\(\int \sec ^6(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx\) [280]

Optimal result

Integrand size = 26, antiderivative size = 88 \[ \int \sec ^6(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {8 i (a+i a \tan (c+d x))^{7/2}}{7 a^3 d}+\frac {8 i (a+i a \tan (c+d x))^{9/2}}{9 a^4 d}-\frac {2 i (a+i a \tan (c+d x))^{11/2}}{11 a^5 d} \]

[Out]

-8/7*I*(a+I*a*tan(d*x+c))^(7/2)/a^3/d+8/9*I*(a+I*a*tan(d*x+c))^(9/2)/a^4/d-2/11*I*(a+I*a*tan(d*x+c))^(11/2)/a^
5/d

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3568, 45} \[ \int \sec ^6(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {2 i (a+i a \tan (c+d x))^{11/2}}{11 a^5 d}+\frac {8 i (a+i a \tan (c+d x))^{9/2}}{9 a^4 d}-\frac {8 i (a+i a \tan (c+d x))^{7/2}}{7 a^3 d} \]

[In]

Int[Sec[c + d*x]^6*Sqrt[a + I*a*Tan[c + d*x]],x]

[Out]

(((-8*I)/7)*(a + I*a*Tan[c + d*x])^(7/2))/(a^3*d) + (((8*I)/9)*(a + I*a*Tan[c + d*x])^(9/2))/(a^4*d) - (((2*I)
/11)*(a + I*a*Tan[c + d*x])^(11/2))/(a^5*d)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.66 \[ \int \sec ^6(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\frac {2 (-i+\tan (c+d x))^3 \sqrt {a+i a \tan (c+d x)} \left (-151+182 i \tan (c+d x)+63 \tan ^2(c+d x)\right )}{693 d} \]

[In]

Integrate[Sec[c + d*x]^6*Sqrt[a + I*a*Tan[c + d*x]],x]

[Out]

(2*(-I + Tan[c + d*x])^3*Sqrt[a + I*a*Tan[c + d*x]]*(-151 + (182*I)*Tan[c + d*x] + 63*Tan[c + d*x]^2))/(693*d)

Maple [A] (verified)

Time = 1.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72

method	result	size
derivativedivides	\(\frac {2 i \left (-\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {11}{2}}}{11}+\frac {4 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {9}{2}}}{9}-\frac {4 a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7}\right )}{d \,a^{5}}\)	\(63\)
default	\(\frac {2 i \left (-\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {11}{2}}}{11}+\frac {4 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {9}{2}}}{9}-\frac {4 a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7}\right )}{d \,a^{5}}\)	\(63\)

[In]

int(sec(d*x+c)^6*(a+I*a*tan(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

2*I/d/a^5*(-1/11*(a+I*a*tan(d*x+c))^(11/2)+4/9*a*(a+I*a*tan(d*x+c))^(9/2)-4/7*a^2*(a+I*a*tan(d*x+c))^(7/2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.35 \[ \int \sec ^6(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {64 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (8 i \, e^{\left (11 i \, d x + 11 i \, c\right )} + 44 i \, e^{\left (9 i \, d x + 9 i \, c\right )} + 99 i \, e^{\left (7 i \, d x + 7 i \, c\right )}\right )}}{693 \, {\left (d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

[In]

integrate(sec(d*x+c)^6*(a+I*a*tan(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

-64/693*sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(8*I*e^(11*I*d*x + 11*I*c) + 44*I*e^(9*I*d*x + 9*I*c) + 99*I
*e^(7*I*d*x + 7*I*c))/(d*e^(10*I*d*x + 10*I*c) + 5*d*e^(8*I*d*x + 8*I*c) + 10*d*e^(6*I*d*x + 6*I*c) + 10*d*e^(
4*I*d*x + 4*I*c) + 5*d*e^(2*I*d*x + 2*I*c) + d)

Sympy [F]

\[ \int \sec ^6(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\int \sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )} \sec ^{6}{\left (c + d x \right )}\, dx \]

[In]

integrate(sec(d*x+c)**6*(a+I*a*tan(d*x+c))**(1/2),x)

[Out]

Integral(sqrt(I*a*(tan(c + d*x) - I))*sec(c + d*x)**6, x)

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.66 \[ \int \sec ^6(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {2 i \, {\left (63 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {11}{2}} - 308 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a + 396 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{2}\right )}}{693 \, a^{5} d} \]

[In]

integrate(sec(d*x+c)^6*(a+I*a*tan(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

-2/693*I*(63*(I*a*tan(d*x + c) + a)^(11/2) - 308*(I*a*tan(d*x + c) + a)^(9/2)*a + 396*(I*a*tan(d*x + c) + a)^(
7/2)*a^2)/(a^5*d)

Giac [F]

\[ \int \sec ^6(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\int { \sqrt {i \, a \tan \left (d x + c\right ) + a} \sec \left (d x + c\right )^{6} \,d x } \]

[In]

integrate(sec(d*x+c)^6*(a+I*a*tan(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(I*a*tan(d*x + c) + a)*sec(d*x + c)^6, x)

Mupad [B] (verification not implemented)

Time = 7.76 (sec) , antiderivative size = 352, normalized size of antiderivative = 4.00 \[ \int \sec ^6(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,512{}\mathrm {i}}{693\,d}-\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,256{}\mathrm {i}}{693\,d\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}-\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,64{}\mathrm {i}}{231\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2}+\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,7232{}\mathrm {i}}{693\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3}-\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,1472{}\mathrm {i}}{99\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^4}+\frac {\sqrt {a-\frac {a\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1}}\,64{}\mathrm {i}}{11\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^5} \]

[In]

int((a + a*tan(c + d*x)*1i)^(1/2)/cos(c + d*x)^6,x)

[Out]

((a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*7232i)/(693*d*(exp(c*2i + d*x*2i) +
1)^3) - ((a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*256i)/(693*d*(exp(c*2i + d*x
*2i) + 1)) - ((a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*64i)/(231*d*(exp(c*2i +
d*x*2i) + 1)^2) - ((a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*512i)/(693*d) - (
(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*1472i)/(99*d*(exp(c*2i + d*x*2i) + 1)
^4) + ((a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*64i)/(11*d*(exp(c*2i + d*x*2i)
+ 1)^5)