3.646 \(\int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=241 \[ \frac {8 b}{3 a^2 d \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}+\frac {2 b^2 \left (5 a^2+8 b^2\right ) \sqrt {\tan (c+d x)}}{3 a^3 d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}-\frac {i \tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d (-b+i a)^{3/2}}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d (b+i a)^{3/2}} \]

[Out]

-I*arctan((I*a-b)^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(1/2))/(I*a-b)^(3/2)/d-I*arctanh((I*a+b)^(1/2)*tan(d
*x+c)^(1/2)/(a+b*tan(d*x+c))^(1/2))/(I*a+b)^(3/2)/d+8/3*b/a^2/d/tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(1/2)+2/3*b^
2*(5*a^2+8*b^2)*tan(d*x+c)^(1/2)/a^3/(a^2+b^2)/d/(a+b*tan(d*x+c))^(1/2)-2/3/a/d/(a+b*tan(d*x+c))^(1/2)/tan(d*x
+c)^(3/2)

________________________________________________________________________________________

Rubi [A]  time = 0.83, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3569, 3649, 3650, 3616, 3615, 93, 203, 206} \[ \frac {2 b^2 \left (5 a^2+8 b^2\right ) \sqrt {\tan (c+d x)}}{3 a^3 d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {8 b}{3 a^2 d \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}-\frac {i \tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d (-b+i a)^{3/2}}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d (b+i a)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(3/2)),x]

[Out]

((-I)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a - b)^(3/2)*d) - (I*ArcTanh[(S
qrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a + b)^(3/2)*d) - 2/(3*a*d*Tan[c + d*x]^(3/2)*
Sqrt[a + b*Tan[c + d*x]]) + (8*b)/(3*a^2*d*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]) + (2*b^2*(5*a^2 + 8*b^
2)*Sqrt[Tan[c + d*x]])/(3*a^3*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 203

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

Rule 206

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

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 3615

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

Rule 3616

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 5.62, size = 223, normalized size = 0.93 \[ \frac {\frac {2 b^2 \left (5 a^2+8 b^2\right ) \tan ^2(c+d x)+8 a b \left (a^2+b^2\right ) \tan (c+d x)-2 a^2 \left (a^2+b^2\right )}{a^3 \left (a^2+b^2\right ) \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}-\frac {3 (-1)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(-a+i b)^{3/2}}+\frac {3 (-1)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(a+i b)^{3/2}}}{3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(3/2)),x]

[Out]

((-3*(-1)^(3/4)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(-a + I*b)^(3
/2) + (3*(-1)^(3/4)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(a + I*b)^
(3/2) + (-2*a^2*(a^2 + b^2) + 8*a*b*(a^2 + b^2)*Tan[c + d*x] + 2*b^2*(5*a^2 + 8*b^2)*Tan[c + d*x]^2)/(a^3*(a^2
+ b^2)*Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]]))/(3*d)

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)^(5/2)/(a+b*tan(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)^(5/2)/(a+b*tan(d*x+c))^(3/2),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [B]  time = 1.44, size = 763924, normalized size = 3169.81 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/tan(d*x+c)^(5/2)/(a+b*tan(d*x+c))^(3/2),x)

[Out]

result too large to display

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \tan \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)^(5/2)/(a+b*tan(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((b*tan(d*x + c) + a)^(3/2)*tan(d*x + c)^(5/2)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\mathrm {tan}\left (c+d\,x\right )}^{5/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(tan(c + d*x)^(5/2)*(a + b*tan(c + d*x))^(3/2)),x)

[Out]

int(1/(tan(c + d*x)^(5/2)*(a + b*tan(c + d*x))^(3/2)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {3}{2}} \tan ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)**(5/2)/(a+b*tan(d*x+c))**(3/2),x)

[Out]

Integral(1/((a + b*tan(c + d*x))**(3/2)*tan(c + d*x)**(5/2)), x)

________________________________________________________________________________________