))*a^2-1/d*B/b/(a^2+b^2)^(5/2)/(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))*a^5+1/d*B*b^3/(a^2+b^2)^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))-2*b*B/(a^2+b^2)/d/(a+b

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2174 vs. \(2 (97) = 194\).

Time = 0.31 (sec) , antiderivative size = 2174, normalized size of antiderivative = 17.67 \[ \int \frac {a B+b B \tan (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \]

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

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

*a*b^2 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^

10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)

*d^2))*log(-(3*B^3*a^2*b - B^3*b^3)*sqrt(b*tan(d*x + c) + a) + ((a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*d^3*sqrt(-

(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b

^10 + b^12)*d^4)) + 2*(3*B^2*a^3*b^2 - B^2*a*b^4)*d)*sqrt(-(B^2*a^3 - 3*B^2*a*b^2 + (a^6 + 3*a^4*b^2 + 3*a^2*b

^4 + b^6)*d^2*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 +

15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))) - ((a^2*b + b^3)*d*tan(d*x

+ c) + (a^3 + a*b^2)*d)*sqrt(-(B^2*a^3 - 3*B^2*a*b^2 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*B^4*a^

4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^1

2)*d^4)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))*log(-(3*B^3*a^2*b - B^3*b^3)*sqrt(b*tan(d*x + c) + a) - (

(a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*d^3*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 +

15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + 2*(3*B^2*a^3*b^2 - B^2*a*b^4)*d)*sqrt(-(B^2*

a^3 - 3*B^2*a*b^2 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((

a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))/((a^6 + 3*a^4*b^2 + 3*a^2

*b^4 + b^6)*d^2))) - ((a^2*b + b^3)*d*tan(d*x + c) + (a^3 + a*b^2)*d)*sqrt(-(B^2*a^3 - 3*B^2*a*b^2 - (a^6 + 3*

a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^

4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))*log(-(3*B^3

*a^2*b - B^3*b^3)*sqrt(b*tan(d*x + c) + a) + ((a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*d^3*sqrt(-(9*B^4*a^4*b^2 - 6

*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))

- 2*(3*B^2*a^3*b^2 - B^2*a*b^4)*d)*sqrt(-(B^2*a^3 - 3*B^2*a*b^2 - (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt

(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2

*b^10 + b^12)*d^4)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))) + ((a^2*b + b^3)*d*tan(d*x + c) + (a^3 + a*b^

2)*d)*sqrt(-(B^2*a^3 - 3*B^2*a*b^2 - (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*

b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))/((a^6 +

3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))*log(-(3*B^3*a^2*b - B^3*b^3)*sqrt(b*tan(d*x + c) + a) - ((a^8 + 2*a^6*b^2 -

2*a^2*b^6 - b^8)*d^3*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^

6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - 2*(3*B^2*a^3*b^2 - B^2*a*b^4)*d)*sqrt(-(B^2*a^3 - 3*B^2*a*b^2

- (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2

+ 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2)))

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

Sympy [F]

\[ \int \frac {a B+b B \tan (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=B \int \frac {1}{a \sqrt {a + b \tan {\left (c + d x \right )}} + b \sqrt {a + b \tan {\left (c + d x \right )}} \tan {\left (c + d x \right )}}\, dx \]

integrate((B*a+b*B*tan(d*x+c))/(a+b*tan(d*x+c))**(5/2),x)

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

Maxima [F(-2)]

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

integrate((B*a+b*B*tan(d*x+c))/(a+b*tan(d*x+c))^(5/2),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. \[ \int \frac {a B+b B \tan (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Timed out} \]

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

Mupad [B] (veriﬁcation not implemented)

Time = 21.21 (sec) , antiderivative size = 9618, normalized size of antiderivative = 78.20 \[ \int \frac {a B+b B \tan (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \]

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

(log(16*B^3*a^4*b^15*d^2 - ((((320*B^4*a^6*b^8*d^4 - 16*B^4*a^4*b^10*d^4 - 1760*B^4*a^8*b^6*d^4 + 1600*B^4*a^1

0*b^4*d^4 - 400*B^4*a^12*b^2*d^4)^(1/2) - 4*B^2*a^7*d^2 - 20*B^2*a^3*b^4*d^2 + 40*B^2*a^5*b^2*d^2)/(a^10*d^4 +

b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2)*(((((320*B^4*a^6*b^8*d^4 -

16*B^4*a^4*b^10*d^4 - 1760*B^4*a^8*b^6*d^4 + 1600*B^4*a^10*b^4*d^4 - 400*B^4*a^12*b^2*d^4)^(1/2) - 4*B^2*a^7*

d^2 - 20*B^2*a^3*b^4*d^2 + 40*B^2*a^5*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*

b^4*d^4 + 5*a^8*b^2*d^4))^(1/2)*(((((320*B^4*a^6*b^8*d^4 - 16*B^4*a^4*b^10*d^4 - 1760*B^4*a^8*b^6*d^4 + 1600*B

^4*a^10*b^4*d^4 - 400*B^4*a^12*b^2*d^4)^(1/2) - 4*B^2*a^7*d^2 - 20*B^2*a^3*b^4*d^2 + 40*B^2*a^5*b^2*d^2)/(a^10

*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2)*(a + b*tan(c + d*x))

^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 7680*a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 16128*

a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^17*b^6*d^5 + 640*a^19*b^4*d^5 + 64*a^21*b^2*d

^5))/4 - 32*B*a*b^21*d^4 - 160*B*a^3*b^19*d^4 - 128*B*a^5*b^17*d^4 + 896*B*a^7*b^15*d^4 + 3136*B*a^9*b^13*d^4

+ 4928*B*a^11*b^11*d^4 + 4480*B*a^13*b^9*d^4 + 2432*B*a^15*b^7*d^4 + 736*B*a^17*b^5*d^4 + 96*B*a^19*b^3*d^4))/

4 - (a + b*tan(c + d*x))^(1/2)*(320*B^2*a^6*b^14*d^3 - 16*B^2*a^2*b^18*d^3 + 1024*B^2*a^8*b^12*d^3 + 1440*B^2*

a^10*b^10*d^3 + 1024*B^2*a^12*b^8*d^3 + 320*B^2*a^14*b^6*d^3 - 16*B^2*a^18*b^2*d^3)))/4 + 96*B^3*a^6*b^13*d^2

+ 240*B^3*a^8*b^11*d^2 + 320*B^3*a^10*b^9*d^2 + 240*B^3*a^12*b^7*d^2 + 96*B^3*a^14*b^5*d^2 + 16*B^3*a^16*b^3*d

^2)*(((320*B^4*a^6*b^8*d^4 - 16*B^4*a^4*b^10*d^4 - 1760*B^4*a^8*b^6*d^4 + 1600*B^4*a^10*b^4*d^4 - 400*B^4*a^12

*b^2*d^4)^(1/2) - 4*B^2*a^7*d^2 - 20*B^2*a^3*b^4*d^2 + 40*B^2*a^5*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^

4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2))/4 + (log(16*B^3*a^4*b^15*d^2 - ((-((320*B^4*a^6*b

^8*d^4 - 16*B^4*a^4*b^10*d^4 - 1760*B^4*a^8*b^6*d^4 + 1600*B^4*a^10*b^4*d^4 - 400*B^4*a^12*b^2*d^4)^(1/2) + 4*

B^2*a^7*d^2 + 20*B^2*a^3*b^4*d^2 - 40*B^2*a^5*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 +

10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2)*(((-((320*B^4*a^6*b^8*d^4 - 16*B^4*a^4*b^10*d^4 - 1760*B^4*a^8*b^6*d^4

+ 1600*B^4*a^10*b^4*d^4 - 400*B^4*a^12*b^2*d^4)^(1/2) + 4*B^2*a^7*d^2 + 20*B^2*a^3*b^4*d^2 - 40*B^2*a^5*b^2*d

^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2)*(((-((320*B

^4*a^6*b^8*d^4 - 16*B^4*a^4*b^10*d^4 - 1760*B^4*a^8*b^6*d^4 + 1600*B^4*a^10*b^4*d^4 - 400*B^4*a^12*b^2*d^4)^(1

/2) + 4*B^2*a^7*d^2 + 20*B^2*a^3*b^4*d^2 - 40*B^2*a^5*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b

^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 +

2880*a^5*b^18*d^5 + 7680*a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680

*a^15*b^8*d^5 + 2880*a^17*b^6*d^5 + 640*a^19*b^4*d^5 + 64*a^21*b^2*d^5))/4 - 32*B*a*b^21*d^4 - 160*B*a^3*b^19*

d^4 - 128*B*a^5*b^17*d^4 + 896*B*a^7*b^15*d^4 + 3136*B*a^9*b^13*d^4 + 4928*B*a^11*b^11*d^4 + 4480*B*a^13*b^9*d

^4 + 2432*B*a^15*b^7*d^4 + 736*B*a^17*b^5*d^4 + 96*B*a^19*b^3*d^4))/4 - (a + b*tan(c + d*x))^(1/2)*(320*B^2*a^

6*b^14*d^3 - 16*B^2*a^2*b^18*d^3 + 1024*B^2*a^8*b^12*d^3 + 1440*B^2*a^10*b^10*d^3 + 1024*B^2*a^12*b^8*d^3 + 32

0*B^2*a^14*b^6*d^3 - 16*B^2*a^18*b^2*d^3)))/4 + 96*B^3*a^6*b^13*d^2 + 240*B^3*a^8*b^11*d^2 + 320*B^3*a^10*b^9*

d^2 + 240*B^3*a^12*b^7*d^2 + 96*B^3*a^14*b^5*d^2 + 16*B^3*a^16*b^3*d^2)*(-((320*B^4*a^6*b^8*d^4 - 16*B^4*a^4*b

^10*d^4 - 1760*B^4*a^8*b^6*d^4 + 1600*B^4*a^10*b^4*d^4 - 400*B^4*a^12*b^2*d^4)^(1/2) + 4*B^2*a^7*d^2 + 20*B^2*

a^3*b^4*d^2 - 40*B^2*a^5*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a

^8*b^2*d^4))^(1/2))/4 - log(16*B^3*a^4*b^15*d^2 - (((320*B^4*a^6*b^8*d^4 - 16*B^4*a^4*b^10*d^4 - 1760*B^4*a^8*

b^6*d^4 + 1600*B^4*a^10*b^4*d^4 - 400*B^4*a^12*b^2*d^4)^(1/2) - 4*B^2*a^7*d^2 - 20*B^2*a^3*b^4*d^2 + 40*B^2*a^

5*b^2*d^2)/(16*a^10*d^4 + 16*b^10*d^4 + 80*a^2*b^8*d^4 + 160*a^4*b^6*d^4 + 160*a^6*b^4*d^4 + 80*a^8*b^2*d^4))^

(1/2)*((((320*B^4*a^6*b^8*d^4 - 16*B^4*a^4*b^10*d^4 - 1760*B^4*a^8*b^6*d^4 + 1600*B^4*a^10*b^4*d^4 - 400*B^4*a

^12*b^2*d^4)^(1/2) - 4*B^2*a^7*d^2 - 20*B^2*a^3*b^4*d^2 + 40*B^2*a^5*b^2*d^2)/(16*a^10*d^4 + 16*b^10*d^4 + 80*

a^2*b^8*d^4 + 160*a^4*b^6*d^4 + 160*a^6*b^4*d^4 + 80*a^8*b^2*d^4))^(1/2)*(896*B*a^7*b^15*d^4 - 32*B*a*b^21*d^4

- 160*B*a^3*b^19*d^4 - 128*B*a^5*b^17*d^4 - (((320*B^4*a^6*b^8*d^4 - 16*B^4*a^4*b^10*d^4 - 1760*B^4*a^8*b^6*d

^4 + 1600*B^4*a^10*b^4*d^4 - 400*B^4*a^12*b^2*d^4)^(1/2) - 4*B^2*a^7*d^2 - 20*B^2*a^3*b^4*d^2 + 40*B^2*a^5*b^2

*d^2)/(16*a^10*d^4 + 16*b^10*d^4 + 80*a^2*b^8*d^4 + 160*a^4*b^6*d^4 + 160*a^6*b^4*d^4 + 80*a^8*b^2*d^4))^(1/2)

*(a + b*tan(c + d*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 7680*a^7*b^16*d^5 + 13440*

a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^17*b^6*d^5 + 640*a^19*b^

4*d^5 + 64*a^21*b^2*d^5) + 3136*B*a^9*b^13*d^4 + 4928*B*a^11*b^11*d^4 + 4480*B*a^13*b^9*d^4 + 2432*B*a^15*b^7*

d^4 + 736*B*a^17*b^5*d^4 + 96*B*a^19*b^3*d^4) + (a + b*tan(c + d*x))^(1/2)*(320*B^2*a^6*b^14*d^3 - 16*B^2*a^2*

b^18*d^3 + 1024*B^2*a^8*b^12*d^3 + 1440*B^2*a^10*b^10*d^3 + 1024*B^2*a^12*b^8*d^3 + 320*B^2*a^14*b^6*d^3 - 16*

B^2*a^18*b^2*d^3)) + 96*B^3*a^6*b^13*d^2 + 240*B^3*a^8*b^11*d^2 + 320*B^3*a^10*b^9*d^2 + 240*B^3*a^12*b^7*d^2

+ 96*B^3*a^14*b^5*d^2 + 16*B^3*a^16*b^3*d^2)*(((320*B^4*a^6*b^8*d^4 - 16*B^4*a^4*b^10*d^4 - 1760*B^4*a^8*b^6*d

^4 + 1600*B^4*a^10*b^4*d^4 - 400*B^4*a^12*b^2*d^4)^(1/2) - 4*B^2*a^7*d^2 - 20*B^2*a^3*b^4*d^2 + 40*B^2*a^5*b^2

*d^2)/(16*a^10*d^4 + 16*b^10*d^4 + 80*a^2*b^8*d^4 + 160*a^4*b^6*d^4 + 160*a^6*b^4*d^4 + 80*a^8*b^2*d^4))^(1/2)

- log(16*B^3*a^4*b^15*d^2 - (-((320*B^4*a^6*b^8*d^4 - 16*B^4*a^4*b^10*d^4 - 1760*B^4*a^8*b^6*d^4 + 1600*B^4*a

^10*b^4*d^4 - 400*B^4*a^12*b^2*d^4)^(1/2) + 4*B^2*a^7*d^2 + 20*B^2*a^3*b^4*d^2 - 40*B^2*a^5*b^2*d^2)/(16*a^10*

d^4 + 16*b^10*d^4 + 80*a^2*b^8*d^4 + 160*a^4*b^6*d^4 + 160*a^6*b^4*d^4 + 80*a^8*b^2*d^4))^(1/2)*((-((320*B^4*a

^6*b^8*d^4 - 16*B^4*a^4*b^10*d^4 - 1760*B^4*a^8*b^6*d^4 + 1600*B^4*a^10*b^4*d^4 - 400*B^4*a^12*b^2*d^4)^(1/2)

+ 4*B^2*a^7*d^2 + 20*B^2*a^3*b^4*d^2 - 40*B^2*a^5*b^2*d^2)/(16*a^10*d^4 + 16*b^10*d^4 + 80*a^2*b^8*d^4 + 160*a

^4*b^6*d^4 + 160*a^6*b^4*d^4 + 80*a^8*b^2*d^4))^(1/2)*(896*B*a^7*b^15*d^4 - 32*B*a*b^21*d^4 - 160*B*a^3*b^19*d

^4 - 128*B*a^5*b^17*d^4 - (-((320*B^4*a^6*b^8*d^4 - 16*B^4*a^4*b^10*d^4 - 1760*B^4*a^8*b^6*d^4 + 1600*B^4*a^10

*b^4*d^4 - 400*B^4*a^12*b^2*d^4)^(1/2) + 4*B^2*a^7*d^2 + 20*B^2*a^3*b^4*d^2 - 40*B^2*a^5*b^2*d^2)/(16*a^10*d^4

+ 16*b^10*d^4 + 80*a^2*b^8*d^4 + 160*a^4*b^6*d^4 + 160*a^6*b^4*d^4 + 80*a^8*b^2*d^4))^(1/2)*(a + b*tan(c + d*

x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 7680*a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 161

28*a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^17*b^6*d^5 + 640*a^19*b^4*d^5 + 64*a^21*b^

2*d^5) + 3136*B*a^9*b^13*d^4 + 4928*B*a^11*b^11*d^4 + 4480*B*a^13*b^9*d^4 + 2432*B*a^15*b^7*d^4 + 736*B*a^17*b

^5*d^4 + 96*B*a^19*b^3*d^4) + (a + b*tan(c + d*x))^(1/2)*(320*B^2*a^6*b^14*d^3 - 16*B^2*a^2*b^18*d^3 + 1024*B^

2*a^8*b^12*d^3 + 1440*B^2*a^10*b^10*d^3 + 1024*B^2*a^12*b^8*d^3 + 320*B^2*a^14*b^6*d^3 - 16*B^2*a^18*b^2*d^3))

+ 96*B^3*a^6*b^13*d^2 + 240*B^3*a^8*b^11*d^2 + 320*B^3*a^10*b^9*d^2 + 240*B^3*a^12*b^7*d^2 + 96*B^3*a^14*b^5*

d^2 + 16*B^3*a^16*b^3*d^2)*(-((320*B^4*a^6*b^8*d^4 - 16*B^4*a^4*b^10*d^4 - 1760*B^4*a^8*b^6*d^4 + 1600*B^4*a^1

0*b^4*d^4 - 400*B^4*a^12*b^2*d^4)^(1/2) + 4*B^2*a^7*d^2 + 20*B^2*a^3*b^4*d^2 - 40*B^2*a^5*b^2*d^2)/(16*a^10*d^

4 + 16*b^10*d^4 + 80*a^2*b^8*d^4 + 160*a^4*b^6*d^4 + 160*a^6*b^4*d^4 + 80*a^8*b^2*d^4))^(1/2) + (log(8*B^3*b^1

9*d^2 - ((((320*B^4*a^2*b^12*d^4 - 16*B^4*b^14*d^4 - 1760*B^4*a^4*b^10*d^4 + 1600*B^4*a^6*b^8*d^4 - 400*B^4*a^

8*b^6*d^4)^(1/2) - 40*B^2*a^3*b^4*d^2 + 4*B^2*a^5*b^2*d^2 + 20*B^2*a*b^6*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8

*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2)*(((((320*B^4*a^2*b^12*d^4 - 16*B^4*b^14*d^4 - 1

760*B^4*a^4*b^10*d^4 + 1600*B^4*a^6*b^8*d^4 - 400*B^4*a^8*b^6*d^4)^(1/2) - 40*B^2*a^3*b^4*d^2 + 4*B^2*a^5*b^2*

d^2 + 20*B^2*a*b^6*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4

))^(1/2)*(((((320*B^4*a^2*b^12*d^4 - 16*B^4*b^14*d^4 - 1760*B^4*a^4*b^10*d^4 + 1600*B^4*a^6*b^8*d^4 - 400*B^4*

a^8*b^6*d^4)^(1/2) - 40*B^2*a^3*b^4*d^2 + 4*B^2*a^5*b^2*d^2 + 20*B^2*a*b^6*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b

^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^22*d^5 + 6

40*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 7680*a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440*a^1

3*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^17*b^6*d^5 + 640*a^19*b^4*d^5 + 64*a^21*b^2*d^5))/4 + 96*B*a*b^21*d^4

+ 736*B*a^3*b^19*d^4 + 2432*B*a^5*b^17*d^4 + 4480*B*a^7*b^15*d^4 + 4928*B*a^9*b^13*d^4 + 3136*B*a^11*b^11*d^4

+ 896*B*a^13*b^9*d^4 - 128*B*a^15*b^7*d^4 - 160*B*a^17*b^5*d^4 - 32*B*a^19*b^3*d^4))/4 + (a + b*tan(c + d*x))^

(1/2)*(320*B^2*a^4*b^16*d^3 - 16*B^2*b^20*d^3 + 1024*B^2*a^6*b^14*d^3 + 1440*B^2*a^8*b^12*d^3 + 1024*B^2*a^10*

b^10*d^3 + 320*B^2*a^12*b^8*d^3 - 16*B^2*a^16*b^4*d^3)))/4 + 40*B^3*a^2*b^17*d^2 + 72*B^3*a^4*b^15*d^2 + 40*B^

3*a^6*b^13*d^2 - 40*B^3*a^8*b^11*d^2 - 72*B^3*a^10*b^9*d^2 - 40*B^3*a^12*b^7*d^2 - 8*B^3*a^14*b^5*d^2)*(((320*

B^4*a^2*b^12*d^4 - 16*B^4*b^14*d^4 - 1760*B^4*a^4*b^10*d^4 + 1600*B^4*a^6*b^8*d^4 - 400*B^4*a^8*b^6*d^4)^(1/2)

- 40*B^2*a^3*b^4*d^2 + 4*B^2*a^5*b^2*d^2 + 20*B^2*a*b^6*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^

6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2))/4 + (log(8*B^3*b^19*d^2 - ((-((320*B^4*a^2*b^12*d^4 - 16*B^4*b

^14*d^4 - 1760*B^4*a^4*b^10*d^4 + 1600*B^4*a^6*b^8*d^4 - 400*B^4*a^8*b^6*d^4)^(1/2) + 40*B^2*a^3*b^4*d^2 - 4*B

^2*a^5*b^2*d^2 - 20*B^2*a*b^6*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*

a^8*b^2*d^4))^(1/2)*(((-((320*B^4*a^2*b^12*d^4 - 16*B^4*b^14*d^4 - 1760*B^4*a^4*b^10*d^4 + 1600*B^4*a^6*b^8*d^

4 - 400*B^4*a^8*b^6*d^4)^(1/2) + 40*B^2*a^3*b^4*d^2 - 4*B^2*a^5*b^2*d^2 - 20*B^2*a*b^6*d^2)/(a^10*d^4 + b^10*d

^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2)*(((-((320*B^4*a^2*b^12*d^4 - 16*B

^4*b^14*d^4 - 1760*B^4*a^4*b^10*d^4 + 1600*B^4*a^6*b^8*d^4 - 400*B^4*a^8*b^6*d^4)^(1/2) + 40*B^2*a^3*b^4*d^2 -

4*B^2*a^5*b^2*d^2 - 20*B^2*a*b^6*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4

+ 5*a^8*b^2*d^4))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 768

0*a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^1

7*b^6*d^5 + 640*a^19*b^4*d^5 + 64*a^21*b^2*d^5))/4 + 96*B*a*b^21*d^4 + 736*B*a^3*b^19*d^4 + 2432*B*a^5*b^17*d^

4 + 4480*B*a^7*b^15*d^4 + 4928*B*a^9*b^13*d^4 + 3136*B*a^11*b^11*d^4 + 896*B*a^13*b^9*d^4 - 128*B*a^15*b^7*d^4

- 160*B*a^17*b^5*d^4 - 32*B*a^19*b^3*d^4))/4 + (a + b*tan(c + d*x))^(1/2)*(320*B^2*a^4*b^16*d^3 - 16*B^2*b^20

*d^3 + 1024*B^2*a^6*b^14*d^3 + 1440*B^2*a^8*b^12*d^3 + 1024*B^2*a^10*b^10*d^3 + 320*B^2*a^12*b^8*d^3 - 16*B^2*

a^16*b^4*d^3)))/4 + 40*B^3*a^2*b^17*d^2 + 72*B^3*a^4*b^15*d^2 + 40*B^3*a^6*b^13*d^2 - 40*B^3*a^8*b^11*d^2 - 72

*B^3*a^10*b^9*d^2 - 40*B^3*a^12*b^7*d^2 - 8*B^3*a^14*b^5*d^2)*(-((320*B^4*a^2*b^12*d^4 - 16*B^4*b^14*d^4 - 176

0*B^4*a^4*b^10*d^4 + 1600*B^4*a^6*b^8*d^4 - 400*B^4*a^8*b^6*d^4)^(1/2) + 40*B^2*a^3*b^4*d^2 - 4*B^2*a^5*b^2*d^

2 - 20*B^2*a*b^6*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))

^(1/2))/4 - log(8*B^3*b^19*d^2 - (((320*B^4*a^2*b^12*d^4 - 16*B^4*b^14*d^4 - 1760*B^4*a^4*b^10*d^4 + 1600*B^4*

a^6*b^8*d^4 - 400*B^4*a^8*b^6*d^4)^(1/2) - 40*B^2*a^3*b^4*d^2 + 4*B^2*a^5*b^2*d^2 + 20*B^2*a*b^6*d^2)/(16*a^10

*d^4 + 16*b^10*d^4 + 80*a^2*b^8*d^4 + 160*a^4*b^6*d^4 + 160*a^6*b^4*d^4 + 80*a^8*b^2*d^4))^(1/2)*((((320*B^4*a

^2*b^12*d^4 - 16*B^4*b^14*d^4 - 1760*B^4*a^4*b^10*d^4 + 1600*B^4*a^6*b^8*d^4 - 400*B^4*a^8*b^6*d^4)^(1/2) - 40

*B^2*a^3*b^4*d^2 + 4*B^2*a^5*b^2*d^2 + 20*B^2*a*b^6*d^2)/(16*a^10*d^4 + 16*b^10*d^4 + 80*a^2*b^8*d^4 + 160*a^4

*b^6*d^4 + 160*a^6*b^4*d^4 + 80*a^8*b^2*d^4))^(1/2)*(96*B*a*b^21*d^4 - (((320*B^4*a^2*b^12*d^4 - 16*B^4*b^14*d

^4 - 1760*B^4*a^4*b^10*d^4 + 1600*B^4*a^6*b^8*d^4 - 400*B^4*a^8*b^6*d^4)^(1/2) - 40*B^2*a^3*b^4*d^2 + 4*B^2*a^

5*b^2*d^2 + 20*B^2*a*b^6*d^2)/(16*a^10*d^4 + 16*b^10*d^4 + 80*a^2*b^8*d^4 + 160*a^4*b^6*d^4 + 160*a^6*b^4*d^4

+ 80*a^8*b^2*d^4))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 76

80*a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^

17*b^6*d^5 + 640*a^19*b^4*d^5 + 64*a^21*b^2*d^5) + 736*B*a^3*b^19*d^4 + 2432*B*a^5*b^17*d^4 + 4480*B*a^7*b^15*

d^4 + 4928*B*a^9*b^13*d^4 + 3136*B*a^11*b^11*d^4 + 896*B*a^13*b^9*d^4 - 128*B*a^15*b^7*d^4 - 160*B*a^17*b^5*d^

4 - 32*B*a^19*b^3*d^4) - (a + b*tan(c + d*x))^(1/2)*(320*B^2*a^4*b^16*d^3 - 16*B^2*b^20*d^3 + 1024*B^2*a^6*b^1

4*d^3 + 1440*B^2*a^8*b^12*d^3 + 1024*B^2*a^10*b^10*d^3 + 320*B^2*a^12*b^8*d^3 - 16*B^2*a^16*b^4*d^3)) + 40*B^3

*a^2*b^17*d^2 + 72*B^3*a^4*b^15*d^2 + 40*B^3*a^6*b^13*d^2 - 40*B^3*a^8*b^11*d^2 - 72*B^3*a^10*b^9*d^2 - 40*B^3

*a^12*b^7*d^2 - 8*B^3*a^14*b^5*d^2)*(((320*B^4*a^2*b^12*d^4 - 16*B^4*b^14*d^4 - 1760*B^4*a^4*b^10*d^4 + 1600*B

^4*a^6*b^8*d^4 - 400*B^4*a^8*b^6*d^4)^(1/2) - 40*B^2*a^3*b^4*d^2 + 4*B^2*a^5*b^2*d^2 + 20*B^2*a*b^6*d^2)/(16*a

^10*d^4 + 16*b^10*d^4 + 80*a^2*b^8*d^4 + 160*a^4*b^6*d^4 + 160*a^6*b^4*d^4 + 80*a^8*b^2*d^4))^(1/2) - log(8*B^

3*b^19*d^2 - (-((320*B^4*a^2*b^12*d^4 - 16*B^4*b^14*d^4 - 1760*B^4*a^4*b^10*d^4 + 1600*B^4*a^6*b^8*d^4 - 400*B

^4*a^8*b^6*d^4)^(1/2) + 40*B^2*a^3*b^4*d^2 - 4*B^2*a^5*b^2*d^2 - 20*B^2*a*b^6*d^2)/(16*a^10*d^4 + 16*b^10*d^4

+ 80*a^2*b^8*d^4 + 160*a^4*b^6*d^4 + 160*a^6*b^4*d^4 + 80*a^8*b^2*d^4))^(1/2)*((-((320*B^4*a^2*b^12*d^4 - 16*B

^4*b^14*d^4 - 1760*B^4*a^4*b^10*d^4 + 1600*B^4*a^6*b^8*d^4 - 400*B^4*a^8*b^6*d^4)^(1/2) + 40*B^2*a^3*b^4*d^2 -

4*B^2*a^5*b^2*d^2 - 20*B^2*a*b^6*d^2)/(16*a^10*d^4 + 16*b^10*d^4 + 80*a^2*b^8*d^4 + 160*a^4*b^6*d^4 + 160*a^6

*b^4*d^4 + 80*a^8*b^2*d^4))^(1/2)*(96*B*a*b^21*d^4 - (-((320*B^4*a^2*b^12*d^4 - 16*B^4*b^14*d^4 - 1760*B^4*a^4

*b^10*d^4 + 1600*B^4*a^6*b^8*d^4 - 400*B^4*a^8*b^6*d^4)^(1/2) + 40*B^2*a^3*b^4*d^2 - 4*B^2*a^5*b^2*d^2 - 20*B^

2*a*b^6*d^2)/(16*a^10*d^4 + 16*b^10*d^4 + 80*a^2*b^8*d^4 + 160*a^4*b^6*d^4 + 160*a^6*b^4*d^4 + 80*a^8*b^2*d^4)

)^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 7680*a^7*b^16*d^5 +

13440*a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^17*b^6*d^5 + 640*

a^19*b^4*d^5 + 64*a^21*b^2*d^5) + 736*B*a^3*b^19*d^4 + 2432*B*a^5*b^17*d^4 + 4480*B*a^7*b^15*d^4 + 4928*B*a^9*

b^13*d^4 + 3136*B*a^11*b^11*d^4 + 896*B*a^13*b^9*d^4 - 128*B*a^15*b^7*d^4 - 160*B*a^17*b^5*d^4 - 32*B*a^19*b^3

*d^4) - (a + b*tan(c + d*x))^(1/2)*(320*B^2*a^4*b^16*d^3 - 16*B^2*b^20*d^3 + 1024*B^2*a^6*b^14*d^3 + 1440*B^2*

a^8*b^12*d^3 + 1024*B^2*a^10*b^10*d^3 + 320*B^2*a^12*b^8*d^3 - 16*B^2*a^16*b^4*d^3)) + 40*B^3*a^2*b^17*d^2 + 7

2*B^3*a^4*b^15*d^2 + 40*B^3*a^6*b^13*d^2 - 40*B^3*a^8*b^11*d^2 - 72*B^3*a^10*b^9*d^2 - 40*B^3*a^12*b^7*d^2 - 8

*B^3*a^14*b^5*d^2)*(-((320*B^4*a^2*b^12*d^4 - 16*B^4*b^14*d^4 - 1760*B^4*a^4*b^10*d^4 + 1600*B^4*a^6*b^8*d^4 -

400*B^4*a^8*b^6*d^4)^(1/2) + 40*B^2*a^3*b^4*d^2 - 4*B^2*a^5*b^2*d^2 - 20*B^2*a*b^6*d^2)/(16*a^10*d^4 + 16*b^1

0*d^4 + 80*a^2*b^8*d^4 + 160*a^4*b^6*d^4 + 160*a^6*b^4*d^4 + 80*a^8*b^2*d^4))^(1/2) - ((2*B*a*b)/(3*(a^2 + b^2

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

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

\(\int \frac {a B+b B \tan (c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx\) [368]

Optimal result