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

Optimal. Leaf size=406 \[ -\frac {b B \log \left (-\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \tan (c+d x)}+\sqrt {a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt {2} d \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}+\frac {b B \log \left (\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \tan (c+d x)}+\sqrt {a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt {2} d \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}+\frac {b B \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a^2+b^2}+a}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} d \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}}}-\frac {b B \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a^2+b^2}+a}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} d \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}}} \]

[Out]

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)

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Rubi [A]  time = 0.33, antiderivative size = 406, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {21, 3485, 708, 1094, 634, 618, 206, 628} \[ -\frac {b B \log \left (-\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \tan (c+d x)}+\sqrt {a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt {2} d \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}+\frac {b B \log \left (\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \tan (c+d x)}+\sqrt {a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt {2} d \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}+\frac {b B \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a^2+b^2}+a}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} d \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}}}-\frac {b B \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a^2+b^2}+a}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} d \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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)

Rule 21

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 +
d*x, a + b*x])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

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]

Rule 628

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]

Rule 634

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]

Rule 708

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]

Rule 1094

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]

Rule 3485

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*} \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 (c+d x)}} \, dx\\ &=\frac {(b B) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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 \tanh ^{-1}\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 \tanh ^{-1}\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*}

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Mathematica [C]  time = 0.07, size = 88, normalized size = 0.22 \[ -\frac {i B \left (\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-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
[a + I*b]]/Sqrt[a + I*b]))/d

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fricas [B]  time = 1.33, size = 2127, normalized size = 5.24 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*sqrt(2)*(a^2 + b^2)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^4*(B^4/((a^2 + b^2)*d^4))^(3/4)*sqrt
((B^2*a^2 + B^2*b^2 + (a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2))*arctan((sqrt(2)*(a^4 + 2*a^2*b
^2 + b^4)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^7*sqrt((sqrt(2)*B^5*b^3*d*sqrt((a*cos(d*x + c) + b*sin
(d*x + c))/cos(d*x + c))*(B^4/((a^2 + b^2)*d^4))^(1/4)*sqrt((B^2*a^2 + B^2*b^2 + (a^3 + a*b^2)*d^2*sqrt(B^4/((
a^2 + b^2)*d^4)))/(B^2*b^2))*cos(d*x + c) + B^6*a*b^2*cos(d*x + c) + B^6*b^3*sin(d*x + c) + (B^4*a^2*b^2 + B^4
*b^4)*d^2*sqrt(B^4/((a^2 + b^2)*d^4))*cos(d*x + c))/cos(d*x + c))*(B^4/((a^2 + b^2)*d^4))^(5/4)*sqrt((B^2*a^2
+ B^2*b^2 + (a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2)) - sqrt(2)*(B^3*a^4*b + 2*B^3*a^2*b^3 + B
^3*b^5)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^7*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(
B^4/((a^2 + b^2)*d^4))^(5/4)*sqrt((B^2*a^2 + B^2*b^2 + (a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2
)) - (B^6*a^4 + 2*B^6*a^2*b^2 + B^6*b^4)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^4*sqrt(B^4/((a^2 + b^2)
*d^4)) - (B^8*a^3 + B^8*a*b^2)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2)/(B^10*b^2)) + 4*sqrt(2)*(a^2 +
b^2)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^4*(B^4/((a^2 + b^2)*d^4))^(3/4)*sqrt((B^2*a^2 + B^2*b^2 +
(a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2))*arctan((sqrt(2)*(a^4 + 2*a^2*b^2 + b^4)*sqrt(B^4*b^2
/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^7*sqrt(-(sqrt(2)*B^5*b^3*d*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x +
c))*(B^4/((a^2 + b^2)*d^4))^(1/4)*sqrt((B^2*a^2 + B^2*b^2 + (a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^
2*b^2))*cos(d*x + c) - B^6*a*b^2*cos(d*x + c) - B^6*b^3*sin(d*x + c) - (B^4*a^2*b^2 + B^4*b^4)*d^2*sqrt(B^4/((
a^2 + b^2)*d^4))*cos(d*x + c))/cos(d*x + c))*(B^4/((a^2 + b^2)*d^4))^(5/4)*sqrt((B^2*a^2 + B^2*b^2 + (a^3 + a*
b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2)) - sqrt(2)*(B^3*a^4*b + 2*B^3*a^2*b^3 + B^3*b^5)*sqrt(B^4*b^2/
((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^7*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^2 + b^2)*d^4)
)^(5/4)*sqrt((B^2*a^2 + B^2*b^2 + (a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2)) + (B^6*a^4 + 2*B^6
*a^2*b^2 + B^6*b^4)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^4*sqrt(B^4/((a^2 + b^2)*d^4)) + (B^8*a^3 + B
^8*a*b^2)*sqrt(B^4*b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2)/(B^10*b^2)) + sqrt(2)*(B^2*a*d^2*sqrt(B^4/((a^2 + b
^2)*d^4)) - B^4)*(B^4/((a^2 + b^2)*d^4))^(1/4)*sqrt((B^2*a^2 + B^2*b^2 + (a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 + b^
2)*d^4)))/(B^2*b^2))*log((sqrt(2)*B^5*b^3*d*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^2 +
b^2)*d^4))^(1/4)*sqrt((B^2*a^2 + B^2*b^2 + (a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2))*cos(d*x +
c) + B^6*a*b^2*cos(d*x + c) + B^6*b^3*sin(d*x + c) + (B^4*a^2*b^2 + B^4*b^4)*d^2*sqrt(B^4/((a^2 + b^2)*d^4))*
cos(d*x + c))/cos(d*x + c)) - sqrt(2)*(B^2*a*d^2*sqrt(B^4/((a^2 + b^2)*d^4)) - B^4)*(B^4/((a^2 + b^2)*d^4))^(1
/4)*sqrt((B^2*a^2 + B^2*b^2 + (a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2))*log(-(sqrt(2)*B^5*b^3*
d*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^2 + b^2)*d^4))^(1/4)*sqrt((B^2*a^2 + B^2*b^2 +
(a^3 + a*b^2)*d^2*sqrt(B^4/((a^2 + b^2)*d^4)))/(B^2*b^2))*cos(d*x + c) - B^6*a*b^2*cos(d*x + c) - B^6*b^3*sin
(d*x + c) - (B^4*a^2*b^2 + B^4*b^4)*d^2*sqrt(B^4/((a^2 + b^2)*d^4))*cos(d*x + c))/cos(d*x + c)))/B^4

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Warning, need to choose a branch for the root of
a polynomial with parameters. This might be wrong.The choice was done assuming [d]=[-13,-93]sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueWarning, need to choose a branch for
the root of a polynomial with parameters. This might be wrong.The choice was done assuming [d]=[50,-99]sym2po
ly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen
& e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m &
i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur &
l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argum
ent Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r
2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,
const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,c
onst vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) E
rror: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument
Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym
(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,cons
t index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const
vecteur & l) Error: Bad Argument ValueWarning, integration of abs or sign assumes constant sign by intervals
(correct if the argument is real):Check [abs(t_nostep^2-1)]Discontinuities at zeroes of t_nostep^2-1 were not
checkedEvaluation time: 81.33Done

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maple [B]  time = 0.25, size = 1575, normalized size = 3.88 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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-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+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-1/4/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+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*tan(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)*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)^(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^4-3/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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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
e details)Is b-a positive, negative or zero?

________________________________________________________________________________________

mupad [B]  time = 10.81, size = 6453, normalized size = 15.89 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ B \int \frac {1}{\sqrt {a + b \tan {\left (c + d x \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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