3.1005 \(\int \frac{(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{7/2}} \, dx\)

Optimal. Leaf size=136 \[ \frac{2 i (c-i c \tan (e+f x))^{3/2}}{105 a^2 f (a+i a \tan (e+f x))^{3/2}}+\frac{2 i (c-i c \tan (e+f x))^{3/2}}{35 a f (a+i a \tan (e+f x))^{5/2}}+\frac{i (c-i c \tan (e+f x))^{3/2}}{7 f (a+i a \tan (e+f x))^{7/2}} \]

[Out]

((I/7)*(c - I*c*Tan[e + f*x])^(3/2))/(f*(a + I*a*Tan[e + f*x])^(7/2)) + (((2*I)/35)*(c - I*c*Tan[e + f*x])^(3/
2))/(a*f*(a + I*a*Tan[e + f*x])^(5/2)) + (((2*I)/105)*(c - I*c*Tan[e + f*x])^(3/2))/(a^2*f*(a + I*a*Tan[e + f*
x])^(3/2))

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Rubi [A]  time = 0.144283, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {3523, 45, 37} \[ \frac{2 i (c-i c \tan (e+f x))^{3/2}}{105 a^2 f (a+i a \tan (e+f x))^{3/2}}+\frac{2 i (c-i c \tan (e+f x))^{3/2}}{35 a f (a+i a \tan (e+f x))^{5/2}}+\frac{i (c-i c \tan (e+f x))^{3/2}}{7 f (a+i a \tan (e+f x))^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I/7)*(c - I*c*Tan[e + f*x])^(3/2))/(f*(a + I*a*Tan[e + f*x])^(7/2)) + (((2*I)/35)*(c - I*c*Tan[e + f*x])^(3/
2))/(a*f*(a + I*a*Tan[e + f*x])^(5/2)) + (((2*I)/105)*(c - I*c*Tan[e + f*x])^(3/2))/(a^2*f*(a + I*a*Tan[e + f*
x])^(3/2))

Rule 3523

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

Rule 45

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{7/2}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{\sqrt{c-i c x}}{(a+i a x)^{9/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{i (c-i c \tan (e+f x))^{3/2}}{7 f (a+i a \tan (e+f x))^{7/2}}+\frac{(2 c) \operatorname{Subst}\left (\int \frac{\sqrt{c-i c x}}{(a+i a x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{7 f}\\ &=\frac{i (c-i c \tan (e+f x))^{3/2}}{7 f (a+i a \tan (e+f x))^{7/2}}+\frac{2 i (c-i c \tan (e+f x))^{3/2}}{35 a f (a+i a \tan (e+f x))^{5/2}}+\frac{(2 c) \operatorname{Subst}\left (\int \frac{\sqrt{c-i c x}}{(a+i a x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{35 a f}\\ &=\frac{i (c-i c \tan (e+f x))^{3/2}}{7 f (a+i a \tan (e+f x))^{7/2}}+\frac{2 i (c-i c \tan (e+f x))^{3/2}}{35 a f (a+i a \tan (e+f x))^{5/2}}+\frac{2 i (c-i c \tan (e+f x))^{3/2}}{105 a^2 f (a+i a \tan (e+f x))^{3/2}}\\ \end{align*}

Mathematica [A]  time = 3.75284, size = 101, normalized size = 0.74 \[ \frac{i c (\tan (e+f x)+i) \sec ^2(e+f x) \sqrt{c-i c \tan (e+f x)} (10 i \sin (2 (e+f x))+25 \cos (2 (e+f x))+21)}{210 a^3 f (\tan (e+f x)-i)^3 \sqrt{a+i a \tan (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I/210)*c*Sec[e + f*x]^2*(21 + 25*Cos[2*(e + f*x)] + (10*I)*Sin[2*(e + f*x)])*(I + Tan[e + f*x])*Sqrt[c - I*c
*Tan[e + f*x]])/(a^3*f*(-I + Tan[e + f*x])^3*Sqrt[a + I*a*Tan[e + f*x]])

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Maple [A]  time = 0.036, size = 85, normalized size = 0.6 \begin{align*} -{\frac{c \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) \left ( 10\,i\tan \left ( fx+e \right ) -2\, \left ( \tan \left ( fx+e \right ) \right ) ^{2}+23 \right ) }{105\,f{a}^{4} \left ( -\tan \left ( fx+e \right ) +i \right ) ^{5}}\sqrt{-c \left ( -1+i\tan \left ( fx+e \right ) \right ) }\sqrt{a \left ( 1+i\tan \left ( fx+e \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105/f*(-c*(-1+I*tan(f*x+e)))^(1/2)*(a*(1+I*tan(f*x+e)))^(1/2)/a^4*c*(1+tan(f*x+e)^2)*(10*I*tan(f*x+e)-2*tan
(f*x+e)^2+23)/(-tan(f*x+e)+I)^5

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Maxima [A]  time = 1.95164, size = 184, normalized size = 1.35 \begin{align*} \frac{{\left (15 i \, c \cos \left (7 \, f x + 7 \, e\right ) + 42 i \, c \cos \left (\frac{5}{7} \, \arctan \left (\sin \left (7 \, f x + 7 \, e\right ), \cos \left (7 \, f x + 7 \, e\right )\right )\right ) + 35 i \, c \cos \left (\frac{3}{7} \, \arctan \left (\sin \left (7 \, f x + 7 \, e\right ), \cos \left (7 \, f x + 7 \, e\right )\right )\right ) + 15 \, c \sin \left (7 \, f x + 7 \, e\right ) + 42 \, c \sin \left (\frac{5}{7} \, \arctan \left (\sin \left (7 \, f x + 7 \, e\right ), \cos \left (7 \, f x + 7 \, e\right )\right )\right ) + 35 \, c \sin \left (\frac{3}{7} \, \arctan \left (\sin \left (7 \, f x + 7 \, e\right ), \cos \left (7 \, f x + 7 \, e\right )\right )\right )\right )} \sqrt{c}}{420 \, a^{\frac{7}{2}} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*(15*I*c*cos(7*f*x + 7*e) + 42*I*c*cos(5/7*arctan2(sin(7*f*x + 7*e), cos(7*f*x + 7*e))) + 35*I*c*cos(3/7*
arctan2(sin(7*f*x + 7*e), cos(7*f*x + 7*e))) + 15*c*sin(7*f*x + 7*e) + 42*c*sin(5/7*arctan2(sin(7*f*x + 7*e),
cos(7*f*x + 7*e))) + 35*c*sin(3/7*arctan2(sin(7*f*x + 7*e), cos(7*f*x + 7*e))))*sqrt(c)/(a^(7/2)*f)

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Fricas [A]  time = 1.35996, size = 350, normalized size = 2.57 \begin{align*} \frac{{\left (-92 i \, c e^{\left (9 i \, f x + 9 i \, e\right )} - 92 i \, c e^{\left (7 i \, f x + 7 i \, e\right )} + 35 i \, c e^{\left (6 i \, f x + 6 i \, e\right )} + 77 i \, c e^{\left (4 i \, f x + 4 i \, e\right )} + 57 i \, c e^{\left (2 i \, f x + 2 i \, e\right )} + 15 i \, c\right )} \sqrt{\frac{a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} e^{\left (-7 i \, f x - 7 i \, e\right )}}{420 \, a^{4} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*(-92*I*c*e^(9*I*f*x + 9*I*e) - 92*I*c*e^(7*I*f*x + 7*I*e) + 35*I*c*e^(6*I*f*x + 6*I*e) + 77*I*c*e^(4*I*f
*x + 4*I*e) + 57*I*c*e^(2*I*f*x + 2*I*e) + 15*I*c)*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*
e) + 1))*e^(-7*I*f*x - 7*I*e)/(a^4*f)

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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((c-I*c*tan(f*x+e))**(3/2)/(a+I*a*tan(f*x+e))**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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