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

Optimal. Leaf size=132 \[ \frac {4 a b}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {(-b+i a) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d (a-i b)^{3/2}}-\frac {(b+i a) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d (a+i b)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.23, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3529, 3539, 3537, 63, 208} \[ \frac {4 a b}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {(-b+i a) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d (a-i b)^{3/2}}-\frac {(b+i a) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d (a+i b)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 63

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,
b, c, d, m, n, x]

Rule 208

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

Rule 3529

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

Rule 3537

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

Rule 3539

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]
&& NeQ[c^2 + d^2, 0] &&  !IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {-a+b \tan (c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx &=\frac {4 a b}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {\int \frac {-a^2+b^2+2 a b \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{a^2+b^2}\\ &=\frac {4 a b}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}-\frac {(a-i b) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (a+i b)}-\frac {(a+i b) \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (a-i b)}\\ &=\frac {4 a b}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {(a+i b) \operatorname {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 {(i a+b) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 (a+i b) d}\\ &=\frac {4 a b}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {(a-i b) \operatorname {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 {(a+i b) \operatorname {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 a-b) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{3/2} d}-\frac {(i a+b) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{3/2} d}+\frac {4 a b}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}\\ \end {align*}

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Mathematica [C]  time = 0.37, size = 154, normalized size = 1.17 \[ -\frac {i \cos (c+d x) (a-b \tan (c+d x)) \left ((a+i b)^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {a+b \tan (c+d x)}{a-i b}\right )-(a-i b)^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {a+b \tan (c+d x)}{a+i b}\right )\right )}{d (a-i b) (a+i b) \sqrt {a+b \tan (c+d x)} (a \cos (c+d x)-b \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*Cos[c + d*x]*((a + I*b)^2*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a - I*b)] - (a - I*b)^2*
Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a + I*b)])*(a - b*Tan[c + d*x]))/((a - I*b)*(a + I*b)*d*
(a*Cos[c + d*x] - b*Sin[c + d*x])*Sqrt[a + b*Tan[c + d*x]])

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fricas [B]  time = 1.58, size = 6318, normalized size = 47.86 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*sqrt(2)*((a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*d^5*cos(d*x + c)^2 + 2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a
*b^7)*d^5*cos(d*x + c)*sin(d*x + c) + (a^6*b^2 + 3*a^4*b^4 + 3*a^2*b^6 + b^8)*d^5)*sqrt((a^10 + 5*a^8*b^2 + 10
*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 + (a^11 - 7*a^9*b^2 - 22*a^7*b^4 - 14*a^5*b^6 + 5*a^3*b^8 + 5*a*b^10)
*d^2*sqrt(1/((a^2 + b^2)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((25*a^8*b^2
- 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((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))*(1/((a^2 + b^2)*d^4))^(3/4)*arctan(((5*a^12 + 10*a^10*b^2 - 9*a^8*b^4 - 36*a^6*b^6 -
29*a^4*b^8 - 6*a^2*b^10 + b^12)*d^4*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((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))*sqrt(1/((a^2 + b^2)*d^4)) + (5*
a^11 + 5*a^9*b^2 - 14*a^7*b^4 - 22*a^5*b^6 - 7*a^3*b^8 + a*b^10)*d^2*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*
b^6 - 20*a^2*b^8 + b^10)/((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))
- sqrt(2)*((3*a^6 + 5*a^4*b^2 + a^2*b^4 - b^6)*d^7*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8
+ b^10)/((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))*sqrt(1/((a^2 + b
^2)*d^4)) + 2*(a^5 + 2*a^3*b^2 + a*b^4)*d^5*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/
((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)))*sqrt((a^10 + 5*a^8*b^2
+ 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 + (a^11 - 7*a^9*b^2 - 22*a^7*b^4 - 14*a^5*b^6 + 5*a^3*b^8 + 5*a*b
^10)*d^2*sqrt(1/((a^2 + b^2)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt(((25*a^
14*b^2 - 25*a^12*b^4 - 115*a^10*b^6 + 35*a^8*b^8 + 171*a^6*b^10 + 53*a^4*b^12 - 17*a^2*b^14 + b^16)*d^2*sqrt(1
/((a^2 + b^2)*d^4))*cos(d*x + c) + sqrt(2)*(2*(25*a^13*b^3 - 50*a^11*b^5 - 65*a^9*b^7 + 100*a^7*b^9 + 71*a^5*b
^11 - 18*a^3*b^13 + a*b^15)*d^3*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) + (75*a^12*b^3 - 250*a^10*b^5 + 105*a^8
*b^7 + 260*a^6*b^9 - 147*a^4*b^11 + 22*a^2*b^13 - b^15)*d*cos(d*x + c))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 +
10*a^4*b^6 + 5*a^2*b^8 + b^10 + (a^11 - 7*a^9*b^2 - 22*a^7*b^4 - 14*a^5*b^6 + 5*a^3*b^8 + 5*a*b^10)*d^2*sqrt(1
/((a^2 + b^2)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((a*cos(d*x + c) + b*si
n(d*x + c))/cos(d*x + c))*(1/((a^2 + b^2)*d^4))^(1/4) + (25*a^13*b^2 - 50*a^11*b^4 - 65*a^9*b^6 + 100*a^7*b^8
+ 71*a^5*b^10 - 18*a^3*b^12 + a*b^14)*cos(d*x + c) + (25*a^12*b^3 - 50*a^10*b^5 - 65*a^8*b^7 + 100*a^6*b^9 + 7
1*a^4*b^11 - 18*a^2*b^13 + b^15)*sin(d*x + c))/cos(d*x + c))*(1/((a^2 + b^2)*d^4))^(3/4) + sqrt(2)*((15*a^12*b
+ 10*a^10*b^3 - 47*a^8*b^5 - 52*a^6*b^7 + a^4*b^9 + 10*a^2*b^11 - b^13)*d^7*sqrt((25*a^8*b^2 - 100*a^6*b^4 +
110*a^4*b^6 - 20*a^2*b^8 + b^10)/((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))*sqrt(1/((a^2 + b^2)*d^4)) + 2*(5*a^11*b + 5*a^9*b^3 - 14*a^7*b^5 - 22*a^5*b^7 - 7*a^3*b^9 + a*b^11)*d
^5*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((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)))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b
^10 + (a^11 - 7*a^9*b^2 - 22*a^7*b^4 - 14*a^5*b^6 + 5*a^3*b^8 + 5*a*b^10)*d^2*sqrt(1/((a^2 + b^2)*d^4)))/(25*a
^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*
(1/((a^2 + b^2)*d^4))^(3/4))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)) + 4*sqrt(2)*((a^8 +
2*a^6*b^2 - 2*a^2*b^6 - b^8)*d^5*cos(d*x + c)^2 + 2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*d^5*cos(d*x + c)*
sin(d*x + c) + (a^6*b^2 + 3*a^4*b^4 + 3*a^2*b^6 + b^8)*d^5)*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 +
5*a^2*b^8 + b^10 + (a^11 - 7*a^9*b^2 - 22*a^7*b^4 - 14*a^5*b^6 + 5*a^3*b^8 + 5*a*b^10)*d^2*sqrt(1/((a^2 + b^2
)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^
4*b^6 - 20*a^2*b^8 + b^10)/((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
))*(1/((a^2 + b^2)*d^4))^(3/4)*arctan(-((5*a^12 + 10*a^10*b^2 - 9*a^8*b^4 - 36*a^6*b^6 - 29*a^4*b^8 - 6*a^2*b^
10 + b^12)*d^4*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((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))*sqrt(1/((a^2 + b^2)*d^4)) + (5*a^11 + 5*a^9*b^2 - 14*
a^7*b^4 - 22*a^5*b^6 - 7*a^3*b^8 + a*b^10)*d^2*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^1
0)/((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)) + sqrt(2)*((3*a^6 + 5
*a^4*b^2 + a^2*b^4 - b^6)*d^7*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^12 + 6*a^1
0*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))*sqrt(1/((a^2 + b^2)*d^4)) + 2*(a^5 + 2
*a^3*b^2 + a*b^4)*d^5*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((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)))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*
b^6 + 5*a^2*b^8 + b^10 + (a^11 - 7*a^9*b^2 - 22*a^7*b^4 - 14*a^5*b^6 + 5*a^3*b^8 + 5*a*b^10)*d^2*sqrt(1/((a^2
+ b^2)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt(((25*a^14*b^2 - 25*a^12*b^4 -
115*a^10*b^6 + 35*a^8*b^8 + 171*a^6*b^10 + 53*a^4*b^12 - 17*a^2*b^14 + b^16)*d^2*sqrt(1/((a^2 + b^2)*d^4))*co
s(d*x + c) - sqrt(2)*(2*(25*a^13*b^3 - 50*a^11*b^5 - 65*a^9*b^7 + 100*a^7*b^9 + 71*a^5*b^11 - 18*a^3*b^13 + a*
b^15)*d^3*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) + (75*a^12*b^3 - 250*a^10*b^5 + 105*a^8*b^7 + 260*a^6*b^9 - 1
47*a^4*b^11 + 22*a^2*b^13 - b^15)*d*cos(d*x + c))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8
+ b^10 + (a^11 - 7*a^9*b^2 - 22*a^7*b^4 - 14*a^5*b^6 + 5*a^3*b^8 + 5*a*b^10)*d^2*sqrt(1/((a^2 + b^2)*d^4)))/(
25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x +
c))*(1/((a^2 + b^2)*d^4))^(1/4) + (25*a^13*b^2 - 50*a^11*b^4 - 65*a^9*b^6 + 100*a^7*b^8 + 71*a^5*b^10 - 18*a^3
*b^12 + a*b^14)*cos(d*x + c) + (25*a^12*b^3 - 50*a^10*b^5 - 65*a^8*b^7 + 100*a^6*b^9 + 71*a^4*b^11 - 18*a^2*b^
13 + b^15)*sin(d*x + c))/cos(d*x + c))*(1/((a^2 + b^2)*d^4))^(3/4) - sqrt(2)*((15*a^12*b + 10*a^10*b^3 - 47*a^
8*b^5 - 52*a^6*b^7 + a^4*b^9 + 10*a^2*b^11 - b^13)*d^7*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b
^8 + b^10)/((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))*sqrt(1/((a^2
+ b^2)*d^4)) + 2*(5*a^11*b + 5*a^9*b^3 - 14*a^7*b^5 - 22*a^5*b^7 - 7*a^3*b^9 + a*b^11)*d^5*sqrt((25*a^8*b^2 -
100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((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)))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 + (a^11 - 7*a^9*b^
2 - 22*a^7*b^4 - 14*a^5*b^6 + 5*a^3*b^8 + 5*a*b^10)*d^2*sqrt(1/((a^2 + b^2)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 +
110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/((a^2 + b^2)*d^4))^
(3/4))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)) + sqrt(2)*((a^6 + a^4*b^2 - a^2*b^4 - b^6
)*d*cos(d*x + c)^2 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d*cos(d*x + c)*sin(d*x + c) + (a^4*b^2 + 2*a^2*b^4 + b^6)*d
- ((a^7 - 11*a^5*b^2 + 15*a^3*b^4 - 5*a*b^6)*d^3*cos(d*x + c)^2 + 2*(a^6*b - 10*a^4*b^3 + 5*a^2*b^5)*d^3*cos(
d*x + c)*sin(d*x + c) + (a^5*b^2 - 10*a^3*b^4 + 5*a*b^6)*d^3)*sqrt(1/((a^2 + b^2)*d^4)))*sqrt((a^10 + 5*a^8*b^
2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 + (a^11 - 7*a^9*b^2 - 22*a^7*b^4 - 14*a^5*b^6 + 5*a^3*b^8 + 5*a
*b^10)*d^2*sqrt(1/((a^2 + b^2)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*(1/((a^2 +
b^2)*d^4))^(1/4)*log(((25*a^14*b^2 - 25*a^12*b^4 - 115*a^10*b^6 + 35*a^8*b^8 + 171*a^6*b^10 + 53*a^4*b^12 - 1
7*a^2*b^14 + b^16)*d^2*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) + sqrt(2)*(2*(25*a^13*b^3 - 50*a^11*b^5 - 65*a^9
*b^7 + 100*a^7*b^9 + 71*a^5*b^11 - 18*a^3*b^13 + a*b^15)*d^3*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) + (75*a^12
*b^3 - 250*a^10*b^5 + 105*a^8*b^7 + 260*a^6*b^9 - 147*a^4*b^11 + 22*a^2*b^13 - b^15)*d*cos(d*x + c))*sqrt((a^1
0 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 + (a^11 - 7*a^9*b^2 - 22*a^7*b^4 - 14*a^5*b^6 + 5*a
^3*b^8 + 5*a*b^10)*d^2*sqrt(1/((a^2 + b^2)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)
)*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/((a^2 + b^2)*d^4))^(1/4) + (25*a^13*b^2 - 50*a^11*b^
4 - 65*a^9*b^6 + 100*a^7*b^8 + 71*a^5*b^10 - 18*a^3*b^12 + a*b^14)*cos(d*x + c) + (25*a^12*b^3 - 50*a^10*b^5 -
65*a^8*b^7 + 100*a^6*b^9 + 71*a^4*b^11 - 18*a^2*b^13 + b^15)*sin(d*x + c))/cos(d*x + c)) - sqrt(2)*((a^6 + a^
4*b^2 - a^2*b^4 - b^6)*d*cos(d*x + c)^2 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d*cos(d*x + c)*sin(d*x + c) + (a^4*b^2
+ 2*a^2*b^4 + b^6)*d - ((a^7 - 11*a^5*b^2 + 15*a^3*b^4 - 5*a*b^6)*d^3*cos(d*x + c)^2 + 2*(a^6*b - 10*a^4*b^3
+ 5*a^2*b^5)*d^3*cos(d*x + c)*sin(d*x + c) + (a^5*b^2 - 10*a^3*b^4 + 5*a*b^6)*d^3)*sqrt(1/((a^2 + b^2)*d^4)))*
sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 + (a^11 - 7*a^9*b^2 - 22*a^7*b^4 - 14*a^5*
b^6 + 5*a^3*b^8 + 5*a*b^10)*d^2*sqrt(1/((a^2 + b^2)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^
8 + b^10))*(1/((a^2 + b^2)*d^4))^(1/4)*log(((25*a^14*b^2 - 25*a^12*b^4 - 115*a^10*b^6 + 35*a^8*b^8 + 171*a^6*b
^10 + 53*a^4*b^12 - 17*a^2*b^14 + b^16)*d^2*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) - sqrt(2)*(2*(25*a^13*b^3 -
50*a^11*b^5 - 65*a^9*b^7 + 100*a^7*b^9 + 71*a^5*b^11 - 18*a^3*b^13 + a*b^15)*d^3*sqrt(1/((a^2 + b^2)*d^4))*co
s(d*x + c) + (75*a^12*b^3 - 250*a^10*b^5 + 105*a^8*b^7 + 260*a^6*b^9 - 147*a^4*b^11 + 22*a^2*b^13 - b^15)*d*co
s(d*x + c))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 + (a^11 - 7*a^9*b^2 - 22*a^7*b
^4 - 14*a^5*b^6 + 5*a^3*b^8 + 5*a*b^10)*d^2*sqrt(1/((a^2 + b^2)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6
- 20*a^2*b^8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/((a^2 + b^2)*d^4))^(1/4) + (25*
a^13*b^2 - 50*a^11*b^4 - 65*a^9*b^6 + 100*a^7*b^8 + 71*a^5*b^10 - 18*a^3*b^12 + a*b^14)*cos(d*x + c) + (25*a^1
2*b^3 - 50*a^10*b^5 - 65*a^8*b^7 + 100*a^6*b^9 + 71*a^4*b^11 - 18*a^2*b^13 + b^15)*sin(d*x + c))/cos(d*x + c))
- 16*((a^4*b + a^2*b^3)*cos(d*x + c)^2 + (a^3*b^2 + a*b^4)*cos(d*x + c)*sin(d*x + c))*sqrt((a*cos(d*x + c) +
b*sin(d*x + c))/cos(d*x + c)))/((a^6 + a^4*b^2 - a^2*b^4 - b^6)*d*cos(d*x + c)^2 + 2*(a^5*b + 2*a^3*b^3 + a*b^
5)*d*cos(d*x + c)*sin(d*x + c) + (a^4*b^2 + 2*a^2*b^4 + b^6)*d)

________________________________________________________________________________________

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*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)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)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)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)Warning, need to choose a branch for the root of a polynomial with parameters. This might be
wrong.The choice was done assuming [d]=[-62,5]sym2poly/r2sym(const 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. Thi
s might be wrong.The choice was done assuming [d]=[-99,-57]sym2poly/r2sym(const gen & e,const index_m & i,cons
t vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Erro
r: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Val
uesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const i
ndex_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const ve
cteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: B
ad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesy
m2poly/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 vecteu
r & l) Error: Bad Argument ValueWarning, integration of abs or sign assumes constant sign by intervals (correc
t if the argument is real):Check [abs(t_nostep^2-1)]Discontinuities at zeroes of t_nostep^2-1 were not checked
Evaluation time: 81.18Done

________________________________________________________________________________________

maple [B]  time = 0.31, size = 2291, normalized size = 17.36 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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))*a^4+1/d/b/(a^2+b^2)^(3/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*t
an(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^4-2/d*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^3+1/2/d*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^3-1/d*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^2+1/d/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^6+4/d*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^4+2/d*b^5/(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))-2/d*b^5/(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))+1/4/d*b^3/(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)-1/4/d*b^3/(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)-1/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))+1/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))+1/4/d/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^4+3/4/d*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)*a-1/4/d/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^5+1/d*b^3/(a^2+b^2)^(5/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arcta
n(((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^2-4/d*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^4-1/2/d*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^3+1/4/d/b/(a^2+b^2)^(5/2)*ln(b*tan(d*x+c)+a+(a+b*ta
n(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^5-3/4/d*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)*a-1/4/d/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^4+2/d*b^3/(a^2+b^2)^2/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arcta
n((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/d*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))*a+2/d*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^3-1/d/b/(a^2+b^2)^(5/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*ar
ctan((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^6+4*a*b/(a^2+b^
2)/d/(a+b*tan(d*x+c))^(1/2)

________________________________________________________________________________________

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*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 = 11.92, size = 5475, normalized size = 41.48 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(- (((((24*a*b^4*d^2 - 8*a^3*b^2*d^2)^2/4 - 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) - 12*a*b^4*d^2 + 4*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)*(32*b^
13*d^4 + 96*a^2*b^11*d^4 + 64*a^4*b^9*d^4 - 64*a^6*b^7*d^4 - 96*a^8*b^5*d^4 - 32*a^10*b^3*d^4 + (a + b*tan(c +
d*x))^(1/2)*((((24*a*b^4*d^2 - 8*a^3*b^2*d^2)^2/4 - 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) - 12*a*b^4*d^2 + 4*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)) +
(a + b*tan(c + d*x))^(1/2)*(16*b^12*d^3 + 32*a^2*b^10*d^3 - 32*a^6*b^6*d^3 - 16*a^8*b^4*d^3))*((((24*a*b^4*d^2
- 8*a^3*b^2*d^2)^2/4 - 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) - 12*a*b^4*d^2
+ 4*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*a*b^11*d^2 - 24*a^3*b^9*d
^2 - 24*a^5*b^7*d^2 - 8*a^7*b^5*d^2)*((((24*a*b^4*d^2 - 8*a^3*b^2*d^2)^2/4 - 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) - 12*a*b^4*d^2 + 4*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(- ((-(((24*a*b^4*d^2 - 8*a^3*b^2*d^2)^2/4 - 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) + 12*a*b^4*d^2 - 4*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)*(32*b^13*d^4 + 96*a^2*b^11*d^4 + 64*a^4*b^9*d^4 - 64*a^6*b^7*d^4 - 96*a^8*b^5*d^4 - 32*a^
10*b^3*d^4 + (a + b*tan(c + d*x))^(1/2)*(-(((24*a*b^4*d^2 - 8*a^3*b^2*d^2)^2/4 - 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) + 12*a*b^4*d^2 - 4*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)) + (a + b*tan(c + d*x))^(1/2)*(16*b^12*d^3 + 32*a^2*b^10*d^3 - 32*a^6*b^6*d^3 - 16*a^
8*b^4*d^3))*(-(((24*a*b^4*d^2 - 8*a^3*b^2*d^2)^2/4 - 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) + 12*a*b^4*d^2 - 4*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*a*b^11*d^2 - 24*a^3*b^9*d^2 - 24*a^5*b^7*d^2 - 8*a^7*b^5*d^2)*(-(((24*a*b^4*d^2 - 8*a^3*b^2*d^2)^2/4 - 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) + 12*a*b^4*d^2 - 4*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(((((96*a^6*b^4*d^4 - 16*a^4*b^6*d^4 - 144*a^8*b^2
*d^4)^(1/2) - 4*a^5*d^2 + 12*a^3*b^2*d^2)/(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*t
an(c + d*x))^(1/2)*(16*a^2*b^10*d^3 + 32*a^4*b^8*d^3 - 32*a^8*b^4*d^3 - 16*a^10*b^2*d^3) + ((((96*a^6*b^4*d^4
- 16*a^4*b^6*d^4 - 144*a^8*b^2*d^4)^(1/2) - 4*a^5*d^2 + 12*a^3*b^2*d^2)/(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^2*b^11*d^4 + 256*a^4*b^9*d^4 + 384*a^6*b^7*d^4 + 256*a^8*b^5*d^4 + 64*a^10*b^3*d^4
- ((((96*a^6*b^4*d^4 - 16*a^4*b^6*d^4 - 144*a^8*b^2*d^4)^(1/2) - 4*a^5*d^2 + 12*a^3*b^2*d^2)/(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)*(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))/4))/4))/4 - 8*a^3*b^9*d^2 - 24*a^5*b^7*d^2
- 24*a^7*b^5*d^2 - 8*a^9*b^3*d^2)*(((96*a^6*b^4*d^4 - 16*a^4*b^6*d^4 - 144*a^8*b^2*d^4)^(1/2) - 4*a^5*d^2 + 12
*a^3*b^2*d^2)/(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4))^(1/2))/4 + (log(((-((96*a^6*b^4*d^4 - 16*a^
4*b^6*d^4 - 144*a^8*b^2*d^4)^(1/2) + 4*a^5*d^2 - 12*a^3*b^2*d^2)/(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)*(16*a^2*b^10*d^3 + 32*a^4*b^8*d^3 - 32*a^8*b^4*d^3 - 16*a^10*b^2*d^3
) + ((-((96*a^6*b^4*d^4 - 16*a^4*b^6*d^4 - 144*a^8*b^2*d^4)^(1/2) + 4*a^5*d^2 - 12*a^3*b^2*d^2)/(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^2*b^11*d^4 + 256*a^4*b^9*d^4 + 384*a^6*b^7*d^4 + 256*a^8*b^
5*d^4 + 64*a^10*b^3*d^4 - ((-((96*a^6*b^4*d^4 - 16*a^4*b^6*d^4 - 144*a^8*b^2*d^4)^(1/2) + 4*a^5*d^2 - 12*a^3*b
^2*d^2)/(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)*(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))/4))/4))/4 - 8*a^3*
b^9*d^2 - 24*a^5*b^7*d^2 - 24*a^7*b^5*d^2 - 8*a^9*b^3*d^2)*(-((96*a^6*b^4*d^4 - 16*a^4*b^6*d^4 - 144*a^8*b^2*d
^4)^(1/2) + 4*a^5*d^2 - 12*a^3*b^2*d^2)/(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4))^(1/2))/4 - log(-
(((96*a^6*b^4*d^4 - 16*a^4*b^6*d^4 - 144*a^8*b^2*d^4)^(1/2) - 4*a^5*d^2 + 12*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*a^2*b^10*d^3 + 32*a^4*b^8*d^3 -
32*a^8*b^4*d^3 - 16*a^10*b^2*d^3) - (((96*a^6*b^4*d^4 - 16*a^4*b^6*d^4 - 144*a^8*b^2*d^4)^(1/2) - 4*a^5*d^2 +
12*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*a^2*b^11*d^4 + 256*a^4
*b^9*d^4 + 384*a^6*b^7*d^4 + 256*a^8*b^5*d^4 + 64*a^10*b^3*d^4 + (((96*a^6*b^4*d^4 - 16*a^4*b^6*d^4 - 144*a^8*
b^2*d^4)^(1/2) - 4*a^5*d^2 + 12*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))) - 8*a^3*b^9*d^2 - 24*a^5*b^7*d^2 - 24*a^7*b^5*d^2 - 8*a^9*b^3*d^2)*(((96*a^6*b^4*d
^4 - 16*a^4*b^6*d^4 - 144*a^8*b^2*d^4)^(1/2) - 4*a^5*d^2 + 12*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(- (-((96*a^6*b^4*d^4 - 16*a^4*b^6*d^4 - 144*a^8*b^2*d^4)^(1/2) + 4*a^5*d
^2 - 12*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*a^2*b^10*d^3 + 32*a^4*b^8*d^3 - 32*a^8*b^4*d^3 - 16*a^10*b^2*d^3) - (-((96*a^6*b^4*d^4 - 16*a^4*b^6*
d^4 - 144*a^8*b^2*d^4)^(1/2) + 4*a^5*d^2 - 12*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*a^2*b^11*d^4 + 256*a^4*b^9*d^4 + 384*a^6*b^7*d^4 + 256*a^8*b^5*d^4 + 64*a^10*b^3*d^4 + (-(
(96*a^6*b^4*d^4 - 16*a^4*b^6*d^4 - 144*a^8*b^2*d^4)^(1/2) + 4*a^5*d^2 - 12*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 + 64
0*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))) - 8*a^3*b^9*d^2 - 24*a^5*b^7*d^2 - 24*a
^7*b^5*d^2 - 8*a^9*b^3*d^2)*(-((96*a^6*b^4*d^4 - 16*a^4*b^6*d^4 - 144*a^8*b^2*d^4)^(1/2) + 4*a^5*d^2 - 12*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^12*d^3 + 32*a^2*b^10*d^3 - 32*a^6*b^6*d^3 - 16*a^8*b^4*d^3) + (((96*a^2*b^8*d^4 - 16*b^10*d^4 - 144*a^4*
b^6*d^4)^(1/2) - 12*a*b^4*d^2 + 4*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)*((((96*a^2*b^8*d^4 - 16*b^10*d^4 - 144*a^4*b^6*d^4)^(1/2) - 12*a*b^4*d^2 + 4*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) - 32*b^13*d^4 - 96*a^2*b^11*d^4 - 6
4*a^4*b^9*d^4 + 64*a^6*b^7*d^4 + 96*a^8*b^5*d^4 + 32*a^10*b^3*d^4))*(((96*a^2*b^8*d^4 - 16*b^10*d^4 - 144*a^4*
b^6*d^4)^(1/2) - 12*a*b^4*d^2 + 4*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) - 8*a*b^11*d^2 - 24*a^3*b^9*d^2 - 24*a^5*b^7*d^2 - 8*a^7*b^5*d^2)*(((96*a^2*b^8*d^4 - 16*b^10*d^4 - 144*a^
4*b^6*d^4)^(1/2) - 12*a*b^4*d^2 + 4*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^12*d^3 + 32*a^2*b^10*d^3 - 32*a^6*b^6*d^3 - 16*a^8*b^4*d^3) + (-
((96*a^2*b^8*d^4 - 16*b^10*d^4 - 144*a^4*b^6*d^4)^(1/2) + 12*a*b^4*d^2 - 4*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)*((-((96*a^2*b^8*d^4 - 16*b^10*d^4 - 144*a^4*b^6*d^4)^(1/2) + 12*a
*b^4*d^2 - 4*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) - 32*b^13*d^4 - 96*a^2*b^11*d^4 - 64*a^4*b^9*d^4 + 64*a^6*b^7*d^4 + 96*a^8*b^5*d^4 + 32*a^10*b^3*d^4))*
(-((96*a^2*b^8*d^4 - 16*b^10*d^4 - 144*a^4*b^6*d^4)^(1/2) + 12*a*b^4*d^2 - 4*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) - 8*a*b^11*d^2 - 24*a^3*b^9*d^2 - 24*a^5*b^7*d^2 - 8*a^7*b^5*d^
2)*(-((96*a^2*b^8*d^4 - 16*b^10*d^4 - 144*a^4*b^6*d^4)^(1/2) + 12*a*b^4*d^2 - 4*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) + (4*a*b)/(d*(a^2 + b^2)*(a + b*tan(c + d*x))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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