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

Optimal. Leaf size=128 \[ \frac{2 (a+b \tan (c+d x))^{3/2}}{3 d}+\frac{2 a \sqrt{a+b \tan (c+d x)}}{d}-\frac{(a-i b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}-\frac{(a+i b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.228609, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3528, 3539, 3537, 63, 208} \[ \frac{2 (a+b \tan (c+d x))^{3/2}}{3 d}+\frac{2 a \sqrt{a+b \tan (c+d x)}}{d}-\frac{(a-i b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}-\frac{(a+i b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3528

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

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

Mathematica [A]  time = 0.315965, size = 116, normalized size = 0.91 \[ \frac{2 \sqrt{a+b \tan (c+d x)} (4 a+b \tan (c+d x))-3 (a-i b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )-3 (a+i b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.031, size = 831, normalized size = 6.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(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 = 6.77177, size = 9161, normalized size = 71.57 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(12*sqrt(2)*d^5*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 - (a^3 - 3*a*b^2)*d^2*sqrt((a^6 + 3*a^4*b^2 + 3*
a^2*b^4 + b^6)/d^4))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)^(3/4)*sqrt((9*a^
4*b^2 - 6*a^2*b^4 + b^6)/d^4)*arctan(((3*a^10 + 11*a^8*b^2 + 14*a^6*b^4 + 6*a^4*b^6 - a^2*b^8 - b^10)*d^4*sqrt
((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) + (3*a^13 + 14*a^11*b^2 + 25
*a^9*b^4 + 20*a^7*b^6 + 5*a^5*b^8 - 2*a^3*b^10 - a*b^12)*d^2*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) + sqrt(2)
*((3*a^5 + 2*a^3*b^2 - a*b^4)*d^7*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)*sqrt((9*a^4*b^2 - 6*a^2*b^4 +
b^6)/d^4) + (3*a^8 + 2*a^6*b^2 - 4*a^4*b^4 - 2*a^2*b^6 + b^8)*d^5*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))*sqr
t((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 - (a^3 - 3*a*b^2)*d^2*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4))/(9*a
^4*b^2 - 6*a^2*b^4 + b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((a^6 + 3*a^4*b^2 + 3*a^2*b^4
+ b^6)/d^4)^(3/4) + sqrt(2)*(a*d^7*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)*sqrt((9*a^4*b^2 - 6*a^2*b^4 +
b^6)/d^4) + (a^4 - b^4)*d^5*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6
- (a^3 - 3*a*b^2)*d^2*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt(((9*a
^8 + 12*a^6*b^2 - 2*a^4*b^4 - 4*a^2*b^6 + b^8)*d^2*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)*cos(d*x + c)
+ sqrt(2)*((9*a^6 - 15*a^4*b^2 + 7*a^2*b^4 - b^6)*d^3*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)*cos(d*x +
c) + (9*a^9 + 12*a^7*b^2 - 2*a^5*b^4 - 4*a^3*b^6 + a*b^8)*d*cos(d*x + c))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 +
b^6 - (a^3 - 3*a*b^2)*d^2*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt((
a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)^(1/4) + (9*a^11 + 21*
a^9*b^2 + 10*a^7*b^4 - 6*a^5*b^6 - 3*a^3*b^8 + a*b^10)*cos(d*x + c) + (9*a^10*b + 21*a^8*b^3 + 10*a^6*b^5 - 6*
a^4*b^7 - 3*a^2*b^9 + b^11)*sin(d*x + c))/((a^2 + b^2)*cos(d*x + c)))*((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4
)^(3/4))/(9*a^14*b^2 + 39*a^12*b^4 + 61*a^10*b^6 + 35*a^8*b^8 - 5*a^6*b^10 - 11*a^4*b^12 - a^2*b^14 + b^16))*c
os(d*x + c) + 12*sqrt(2)*d^5*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 - (a^3 - 3*a*b^2)*d^2*sqrt((a^6 + 3*a^4*b
^2 + 3*a^2*b^4 + b^6)/d^4))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)^(3/4)*sqr
t((9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4)*arctan(-((3*a^10 + 11*a^8*b^2 + 14*a^6*b^4 + 6*a^4*b^6 - a^2*b^8 - b^10)*
d^4*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) + (3*a^13 + 14*a^11*
b^2 + 25*a^9*b^4 + 20*a^7*b^6 + 5*a^5*b^8 - 2*a^3*b^10 - a*b^12)*d^2*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) -
sqrt(2)*((3*a^5 + 2*a^3*b^2 - a*b^4)*d^7*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)*sqrt((9*a^4*b^2 - 6*a^
2*b^4 + b^6)/d^4) + (3*a^8 + 2*a^6*b^2 - 4*a^4*b^4 - 2*a^2*b^6 + b^8)*d^5*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/d
^4))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 - (a^3 - 3*a*b^2)*d^2*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^
4))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((a^6 + 3*a^4*b^2 + 3*
a^2*b^4 + b^6)/d^4)^(3/4) - sqrt(2)*(a*d^7*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)*sqrt((9*a^4*b^2 - 6*a
^2*b^4 + b^6)/d^4) + (a^4 - b^4)*d^5*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^
4 + b^6 - (a^3 - 3*a*b^2)*d^2*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sq
rt(((9*a^8 + 12*a^6*b^2 - 2*a^4*b^4 - 4*a^2*b^6 + b^8)*d^2*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)*cos(d
*x + c) - sqrt(2)*((9*a^6 - 15*a^4*b^2 + 7*a^2*b^4 - b^6)*d^3*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)*co
s(d*x + c) + (9*a^9 + 12*a^7*b^2 - 2*a^5*b^4 - 4*a^3*b^6 + a*b^8)*d*cos(d*x + c))*sqrt((a^6 + 3*a^4*b^2 + 3*a^
2*b^4 + b^6 - (a^3 - 3*a*b^2)*d^2*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4))/(9*a^4*b^2 - 6*a^2*b^4 + b^6)
)*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)^(1/4) + (9*a^
11 + 21*a^9*b^2 + 10*a^7*b^4 - 6*a^5*b^6 - 3*a^3*b^8 + a*b^10)*cos(d*x + c) + (9*a^10*b + 21*a^8*b^3 + 10*a^6*
b^5 - 6*a^4*b^7 - 3*a^2*b^9 + b^11)*sin(d*x + c))/((a^2 + b^2)*cos(d*x + c)))*((a^6 + 3*a^4*b^2 + 3*a^2*b^4 +
b^6)/d^4)^(3/4))/(9*a^14*b^2 + 39*a^12*b^4 + 61*a^10*b^6 + 35*a^8*b^8 - 5*a^6*b^10 - 11*a^4*b^12 - a^2*b^14 +
b^16))*cos(d*x + c) + 3*sqrt(2)*((a^3 - 3*a*b^2)*d^3*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)*cos(d*x + c
) + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d*cos(d*x + c))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 - (a^3 - 3*a*b
^2)*d^2*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*((a^6 + 3*a^4*b^2 + 3*a^
2*b^4 + b^6)/d^4)^(1/4)*log(((9*a^8 + 12*a^6*b^2 - 2*a^4*b^4 - 4*a^2*b^6 + b^8)*d^2*sqrt((a^6 + 3*a^4*b^2 + 3*
a^2*b^4 + b^6)/d^4)*cos(d*x + c) + sqrt(2)*((9*a^6 - 15*a^4*b^2 + 7*a^2*b^4 - b^6)*d^3*sqrt((a^6 + 3*a^4*b^2 +
3*a^2*b^4 + b^6)/d^4)*cos(d*x + c) + (9*a^9 + 12*a^7*b^2 - 2*a^5*b^4 - 4*a^3*b^6 + a*b^8)*d*cos(d*x + c))*sqr
t((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 - (a^3 - 3*a*b^2)*d^2*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4))/(9*a
^4*b^2 - 6*a^2*b^4 + b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((a^6 + 3*a^4*b^2 + 3*a^2*b^4
+ b^6)/d^4)^(1/4) + (9*a^11 + 21*a^9*b^2 + 10*a^7*b^4 - 6*a^5*b^6 - 3*a^3*b^8 + a*b^10)*cos(d*x + c) + (9*a^10
*b + 21*a^8*b^3 + 10*a^6*b^5 - 6*a^4*b^7 - 3*a^2*b^9 + b^11)*sin(d*x + c))/((a^2 + b^2)*cos(d*x + c))) - 3*sqr
t(2)*((a^3 - 3*a*b^2)*d^3*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)*cos(d*x + c) + (a^6 + 3*a^4*b^2 + 3*a^
2*b^4 + b^6)*d*cos(d*x + c))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 - (a^3 - 3*a*b^2)*d^2*sqrt((a^6 + 3*a^4*b
^2 + 3*a^2*b^4 + b^6)/d^4))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)^(1/4)*log
(((9*a^8 + 12*a^6*b^2 - 2*a^4*b^4 - 4*a^2*b^6 + b^8)*d^2*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)*cos(d*x
+ c) - sqrt(2)*((9*a^6 - 15*a^4*b^2 + 7*a^2*b^4 - b^6)*d^3*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)*cos(
d*x + c) + (9*a^9 + 12*a^7*b^2 - 2*a^5*b^4 - 4*a^3*b^6 + a*b^8)*d*cos(d*x + c))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*
b^4 + b^6 - (a^3 - 3*a*b^2)*d^2*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*
sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)/d^4)^(1/4) + (9*a^11
+ 21*a^9*b^2 + 10*a^7*b^4 - 6*a^5*b^6 - 3*a^3*b^8 + a*b^10)*cos(d*x + c) + (9*a^10*b + 21*a^8*b^3 + 10*a^6*b^
5 - 6*a^4*b^7 - 3*a^2*b^9 + b^11)*sin(d*x + c))/((a^2 + b^2)*cos(d*x + c))) - 8*(4*(a^7 + 3*a^5*b^2 + 3*a^3*b^
4 + a*b^6)*cos(d*x + c) + (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*sin(d*x + c))*sqrt((a*cos(d*x + c) + b*sin(d*x
+ c))/cos(d*x + c)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d*cos(d*x + c))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out