3.324 \(\int \cos ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx\)

Optimal. Leaf size=137 \[ -\frac{i a^5 \sqrt{a+i a \tan (c+d x)}}{2 d (a-i a \tan (c+d x))^2}+\frac{i a^4 \sqrt{a+i a \tan (c+d x)}}{8 d (a-i a \tan (c+d x))}+\frac{i a^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{8 \sqrt{2} d} \]

[Out]

((I/8)*a^(7/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*d) - ((I/2)*a^5*Sqrt[a + I*a*Ta
n[c + d*x]])/(d*(a - I*a*Tan[c + d*x])^2) + ((I/8)*a^4*Sqrt[a + I*a*Tan[c + d*x]])/(d*(a - I*a*Tan[c + d*x]))

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Rubi [A]  time = 0.0937976, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {3487, 47, 51, 63, 206} \[ -\frac{i a^5 \sqrt{a+i a \tan (c+d x)}}{2 d (a-i a \tan (c+d x))^2}+\frac{i a^4 \sqrt{a+i a \tan (c+d x)}}{8 d (a-i a \tan (c+d x))}+\frac{i a^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{8 \sqrt{2} d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^(7/2),x]

[Out]

((I/8)*a^(7/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*d) - ((I/2)*a^5*Sqrt[a + I*a*Ta
n[c + d*x]])/(d*(a - I*a*Tan[c + d*x])^2) + ((I/8)*a^4*Sqrt[a + I*a*Tan[c + d*x]])/(d*(a - I*a*Tan[c + d*x]))

Rule 3487

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 47

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},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 51

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*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

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 206

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

Rubi steps

\begin{align*} \int \cos ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx &=-\frac{\left (i a^5\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+x}}{(a-x)^3} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{i a^5 \sqrt{a+i a \tan (c+d x)}}{2 d (a-i a \tan (c+d x))^2}+\frac{\left (i a^5\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x)^2 \sqrt{a+x}} \, dx,x,i a \tan (c+d x)\right )}{4 d}\\ &=-\frac{i a^5 \sqrt{a+i a \tan (c+d x)}}{2 d (a-i a \tan (c+d x))^2}+\frac{i a^4 \sqrt{a+i a \tan (c+d x)}}{8 d (a-i a \tan (c+d x))}+\frac{\left (i a^4\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x) \sqrt{a+x}} \, dx,x,i a \tan (c+d x)\right )}{16 d}\\ &=-\frac{i a^5 \sqrt{a+i a \tan (c+d x)}}{2 d (a-i a \tan (c+d x))^2}+\frac{i a^4 \sqrt{a+i a \tan (c+d x)}}{8 d (a-i a \tan (c+d x))}+\frac{\left (i a^4\right ) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\sqrt{a+i a \tan (c+d x)}\right )}{8 d}\\ &=\frac{i a^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{8 \sqrt{2} d}-\frac{i a^5 \sqrt{a+i a \tan (c+d x)}}{2 d (a-i a \tan (c+d x))^2}+\frac{i a^4 \sqrt{a+i a \tan (c+d x)}}{8 d (a-i a \tan (c+d x))}\\ \end{align*}

Mathematica [A]  time = 1.34732, size = 152, normalized size = 1.11 \[ -\frac{i e^{-4 i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left (e^{i (c+d x)}+3 e^{3 i (c+d x)}+2 e^{5 i (c+d x)}-\sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left (e^{i (c+d x)}\right )\right ) (a+i a \tan (c+d x))^{7/2}}{8 \sqrt{2} d \sec ^{\frac{7}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^(7/2),x]

[Out]

((-I/8)*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(E^(I*(c + d*x)) + 3*E^((3*I)*(c + d*x)) + 2*E^((5*I)*
(c + d*x)) - Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))])*(a + I*a*Tan[c + d*x])^(7/2))/(Sqrt[2]*d*
E^((4*I)*(c + d*x))*Sec[c + d*x]^(7/2))

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Maple [B]  time = 0.344, size = 742, normalized size = 5.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*(a+I*a*tan(d*x+c))^(7/2),x)

[Out]

-1/128/d*a^3*(256*I*cos(d*x+c)^8+2^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*(-2*cos(d*x+
c)/(cos(d*x+c)+1))^(7/2)*cos(d*x+c)^3*sin(d*x+c)-224*I*cos(d*x+c)^6+3*2^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c
)/(cos(d*x+c)+1))^(1/2))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)*cos(d*x+c)^2*sin(d*x+c)+3*I*(-2*cos(d*x+c)/(cos(
d*x+c)+1))^(7/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*2^(1/2)*cos(d
*x+c)*sin(d*x+c)+3*2^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*(-2*cos(d*x+c)/(cos(d*x+c)
+1))^(7/2)*cos(d*x+c)*sin(d*x+c)-128*I*cos(d*x+c)^7+2^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^
(1/2))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)*sin(d*x+c)+I*2^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c
)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)*sin(d*x+c)+3*I*(-2*cos(d*x+c)/(cos(d*x
+c)+1))^(7/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*2^(1/2)*cos(d*x+
c)^2*sin(d*x+c)-256*sin(d*x+c)*cos(d*x+c)^7+I*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)*arctanh(1/2*2^(1/2)*(-2*cos
(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*2^(1/2)*cos(d*x+c)^3*sin(d*x+c)+128*cos(d*x+c)^6*sin(d*x+
c)+80*I*cos(d*x+c)^5+96*cos(d*x+c)^5*sin(d*x+c)+16*I*cos(d*x+c)^4-16*sin(d*x+c)*cos(d*x+c)^4)*(a*(I*sin(d*x+c)
+cos(d*x+c))/cos(d*x+c))^(1/2)/(I*sin(d*x+c)+cos(d*x+c)-1)/cos(d*x+c)^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(a+I*a*tan(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.27462, size = 737, normalized size = 5.38 \begin{align*} \frac{\sqrt{2}{\left (-2 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 3 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{3}\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )} - \sqrt{\frac{1}{2}} \sqrt{-\frac{a^{7}}{d^{2}}} d \log \left (\frac{{\left (2 i \, \sqrt{\frac{1}{2}} \sqrt{-\frac{a^{7}}{d^{2}}} d e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt{2}{\left (a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3}\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{a^{3}}\right ) + \sqrt{\frac{1}{2}} \sqrt{-\frac{a^{7}}{d^{2}}} d \log \left (\frac{{\left (-2 i \, \sqrt{\frac{1}{2}} \sqrt{-\frac{a^{7}}{d^{2}}} d e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt{2}{\left (a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3}\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{a^{3}}\right )}{16 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(a+I*a*tan(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

1/16*(sqrt(2)*(-2*I*a^3*e^(4*I*d*x + 4*I*c) - 3*I*a^3*e^(2*I*d*x + 2*I*c) - I*a^3)*sqrt(a/(e^(2*I*d*x + 2*I*c)
+ 1))*e^(I*d*x + I*c) - sqrt(1/2)*sqrt(-a^7/d^2)*d*log((2*I*sqrt(1/2)*sqrt(-a^7/d^2)*d*e^(2*I*d*x + 2*I*c) +
sqrt(2)*(a^3*e^(2*I*d*x + 2*I*c) + a^3)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(I*d*x + I*c))*e^(-2*I*d*x - 2*I*c
)/a^3) + sqrt(1/2)*sqrt(-a^7/d^2)*d*log((-2*I*sqrt(1/2)*sqrt(-a^7/d^2)*d*e^(2*I*d*x + 2*I*c) + sqrt(2)*(a^3*e^
(2*I*d*x + 2*I*c) + a^3)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(I*d*x + I*c))*e^(-2*I*d*x - 2*I*c)/a^3))/d

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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(cos(d*x+c)**4*(a+I*a*tan(d*x+c))**(7/2),x)

[Out]

Timed out

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Giac [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(cos(d*x+c)^4*(a+I*a*tan(d*x+c))^(7/2),x, algorithm="giac")

[Out]

Timed out