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

Optimal. Leaf size=152 \[ -\frac{B+i A}{f (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}+\frac{(B+2 i A) \sqrt{c-i c \tan (e+f x)}}{3 a c f \sqrt{a+i a \tan (e+f x)}}+\frac{(B+2 i A) \sqrt{c-i c \tan (e+f x)}}{3 c f (a+i a \tan (e+f x))^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.246077, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.089, Rules used = {3588, 78, 45, 37} \[ -\frac{B+i A}{f (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}+\frac{(B+2 i A) \sqrt{c-i c \tan (e+f x)}}{3 a c f \sqrt{a+i a \tan (e+f x)}}+\frac{(B+2 i A) \sqrt{c-i c \tan (e+f x)}}{3 c f (a+i a \tan (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

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{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{A+B x}{(a+i a x)^{5/2} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{i A+B}{f (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}+\frac{(a (2 A-i B)) \operatorname{Subst}\left (\int \frac{1}{(a+i a x)^{5/2} \sqrt{c-i c x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{i A+B}{f (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}+\frac{(2 i A+B) \sqrt{c-i c \tan (e+f x)}}{3 c f (a+i a \tan (e+f x))^{3/2}}+\frac{(2 A-i B) \operatorname{Subst}\left (\int \frac{1}{(a+i a x)^{3/2} \sqrt{c-i c x}} \, dx,x,\tan (e+f x)\right )}{3 f}\\ &=-\frac{i A+B}{f (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}+\frac{(2 i A+B) \sqrt{c-i c \tan (e+f x)}}{3 c f (a+i a \tan (e+f x))^{3/2}}+\frac{(2 i A+B) \sqrt{c-i c \tan (e+f x)}}{3 a c f \sqrt{a+i a \tan (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 4.81573, size = 85, normalized size = 0.56 \[ -\frac{i \sqrt{c-i c \tan (e+f x)} ((B+2 i A) \sin (2 (e+f x))+(A-2 i B) \cos (2 (e+f x))-3 A)}{6 a c f \sqrt{a+i a \tan (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I/6)*(-3*A + (A - (2*I)*B)*Cos[2*(e + f*x)] + ((2*I)*A + B)*Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan[e + f*x]])/(
a*c*f*Sqrt[a + I*a*Tan[e + f*x]])

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Maple [A]  time = 0.187, size = 152, normalized size = 1. \begin{align*}{\frac{{\frac{i}{3}} \left ( 2\,iA \left ( \tan \left ( fx+e \right ) \right ) ^{4}-iB \left ( \tan \left ( fx+e \right ) \right ) ^{3}+B \left ( \tan \left ( fx+e \right ) \right ) ^{4}+3\,iA \left ( \tan \left ( fx+e \right ) \right ) ^{2}+2\,A \left ( \tan \left ( fx+e \right ) \right ) ^{3}-iB\tan \left ( fx+e \right ) +iA+2\,A\tan \left ( fx+e \right ) -B \right ) }{f{a}^{2}c \left ( -\tan \left ( fx+e \right ) +i \right ) ^{3} \left ( \tan \left ( fx+e \right ) +i \right ) ^{2}}\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((A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^(3/2),x)

[Out]

1/3*I/f*(-c*(-1+I*tan(f*x+e)))^(1/2)*(a*(1+I*tan(f*x+e)))^(1/2)/a^2/c*(2*I*A*tan(f*x+e)^4-I*B*tan(f*x+e)^3+B*t
an(f*x+e)^4+3*I*A*tan(f*x+e)^2+2*A*tan(f*x+e)^3-I*B*tan(f*x+e)+I*A+2*A*tan(f*x+e)-B)/(-tan(f*x+e)+I)^3/(tan(f*
x+e)+I)^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.3744, size = 400, normalized size = 2.63 \begin{align*} \frac{{\left ({\left (-3 i \, A - 3 \, B\right )} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-4 i \, A + 4 \, B\right )} e^{\left (5 i \, f x + 5 i \, e\right )} +{\left (3 i \, A - 3 \, B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (-4 i \, A + 4 \, B\right )} e^{\left (3 i \, f x + 3 i \, e\right )} +{\left (7 i \, A - B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, A - B\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 (-3 i \, f x - 3 i \, e\right )}}{12 \, a^{2} c f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

1/12*((-3*I*A - 3*B)*e^(6*I*f*x + 6*I*e) + (-4*I*A + 4*B)*e^(5*I*f*x + 5*I*e) + (3*I*A - 3*B)*e^(4*I*f*x + 4*I
*e) + (-4*I*A + 4*B)*e^(3*I*f*x + 3*I*e) + (7*I*A - B)*e^(2*I*f*x + 2*I*e) + I*A - B)*sqrt(a/(e^(2*I*f*x + 2*I
*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1))*e^(-3*I*f*x - 3*I*e)/(a^2*c*f)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))**(1/2)/(a+I*a*tan(f*x+e))**(3/2),x)

[Out]

Exception raised: AttributeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^(3/2),x, algorithm="giac")

[Out]

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