\(\int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx\) [321]

Optimal result

Integrand size = 26, antiderivative size = 59 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {4 i (a+i a \tan (c+d x))^{11/2}}{11 a^2 d}+\frac {2 i (a+i a \tan (c+d x))^{13/2}}{13 a^3 d} \]

[Out]

-4/11*I*(a+I*a*tan(d*x+c))^(11/2)/a^2/d+2/13*I*(a+I*a*tan(d*x+c))^(13/2)/a^3/d

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 59, 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} \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\frac {2 i (a+i a \tan (c+d x))^{13/2}}{13 a^3 d}-\frac {4 i (a+i a \tan (c+d x))^{11/2}}{11 a^2 d} \]

[In]

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

[Out]

(((-4*I)/11)*(a + I*a*Tan[c + d*x])^(11/2))/(a^2*d) + (((2*I)/13)*(a + I*a*Tan[c + d*x])^(13/2))/(a^3*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) (a+x)^{9/2} \, dx,x,i a \tan (c+d x)\right )}{a^3 d} \\ & = -\frac {i \text {Subst}\left (\int \left (2 a (a+x)^{9/2}-(a+x)^{11/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d} \\ & = -\frac {4 i (a+i a \tan (c+d x))^{11/2}}{11 a^2 d}+\frac {2 i (a+i a \tan (c+d x))^{13/2}}{13 a^3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.86 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\frac {2 a^3 (15-11 i \tan (c+d x)) (-i+\tan (c+d x))^5 \sqrt {a+i a \tan (c+d x)}}{143 d} \]

[In]

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

[Out]

(2*a^3*(15 - (11*I)*Tan[c + d*x])*(-I + Tan[c + d*x])^5*Sqrt[a + I*a*Tan[c + d*x]])/(143*d)

Maple [A] (verified)

Time = 187.49 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.75

method	result	size
derivativedivides	\(\frac {2 i \left (\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {13}{2}}}{13}-\frac {2 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {11}{2}}}{11}\right )}{d \,a^{3}}\)	\(44\)
default	\(\frac {2 i \left (\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {13}{2}}}{13}-\frac {2 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {11}{2}}}{11}\right )}{d \,a^{3}}\)	\(44\)

[In]

int(sec(d*x+c)^4*(a+I*a*tan(d*x+c))^(7/2),x,method=_RETURNVERBOSE)

[Out]

2*I/d/a^3*(1/13*(a+I*a*tan(d*x+c))^(13/2)-2/11*a*(a+I*a*tan(d*x+c))^(11/2))

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. 126 vs. \(2 (43) = 86\).

Time = 0.26 (sec) , antiderivative size = 126, normalized size of antiderivative = 2.14 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {128 \, \sqrt {2} {\left (2 i \, a^{3} e^{\left (13 i \, d x + 13 i \, c\right )} + 13 i \, a^{3} e^{\left (11 i \, d x + 11 i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{143 \, {\left (d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

[In]

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

[Out]

-128/143*sqrt(2)*(2*I*a^3*e^(13*I*d*x + 13*I*c) + 13*I*a^3*e^(11*I*d*x + 11*I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c)
+ 1))/(d*e^(12*I*d*x + 12*I*c) + 6*d*e^(10*I*d*x + 10*I*c) + 15*d*e^(8*I*d*x + 8*I*c) + 20*d*e^(6*I*d*x + 6*I*
c) + 15*d*e^(4*I*d*x + 4*I*c) + 6*d*e^(2*I*d*x + 2*I*c) + d)

Sympy [F(-1)]

Timed out. \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.68 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\frac {2 i \, {\left (11 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {13}{2}} - 26 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {11}{2}} a\right )}}{143 \, a^{3} d} \]

[In]

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

[Out]

2/143*I*(11*(I*a*tan(d*x + c) + a)^(13/2) - 26*(I*a*tan(d*x + c) + a)^(11/2)*a)/(a^3*d)

Giac [F]

\[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \sec \left (d x + c\right )^{4} \,d x } \]

[In]

integrate(sec(d*x+c)^4*(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)^4, x)

Mupad [B] (verification not implemented)

Time = 8.68 (sec) , antiderivative size = 434, normalized size of antiderivative = 7.36 \[ \int \sec ^4(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}}\,256{}\mathrm {i}}{143\,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}}{143\,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}}\,4480{}\mathrm {i}}{143\,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}}\,11520{}\mathrm {i}}{143\,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}}\,12800{}\mathrm {i}}{143\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^4}-\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}}\,6784{}\mathrm {i}}{143\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^5}+\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}}{13\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^6} \]

[In]

int((a + a*tan(c + d*x)*1i)^(7/2)/cos(c + d*x)^4,x)

[Out]

(a^3*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*4480i)/(143*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)/(143*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)*256i)/(143*
d) - (a^3*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*11520i)/(143*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)*12800i)/(143*d
*(exp(c*2i + d*x*2i) + 1)^4) - (a^3*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*6
784i)/(143*d*(exp(c*2i + d*x*2i) + 1)^5) + (a^3*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) +
1))^(1/2)*128i)/(13*d*(exp(c*2i + d*x*2i) + 1)^6)