\(\int \sec ^2(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx\) [322]

Optimal result

Integrand size = 26, antiderivative size = 29 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {2 i (a+i a \tan (c+d x))^{9/2}}{9 a d} \]

[Out]

-2/9*I*(a+I*a*tan(d*x+c))^(9/2)/a/d

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3568, 32} \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {2 i (a+i a \tan (c+d x))^{9/2}}{9 a d} \]

[In]

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

[Out]

(((-2*I)/9)*(a + I*a*Tan[c + d*x])^(9/2))/(a*d)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {Subst}\left (\int (a+x)^{7/2} \, dx,x,i a \tan (c+d x)\right )}{a d} \\ & = -\frac {2 i (a+i a \tan (c+d x))^{9/2}}{9 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {2 i (a+i a \tan (c+d x))^{9/2}}{9 a d} \]

[In]

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

[Out]

(((-2*I)/9)*(a + I*a*Tan[c + d*x])^(9/2))/(a*d)

Maple [A] (verified)

Time = 4.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(-\frac {2 i \left (a +i a \tan \left (d x +c \right )\right )^{\frac {9}{2}}}{9 a d}\)	\(24\)
default	\(-\frac {2 i \left (a +i a \tan \left (d x +c \right )\right )^{\frac {9}{2}}}{9 a d}\)	\(24\)

[In]

int(sec(d*x+c)^2*(a+I*a*tan(d*x+c))^(7/2),x,method=_RETURNVERBOSE)

[Out]

-2/9*I*(a+I*a*tan(d*x+c))^(9/2)/a/d

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. 85 vs. \(2 (21) = 42\).

Time = 0.25 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.93 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {32 i \, \sqrt {2} a^{3} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (9 i \, d x + 9 i \, c\right )}}{9 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

[In]

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

[Out]

-32/9*I*sqrt(2)*a^3*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(9*I*d*x + 9*I*c)/(d*e^(8*I*d*x + 8*I*c) + 4*d*e^(6*I*
d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) + 4*d*e^(2*I*d*x + 2*I*c) + d)

Sympy [F(-1)]

Timed out. \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**2*(a+I*a*tan(d*x+c))**(7/2),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {2 i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {9}{2}}}{9 \, a d} \]

[In]

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

[Out]

-2/9*I*(I*a*tan(d*x + c) + a)^(9/2)/(a*d)

Giac [F]

\[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \sec \left (d x + c\right )^{2} \,d x } \]

[In]

integrate(sec(d*x+c)^2*(a+I*a*tan(d*x+c))^(7/2),x, algorithm="giac")

[Out]

integrate((I*a*tan(d*x + c) + a)^(7/2)*sec(d*x + c)^2, x)

Mupad [B] (verification not implemented)

Time = 6.92 (sec) , antiderivative size = 306, normalized size of antiderivative = 10.55 \[ \int \sec ^2(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {a^3\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,32{}\mathrm {i}}{9\,d}+\frac {a^3\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,128{}\mathrm {i}}{9\,d\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}-\frac {a^3\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,64{}\mathrm {i}}{3\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2}+\frac {a^3\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,128{}\mathrm {i}}{9\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3}-\frac {a^3\,\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,32{}\mathrm {i}}{9\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^4} \]

[In]

int((a + a*tan(c + d*x)*1i)^(7/2)/cos(c + d*x)^2,x)

[Out]

(a^3*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*128i)/(9*d*(exp(c*2i + d*x*2i) +
1)) - (a^3*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*32i)/(9*d) - (a^3*(a - (a
*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*64i)/(3*d*(exp(c*2i + d*x*2i) + 1)^2) + (a^3
*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*128i)/(9*d*(exp(c*2i + d*x*2i) + 1)^
3) - (a^3*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*32i)/(9*d*(exp(c*2i + d*x*2
i) + 1)^4)