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[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]])

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Rubi [A]  time = 0.225423, antiderivative size = 132, normalized size of antiderivative = 1., 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 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 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]

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 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]

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*}

Mathematica [C]  time = 0.321831, size = 154, normalized size = 1.17 \[ -\frac{i \cos (c+d x) (a-b \tan (c+d x)) \left ((a+i b)^2 \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},\frac{a+b \tan (c+d x)}{a-i b}\right )-(a-i b)^2 \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},\frac{a+b \tan (c+d x)}{a+i b}\right )\right )}{d (a-i b) (a+i b) \sqrt{a+b \tan (c+d x)} (a \cos (c+d x)-b \sin (c+d x))} \]

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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Maple [B]  time = 0.097, size = 2291, normalized size = 17.4 \begin{align*} \text{result too large to display} \end{align*}

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^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^2-1/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^4-2/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(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((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)*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)^(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^4+2/d*b/(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^3-2/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^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))*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+1/4/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^5+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/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^6-1/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^6+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^3-4/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^4-2/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/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))+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))+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(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/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^4-1/4/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)+1/4/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)+4*a*b/(a^2+b^2
)/d/(a+b*tan(d*x+c))^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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

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Fricas [B]  time = 3.5052, size = 14344, normalized size = 108.67 \begin{align*} \text{result too large to display} \end{align*}

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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \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 - \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 )}}\, dx \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \tan \left (d x + c\right ) - a}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

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]

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