3.7.6 \(\int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx\) [606]

Optimal. Leaf size=444 \[ \frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \]

[Out]

-1/4*b^(5/2)*(63*a^4+46*a^2*b^2+15*b^4)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))/a^(7/2)/(a^2+b^2)^3/d-1/2*(a-
b)*(a^2+4*a*b+b^2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)-1/2*(a-b)*(a^2+4*a*b+b^2)*arctan(
1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)-1/4*(a+b)*(a^2-4*a*b+b^2)*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(
d*x+c))/(a^2+b^2)^3/d*2^(1/2)+1/4*(a+b)*(a^2-4*a*b+b^2)*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)^3/
d*2^(1/2)+1/4*(-8*a^4-31*a^2*b^2-15*b^4)/a^3/(a^2+b^2)^2/d/tan(d*x+c)^(1/2)+1/2*b^2/a/(a^2+b^2)/d/tan(d*x+c)^(
1/2)/(a+b*tan(d*x+c))^2+1/4*b^2*(13*a^2+5*b^2)/a^2/(a^2+b^2)^2/d/tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))

________________________________________________________________________________________

Rubi [A]
time = 0.74, antiderivative size = 444, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3650, 3730, 3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \begin {gather*} \frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {ArcTan}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 d \left (a^2+b^2\right )^2 \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{7/2} d \left (a^2+b^2\right )^3}-\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 d \left (a^2+b^2\right )^2 \sqrt {\tan (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^3),x]

[Out]

((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((a - b)*(a^2
+ 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - (b^(5/2)*(63*a^4 + 46*a^2*
b^2 + 15*b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*a^(7/2)*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*
b + b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + ((a + b)*(a^2 - 4*a
*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - (8*a^4 + 31*a^2*b^
2 + 15*b^4)/(4*a^3*(a^2 + b^2)^2*d*Sqrt[Tan[c + d*x]]) + b^2/(2*a*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[
c + d*x])^2) + (b^2*(13*a^2 + 5*b^2))/(4*a^2*(a^2 + b^2)^2*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x]))

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 210

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

Rule 211

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

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1182

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(-a)*c]

Rule 3615

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[c^2 + d^2, 0]

Rule 3650

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

Rule 3715

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

Rule 3730

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Ta
n[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3734

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

Rubi steps

\begin {align*} \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx &=\frac {b^2}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {\int \frac {\frac {1}{2} \left (4 a^2+5 b^2\right )-2 a b \tan (c+d x)+\frac {5}{2} b^2 \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx}{2 a \left (a^2+b^2\right )}\\ &=\frac {b^2}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}+\frac {\int \frac {\frac {1}{4} \left (8 a^4+31 a^2 b^2+15 b^4\right )-4 a^3 b \tan (c+d x)+\frac {3}{4} b^2 \left (13 a^2+5 b^2\right ) \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))} \, dx}{2 a^2 \left (a^2+b^2\right )^2}\\ &=-\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {\int \frac {\frac {1}{8} b \left (24 a^4+31 a^2 b^2+15 b^4\right )+a^3 \left (a^2-b^2\right ) \tan (c+d x)+\frac {1}{8} b \left (8 a^4+31 a^2 b^2+15 b^4\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{a^3 \left (a^2+b^2\right )^2}\\ &=-\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {\int \frac {a^3 b \left (3 a^2-b^2\right )+a^4 \left (a^2-3 b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{a^3 \left (a^2+b^2\right )^3}-\frac {\left (b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right )\right ) \int \frac {1+\tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{8 a^3 \left (a^2+b^2\right )^3}\\ &=-\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {2 \text {Subst}\left (\int \frac {a^3 b \left (3 a^2-b^2\right )+a^4 \left (a^2-3 b^2\right ) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a^3 \left (a^2+b^2\right )^3 d}-\frac {\left (b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{8 a^3 \left (a^2+b^2\right )^3 d}\\ &=-\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right )^3 d}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right )^3 d}-\frac {\left (b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{4 a^3 \left (a^2+b^2\right )^3 d}\\ &=-\frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d}-\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right )^3 d}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right )^3 d}\\ &=-\frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}\\ &=\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 4.17, size = 358, normalized size = 0.81 \begin {gather*} \frac {\frac {-8 a^{13/2}-39 a^{9/2} b^2-46 a^{5/2} b^4-15 \sqrt {a} b^6-4 (-1)^{3/4} a^{7/2} (a+i b)^3 \text {ArcTan}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sqrt {\tan (c+d x)}-63 a^4 b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {\tan (c+d x)}-46 a^2 b^{9/2} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {\tan (c+d x)}-15 b^{13/2} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {\tan (c+d x)}-4 \sqrt [4]{-1} a^{7/2} (i a+b)^3 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sqrt {\tan (c+d x)}}{a^{5/2} \left (a^2+b^2\right )^2}+\frac {2 b^2}{(a+b \tan (c+d x))^2}+\frac {13 a^2 b^2+5 b^4}{a \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^3),x]

[Out]

((-8*a^(13/2) - 39*a^(9/2)*b^2 - 46*a^(5/2)*b^4 - 15*Sqrt[a]*b^6 - 4*(-1)^(3/4)*a^(7/2)*(a + I*b)^3*ArcTan[(-1
)^(3/4)*Sqrt[Tan[c + d*x]]]*Sqrt[Tan[c + d*x]] - 63*a^4*b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*S
qrt[Tan[c + d*x]] - 46*a^2*b^(9/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[Tan[c + d*x]] - 15*b^(13/
2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[Tan[c + d*x]] - 4*(-1)^(1/4)*a^(7/2)*(I*a + b)^3*ArcTanh[
(-1)^(3/4)*Sqrt[Tan[c + d*x]]]*Sqrt[Tan[c + d*x]])/(a^(5/2)*(a^2 + b^2)^2) + (2*b^2)/(a + b*Tan[c + d*x])^2 +
(13*a^2*b^2 + 5*b^4)/(a*(a^2 + b^2)*(a + b*Tan[c + d*x])))/(4*a*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]])

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Maple [A]
time = 0.15, size = 356, normalized size = 0.80

method	result	size
derivativedivides	\(\frac {-\frac {2}{a^{3} \sqrt {\tan \left (d x +c \right )}}-\frac {2 b^{3} \left (\frac {\left (\frac {15}{8} a^{4} b +\frac {11}{4} a^{2} b^{3}+\frac {7}{8} b^{5}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )+\frac {a \left (17 a^{4}+26 a^{2} b^{2}+9 b^{4}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{8}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\left (63 a^{4}+46 a^{2} b^{2}+15 b^{4}\right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right )^{3} a^{3}}+\frac {\frac {\left (-3 a^{2} b +b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-a^{3}+3 b^{2} a \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{3}}}{d}\)	\(356\)
default	\(\frac {-\frac {2}{a^{3} \sqrt {\tan \left (d x +c \right )}}-\frac {2 b^{3} \left (\frac {\left (\frac {15}{8} a^{4} b +\frac {11}{4} a^{2} b^{3}+\frac {7}{8} b^{5}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )+\frac {a \left (17 a^{4}+26 a^{2} b^{2}+9 b^{4}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{8}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\left (63 a^{4}+46 a^{2} b^{2}+15 b^{4}\right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right )^{3} a^{3}}+\frac {\frac {\left (-3 a^{2} b +b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-a^{3}+3 b^{2} a \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{3}}}{d}\)	\(356\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-2/a^3/tan(d*x+c)^(1/2)-2*b^3/(a^2+b^2)^3/a^3*(((15/8*a^4*b+11/4*a^2*b^3+7/8*b^5)*tan(d*x+c)^(3/2)+1/8*a*
(17*a^4+26*a^2*b^2+9*b^4)*tan(d*x+c)^(1/2))/(a+b*tan(d*x+c))^2+1/8*(63*a^4+46*a^2*b^2+15*b^4)/(a*b)^(1/2)*arct
an(b*tan(d*x+c)^(1/2)/(a*b)^(1/2)))+2/(a^2+b^2)^3*(1/8*(-3*a^2*b+b^3)*2^(1/2)*(ln((1+2^(1/2)*tan(d*x+c)^(1/2)+
tan(d*x+c))/(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*
tan(d*x+c)^(1/2)))+1/8*(-a^3+3*a*b^2)*2^(1/2)*(ln((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1/2)*tan(d*x+c
)^(1/2)+tan(d*x+c)))+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2)))))

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Maxima [A]
time = 0.51, size = 464, normalized size = 1.05 \begin {gather*} -\frac {\frac {{\left (63 \, a^{4} b^{3} + 46 \, a^{2} b^{5} + 15 \, b^{7}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6}\right )} \sqrt {a b}} + \frac {8 \, a^{6} + 16 \, a^{4} b^{2} + 8 \, a^{2} b^{4} + {\left (8 \, a^{4} b^{2} + 31 \, a^{2} b^{4} + 15 \, b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left (16 \, a^{5} b + 49 \, a^{3} b^{3} + 25 \, a b^{5}\right )} \tan \left (d x + c\right )}{{\left (a^{7} b^{2} + 2 \, a^{5} b^{4} + a^{3} b^{6}\right )} \tan \left (d x + c\right )^{\frac {5}{2}} + 2 \, {\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + {\left (a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} \sqrt {\tan \left (d x + c\right )}} + \frac {2 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}}}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/4*((63*a^4*b^3 + 46*a^2*b^5 + 15*b^7)*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^9 + 3*a^7*b^2 + 3*a^5*b^4
+ a^3*b^6)*sqrt(a*b)) + (8*a^6 + 16*a^4*b^2 + 8*a^2*b^4 + (8*a^4*b^2 + 31*a^2*b^4 + 15*b^6)*tan(d*x + c)^2 + (
16*a^5*b + 49*a^3*b^3 + 25*a*b^5)*tan(d*x + c))/((a^7*b^2 + 2*a^5*b^4 + a^3*b^6)*tan(d*x + c)^(5/2) + 2*(a^8*b
+ 2*a^6*b^3 + a^4*b^5)*tan(d*x + c)^(3/2) + (a^9 + 2*a^7*b^2 + a^5*b^4)*sqrt(tan(d*x + c))) + (2*sqrt(2)*(a^3
+ 3*a^2*b - 3*a*b^2 - b^3)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*(a^3 + 3*a^2*b -
3*a*b^2 - b^3)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) - sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)
*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) + sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*log(-sqrt(2)*sqr
t(tan(d*x + c)) + tan(d*x + c) + 1))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6))/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 12694 vs. \(2 (392) = 784\).
time = 19.74, size = 25393, normalized size = 57.19 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

[-1/16*(16*sqrt(2)*((a^25 + 3*a^23*b^2 - 17*a^21*b^4 - 123*a^19*b^6 - 342*a^17*b^8 - 546*a^15*b^10 - 546*a^13*
b^12 - 342*a^11*b^14 - 123*a^9*b^16 - 17*a^7*b^18 + 3*a^5*b^20 + a^3*b^22)*d^5*cos(d*x + c)^6 - (a^25 - 3*a^23
*b^2 - 69*a^21*b^4 - 321*a^19*b^6 - 774*a^17*b^8 - 1134*a^15*b^10 - 1050*a^13*b^12 - 594*a^11*b^14 - 171*a^9*b
^16 + a^7*b^18 + 15*a^5*b^20 + 3*a^3*b^22)*d^5*cos(d*x + c)^4 - 3*(2*a^23*b^2 + 17*a^21*b^4 + 63*a^19*b^6 + 13
2*a^17*b^8 + 168*a^15*b^10 + 126*a^13*b^12 + 42*a^11*b^14 - 12*a^9*b^16 - 18*a^7*b^18 - 7*a^5*b^20 - a^3*b^22)
*d^5*cos(d*x + c)^2 - (a^21*b^4 + 9*a^19*b^6 + 36*a^17*b^8 + 84*a^15*b^10 + 126*a^13*b^12 + 126*a^11*b^14 + 84
*a^9*b^16 + 36*a^7*b^18 + 9*a^5*b^20 + a^3*b^22)*d^5 + 4*((a^24*b + 8*a^22*b^3 + 27*a^20*b^5 + 48*a^18*b^7 + 4
2*a^16*b^9 - 42*a^12*b^13 - 48*a^10*b^15 - 27*a^8*b^17 - 8*a^6*b^19 - a^4*b^21)*d^5*cos(d*x + c)^5 - (a^24*b +
7*a^22*b^3 + 18*a^20*b^5 + 12*a^18*b^7 - 42*a^16*b^9 - 126*a^14*b^11 - 168*a^12*b^13 - 132*a^10*b^15 - 63*a^8
*b^17 - 17*a^6*b^19 - 2*a^4*b^21)*d^5*cos(d*x + c)^3 - (a^22*b^3 + 9*a^20*b^5 + 36*a^18*b^7 + 84*a^16*b^9 + 12
6*a^14*b^11 + 126*a^12*b^13 + 84*a^10*b^15 + 36*a^8*b^17 + 9*a^6*b^19 + a^4*b^21)*d^5*cos(d*x + c))*sin(d*x +
c))*sqrt((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 + 2*(3*a^17*b + 8*a^15*
b^3 - 12*a^13*b^5 - 72*a^11*b^7 - 110*a^9*b^9 - 72*a^7*b^11 - 12*a^5*b^13 + 8*a^3*b^15 + 3*a*b^17)*d^2*sqrt(1/
((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^12 - 30*a^10*b^2 + 2
55*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12))*sqrt((a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6
*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/((a^24 + 12*a^22*b^2 + 66*a^20*b^4 + 220*a^18*b^6 + 495*a^16*b^8 + 79
2*a^14*b^10 + 924*a^12*b^12 + 792*a^10*b^14 + 495*a^8*b^16 + 220*a^6*b^18 + 66*a^4*b^20 + 12*a^2*b^22 + b^24)*
d^4))*(1/((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))^(3/4)*arctan(((
a^24 - 6*a^22*b^2 - 84*a^20*b^4 - 322*a^18*b^6 - 603*a^16*b^8 - 540*a^14*b^10 + 540*a^10*b^14 + 603*a^8*b^16 +
322*a^6*b^18 + 84*a^4*b^20 + 6*a^2*b^22 - b^24)*d^4*sqrt((a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 25
5*a^4*b^8 - 30*a^2*b^10 + b^12)/((a^24 + 12*a^22*b^2 + 66*a^20*b^4 + 220*a^18*b^6 + 495*a^16*b^8 + 792*a^14*b^
10 + 924*a^12*b^12 + 792*a^10*b^14 + 495*a^8*b^16 + 220*a^6*b^18 + 66*a^4*b^20 + 12*a^2*b^22 + b^24)*d^4))*sqr
t(1/((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^26*b
+ 35*a^24*b^3 + 186*a^22*b^5 + 594*a^20*b^7 + 1265*a^18*b^9 + 1881*a^16*b^11 + 1980*a^14*b^13 + 1452*a^12*b^1
5 + 693*a^10*b^17 + 165*a^8*b^19 - 22*a^6*b^21 - 30*a^4*b^23 - 9*a^2*b^25 - b^27)*d^7*sqrt((a^12 - 30*a^10*b^2
+ 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/((a^24 + 12*a^22*b^2 + 66*a^20*b^4 + 220*a^18
*b^6 + 495*a^16*b^8 + 792*a^14*b^10 + 924*a^12*b^12 + 792*a^10*b^14 + 495*a^8*b^16 + 220*a^6*b^18 + 66*a^4*b^2
0 + 12*a^2*b^22 + b^24)*d^4))*sqrt(1/((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^21 + 6*a^19*b^2 + 9*a^17*b^4 - 24*a^15*b^6 - 126*a^13*b^8 - 252*a^11*b^10 - 294*a^9*b^12 - 2
16*a^7*b^14 - 99*a^5*b^16 - 26*a^3*b^18 - 3*a*b^20)*d^5*sqrt((a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 +
255*a^4*b^8 - 30*a^2*b^10 + b^12)/((a^24 + 12*a^22*b^2 + 66*a^20*b^4 + 220*a^18*b^6 + 495*a^16*b^8 + 792*a^14
*b^10 + 924*a^12*b^12 + 792*a^10*b^14 + 495*a^8*b^16 + 220*a^6*b^18 + 66*a^4*b^20 + 12*a^2*b^22 + b^24)*d^4)))
*sqrt((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 + 2*(3*a^17*b + 8*a^15*b^3
- 12*a^13*b^5 - 72*a^11*b^7 - 110*a^9*b^9 - 72*a^7*b^11 - 12*a^5*b^13 + 8*a^3*b^15 + 3*a*b^17)*d^2*sqrt(1/((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^12 - 30*a^10*b^2 + 255*
a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12))*sqrt(((a^18 - 27*a^16*b^2 + 168*a^14*b^4 + 224*a^12
*b^6 - 366*a^10*b^8 - 366*a^8*b^10 + 224*a^6*b^12 + 168*a^4*b^14 - 27*a^2*b^16 + b^18)*d^2*sqrt(1/((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))*cos(d*x + c) + sqrt(2)*((a^21 - 30*a
^19*b^2 + 249*a^17*b^4 - 280*a^15*b^6 - 1038*a^13*b^8 + 732*a^11*b^10 + 1322*a^9*b^12 - 504*a^7*b^14 - 531*a^5
*b^16 + 82*a^3*b^18 - 3*a*b^20)*d^3*sqrt(1/((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))*cos(d*x + c) - (3*a^14*b - 91*a^12*b^3 + 795*a^10*b^5 - 1611*a^8*b^7 + 1217*a^6*b^9 - 345*a
^4*b^11 + 33*a^2*b^13 - b^15)*d*cos(d*x + c))*sqrt((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 + 2*(3*a^17*b + 8*a^15*b^3 - 12*a^13*b^5 - 72*a^11*b^7 - 110*a^9*b^9 - 72*a^7*b^11 - 12*a^5
*b^13 + 8*a^3*b^15 + 3*a*b^17)*d^2*sqrt(1/((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^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12))*sqrt(s
in(d*x + c)/cos(d*x + c))*(1/((a^12 + 6*a^10*b^...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \tan {\left (c + d x \right )}\right )^{3} \tan ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)**(3/2)/(a+b*tan(d*x+c))**3,x)

[Out]

Integral(1/((a + b*tan(c + d*x))**3*tan(c + d*x)**(3/2)), x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^3,x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 13.71, size = 2500, normalized size = 5.63 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(tan(c + d*x)^(3/2)*(a + b*tan(c + d*x))^3),x)

[Out]

(log(29491200*a^22*b^35*d^4 - ((((-1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i -
20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2)*(((((-1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^
4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2)*(251658240*a^24*b^45*d^8 + 5049942016*a^26*b^43*d^8 + 483
68713728*a^28*b^41*d^8 + 293819383808*a^30*b^39*d^8 + 1268458192896*a^32*b^37*d^8 + 4132731617280*a^34*b^35*d^
8 + 10531192700928*a^36*b^33*d^8 + 21462823993344*a^38*b^31*d^8 + 35469618315264*a^40*b^29*d^8 + 4789690485964
8*a^42*b^27*d^8 + 52983958077440*a^44*b^25*d^8 + 47896904859648*a^46*b^23*d^8 + 35090285461504*a^48*b^21*d^8 +
20487396655104*a^50*b^19*d^8 + 9230622916608*a^52*b^17*d^8 + 2994733056000*a^54*b^15*d^8 + 565576728576*a^56*
b^13*d^8 - 18572378112*a^58*b^11*d^8 - 50281316352*a^60*b^9*d^8 - 16089350144*a^62*b^7*d^8 - 2516582400*a^64*b
^5*d^8 - 167772160*a^66*b^3*d^8 + (tan(c + d*x)^(1/2)*(-1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2
- a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2)*(134217728*a^27*b^45*d^9 + 2550136832*a^29*b^43*
d^9 + 22817013760*a^31*b^41*d^9 + 127506841600*a^33*b^39*d^9 + 497276682240*a^35*b^37*d^9 + 1430626762752*a^37
*b^35*d^9 + 3121367482368*a^39*b^33*d^9 + 5202279137280*a^41*b^31*d^9 + 6502848921600*a^43*b^29*d^9 + 56358023
98720*a^45*b^27*d^9 + 2254320959488*a^47*b^25*d^9 - 2254320959488*a^49*b^23*d^9 - 5635802398720*a^51*b^21*d^9
- 6502848921600*a^53*b^19*d^9 - 5202279137280*a^55*b^17*d^9 - 3121367482368*a^57*b^15*d^9 - 1430626762752*a^59
*b^13*d^9 - 497276682240*a^61*b^11*d^9 - 127506841600*a^63*b^9*d^9 - 22817013760*a^65*b^7*d^9 - 2550136832*a^6
7*b^5*d^9 - 134217728*a^69*b^3*d^9))/2))/2 - tan(c + d*x)^(1/2)*(471859200*a^22*b^44*d^7 + 9500098560*a^24*b^4
2*d^7 + 91857354752*a^26*b^40*d^7 + 564502986752*a^28*b^38*d^7 + 2464648527872*a^30*b^36*d^7 + 8104469069824*a
^32*b^34*d^7 + 20769933361152*a^34*b^32*d^7 + 42351565209600*a^36*b^30*d^7 + 69534945902592*a^38*b^28*d^7 + 92
434029608960*a^40*b^26*d^7 + 99508717355008*a^42*b^24*d^7 + 86342935511040*a^44*b^22*d^7 + 59767095558144*a^46
*b^20*d^7 + 32432589897728*a^48*b^18*d^7 + 13411815522304*a^50*b^16*d^7 + 4030457708544*a^52*b^14*d^7 + 805425
905664*a^54*b^12*d^7 + 86608183296*a^56*b^10*d^7 + 1612709888*a^58*b^8*d^7 + 16777216*a^60*b^6*d^7 + 167772160
*a^62*b^4*d^7 + 16777216*a^64*b^2*d^7))*(-1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2
*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2))/2 - 117964800*a^21*b^42*d^6 - 841482240*a^23*b^40*d^6 + 38293
99552*a^25*b^38*d^6 + 78068580352*a^27*b^36*d^6 + 497438162944*a^29*b^34*d^6 + 1899895980032*a^31*b^32*d^6 + 4
972695519232*a^33*b^30*d^6 + 9371195015168*a^35*b^28*d^6 + 12890720436224*a^37*b^26*d^6 + 12726089809920*a^39*
b^24*d^6 + 8366961197056*a^41*b^22*d^6 + 2597662490624*a^43*b^20*d^6 - 1171836108800*a^45*b^18*d^6 - 198688165
0688*a^47*b^16*d^6 - 1237583921152*a^49*b^14*d^6 - 449507753984*a^51*b^12*d^6 - 97476149248*a^53*b^10*d^6 - 11
931222016*a^55*b^8*d^6 - 1006632960*a^57*b^6*d^6 - 134217728*a^59*b^4*d^6 - 8388608*a^61*b^2*d^6))/2 + tan(c +
d*x)^(1/2)*(7610564608*a^27*b^33*d^5 - 597688320*a^23*b^37*d^5 - 1671430144*a^25*b^35*d^5 - 58982400*a^21*b^3
9*d^5 + 85774565376*a^29*b^31*d^5 + 385487994880*a^31*b^29*d^5 + 1104303620096*a^33*b^27*d^5 + 2240523796480*a
^35*b^25*d^5 + 3345249468416*a^37*b^23*d^5 + 3717287903232*a^39*b^21*d^5 + 3053967114240*a^41*b^19*d^5 + 18074
74491392*a^43*b^17*d^5 + 726513221632*a^45*b^15*d^5 + 170768990208*a^47*b^13*d^5 + 10492051456*a^49*b^11*d^5 -
4917821440*a^51*b^9*d^5 - 923009024*a^53*b^7*d^5 + 8388608*a^55*b^5*d^5))*(-1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*
b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2))/2 + 460062720*a^24*b^33*d^
4 + 3439722496*a^26*b^31*d^4 + 16227237888*a^28*b^29*d^4 + 53669396480*a^30*b^27*d^4 + 131031367680*a^32*b^25*
d^4 + 242529730560*a^34*b^23*d^4 + 344454070272*a^36*b^21*d^4 + 375993532416*a^38*b^19*d^4 + 313043189760*a^40
*b^17*d^4 + 195253370880*a^42*b^15*d^4 + 88318935040*a^44*b^13*d^4 + 27352498176*a^46*b^11*d^4 + 5187043328*a^
48*b^9*d^4 + 454164480*a^50*b^7*d^4)*(-1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15
i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2))/2 - log(29491200*a^22*b^35*d^4 - ((-1/(4*(b^6*d^2*1i - a^6*d^2*1
i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i)))^(1/2)*(((-1/(4*(b^6*d^2*
1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i)))^(1/2)*(251
658240*a^24*b^45*d^8 + 5049942016*a^26*b^43*d^8 + 48368713728*a^28*b^41*d^8 + 293819383808*a^30*b^39*d^8 + 126
8458192896*a^32*b^37*d^8 + 4132731617280*a^34*b^35*d^8 + 10531192700928*a^36*b^33*d^8 + 21462823993344*a^38*b^
31*d^8 + 35469618315264*a^40*b^29*d^8 + 47896904859648*a^42*b^27*d^8 + 52983958077440*a^44*b^25*d^8 + 47896904
859648*a^46*b^23*d^8 + 35090285461504*a^48*b^21*d^8 + 20487396655104*a^50*b^19*d^8 + 9230622916608*a^52*b^17*d
^8 + 2994733056000*a^54*b^15*d^8 + 565576728576...

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