3.590 \(\int \frac{1}{\tan ^{\frac{7}{2}}(c+d x) (a+b \tan (c+d x))} \, dx\)

Optimal. Leaf size=300 \[ -\frac{2 b^{9/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{a^{7/2} d \left (a^2+b^2\right )}-\frac{(a+b) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d \left (a^2+b^2\right )}+\frac{(a+b) \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} d \left (a^2+b^2\right )}+\frac{2 \left (a^2-b^2\right )}{a^3 d \sqrt{\tan (c+d x)}}+\frac{(a-b) \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )}-\frac{(a-b) \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )}+\frac{2 b}{3 a^2 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2}{5 a d \tan ^{\frac{5}{2}}(c+d x)} \]

[Out]

-(((a + b)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d)) + ((a + b)*ArcTan[1 + Sqrt[2]*Sqrt
[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*b^(9/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(7/2)*(
a^2 + b^2)*d) + ((a - b)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((a -
b)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - 2/(5*a*d*Tan[c + d*x]^(5/2
)) + (2*b)/(3*a^2*d*Tan[c + d*x]^(3/2)) + (2*(a^2 - b^2))/(a^3*d*Sqrt[Tan[c + d*x]])

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Rubi [A]  time = 0.794575, antiderivative size = 300, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 14, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.609, Rules used = {3569, 3649, 3650, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ -\frac{2 b^{9/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{a^{7/2} d \left (a^2+b^2\right )}-\frac{(a+b) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d \left (a^2+b^2\right )}+\frac{(a+b) \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} d \left (a^2+b^2\right )}+\frac{2 \left (a^2-b^2\right )}{a^3 d \sqrt{\tan (c+d x)}}+\frac{(a-b) \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )}-\frac{(a-b) \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )}+\frac{2 b}{3 a^2 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2}{5 a d \tan ^{\frac{5}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[1/(Tan[c + d*x]^(7/2)*(a + b*Tan[c + d*x])),x]

[Out]

-(((a + b)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d)) + ((a + b)*ArcTan[1 + Sqrt[2]*Sqrt
[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*b^(9/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(7/2)*(
a^2 + b^2)*d) + ((a - b)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((a -
b)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - 2/(5*a*d*Tan[c + d*x]^(5/2
)) + (2*b)/(3*a^2*d*Tan[c + d*x]^(3/2)) + (2*(a^2 - b^2))/(a^3*d*Sqrt[Tan[c + d*x]])

Rule 3569

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 3649

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*T
an[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 3650

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

Rule 3653

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]

Rule 3534

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 1168

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 1162

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 617

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 204

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

Rule 1165

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 628

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 3634

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 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 205

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

Rubi steps

\begin{align*} \int \frac{1}{\tan ^{\frac{7}{2}}(c+d x) (a+b \tan (c+d x))} \, dx &=-\frac{2}{5 a d \tan ^{\frac{5}{2}}(c+d x)}-\frac{2 \int \frac{\frac{5 b}{2}+\frac{5}{2} a \tan (c+d x)+\frac{5}{2} b \tan ^2(c+d x)}{\tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx}{5 a}\\ &=-\frac{2}{5 a d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 b}{3 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{4 \int \frac{-\frac{15}{4} \left (a^2-b^2\right )+\frac{15}{4} b^2 \tan ^2(c+d x)}{\tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))} \, dx}{15 a^2}\\ &=-\frac{2}{5 a d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 b}{3 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (a^2-b^2\right )}{a^3 d \sqrt{\tan (c+d x)}}-\frac{8 \int \frac{-\frac{15}{8} b \left (a^2-b^2\right )-\frac{15}{8} a^3 \tan (c+d x)-\frac{15}{8} b \left (a^2-b^2\right ) \tan ^2(c+d x)}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{15 a^3}\\ &=-\frac{2}{5 a d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 b}{3 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (a^2-b^2\right )}{a^3 d \sqrt{\tan (c+d x)}}-\frac{8 \int \frac{-\frac{15 a^3 b}{8}-\frac{15}{8} a^4 \tan (c+d x)}{\sqrt{\tan (c+d x)}} \, dx}{15 a^3 \left (a^2+b^2\right )}-\frac{b^5 \int \frac{1+\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 )}\\ &=-\frac{2}{5 a d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 b}{3 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (a^2-b^2\right )}{a^3 d \sqrt{\tan (c+d x)}}-\frac{16 \operatorname{Subst}\left (\int \frac{-\frac{15 a^3 b}{8}-\frac{15 a^4 x^2}{8}}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{15 a^3 \left (a^2+b^2\right ) d}-\frac{b^5 \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{a^3 \left (a^2+b^2\right ) d}\\ &=-\frac{2}{5 a d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 b}{3 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (a^2-b^2\right )}{a^3 d \sqrt{\tan (c+d x)}}-\frac{(a-b) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac{\left (2 b^5\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac{(a+b) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}\\ &=-\frac{2 b^{9/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{a^{7/2} \left (a^2+b^2\right ) d}-\frac{2}{5 a d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 b}{3 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (a^2-b^2\right )}{a^3 d \sqrt{\tan (c+d x)}}+\frac{(a-b) \operatorname{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 ) d}+\frac{(a-b) \operatorname{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 ) d}+\frac{(a+b) \operatorname{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 ) d}+\frac{(a+b) \operatorname{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 ) d}\\ &=-\frac{2 b^{9/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{a^{7/2} \left (a^2+b^2\right ) d}+\frac{(a-b) \log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right ) d}-\frac{(a-b) \log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right ) d}-\frac{2}{5 a d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 b}{3 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (a^2-b^2\right )}{a^3 d \sqrt{\tan (c+d x)}}+\frac{(a+b) \operatorname{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 ) d}-\frac{(a+b) \operatorname{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 ) d}\\ &=-\frac{(a+b) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right ) d}+\frac{(a+b) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right ) d}-\frac{2 b^{9/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{a^{7/2} \left (a^2+b^2\right ) d}+\frac{(a-b) \log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right ) d}-\frac{(a-b) \log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right ) d}-\frac{2}{5 a d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 b}{3 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (a^2-b^2\right )}{a^3 d \sqrt{\tan (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 3.60333, size = 248, normalized size = 0.83 \[ \frac{-15 \left (\frac{8 b^{9/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{a^{3/2} \left (a^2+b^2\right )}+\frac{2 \sqrt{2} a^2 (a+b) \left (\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )-\tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )\right )}{a^2+b^2}-\frac{\sqrt{2} a^2 (a-b) \left (\log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )-\log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )\right )}{a^2+b^2}-\frac{8 (a-b) (a+b)}{a \sqrt{\tan (c+d x)}}\right )-\frac{24 a}{\tan ^{\frac{5}{2}}(c+d x)}+\frac{40 b}{\tan ^{\frac{3}{2}}(c+d x)}}{60 a^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Tan[c + d*x]^(7/2)*(a + b*Tan[c + d*x])),x]

[Out]

(-15*((2*Sqrt[2]*a^2*(a + b)*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])
)/(a^2 + b^2) + (8*b^(9/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(3/2)*(a^2 + b^2)) - (Sqrt[2]*a^2*
(a - b)*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x
]]))/(a^2 + b^2) - (8*(a - b)*(a + b))/(a*Sqrt[Tan[c + d*x]])) - (24*a)/Tan[c + d*x]^(5/2) + (40*b)/Tan[c + d*
x]^(3/2))/(60*a^2*d)

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Maple [A]  time = 0.032, size = 369, normalized size = 1.2 \begin{align*} -2\,{\frac{{b}^{5}}{d{a}^{3} \left ({a}^{2}+{b}^{2} \right ) \sqrt{ab}}\arctan \left ({\frac{\sqrt{\tan \left ( dx+c \right ) }b}{\sqrt{ab}}} \right ) }-{\frac{2}{5\,ad} \left ( \tan \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}}+2\,{\frac{1}{ad\sqrt{\tan \left ( dx+c \right ) }}}-2\,{\frac{{b}^{2}}{d{a}^{3}\sqrt{\tan \left ( dx+c \right ) }}}+{\frac{2\,b}{3\,{a}^{2}d} \left ( \tan \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}}+{\frac{b\sqrt{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) }\arctan \left ( 1+\sqrt{2}\sqrt{\tan \left ( dx+c \right ) } \right ) }+{\frac{b\sqrt{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) }\arctan \left ( -1+\sqrt{2}\sqrt{\tan \left ( dx+c \right ) } \right ) }+{\frac{b\sqrt{2}}{4\,d \left ({a}^{2}+{b}^{2} \right ) }\ln \left ({ \left ( 1+\sqrt{2}\sqrt{\tan \left ( dx+c \right ) }+\tan \left ( dx+c \right ) \right ) \left ( 1-\sqrt{2}\sqrt{\tan \left ( dx+c \right ) }+\tan \left ( dx+c \right ) \right ) ^{-1}} \right ) }+{\frac{a\sqrt{2}}{4\,d \left ({a}^{2}+{b}^{2} \right ) }\ln \left ({ \left ( 1-\sqrt{2}\sqrt{\tan \left ( dx+c \right ) }+\tan \left ( dx+c \right ) \right ) \left ( 1+\sqrt{2}\sqrt{\tan \left ( dx+c \right ) }+\tan \left ( dx+c \right ) \right ) ^{-1}} \right ) }+{\frac{a\sqrt{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) }\arctan \left ( 1+\sqrt{2}\sqrt{\tan \left ( dx+c \right ) } \right ) }+{\frac{a\sqrt{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) }\arctan \left ( -1+\sqrt{2}\sqrt{\tan \left ( dx+c \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/tan(d*x+c)^(7/2)/(a+b*tan(d*x+c)),x)

[Out]

-2/d/a^3*b^5/(a^2+b^2)/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))-2/5/a/d/tan(d*x+c)^(5/2)+2/a/d/tan(d
*x+c)^(1/2)-2/d/a^3/tan(d*x+c)^(1/2)*b^2+2/3*b/a^2/d/tan(d*x+c)^(3/2)+1/2/d/(a^2+b^2)*b*2^(1/2)*arctan(1+2^(1/
2)*tan(d*x+c)^(1/2))+1/2/d/(a^2+b^2)*b*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))+1/4/d/(a^2+b^2)*b*2^(1/2)*l
n((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)))+1/4/d/(a^2+b^2)*a*2^(1/2)*l
n((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)))+1/2/d/(a^2+b^2)*a*2^(1/2)*a
rctan(1+2^(1/2)*tan(d*x+c)^(1/2))+1/2/d/(a^2+b^2)*a*2^(1/2)*arctan(-1+2^(1/2)*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(1/tan(d*x+c)^(7/2)/(a+b*tan(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 90.4018, size = 18077, normalized size = 60.26 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)^(7/2)/(a+b*tan(d*x+c)),x, algorithm="fricas")

[Out]

[-1/60*(60*sqrt(2)*((a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6)*d^5*cos(d*x + c)^4 - 2*(a^9 + 3*a^7*b^2 + 3*a^5*b^
4 + a^3*b^6)*d^5*cos(d*x + c)^2 + (a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6)*d^5)*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2
*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt((a^4 - 2
*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4)
*arctan(-((a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*d^4*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 +
4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - sqrt(2)*((a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^
2*b^7 + b^9)*d^7*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((
a^4 + 2*a^2*b^2 + b^4)*d^4)) - (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^5*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 +
4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^
2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(((a^6 - a^4*b^2 - a^2*b^4 + b^6)*d^2*sq
rt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + sqrt(2)*((a^7 - a^5*b^2 - a^3*b^4 + a*b^6)*d^3*sqrt(1/((a^4
+ 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - (a^4*b - 2*a^2*b^3 + b^5)*d*cos(d*x + c))*sqrt((a^4 + 2*a^2*b^2 + b^4
+ 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(
d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (a^4 - 2*a^2*b^2 + b^4)*sin(d*x + c))/cos(d*x
+ c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4) - sqrt(2)*((a^10*b + 3*a^8*b^3 + 2*a^6*b^5 - 2*a^4*b^7 - 3*a^2*
b^9 - b^11)*d^7*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a
^4 + 2*a^2*b^2 + b^4)*d^4)) - (a^9 + 2*a^7*b^2 - 2*a^3*b^6 - a*b^8)*d^5*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4
*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2
*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*
a^2*b^2 + b^4)*d^4))^(3/4))/(a^4 - 2*a^2*b^2 + b^4)) + 60*sqrt(2)*((a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6)*d^5
*cos(d*x + c)^4 - 2*(a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6)*d^5*cos(d*x + c)^2 + (a^9 + 3*a^7*b^2 + 3*a^5*b^4
+ a^3*b^6)*d^5)*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4
)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8
)*d^4))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4)*arctan(((a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*d^4*sqrt((a^4 - 2*
a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) +
sqrt(2)*((a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*d^7*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2
+ 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*
b^6)*d^5*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^
2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4)
)*sqrt(((a^6 - a^4*b^2 - a^2*b^4 + b^6)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - sqrt(2)*((a^7
- a^5*b^2 - a^3*b^4 + a*b^6)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - (a^4*b - 2*a^2*b^3 + b^
5)*d*cos(d*x + c))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 +
b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) +
(a^4 - 2*a^2*b^2 + b^4)*sin(d*x + c))/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4) + sqrt(2)*((a^10*
b + 3*a^8*b^3 + 2*a^6*b^5 - 2*a^4*b^7 - 3*a^2*b^9 - b^11)*d^7*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 +
6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - (a^9 + 2*a^7*b^2 - 2*a^3*b^6 - a*b
^8)*d^5*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2
*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))
*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4))/(a^4 - 2*a^2*b^2 + b^4)) + 15*sqrt(2
)*((a^5 + a^3*b^2)*d*cos(d*x + c)^4 - 2*(a^5 + a^3*b^2)*d*cos(d*x + c)^2 + (a^5 + a^3*b^2)*d - 2*((a^6*b + a^4
*b^3)*d^3*cos(d*x + c)^4 - 2*(a^6*b + a^4*b^3)*d^3*cos(d*x + c)^2 + (a^6*b + a^4*b^3)*d^3)*sqrt(1/((a^4 + 2*a^
2*b^2 + b^4)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 +
b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4)*log(((a^6 - a^4*b^2 - a^2*b^4 +
b^6)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + sqrt(2)*((a^7 - a^5*b^2 - a^3*b^4 + a*b^6)*d^3*s
qrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - (a^4*b - 2*a^2*b^3 + b^5)*d*cos(d*x + c))*sqrt((a^4 + 2*a^
2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4)
)*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (a^4 - 2*a^2*b^2 + b^4)*sin(d*x +
c))/cos(d*x + c)) - 15*sqrt(2)*((a^5 + a^3*b^2)*d*cos(d*x + c)^4 - 2*(a^5 + a^3*b^2)*d*cos(d*x + c)^2 + (a^5 +
a^3*b^2)*d - 2*((a^6*b + a^4*b^3)*d^3*cos(d*x + c)^4 - 2*(a^6*b + a^4*b^3)*d^3*cos(d*x + c)^2 + (a^6*b + a^4*
b^3)*d^3)*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d
^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4)*log
(((a^6 - a^4*b^2 - a^2*b^4 + b^6)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - sqrt(2)*((a^7 - a^5
*b^2 - a^3*b^4 + a*b^6)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - (a^4*b - 2*a^2*b^3 + b^5)*d*c
os(d*x + c))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d
^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (a^4
- 2*a^2*b^2 + b^4)*sin(d*x + c))/cos(d*x + c)) - 30*(b^4*cos(d*x + c)^4 - 2*b^4*cos(d*x + c)^2 + b^4)*sqrt(-b/
a)*log(-(6*a*b*cos(d*x + c)*sin(d*x + c) - (a^2 - b^2)*cos(d*x + c)^2 - b^2 - 4*(a^2*cos(d*x + c)^2 - a*b*cos(
d*x + c)*sin(d*x + c))*sqrt(-b/a)*sqrt(sin(d*x + c)/cos(d*x + c)))/(2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b
^2)*cos(d*x + c)^2 + b^2)) + 8*(5*(a^3*b + a*b^3)*cos(d*x + c)^4 - 5*(a^3*b + a*b^3)*cos(d*x + c)^2 + 3*((6*a^
4 + a^2*b^2 - 5*b^4)*cos(d*x + c)^3 - 5*(a^4 - b^4)*cos(d*x + c))*sin(d*x + c))*sqrt(sin(d*x + c)/cos(d*x + c)
))/((a^5 + a^3*b^2)*d*cos(d*x + c)^4 - 2*(a^5 + a^3*b^2)*d*cos(d*x + c)^2 + (a^5 + a^3*b^2)*d), -1/60*(60*sqrt
(2)*((a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6)*d^5*cos(d*x + c)^4 - 2*(a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6)*d^
5*cos(d*x + c)^2 + (a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6)*d^5)*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3
*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt((a^4 - 2*a^2*b^2 + b^4)
/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4)*arctan(-((a^8
+ 2*a^6*b^2 - 2*a^2*b^6 - b^8)*d^4*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^
8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - sqrt(2)*((a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*d^
7*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2
+ b^4)*d^4)) - (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^5*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a
^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4
+ 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(((a^6 - a^4*b^2 - a^2*b^4 + b^6)*d^2*sqrt(1/((a^4 + 2*
a^2*b^2 + b^4)*d^4))*cos(d*x + c) + sqrt(2)*((a^7 - a^5*b^2 - a^3*b^4 + a*b^6)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 +
b^4)*d^4))*cos(d*x + c) - (a^4*b - 2*a^2*b^3 + b^5)*d*cos(d*x + c))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2
*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*
x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (a^4 - 2*a^2*b^2 + b^4)*sin(d*x + c))/cos(d*x + c))*(1/((a^4
+ 2*a^2*b^2 + b^4)*d^4))^(3/4) - sqrt(2)*((a^10*b + 3*a^8*b^3 + 2*a^6*b^5 - 2*a^4*b^7 - 3*a^2*b^9 - b^11)*d^7
*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2
+ b^4)*d^4)) - (a^9 + 2*a^7*b^2 - 2*a^3*b^6 - a*b^8)*d^5*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^
4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 +
2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*
d^4))^(3/4))/(a^4 - 2*a^2*b^2 + b^4)) + 60*sqrt(2)*((a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6)*d^5*cos(d*x + c)^4
- 2*(a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6)*d^5*cos(d*x + c)^2 + (a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6)*d^5)
*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 -
2*a^2*b^2 + b^4))*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*(1/((a^
4 + 2*a^2*b^2 + b^4)*d^4))^(3/4)*arctan(((a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*d^4*sqrt((a^4 - 2*a^2*b^2 + b^4)/
((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + sqrt(2)*((a^8*b
+ 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*d^7*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4
*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^5*sqrt((
a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2
*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(((a^6 -
a^4*b^2 - a^2*b^4 + b^6)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - sqrt(2)*((a^7 - a^5*b^2 - a^
3*b^4 + a*b^6)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - (a^4*b - 2*a^2*b^3 + b^5)*d*cos(d*x +
c))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^
4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (a^4 - 2*a^2*b
^2 + b^4)*sin(d*x + c))/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4) + sqrt(2)*((a^10*b + 3*a^8*b^3 +
2*a^6*b^5 - 2*a^4*b^7 - 3*a^2*b^9 - b^11)*d^7*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*
a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - (a^9 + 2*a^7*b^2 - 2*a^3*b^6 - a*b^8)*d^5*sqrt((a
^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*
(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x +
c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4))/(a^4 - 2*a^2*b^2 + b^4)) + 15*sqrt(2)*((a^5 + a^3*b
^2)*d*cos(d*x + c)^4 - 2*(a^5 + a^3*b^2)*d*cos(d*x + c)^2 + (a^5 + a^3*b^2)*d - 2*((a^6*b + a^4*b^3)*d^3*cos(d
*x + c)^4 - 2*(a^6*b + a^4*b^3)*d^3*cos(d*x + c)^2 + (a^6*b + a^4*b^3)*d^3)*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^
4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a
^4 - 2*a^2*b^2 + b^4))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4)*log(((a^6 - a^4*b^2 - a^2*b^4 + b^6)*d^2*sqrt(1
/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + sqrt(2)*((a^7 - a^5*b^2 - a^3*b^4 + a*b^6)*d^3*sqrt(1/((a^4 + 2
*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - (a^4*b - 2*a^2*b^3 + b^5)*d*cos(d*x + c))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2
*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x
+ c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (a^4 - 2*a^2*b^2 + b^4)*sin(d*x + c))/cos(d*x + c
)) - 15*sqrt(2)*((a^5 + a^3*b^2)*d*cos(d*x + c)^4 - 2*(a^5 + a^3*b^2)*d*cos(d*x + c)^2 + (a^5 + a^3*b^2)*d - 2
*((a^6*b + a^4*b^3)*d^3*cos(d*x + c)^4 - 2*(a^6*b + a^4*b^3)*d^3*cos(d*x + c)^2 + (a^6*b + a^4*b^3)*d^3)*sqrt(
1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4
+ 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4)*log(((a^6 - a^4*b^
2 - a^2*b^4 + b^6)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - sqrt(2)*((a^7 - a^5*b^2 - a^3*b^4
+ a*b^6)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - (a^4*b - 2*a^2*b^3 + b^5)*d*cos(d*x + c))*sq
rt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*
a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (a^4 - 2*a^2*b^2 + b
^4)*sin(d*x + c))/cos(d*x + c)) + 120*(b^4*cos(d*x + c)^4 - 2*b^4*cos(d*x + c)^2 + b^4)*sqrt(b/a)*arctan((2*a^
2*b*cos(d*x + c)^2*sin(d*x + c) + a*b^2*cos(d*x + c) + (a^3 - a*b^2)*cos(d*x + c)^3)*sqrt(b/a)*sqrt(sin(d*x +
c)/cos(d*x + c))/(2*a*b^2*cos(d*x + c)^3 - 2*a*b^2*cos(d*x + c) - (b^3 + (a^2*b - b^3)*cos(d*x + c)^2)*sin(d*x
+ c))) + 8*(5*(a^3*b + a*b^3)*cos(d*x + c)^4 - 5*(a^3*b + a*b^3)*cos(d*x + c)^2 + 3*((6*a^4 + a^2*b^2 - 5*b^4
)*cos(d*x + c)^3 - 5*(a^4 - b^4)*cos(d*x + c))*sin(d*x + c))*sqrt(sin(d*x + c)/cos(d*x + c)))/((a^5 + a^3*b^2)
*d*cos(d*x + c)^4 - 2*(a^5 + a^3*b^2)*d*cos(d*x + c)^2 + (a^5 + a^3*b^2)*d)]

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Sympy [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(1/tan(d*x+c)**(7/2)/(a+b*tan(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.19236, size = 392, normalized size = 1.31 \begin{align*} -\frac{2 \, b^{5} \arctan \left (\frac{b \sqrt{\tan \left (d x + c\right )}}{\sqrt{a b}}\right )}{{\left (a^{5} d + a^{3} b^{2} d\right )} \sqrt{a b}} + \frac{{\left (\sqrt{2} a + \sqrt{2} b\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{\tan \left (d x + c\right )}\right )}\right )}{2 \,{\left (a^{2} d + b^{2} d\right )}} + \frac{{\left (\sqrt{2} a + \sqrt{2} b\right )} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{\tan \left (d x + c\right )}\right )}\right )}{2 \,{\left (a^{2} d + b^{2} d\right )}} - \frac{{\left (\sqrt{2} a - \sqrt{2} b\right )} \log \left (\sqrt{2} \sqrt{\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{4 \,{\left (a^{2} d + b^{2} d\right )}} + \frac{{\left (\sqrt{2} a - \sqrt{2} b\right )} \log \left (-\sqrt{2} \sqrt{\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{4 \,{\left (a^{2} d + b^{2} d\right )}} + \frac{2 \,{\left (15 \, a^{2} \tan \left (d x + c\right )^{2} - 15 \, b^{2} \tan \left (d x + c\right )^{2} + 5 \, a b \tan \left (d x + c\right ) - 3 \, a^{2}\right )}}{15 \, a^{3} d \tan \left (d x + c\right )^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)^(7/2)/(a+b*tan(d*x+c)),x, algorithm="giac")

[Out]

-2*b^5*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^5*d + a^3*b^2*d)*sqrt(a*b)) + 1/2*(sqrt(2)*a + sqrt(2)*b)*ar
ctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c))))/(a^2*d + b^2*d) + 1/2*(sqrt(2)*a + sqrt(2)*b)*arctan(-1/2*s
qrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c))))/(a^2*d + b^2*d) - 1/4*(sqrt(2)*a - sqrt(2)*b)*log(sqrt(2)*sqrt(tan(d*
x + c)) + tan(d*x + c) + 1)/(a^2*d + b^2*d) + 1/4*(sqrt(2)*a - sqrt(2)*b)*log(-sqrt(2)*sqrt(tan(d*x + c)) + ta
n(d*x + c) + 1)/(a^2*d + b^2*d) + 2/15*(15*a^2*tan(d*x + c)^2 - 15*b^2*tan(d*x + c)^2 + 5*a*b*tan(d*x + c) - 3
*a^2)/(a^3*d*tan(d*x + c)^(5/2))