\(\int \frac {(a+b \tan (e+f x))^{7/2}}{(c+d \tan (e+f x))^{3/2}} \, dx\) [1289]

Optimal result

Integrand size = 29, antiderivative size = 356 \[ \int \frac {(a+b \tan (e+f x))^{7/2}}{(c+d \tan (e+f x))^{3/2}} \, dx=-\frac {i (a-i b)^{7/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(c-i d)^{3/2} f}+\frac {i (a+i b)^{7/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(c+i d)^{3/2} f}-\frac {b^{5/2} (3 b c-7 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{d^{5/2} f}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{3/2}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}-\frac {b \left (2 a d (2 b c-a d)-b^2 \left (3 c^2+d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d^2 \left (c^2+d^2\right ) f} \]

[Out]

-I*(a-I*b)^(7/2)*arctanh((c-I*d)^(1/2)*(a+b*tan(f*x+e))^(1/2)/(a-I*b)^(1/2)/(c+d*tan(f*x+e))^(1/2))/(c-I*d)^(3
/2)/f+I*(a+I*b)^(7/2)*arctanh((c+I*d)^(1/2)*(a+b*tan(f*x+e))^(1/2)/(a+I*b)^(1/2)/(c+d*tan(f*x+e))^(1/2))/(c+I*
d)^(3/2)/f-b^(5/2)*(-7*a*d+3*b*c)*arctanh(d^(1/2)*(a+b*tan(f*x+e))^(1/2)/b^(1/2)/(c+d*tan(f*x+e))^(1/2))/d^(5/
2)/f-b*(2*a*d*(-a*d+2*b*c)-b^2*(3*c^2+d^2))*(a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(1/2)/d^2/(c^2+d^2)/f-2*(-
a*d+b*c)^2*(a+b*tan(f*x+e))^(3/2)/d/(c^2+d^2)/f/(c+d*tan(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 5.31 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {3646, 3728, 3736, 6857, 65, 223, 212, 95, 214} \[ \int \frac {(a+b \tan (e+f x))^{7/2}}{(c+d \tan (e+f x))^{3/2}} \, dx=-\frac {b^{5/2} (3 b c-7 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{d^{5/2} f}-\frac {i (a-i b)^{7/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (c-i d)^{3/2}}+\frac {i (a+i b)^{7/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (c+i d)^{3/2}}-\frac {b \left (2 a d (2 b c-a d)-b^2 \left (3 c^2+d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d^2 f \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{3/2}}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}} \]

[In]

Int[(a + b*Tan[e + f*x])^(7/2)/(c + d*Tan[e + f*x])^(3/2),x]

[Out]

((-I)*(a - I*b)^(7/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]]
)])/((c - I*d)^(3/2)*f) + (I*(a + I*b)^(7/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*S
qrt[c + d*Tan[e + f*x]])])/((c + I*d)^(3/2)*f) - (b^(5/2)*(3*b*c - 7*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e +
f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(d^(5/2)*f) - (2*(b*c - a*d)^2*(a + b*Tan[e + f*x])^(3/2))/(d*(c^2
+ d^2)*f*Sqrt[c + d*Tan[e + f*x]]) - (b*(2*a*d*(2*b*c - a*d) - b^2*(3*c^2 + d^2))*Sqrt[a + b*Tan[e + f*x]]*Sq
rt[c + d*Tan[e + f*x]])/(d^2*(c^2 + d^2)*f)

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 95

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 212

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

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

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 3728

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

Rule 3736

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] :> With[{ff = FreeFactors[Tan[e + f*x], x
]}, Dist[ff/f, Subst[Int[(a + b*ff*x)^m*(c + d*ff*x)^n*((A + B*ff*x + C*ff^2*x^2)/(1 + ff^2*x^2)), x], x, Tan[
e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] &&
NeQ[c^2 + d^2, 0]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{3/2}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {2 \int \frac {\sqrt {a+b \tan (e+f x)} \left (\frac {1}{2} \left (3 b^3 c^2+a^3 c d-7 a b^2 c d+5 a^2 b d^2\right )+\frac {1}{2} d \left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right ) \tan (e+f x)-\frac {1}{2} b \left (2 a d (2 b c-a d)-b^2 \left (3 c^2+d^2\right )\right ) \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx}{d \left (c^2+d^2\right )} \\ & = -\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{3/2}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}-\frac {b \left (2 a d (2 b c-a d)-b^2 \left (3 c^2+d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d^2 \left (c^2+d^2\right ) f}+\frac {2 \int \frac {\frac {1}{4} \left (2 a^4 c d^2-12 a^2 b^2 c d^2+8 a^3 b d^3+a b^3 d \left (7 c^2-d^2\right )-b^4 c \left (3 c^2+d^2\right )\right )+\frac {1}{2} d^2 \left (4 a^3 b c-4 a b^3 c-a^4 d+6 a^2 b^2 d-b^4 d\right ) \tan (e+f x)-\frac {1}{4} b^3 (3 b c-7 a d) \left (c^2+d^2\right ) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{d^2 \left (c^2+d^2\right )} \\ & = -\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{3/2}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}-\frac {b \left (2 a d (2 b c-a d)-b^2 \left (3 c^2+d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d^2 \left (c^2+d^2\right ) f}+\frac {2 \text {Subst}\left (\int \frac {\frac {1}{4} \left (2 a^4 c d^2-12 a^2 b^2 c d^2+8 a^3 b d^3+a b^3 d \left (7 c^2-d^2\right )-b^4 c \left (3 c^2+d^2\right )\right )+\frac {1}{2} d^2 \left (4 a^3 b c-4 a b^3 c-a^4 d+6 a^2 b^2 d-b^4 d\right ) x-\frac {1}{4} b^3 (3 b c-7 a d) \left (c^2+d^2\right ) x^2}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{d^2 \left (c^2+d^2\right ) f} \\ & = -\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{3/2}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}-\frac {b \left (2 a d (2 b c-a d)-b^2 \left (3 c^2+d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d^2 \left (c^2+d^2\right ) f}+\frac {2 \text {Subst}\left (\int \left (-\frac {b^3 (3 b c-7 a d) \left (c^2+d^2\right )}{4 \sqrt {a+b x} \sqrt {c+d x}}+\frac {d^2 \left (a^4 c-6 a^2 b^2 c+b^4 c+4 a^3 b d-4 a b^3 d\right )+d^2 \left (4 a^3 b c-4 a b^3 c-a^4 d+6 a^2 b^2 d-b^4 d\right ) x}{2 \sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{d^2 \left (c^2+d^2\right ) f} \\ & = -\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{3/2}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}-\frac {b \left (2 a d (2 b c-a d)-b^2 \left (3 c^2+d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d^2 \left (c^2+d^2\right ) f}-\frac {\left (b^3 (3 b c-7 a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 d^2 f}+\frac {\text {Subst}\left (\int \frac {d^2 \left (a^4 c-6 a^2 b^2 c+b^4 c+4 a^3 b d-4 a b^3 d\right )+d^2 \left (4 a^3 b c-4 a b^3 c-a^4 d+6 a^2 b^2 d-b^4 d\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{d^2 \left (c^2+d^2\right ) f} \\ & = -\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{3/2}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}-\frac {b \left (2 a d (2 b c-a d)-b^2 \left (3 c^2+d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d^2 \left (c^2+d^2\right ) f}-\frac {\left (b^2 (3 b c-7 a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b \tan (e+f x)}\right )}{d^2 f}+\frac {\text {Subst}\left (\int \left (\frac {i d^2 \left (a^4 c-6 a^2 b^2 c+b^4 c+4 a^3 b d-4 a b^3 d\right )-d^2 \left (4 a^3 b c-4 a b^3 c-a^4 d+6 a^2 b^2 d-b^4 d\right )}{2 (i-x) \sqrt {a+b x} \sqrt {c+d x}}+\frac {i d^2 \left (a^4 c-6 a^2 b^2 c+b^4 c+4 a^3 b d-4 a b^3 d\right )+d^2 \left (4 a^3 b c-4 a b^3 c-a^4 d+6 a^2 b^2 d-b^4 d\right )}{2 (i+x) \sqrt {a+b x} \sqrt {c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{d^2 \left (c^2+d^2\right ) f} \\ & = -\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{3/2}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}-\frac {b \left (2 a d (2 b c-a d)-b^2 \left (3 c^2+d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d^2 \left (c^2+d^2\right ) f}-\frac {(a+i b)^4 \text {Subst}\left (\int \frac {1}{(i-x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (i c-d) f}-\frac {(a-i b)^4 \text {Subst}\left (\int \frac {1}{(i+x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (i c+d) f}-\frac {\left (b^2 (3 b c-7 a d)\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{d^2 f} \\ & = -\frac {b^{5/2} (3 b c-7 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{d^{5/2} f}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{3/2}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}-\frac {b \left (2 a d (2 b c-a d)-b^2 \left (3 c^2+d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d^2 \left (c^2+d^2\right ) f}-\frac {(a+i b)^4 \text {Subst}\left (\int \frac {1}{a+i b-(c+i d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{(i c-d) f}-\frac {(a-i b)^4 \text {Subst}\left (\int \frac {1}{-a+i b-(-c+i d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{(i c+d) f} \\ & = -\frac {i (a-i b)^{7/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(c-i d)^{3/2} f}+\frac {i (a+i b)^{7/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(c+i d)^{3/2} f}-\frac {b^{5/2} (3 b c-7 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{d^{5/2} f}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{3/2}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}-\frac {b \left (2 a d (2 b c-a d)-b^2 \left (3 c^2+d^2\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d^2 \left (c^2+d^2\right ) f} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 6.95 (sec) , antiderivative size = 1877, normalized size of antiderivative = 5.27 \[ \int \frac {(a+b \tan (e+f x))^{7/2}}{(c+d \tan (e+f x))^{3/2}} \, dx=-\frac {i (-a-i b) \left (-\frac {2 b \left (\frac {b}{\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {5}{2},\frac {7}{2},-\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}\right ) (a+b \tan (e+f x))^{5/2} \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}{5 (b c-a d) \sqrt {c+d \tan (e+f x)}}-(-a-i b) \left (-\left ((-a-i b) \left (-\frac {2 \sqrt {a+i b} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(-c-i d) \sqrt {c+i d}}+\frac {2 \sqrt {a+b \tan (e+f x)}}{(-c-i d) \sqrt {c+d \tan (e+f x)}}\right )\right )-\frac {2 (b c-a d) \left (\frac {b}{\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}\right )^{3/2} \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )^2 \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}} \left (-1-\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}\right ) \left (-\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right ) \left (-1-\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}\right )}-\frac {\sqrt {b} \sqrt {d} \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}}\right ) \sqrt {a+b \tan (e+f x)}}{\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}} \sqrt {1+\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}}}\right )}{b d^2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \sqrt {1+\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}}}\right )\right )}{2 f}-\frac {i (-a+i b) \left (\frac {2 b \left (\frac {b}{\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {5}{2},\frac {7}{2},-\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}\right ) (a+b \tan (e+f x))^{5/2} \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}{5 (b c-a d) \sqrt {c+d \tan (e+f x)}}-(-a+i b) \left (-\left ((-a+i b) \left (-\frac {2 \sqrt {-a+i b} \text {arctanh}\left (\frac {\sqrt {-c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {-a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(c-i d) \sqrt {-c+i d}}+\frac {2 \sqrt {a+b \tan (e+f x)}}{(c-i d) \sqrt {c+d \tan (e+f x)}}\right )\right )+\frac {2 (b c-a d) \left (\frac {b}{\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}\right )^{3/2} \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )^2 \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}} \left (-1-\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}\right ) \left (-\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right ) \left (-1-\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}\right )}-\frac {\sqrt {b} \sqrt {d} \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}}\right ) \sqrt {a+b \tan (e+f x)}}{\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}} \sqrt {1+\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}}}\right )}{b d^2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \sqrt {1+\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}}}\right )\right )}{2 f} \]

[In]

Integrate[(a + b*Tan[e + f*x])^(7/2)/(c + d*Tan[e + f*x])^(3/2),x]

[Out]

((-1/2*I)*(-a - I*b)*((-2*b*(b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))^(3/2)*Hypergeometric2F1[3/2, 5/2,
7/2, -((b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))]*(a + b*Tan[e + f
*x])^(5/2)*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(5*(b*c - a*d)*Sqrt[c + d*Tan[e + f*x]]) - (-a - I*b)*(
-((-a - I*b)*((-2*Sqrt[a + I*b]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan
[e + f*x]])])/((-c - I*d)*Sqrt[c + I*d]) + (2*Sqrt[a + b*Tan[e + f*x]])/((-c - I*d)*Sqrt[c + d*Tan[e + f*x]]))
) - (2*(b*c - a*d)*(b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))^(3/2)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c -
a*d))^2*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)]*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*
c - a*d) - (a*b*d)/(b*c - a*d))))*(-((b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b
*c - a*d))*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))))) - (Sq
rt[b]*Sqrt[d]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (
a*b*d)/(b*c - a*d)])]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d
)]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))])))/(b*d^2*Sq
rt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c
- a*d) - (a*b*d)/(b*c - a*d)))]))))/f - ((I/2)*(-a + I*b)*((2*b*(b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)
))^(3/2)*Hypergeometric2F1[3/2, 5/2, 7/2, -((b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*
b*d)/(b*c - a*d))))]*(a + b*Tan[e + f*x])^(5/2)*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(5*(b*c - a*d)*Sqr
t[c + d*Tan[e + f*x]]) - (-a + I*b)*(-((-a + I*b)*((-2*Sqrt[-a + I*b]*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e
+ f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c - I*d)*Sqrt[-c + I*d]) + (2*Sqrt[a + b*Tan[e + f*x]]
)/((c - I*d)*Sqrt[c + d*Tan[e + f*x]]))) + (2*(b*c - a*d)*(b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))^(3/2
)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))^2*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)]*(-1 - (b*d*(a + b*T
an[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))*(-((b*d*(a + b*Tan[e + f*x]))/((b*c -
a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c
- a*d) - (a*b*d)/(b*c - a*d)))))) - (Sqrt[b]*Sqrt[d]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[
b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(
b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d
) - (a*b*d)/(b*c - a*d)))])))/(b*d^2*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]*Sqrt[1 + (b*d*(a + b*Ta
n[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))]))))/f

Maple [F(-1)]

Timed out.

\[\int \frac {\left (a +b \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{\left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}d x\]

[In]

int((a+b*tan(f*x+e))^(7/2)/(c+d*tan(f*x+e))^(3/2),x)

[Out]

int((a+b*tan(f*x+e))^(7/2)/(c+d*tan(f*x+e))^(3/2),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 27489 vs. \(2 (290) = 580\).

Time = 74.62 (sec) , antiderivative size = 55004, normalized size of antiderivative = 154.51 \[ \int \frac {(a+b \tan (e+f x))^{7/2}}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

integrate((a+b*tan(f*x+e))^(7/2)/(c+d*tan(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (e+f x))^{7/2}}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+b*tan(f*x+e))**(7/2)/(c+d*tan(f*x+e))**(3/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(a+b \tan (e+f x))^{7/2}}{(c+d \tan (e+f x))^{3/2}} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((a+b*tan(f*x+e))^(7/2)/(c+d*tan(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*tan(f*x + e) + a)^(7/2)/(d*tan(f*x + e) + c)^(3/2), x)

Giac [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (e+f x))^{7/2}}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+b*tan(f*x+e))^(7/2)/(c+d*tan(f*x+e))^(3/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (e+f x))^{7/2}}{(c+d \tan (e+f x))^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{7/2}}{{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}} \,d x \]

[In]

int((a + b*tan(e + f*x))^(7/2)/(c + d*tan(e + f*x))^(3/2),x)

[Out]

int((a + b*tan(e + f*x))^(7/2)/(c + d*tan(e + f*x))^(3/2), x)