Integrand size = 26, antiderivative size = 84 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {8 i}{a^3 d \sqrt {a+i a \tan (c+d x)}}+\frac {8 i \sqrt {a+i a \tan (c+d x)}}{a^4 d}-\frac {2 i (a+i a \tan (c+d x))^{3/2}}{3 a^5 d} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 84, 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} \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=-\frac {2 i (a+i a \tan (c+d x))^{3/2}}{3 a^5 d}+\frac {8 i \sqrt {a+i a \tan (c+d x)}}{a^4 d}+\frac {8 i}{a^3 d \sqrt {a+i a \tan (c+d x)}} \]
[In]
[Out]
Rule 45
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {Subst}\left (\int \frac {(a-x)^2}{(a+x)^{3/2}} \, dx,x,i a \tan (c+d x)\right )}{a^5 d} \\ & = -\frac {i \text {Subst}\left (\int \left (\frac {4 a^2}{(a+x)^{3/2}}-\frac {4 a}{\sqrt {a+x}}+\sqrt {a+x}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^5 d} \\ & = \frac {8 i}{a^3 d \sqrt {a+i a \tan (c+d x)}}+\frac {8 i \sqrt {a+i a \tan (c+d x)}}{a^4 d}-\frac {2 i (a+i a \tan (c+d x))^{3/2}}{3 a^5 d} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.58 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {2 i \left (23+10 i \tan (c+d x)+\tan ^2(c+d x)\right )}{3 a^3 d \sqrt {a+i a \tan (c+d x)}} \]
[In]
[Out]
Time = 1.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {2 i \left (-\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+4 a \sqrt {a +i a \tan \left (d x +c \right )}+\frac {4 a^{2}}{\sqrt {a +i a \tan \left (d x +c \right )}}\right )}{d \,a^{5}}\) | \(63\) |
default | \(\frac {2 i \left (-\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+4 a \sqrt {a +i a \tan \left (d x +c \right )}+\frac {4 a^{2}}{\sqrt {a +i a \tan \left (d x +c \right )}}\right )}{d \,a^{5}}\) | \(63\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.92 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=-\frac {4 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-8 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 12 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 3 i\right )}}{3 \, {\left (a^{4} d e^{\left (3 i \, d x + 3 i \, c\right )} + a^{4} d e^{\left (i \, d x + i \, c\right )}\right )}} \]
[In]
[Out]
\[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\int \frac {\sec ^{6}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {7}{2}}}\, dx \]
[In]
[Out]
none
Time = 0.38 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.74 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {2 i \, {\left (\frac {12}{\sqrt {i \, a \tan \left (d x + c\right ) + a} a^{2}} - \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} - 12 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} a}{a^{4}}\right )}}{3 \, a d} \]
[In]
[Out]
\[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{6}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
[In]
[Out]
Time = 0.86 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.31 \[ \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {2\,\sqrt {\frac {a\,\left (\cos \left (2\,c+2\,d\,x\right )+1+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,c+2\,d\,x\right )+1}}\,\left (\cos \left (2\,c+2\,d\,x\right )\,23{}\mathrm {i}+\cos \left (4\,c+4\,d\,x\right )\,3{}\mathrm {i}+7\,\sin \left (2\,c+2\,d\,x\right )+3\,\sin \left (4\,c+4\,d\,x\right )+20{}\mathrm {i}\right )}{3\,a^4\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]
[In]
[Out]