Integrand size = 26, antiderivative size = 59 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {4 i (a+i a \tan (c+d x))^{11/2}}{11 a^2 d}+\frac {2 i (a+i a \tan (c+d x))^{13/2}}{13 a^3 d} \] Output:
-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
Time = 0.23 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.86 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\frac {2 a^3 (15-11 i \tan (c+d x)) (-i+\tan (c+d x))^5 \sqrt {a+i a \tan (c+d x)}}{143 d} \] Input:
Integrate[Sec[c + d*x]^4*(a + I*a*Tan[c + d*x])^(7/2),x]
Output:
(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)
Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3042, 3968, 53, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sec (c+d x)^4 (a+i a \tan (c+d x))^{7/2}dx\) |
\(\Big \downarrow \) 3968 |
\(\displaystyle -\frac {i \int (a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{9/2}d(i a \tan (c+d x))}{a^3 d}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\frac {i \int \left (2 a (i \tan (c+d x) a+a)^{9/2}-(i \tan (c+d x) a+a)^{11/2}\right )d(i a \tan (c+d x))}{a^3 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {i \left (\frac {4}{11} a (a+i a \tan (c+d x))^{11/2}-\frac {2}{13} (a+i a \tan (c+d x))^{13/2}\right )}{a^3 d}\) |
Input:
Int[Sec[c + d*x]^4*(a + I*a*Tan[c + d*x])^(7/2),x]
Output:
((-I)*((4*a*(a + I*a*Tan[c + d*x])^(11/2))/11 - (2*(a + I*a*Tan[c + d*x])^ (13/2))/13))/(a^3*d)
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] && 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])
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ ), x_Symbol] :> Simp[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]
Time = 100.94 (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 | \(\frac {2 i \left (\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {13}{2}}}{13}-\frac {2 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {11}{2}}}{11}\right )}{d \,a^{3}}\) | \(44\) |
Input:
int(sec(d*x+c)^4*(a+I*a*tan(d*x+c))^(7/2),x,method=_RETURNVERBOSE)
Output:
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) )
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 = 0.10 (sec) , antiderivative size = 126, normalized size of antiderivative = 2.14 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {128 \, \sqrt {2} {\left (2 i \, a^{3} e^{\left (13 i \, d x + 13 i \, c\right )} + 13 i \, a^{3} e^{\left (11 i \, d x + 11 i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{143 \, {\left (d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \] Input:
integrate(sec(d*x+c)^4*(a+I*a*tan(d*x+c))^(7/2),x, algorithm="fricas")
Output:
-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 + 1 1*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)
Timed out. \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)**4*(a+I*a*tan(d*x+c))**(7/2),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.68 \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\frac {2 i \, {\left (11 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {13}{2}} - 26 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {11}{2}} a\right )}}{143 \, a^{3} d} \] Input:
integrate(sec(d*x+c)^4*(a+I*a*tan(d*x+c))^(7/2),x, algorithm="maxima")
Output:
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)
Exception generated. \[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(sec(d*x+c)^4*(a+I*a*tan(d*x+c))^(7/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad Argument TypeDone
Time = 4.97 (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} \] Input:
int((a + a*tan(c + d*x)*1i)^(7/2)/cos(c + d*x)^4,x)
Output:
(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)*6784i)/ (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)
\[ \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx=\frac {\sqrt {a}\, a^{3} \left (-2 \sqrt {\tan \left (d x +c \right ) i +1}\, \sec \left (d x +c \right )^{4} i -\left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \sec \left (d x +c \right )^{4} \tan \left (d x +c \right )^{3}d x \right ) d i -3 \left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \sec \left (d x +c \right )^{4} \tan \left (d x +c \right )^{2}d x \right ) d +12 \left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \sec \left (d x +c \right )^{4} \tan \left (d x +c \right )d x \right ) d i \right )}{d} \] Input:
int(sec(d*x+c)^4*(a+I*a*tan(d*x+c))^(7/2),x)
Output:
(sqrt(a)*a**3*( - 2*sqrt(tan(c + d*x)*i + 1)*sec(c + d*x)**4*i - int(sqrt( tan(c + d*x)*i + 1)*sec(c + d*x)**4*tan(c + d*x)**3,x)*d*i - 3*int(sqrt(ta n(c + d*x)*i + 1)*sec(c + d*x)**4*tan(c + d*x)**2,x)*d + 12*int(sqrt(tan(c + d*x)*i + 1)*sec(c + d*x)**4*tan(c + d*x),x)*d*i))/d