3.757 \(\int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx\)

Optimal. Leaf size=144 \[ \frac {2 a^3 (5 B+i A) (c-i c \tan (e+f x))^{9/2}}{9 c^2 f}-\frac {8 a^3 (2 B+i A) (c-i c \tan (e+f x))^{7/2}}{7 c f}+\frac {8 a^3 (B+i A) (c-i c \tan (e+f x))^{5/2}}{5 f}-\frac {2 a^3 B (c-i c \tan (e+f x))^{11/2}}{11 c^3 f} \]

[Out]

8/5*a^3*(I*A+B)*(c-I*c*tan(f*x+e))^(5/2)/f-8/7*a^3*(I*A+2*B)*(c-I*c*tan(f*x+e))^(7/2)/c/f+2/9*a^3*(I*A+5*B)*(c
-I*c*tan(f*x+e))^(9/2)/c^2/f-2/11*a^3*B*(c-I*c*tan(f*x+e))^(11/2)/c^3/f

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Rubi [A]  time = 0.20, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {3588, 77} \[ \frac {2 a^3 (5 B+i A) (c-i c \tan (e+f x))^{9/2}}{9 c^2 f}-\frac {8 a^3 (2 B+i A) (c-i c \tan (e+f x))^{7/2}}{7 c f}+\frac {8 a^3 (B+i A) (c-i c \tan (e+f x))^{5/2}}{5 f}-\frac {2 a^3 B (c-i c \tan (e+f x))^{11/2}}{11 c^3 f} \]

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2),x]

[Out]

(8*a^3*(I*A + B)*(c - I*c*Tan[e + f*x])^(5/2))/(5*f) - (8*a^3*(I*A + 2*B)*(c - I*c*Tan[e + f*x])^(7/2))/(7*c*f
) + (2*a^3*(I*A + 5*B)*(c - I*c*Tan[e + f*x])^(9/2))/(9*c^2*f) - (2*a^3*B*(c - I*c*Tan[e + f*x])^(11/2))/(11*c
^3*f)

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 3588

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*c)/f, Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^(n - 1)*(A + B*x), x
], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]

Rubi steps

\begin {align*} \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int (a+i a x)^2 (A+B x) (c-i c x)^{3/2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \operatorname {Subst}\left (\int \left (4 a^2 (A-i B) (c-i c x)^{3/2}-\frac {4 a^2 (A-2 i B) (c-i c x)^{5/2}}{c}+\frac {a^2 (A-5 i B) (c-i c x)^{7/2}}{c^2}+\frac {i a^2 B (c-i c x)^{9/2}}{c^3}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {8 a^3 (i A+B) (c-i c \tan (e+f x))^{5/2}}{5 f}-\frac {8 a^3 (i A+2 B) (c-i c \tan (e+f x))^{7/2}}{7 c f}+\frac {2 a^3 (i A+5 B) (c-i c \tan (e+f x))^{9/2}}{9 c^2 f}-\frac {2 a^3 B (c-i c \tan (e+f x))^{11/2}}{11 c^3 f}\\ \end {align*}

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Mathematica [A]  time = 12.08, size = 139, normalized size = 0.97 \[ -\frac {2 a^3 c^2 \sec ^4(e+f x) \sqrt {c-i c \tan (e+f x)} (\cos (2 e-f x)-i \sin (2 e-f x)) (5 (121 A-74 i B) \tan (e+f x)+\cos (2 (e+f x)) ((605 A-685 i B) \tan (e+f x)-781 i A-701 B)+9 (31 B-44 i A))}{3465 f (\cos (f x)+i \sin (f x))^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2),x]

[Out]

(-2*a^3*c^2*Sec[e + f*x]^4*(Cos[2*e - f*x] - I*Sin[2*e - f*x])*Sqrt[c - I*c*Tan[e + f*x]]*(9*((-44*I)*A + 31*B
) + 5*(121*A - (74*I)*B)*Tan[e + f*x] + Cos[2*(e + f*x)]*((-781*I)*A - 701*B + (605*A - (685*I)*B)*Tan[e + f*x
])))/(3465*f*(Cos[f*x] + I*Sin[f*x])^3)

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fricas [A]  time = 2.05, size = 169, normalized size = 1.17 \[ \frac {\sqrt {2} {\left ({\left (22176 i \, A + 22176 \, B\right )} a^{3} c^{2} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (34848 i \, A + 3168 \, B\right )} a^{3} c^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (15488 i \, A + 1408 \, B\right )} a^{3} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (2816 i \, A + 256 \, B\right )} a^{3} c^{2}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{3465 \, {\left (f e^{\left (10 i \, f x + 10 i \, e\right )} + 5 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))^3*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

1/3465*sqrt(2)*((22176*I*A + 22176*B)*a^3*c^2*e^(6*I*f*x + 6*I*e) + (34848*I*A + 3168*B)*a^3*c^2*e^(4*I*f*x +
4*I*e) + (15488*I*A + 1408*B)*a^3*c^2*e^(2*I*f*x + 2*I*e) + (2816*I*A + 256*B)*a^3*c^2)*sqrt(c/(e^(2*I*f*x + 2
*I*e) + 1))/(f*e^(10*I*f*x + 10*I*e) + 5*f*e^(8*I*f*x + 8*I*e) + 10*f*e^(6*I*f*x + 6*I*e) + 10*f*e^(4*I*f*x +
4*I*e) + 5*f*e^(2*I*f*x + 2*I*e) + f)

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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((a+I*a*tan(f*x+e))^3*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(5/2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.37, size = 121, normalized size = 0.84 \[ \frac {2 i a^{3} \left (\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {11}{2}}}{11}+\frac {\left (-5 i B c +c A \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{9}+\frac {\left (-4 \left (-i B c +c A \right ) c +4 i B \,c^{2}\right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+\frac {4 \left (-i B c +c A \right ) c^{2} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}\right )}{f \,c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+I*a*tan(f*x+e))^3*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(5/2),x)

[Out]

2*I/f*a^3/c^3*(1/11*I*B*(c-I*c*tan(f*x+e))^(11/2)+1/9*(-5*I*B*c+c*A)*(c-I*c*tan(f*x+e))^(9/2)+1/7*(-4*(-I*B*c+
c*A)*c+4*I*B*c^2)*(c-I*c*tan(f*x+e))^(7/2)+4/5*(-I*B*c+c*A)*c^2*(c-I*c*tan(f*x+e))^(5/2))

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maxima [A]  time = 0.67, size = 104, normalized size = 0.72 \[ \frac {2 i \, {\left (315 i \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {11}{2}} B a^{3} + 385 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {9}{2}} {\left (A - 5 i \, B\right )} a^{3} c - 1980 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}} {\left (A - 2 i \, B\right )} a^{3} c^{2} + 2772 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (A - i \, B\right )} a^{3} c^{3}\right )}}{3465 \, c^{3} f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))^3*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

2/3465*I*(315*I*(-I*c*tan(f*x + e) + c)^(11/2)*B*a^3 + 385*(-I*c*tan(f*x + e) + c)^(9/2)*(A - 5*I*B)*a^3*c - 1
980*(-I*c*tan(f*x + e) + c)^(7/2)*(A - 2*I*B)*a^3*c^2 + 2772*(-I*c*tan(f*x + e) + c)^(5/2)*(A - I*B)*a^3*c^3)/
(c^3*f)

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mupad [B]  time = 14.48, size = 349, normalized size = 2.42 \[ -\frac {\left (\frac {a^3\,c^2\,\left (A-B\,1{}\mathrm {i}\right )\,32{}\mathrm {i}}{7\,f}+\frac {a^3\,c^2\,\left (A-B\,3{}\mathrm {i}\right )\,32{}\mathrm {i}}{7\,f}\right )\,\sqrt {c+\frac {c\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}}{{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^3}-\frac {\left (\frac {a^3\,c^2\,\left (A-B\,1{}\mathrm {i}\right )\,32{}\mathrm {i}}{11\,f}-\frac {a^3\,c^2\,\left (A+B\,1{}\mathrm {i}\right )\,32{}\mathrm {i}}{11\,f}\right )\,\sqrt {c+\frac {c\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}}{{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^5}+\frac {\left (\frac {128\,B\,a^3\,c^2}{9\,f}+\frac {a^3\,c^2\,\left (A-B\,1{}\mathrm {i}\right )\,32{}\mathrm {i}}{9\,f}\right )\,\sqrt {c+\frac {c\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}}{{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^4}+\frac {a^3\,c^2\,\left (A-B\,1{}\mathrm {i}\right )\,\sqrt {c+\frac {c\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,32{}\mathrm {i}}{5\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*tan(e + f*x))*(a + a*tan(e + f*x)*1i)^3*(c - c*tan(e + f*x)*1i)^(5/2),x)

[Out]

(((a^3*c^2*(A - B*1i)*32i)/(9*f) + (128*B*a^3*c^2)/(9*f))*(c + (c*(exp(e*2i + f*x*2i)*1i - 1i)*1i)/(exp(e*2i +
f*x*2i) + 1))^(1/2))/(exp(e*2i + f*x*2i) + 1)^4 - (((a^3*c^2*(A - B*1i)*32i)/(11*f) - (a^3*c^2*(A + B*1i)*32i
)/(11*f))*(c + (c*(exp(e*2i + f*x*2i)*1i - 1i)*1i)/(exp(e*2i + f*x*2i) + 1))^(1/2))/(exp(e*2i + f*x*2i) + 1)^5
- (((a^3*c^2*(A - B*1i)*32i)/(7*f) + (a^3*c^2*(A - B*3i)*32i)/(7*f))*(c + (c*(exp(e*2i + f*x*2i)*1i - 1i)*1i)
/(exp(e*2i + f*x*2i) + 1))^(1/2))/(exp(e*2i + f*x*2i) + 1)^3 + (a^3*c^2*(A - B*1i)*(c + (c*(exp(e*2i + f*x*2i)
*1i - 1i)*1i)/(exp(e*2i + f*x*2i) + 1))^(1/2)*32i)/(5*f*(exp(e*2i + f*x*2i) + 1)^2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - i a^{3} \left (\int i A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}\, dx + \int \left (- A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}\right )\, dx + \int \left (- 2 A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}\right )\, dx + \int \left (- A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{5}{\left (e + f x \right )}\right )\, dx + \int \left (- B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}\right )\, dx + \int \left (- 2 B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{4}{\left (e + f x \right )}\right )\, dx + \int \left (- B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{6}{\left (e + f x \right )}\right )\, dx + \int 2 i A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}\, dx + \int i A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{4}{\left (e + f x \right )}\, dx + \int i B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}\, dx + \int 2 i B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}\, dx + \int i B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{5}{\left (e + f x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))**3*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))**(5/2),x)

[Out]

-I*a**3*(Integral(I*A*c**2*sqrt(-I*c*tan(e + f*x) + c), x) + Integral(-A*c**2*sqrt(-I*c*tan(e + f*x) + c)*tan(
e + f*x), x) + Integral(-2*A*c**2*sqrt(-I*c*tan(e + f*x) + c)*tan(e + f*x)**3, x) + Integral(-A*c**2*sqrt(-I*c
*tan(e + f*x) + c)*tan(e + f*x)**5, x) + Integral(-B*c**2*sqrt(-I*c*tan(e + f*x) + c)*tan(e + f*x)**2, x) + In
tegral(-2*B*c**2*sqrt(-I*c*tan(e + f*x) + c)*tan(e + f*x)**4, x) + Integral(-B*c**2*sqrt(-I*c*tan(e + f*x) + c
)*tan(e + f*x)**6, x) + Integral(2*I*A*c**2*sqrt(-I*c*tan(e + f*x) + c)*tan(e + f*x)**2, x) + Integral(I*A*c**
2*sqrt(-I*c*tan(e + f*x) + c)*tan(e + f*x)**4, x) + Integral(I*B*c**2*sqrt(-I*c*tan(e + f*x) + c)*tan(e + f*x)
, x) + Integral(2*I*B*c**2*sqrt(-I*c*tan(e + f*x) + c)*tan(e + f*x)**3, x) + Integral(I*B*c**2*sqrt(-I*c*tan(e
+ f*x) + c)*tan(e + f*x)**5, x))

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