3.4.19 \(\int \sec ^8(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx\) [319]

Optimal. Leaf size=117 \[ -\frac {16 i (a+i a \tan (c+d x))^{15/2}}{15 a^4 d}+\frac {24 i (a+i a \tan (c+d x))^{17/2}}{17 a^5 d}-\frac {12 i (a+i a \tan (c+d x))^{19/2}}{19 a^6 d}+\frac {2 i (a+i a \tan (c+d x))^{21/2}}{21 a^7 d} \]

[Out]

-16/15*I*(a+I*a*tan(d*x+c))^(15/2)/a^4/d+24/17*I*(a+I*a*tan(d*x+c))^(17/2)/a^5/d-12/19*I*(a+I*a*tan(d*x+c))^(1
9/2)/a^6/d+2/21*I*(a+I*a*tan(d*x+c))^(21/2)/a^7/d

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3568, 45} \begin {gather*} \frac {2 i (a+i a \tan (c+d x))^{21/2}}{21 a^7 d}-\frac {12 i (a+i a \tan (c+d x))^{19/2}}{19 a^6 d}+\frac {24 i (a+i a \tan (c+d x))^{17/2}}{17 a^5 d}-\frac {16 i (a+i a \tan (c+d x))^{15/2}}{15 a^4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^8*(a + I*a*Tan[c + d*x])^(7/2),x]

[Out]

(((-16*I)/15)*(a + I*a*Tan[c + d*x])^(15/2))/(a^4*d) + (((24*I)/17)*(a + I*a*Tan[c + d*x])^(17/2))/(a^5*d) - (
((12*I)/19)*(a + I*a*Tan[c + d*x])^(19/2))/(a^6*d) + (((2*I)/21)*(a + I*a*Tan[c + d*x])^(21/2))/(a^7*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*} \int \sec ^8(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx &=-\frac {i \text {Subst}\left (\int (a-x)^3 (a+x)^{13/2} \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac {i \text {Subst}\left (\int \left (8 a^3 (a+x)^{13/2}-12 a^2 (a+x)^{15/2}+6 a (a+x)^{17/2}-(a+x)^{19/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac {16 i (a+i a \tan (c+d x))^{15/2}}{15 a^4 d}+\frac {24 i (a+i a \tan (c+d x))^{17/2}}{17 a^5 d}-\frac {12 i (a+i a \tan (c+d x))^{19/2}}{19 a^6 d}+\frac {2 i (a+i a \tan (c+d x))^{21/2}}{21 a^7 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.97, size = 113, normalized size = 0.97 \begin {gather*} -\frac {2 a^3 \sec ^9(c+d x) (\cos (7 c+10 d x)+i \sin (7 c+10 d x)) (-1311 i+4554 i \cos (2 (c+d x))+2245 \sec (c+d x) \sin (3 (c+d x))+630 \tan (c+d x)) \sqrt {a+i a \tan (c+d x)}}{33915 d (\cos (d x)+i \sin (d x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^8*(a + I*a*Tan[c + d*x])^(7/2),x]

[Out]

(-2*a^3*Sec[c + d*x]^9*(Cos[7*c + 10*d*x] + I*Sin[7*c + 10*d*x])*(-1311*I + (4554*I)*Cos[2*(c + d*x)] + 2245*S
ec[c + d*x]*Sin[3*(c + d*x)] + 630*Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])/(33915*d*(Cos[d*x] + I*Sin[d*x])^
3)

________________________________________________________________________________________

Maple [A]
time = 32.62, size = 181, normalized size = 1.55

method	result	size
default	\(-\frac {2 \left (8192 i \left (\cos ^{10}\left (d x +c \right )\right )-8192 \left (\cos ^{9}\left (d x +c \right )\right ) \sin \left (d x +c \right )+1024 i \left (\cos ^{8}\left (d x +c \right )\right )-5120 \sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )+448 i \left (\cos ^{6}\left (d x +c \right )\right )-4032 \sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )+264 i \left (\cos ^{4}\left (d x +c \right )\right )-3432 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )-8300 i \left (\cos ^{2}\left (d x +c \right )\right )+5440 \sin \left (d x +c \right ) \cos \left (d x +c \right )+1615 i\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a^{3}}{33915 d \cos \left (d x +c \right )^{10}}\)	\(181\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^8*(a+I*a*tan(d*x+c))^(7/2),x,method=_RETURNVERBOSE)

[Out]

-2/33915/d*(8192*I*cos(d*x+c)^10-8192*cos(d*x+c)^9*sin(d*x+c)+1024*I*cos(d*x+c)^8-5120*sin(d*x+c)*cos(d*x+c)^7
+448*I*cos(d*x+c)^6-4032*sin(d*x+c)*cos(d*x+c)^5+264*I*cos(d*x+c)^4-3432*sin(d*x+c)*cos(d*x+c)^3-8300*I*cos(d*
x+c)^2+5440*sin(d*x+c)*cos(d*x+c)+1615*I)*(a*(I*sin(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)/cos(d*x+c)^10*a^3

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 76, normalized size = 0.65 \begin {gather*} \frac {2 i \, {\left (1615 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {21}{2}} - 10710 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {19}{2}} a + 23940 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {17}{2}} a^{2} - 18088 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {15}{2}} a^{3}\right )}}{33915 \, a^{7} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^8*(a+I*a*tan(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

2/33915*I*(1615*(I*a*tan(d*x + c) + a)^(21/2) - 10710*(I*a*tan(d*x + c) + a)^(19/2)*a + 23940*(I*a*tan(d*x + c
) + a)^(17/2)*a^2 - 18088*(I*a*tan(d*x + c) + a)^(15/2)*a^3)/(a^7*d)

________________________________________________________________________________________

Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (85) = 170\).
time = 0.41, size = 202, normalized size = 1.73 \begin {gather*} -\frac {2048 \, \sqrt {2} {\left (16 i \, a^{3} e^{\left (21 i \, d x + 21 i \, c\right )} + 168 i \, a^{3} e^{\left (19 i \, d x + 19 i \, c\right )} + 798 i \, a^{3} e^{\left (17 i \, d x + 17 i \, c\right )} + 2261 i \, a^{3} e^{\left (15 i \, d x + 15 i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{33915 \, {\left (d e^{\left (20 i \, d x + 20 i \, c\right )} + 10 \, d e^{\left (18 i \, d x + 18 i \, c\right )} + 45 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 120 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 210 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 252 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 210 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 120 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 45 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 10 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^8*(a+I*a*tan(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

-2048/33915*sqrt(2)*(16*I*a^3*e^(21*I*d*x + 21*I*c) + 168*I*a^3*e^(19*I*d*x + 19*I*c) + 798*I*a^3*e^(17*I*d*x
+ 17*I*c) + 2261*I*a^3*e^(15*I*d*x + 15*I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))/(d*e^(20*I*d*x + 20*I*c) + 10*
d*e^(18*I*d*x + 18*I*c) + 45*d*e^(16*I*d*x + 16*I*c) + 120*d*e^(14*I*d*x + 14*I*c) + 210*d*e^(12*I*d*x + 12*I*
c) + 252*d*e^(10*I*d*x + 10*I*c) + 210*d*e^(8*I*d*x + 8*I*c) + 120*d*e^(6*I*d*x + 6*I*c) + 45*d*e^(4*I*d*x + 4
*I*c) + 10*d*e^(2*I*d*x + 2*I*c) + d)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**8*(a+I*a*tan(d*x+c))**(7/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^8*(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)^8, x)

________________________________________________________________________________________

Mupad [B]
time = 15.04, size = 690, normalized size = 5.90 \begin {gather*} -\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}}\,32768{}\mathrm {i}}{33915\,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}}\,16384{}\mathrm {i}}{33915\,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}}\,4096{}\mathrm {i}}{11305\,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}}\,2048{}\mathrm {i}}{6783\,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}}\,247808{}\mathrm {i}}{969\,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}}\,1943552{}\mathrm {i}}{1615\,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}}\,12019712{}\mathrm {i}}{4845\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^6}-\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}}\,95516672{}\mathrm {i}}{33915\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^7}+\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}}\,4159488{}\mathrm {i}}{2261\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^8}-\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}}\,260096{}\mathrm {i}}{399\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^9}+\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}}\,2048{}\mathrm {i}}{21\,d\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*tan(c + d*x)*1i)^(7/2)/cos(c + d*x)^8,x)

[Out]

(a^3*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*247808i)/(969*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)*16384i)/(33915*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)*4096i
)/(11305*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)*2048i)/(6783*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)*32768i)/(33915*d) - (a^3*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) +
1))^(1/2)*1943552i)/(1615*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)*12019712i)/(4845*d*(exp(c*2i + d*x*2i) + 1)^6) - (a^3*(a - (a*(exp(c*2i + d*x*2i)*1i
- 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*95516672i)/(33915*d*(exp(c*2i + d*x*2i) + 1)^7) + (a^3*(a - (a*(exp
(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*4159488i)/(2261*d*(exp(c*2i + d*x*2i) + 1)^8) - (
a^3*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*260096i)/(399*d*(exp(c*2i + d*x*2
i) + 1)^9) + (a^3*(a - (a*(exp(c*2i + d*x*2i)*1i - 1i)*1i)/(exp(c*2i + d*x*2i) + 1))^(1/2)*2048i)/(21*d*(exp(c
*2i + d*x*2i) + 1)^10)

________________________________________________________________________________________