\(\int \frac {(c-i c \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx\) [971]

Optimal result

Integrand size = 33, antiderivative size = 125 \[ \int \frac {(c-i c \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx=-\frac {3 i \sqrt {2} c^{5/2} \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{a f}+\frac {3 i c^2 \sqrt {c-i c \tan (e+f x)}}{a f}+\frac {i c^2 (c-i c \tan (e+f x))^{3/2}}{a f (c+i c \tan (e+f x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3603, 3568, 43, 52, 65, 212} \[ \int \frac {(c-i c \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx=-\frac {3 i \sqrt {2} c^{5/2} \text {arctanh}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{a f}+\frac {3 i c^2 \sqrt {c-i c \tan (e+f x)}}{a f}+\frac {i c^2 (c-i c \tan (e+f x))^{3/2}}{a f (c+i c \tan (e+f x))} \]

[In]

Int[(c - I*c*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x]),x]

[Out]

((-3*I)*Sqrt[2]*c^(5/2)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(a*f) + ((3*I)*c^2*Sqrt[c - I*c
*Tan[e + f*x]])/(a*f) + (I*c^2*(c - I*c*Tan[e + f*x])^(3/2))/(a*f*(c + I*c*Tan[e + f*x]))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, 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 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 3568

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(a^(m - 2)*b
*f), Subst[Int[(a - x)^(m/2 - 1)*(a + x)^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x
] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]

Rule 3603

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Di
st[a^m*c^m, Int[Sec[e + f*x]^(2*m)*(c + d*Tan[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0] && IntegerQ[m] &&  !(IGtQ[n, 0] && (LtQ[m, 0] || GtQ[m, n]))

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

Mathematica [C] (verified)

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

Time = 1.15 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.42 \[ \int \frac {(c-i c \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx=\frac {i \operatorname {Hypergeometric2F1}\left (2,\frac {5}{2},\frac {7}{2},-\frac {1}{2} i (i+\tan (e+f x))\right ) (c-i c \tan (e+f x))^{5/2}}{10 a f} \]

[In]

Integrate[(c - I*c*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x]),x]

[Out]

((I/10)*Hypergeometric2F1[2, 5/2, 7/2, (-1/2*I)*(I + Tan[e + f*x])]*(c - I*c*Tan[e + f*x])^(5/2))/(a*f)

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.76

method	result	size
derivativedivides	\(\frac {2 i c^{2} \left (\sqrt {c -i c \tan \left (f x +e \right )}-4 c \left (-\frac {\sqrt {c -i c \tan \left (f x +e \right )}}{8 \left (\frac {c}{2}+\frac {i c \tan \left (f x +e \right )}{2}\right )}+\frac {3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{8 \sqrt {c}}\right )\right )}{f a}\)	\(95\)
default	\(\frac {2 i c^{2} \left (\sqrt {c -i c \tan \left (f x +e \right )}-4 c \left (-\frac {\sqrt {c -i c \tan \left (f x +e \right )}}{8 \left (\frac {c}{2}+\frac {i c \tan \left (f x +e \right )}{2}\right )}+\frac {3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{8 \sqrt {c}}\right )\right )}{f a}\)	\(95\)

[In]

int((c-I*c*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

2*I/f/a*c^2*((c-I*c*tan(f*x+e))^(1/2)-4*c*(-1/8*(c-I*c*tan(f*x+e))^(1/2)/(1/2*c+1/2*I*c*tan(f*x+e))+3/8*2^(1/2
)/c^(1/2)*arctanh(1/2*(c-I*c*tan(f*x+e))^(1/2)*2^(1/2)/c^(1/2))))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 261 vs. \(2 (98) = 196\).

Time = 0.25 (sec) , antiderivative size = 261, normalized size of antiderivative = 2.09 \[ \int \frac {(c-i c \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx=-\frac {{\left (3 \, \sqrt {2} a \sqrt {-\frac {c^{5}}{a^{2} f^{2}}} f e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (-\frac {12 \, {\left (i \, c^{3} + {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt {-\frac {c^{5}}{a^{2} f^{2}}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a f}\right ) - 3 \, \sqrt {2} a \sqrt {-\frac {c^{5}}{a^{2} f^{2}}} f e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (-\frac {12 \, {\left (i \, c^{3} - {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt {-\frac {c^{5}}{a^{2} f^{2}}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a f}\right ) + 2 \, \sqrt {2} {\left (-3 i \, c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - i \, c^{2}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{2 \, a f} \]

[In]

integrate((c-I*c*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e)),x, algorithm="fricas")

[Out]

-1/2*(3*sqrt(2)*a*sqrt(-c^5/(a^2*f^2))*f*e^(2*I*f*x + 2*I*e)*log(-12*(I*c^3 + (a*f*e^(2*I*f*x + 2*I*e) + a*f)*
sqrt(-c^5/(a^2*f^2))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1)))*e^(-I*f*x - I*e)/(a*f)) - 3*sqrt(2)*a*sqrt(-c^5/(a^2*f
^2))*f*e^(2*I*f*x + 2*I*e)*log(-12*(I*c^3 - (a*f*e^(2*I*f*x + 2*I*e) + a*f)*sqrt(-c^5/(a^2*f^2))*sqrt(c/(e^(2*
I*f*x + 2*I*e) + 1)))*e^(-I*f*x - I*e)/(a*f)) + 2*sqrt(2)*(-3*I*c^2*e^(2*I*f*x + 2*I*e) - I*c^2)*sqrt(c/(e^(2*
I*f*x + 2*I*e) + 1)))*e^(-2*I*f*x - 2*I*e)/(a*f)

Sympy [F]

\[ \int \frac {(c-i c \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx=- \frac {i \left (\int \frac {c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}}{\tan {\left (e + f x \right )} - i}\, dx + \int \left (- \frac {c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}}{\tan {\left (e + f x \right )} - i}\right )\, dx + \int \left (- \frac {2 i c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}}{\tan {\left (e + f x \right )} - i}\right )\, dx\right )}{a} \]

[In]

integrate((c-I*c*tan(f*x+e))**(5/2)/(a+I*a*tan(f*x+e)),x)

[Out]

-I*(Integral(c**2*sqrt(-I*c*tan(e + f*x) + c)/(tan(e + f*x) - I), x) + Integral(-c**2*sqrt(-I*c*tan(e + f*x) +
c)*tan(e + f*x)**2/(tan(e + f*x) - I), x) + Integral(-2*I*c**2*sqrt(-I*c*tan(e + f*x) + c)*tan(e + f*x)/(tan(
e + f*x) - I), x))/a

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.02 \[ \int \frac {(c-i c \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx=\frac {i \, {\left (\frac {3 \, \sqrt {2} c^{\frac {7}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-i \, c \tan \left (f x + e\right ) + c}}{\sqrt {2} \sqrt {c} + \sqrt {-i \, c \tan \left (f x + e\right ) + c}}\right )}{a} - \frac {4 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} c^{4}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )} a - 2 \, a c} + \frac {4 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} c^{3}}{a}\right )}}{2 \, c f} \]

[In]

integrate((c-I*c*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e)),x, algorithm="maxima")

[Out]

1/2*I*(3*sqrt(2)*c^(7/2)*log(-(sqrt(2)*sqrt(c) - sqrt(-I*c*tan(f*x + e) + c))/(sqrt(2)*sqrt(c) + sqrt(-I*c*tan
(f*x + e) + c)))/a - 4*sqrt(-I*c*tan(f*x + e) + c)*c^4/((-I*c*tan(f*x + e) + c)*a - 2*a*c) + 4*sqrt(-I*c*tan(f
*x + e) + c)*c^3/a)/(c*f)

Giac [F]

\[ \int \frac {(c-i c \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx=\int { \frac {{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{i \, a \tan \left (f x + e\right ) + a} \,d x } \]

[In]

integrate((c-I*c*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e)),x, algorithm="giac")

[Out]

integrate((-I*c*tan(f*x + e) + c)^(5/2)/(I*a*tan(f*x + e) + a), x)

Mupad [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.87 \[ \int \frac {(c-i c \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx=\frac {c^2\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,2{}\mathrm {i}}{a\,f}-\frac {\sqrt {2}\,{\left (-c\right )}^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {-c}}\right )\,3{}\mathrm {i}}{a\,f}+\frac {c^3\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,2{}\mathrm {i}}{a\,f\,\left (c+c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )} \]

[In]

int((c - c*tan(e + f*x)*1i)^(5/2)/(a + a*tan(e + f*x)*1i),x)

[Out]

(c^2*(c - c*tan(e + f*x)*1i)^(1/2)*2i)/(a*f) - (2^(1/2)*(-c)^(5/2)*atan((2^(1/2)*(c - c*tan(e + f*x)*1i)^(1/2)
)/(2*(-c)^(1/2)))*3i)/(a*f) + (c^3*(c - c*tan(e + f*x)*1i)^(1/2)*2i)/(a*f*(c + c*tan(e + f*x)*1i))