\(\int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx\) [539]

Optimal result

Integrand size = 23, antiderivative size = 165 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{3/2} d}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{3/2} d}-\frac {2 a^2 \tan (c+d x)}{b \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {2 \left (2 a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}{b^2 \left (a^2+b^2\right ) d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3646, 3711, 3620, 3618, 65, 214} \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=-\frac {2 a^2 \tan (c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {2 \left (2 a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}{b^2 d \left (a^2+b^2\right )}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d (a-i b)^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d (a+i b)^{3/2}} \]

[In]

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

[Out]

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

Rule 65

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 214

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

Rule 3618

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

Rule 3620

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 3646

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

Rule 3711

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])
^m*Simp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0]
&&  !LeQ[m, -1]

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

Mathematica [C] (verified)

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

Time = 1.19 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.47 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\frac {\frac {i \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b}}-\frac {i \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b}}+\frac {4 a}{b \sqrt {a+b \tan (c+d x)}}-\frac {a \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {a+b \tan (c+d x)}{a-i b}\right )}{(i a+b) \sqrt {a+b \tan (c+d x)}}+\frac {a \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {a+b \tan (c+d x)}{a+i b}\right )}{(i a-b) \sqrt {a+b \tan (c+d x)}}+\frac {2 \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}}{b d} \]

[In]

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

[Out]

((I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/Sqrt[a - I*b] - (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[
a + I*b]])/Sqrt[a + I*b] + (4*a)/(b*Sqrt[a + b*Tan[c + d*x]]) - (a*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[
c + d*x])/(a - I*b)])/((I*a + b)*Sqrt[a + b*Tan[c + d*x]]) + (a*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c +
d*x])/(a + I*b)])/((I*a - b)*Sqrt[a + b*Tan[c + d*x]]) + (2*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]])/(b*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1771\) vs. \(2(145)=290\).

Time = 0.09 (sec) , antiderivative size = 1772, normalized size of antiderivative = 10.74

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1772\)
default	\(\text {Expression too large to display}\)	\(1772\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2011 vs. \(2 (141) = 282\).

Time = 0.28 (sec) , antiderivative size = 2011, normalized size of antiderivative = 12.19 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*(((a^2*b^3 + b^5)*d*tan(d*x + c) + (a^3*b^2 + a*b^4)*d)*sqrt(((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt
(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12
)*d^4)) + a^3 - 3*a*b^2)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))*log(-(3*a^2 - b^2)*sqrt(b*tan(d*x + c) + a
) + (2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^3*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*
a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - (3*a^4 - 4*a^2*b^2 + b^4)*d)*sqrt(((a^6 + 3*a^4
*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6
+ 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + a^3 - 3*a*b^2)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))) - ((a^2*b
^3 + b^5)*d*tan(d*x + c) + (a^3*b^2 + a*b^4)*d)*sqrt(((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*a^4*b^2
- 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + a^
3 - 3*a*b^2)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))*log(-(3*a^2 - b^2)*sqrt(b*tan(d*x + c) + a) - (2*(a^7
+ 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^3*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20
*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - (3*a^4 - 4*a^2*b^2 + b^4)*d)*sqrt(((a^6 + 3*a^4*b^2 + 3*a^2
*b^4 + b^6)*d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8
+ 6*a^2*b^10 + b^12)*d^4)) + a^3 - 3*a*b^2)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))) - ((a^2*b^3 + b^5)*d*
tan(d*x + c) + (a^3*b^2 + a*b^4)*d)*sqrt(-((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^
4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - a^3 + 3*a*b^2
)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))*log(-(3*a^2 - b^2)*sqrt(b*tan(d*x + c) + a) + (2*(a^7 + 3*a^5*b^2
+ 3*a^3*b^4 + a*b^6)*d^3*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 +
15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (3*a^4 - 4*a^2*b^2 + b^4)*d)*sqrt(-((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6
)*d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b
^10 + b^12)*d^4)) - a^3 + 3*a*b^2)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))) + ((a^2*b^3 + b^5)*d*tan(d*x +
c) + (a^3*b^2 + a*b^4)*d)*sqrt(-((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/(
(a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - a^3 + 3*a*b^2)/((a^6 +
3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))*log(-(3*a^2 - b^2)*sqrt(b*tan(d*x + c) + a) - (2*(a^7 + 3*a^5*b^2 + 3*a^3*b
^4 + a*b^6)*d^3*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8
+ 6*a^2*b^10 + b^12)*d^4)) + (3*a^4 - 4*a^2*b^2 + b^4)*d)*sqrt(-((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt
(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12
)*d^4)) - a^3 + 3*a*b^2)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))) + 4*(2*a^3 + a*b^2 + (a^2*b + b^3)*tan(d*
x + c))*sqrt(b*tan(d*x + c) + a))/((a^2*b^3 + b^5)*d*tan(d*x + c) + (a^3*b^2 + a*b^4)*d)

Sympy [F]

\[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\int \frac {\tan ^{3}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-1)]

Timed out. \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Giac [F(-1)]

Timed out. \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 7.42 (sec) , antiderivative size = 2869, normalized size of antiderivative = 17.39 \[ \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

atan((((1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2)*((1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^
2*d^2*3i - 3*a^2*b*d^2)))^(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^12*d^4 - 96*a^2*b^10*d^4 - 64*a^4*b^8*d^4 + 64*
a^6*b^6*d^4 + 96*a^8*b^4*d^4 + 32*a^10*b^2*d^4) + (a + b*tan(c + d*x))^(1/2)*(16*b^10*d^3 + 32*a^2*b^8*d^3 - 3
2*a^6*b^4*d^3 - 16*a^8*b^2*d^3))*(1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2)*1i + ((1i/
(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2)*(32*b^12*d^4 + (1i/(4*(a^3*d^2*1i + b^3*d^2 - a
*b^2*d^2*3i - 3*a^2*b*d^2)))^(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) + 96*a^2*b^10*d^4 + 64*a^4*b^8*d^4 - 64*a^6*b^6*d^4
- 96*a^8*b^4*d^4 - 32*a^10*b^2*d^4) + (a + b*tan(c + d*x))^(1/2)*(16*b^10*d^3 + 32*a^2*b^8*d^3 - 32*a^6*b^4*d
^3 - 16*a^8*b^2*d^3))*(1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2)*1i)/(((1i/(4*(a^3*d^2
*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2)*(32*b^12*d^4 + (1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i
- 3*a^2*b*d^2)))^(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) + 96*a^2*b^10*d^4 + 64*a^4*b^8*d^4 - 64*a^6*b^6*d^4 - 96*a^8*b
^4*d^4 - 32*a^10*b^2*d^4) + (a + b*tan(c + d*x))^(1/2)*(16*b^10*d^3 + 32*a^2*b^8*d^3 - 32*a^6*b^4*d^3 - 16*a^8
*b^2*d^3))*(1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2) - ((1i/(4*(a^3*d^2*1i + b^3*d^2
- a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2)*((1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(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^12*d^4 - 96*a^2*b^10*d^4 - 64*a^4*b^8*d^4 + 64*a^6*b^6*d^4 + 96*a^8*b^4*d^4 + 32*a^
10*b^2*d^4) + (a + b*tan(c + d*x))^(1/2)*(16*b^10*d^3 + 32*a^2*b^8*d^3 - 32*a^6*b^4*d^3 - 16*a^8*b^2*d^3))*(1i
/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2) + 16*a*b^8*d^2 + 48*a^3*b^6*d^2 + 48*a^5*b^4*d
^2 + 16*a^7*b^2*d^2))*(1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2)*2i + atan(((1/(a^3*d^
2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^(1/2)*(((1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^
(1/2)*(32*a^6*b^6*d^4 - 48*a^2*b^10*d^4 - 32*a^4*b^8*d^4 - 16*b^12*d^4 + 48*a^8*b^4*d^4 + 16*a^10*b^2*d^4 + ((
1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^(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))/2 + ((a + b*tan(c +
d*x))^(1/2)*(16*b^10*d^3 + 32*a^2*b^8*d^3 - 32*a^6*b^4*d^3 - 16*a^8*b^2*d^3))/2)*1i + (1/(a^3*d^2 + b^3*d^2*1i
- 3*a*b^2*d^2 - a^2*b*d^2*3i))^(1/2)*(((1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^(1/2)*(16*b^12
*d^4 + 48*a^2*b^10*d^4 + 32*a^4*b^8*d^4 - 32*a^6*b^6*d^4 - 48*a^8*b^4*d^4 - 16*a^10*b^2*d^4 + ((1/(a^3*d^2 + b
^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^(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))/2 + ((a + b*tan(c + d*x))^(1/2)*(1
6*b^10*d^3 + 32*a^2*b^8*d^3 - 32*a^6*b^4*d^3 - 16*a^8*b^2*d^3))/2)*1i)/((1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2
- a^2*b*d^2*3i))^(1/2)*(((1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^(1/2)*(16*b^12*d^4 + 48*a^2*
b^10*d^4 + 32*a^4*b^8*d^4 - 32*a^6*b^6*d^4 - 48*a^8*b^4*d^4 - 16*a^10*b^2*d^4 + ((1/(a^3*d^2 + b^3*d^2*1i - 3*
a*b^2*d^2 - a^2*b*d^2*3i))^(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))/2 + ((a + b*tan(c + d*x))^(1/2)*(16*b^10*d^3 + 3
2*a^2*b^8*d^3 - 32*a^6*b^4*d^3 - 16*a^8*b^2*d^3))/2) - (1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))
^(1/2)*(((1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^(1/2)*(32*a^6*b^6*d^4 - 48*a^2*b^10*d^4 - 32*
a^4*b^8*d^4 - 16*b^12*d^4 + 48*a^8*b^4*d^4 + 16*a^10*b^2*d^4 + ((1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b
*d^2*3i))^(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))/2 + ((a + b*tan(c + d*x))^(1/2)*(16*b^10*d^3 + 32*a^2*b^8*d^3 - 3
2*a^6*b^4*d^3 - 16*a^8*b^2*d^3))/2) + 16*a*b^8*d^2 + 48*a^3*b^6*d^2 + 48*a^5*b^4*d^2 + 16*a^7*b^2*d^2))*(1/(a^
3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^(1/2)*1i + (2*(a + b*tan(c + d*x))^(1/2))/(b^2*d) + (2*a^3)/
(b^2*d*(a^2 + b^2)*(a + b*tan(c + d*x))^(1/2))