Integrand size = 34, antiderivative size = 192 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx=-\frac {2 A \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {(A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{4 \sqrt {2} a^{5/2} d}+\frac {A+i B}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {3 A+i B}{6 a d (a+i a \tan (c+d x))^{3/2}}+\frac {7 A+i B}{4 a^2 d \sqrt {a+i a \tan (c+d x)}} \]
[Out]
Time = 1.11 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {3677, 3681, 3561, 212, 3680, 65, 214} \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {(A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{4 \sqrt {2} a^{5/2} d}-\frac {2 A \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {7 A+i B}{4 a^2 d \sqrt {a+i a \tan (c+d x)}}+\frac {A+i B}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {3 A+i B}{6 a d (a+i a \tan (c+d x))^{3/2}} \]
[In]
[Out]
Rule 65
Rule 212
Rule 214
Rule 3561
Rule 3677
Rule 3680
Rule 3681
Rubi steps \begin{align*} \text {integral}& = \frac {A+i B}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {\int \frac {\cot (c+d x) \left (5 a A-\frac {5}{2} a (i A-B) \tan (c+d x)\right )}{(a+i a \tan (c+d x))^{3/2}} \, dx}{5 a^2} \\ & = \frac {A+i B}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {3 A+i B}{6 a d (a+i a \tan (c+d x))^{3/2}}+\frac {\int \frac {\cot (c+d x) \left (15 a^2 A-\frac {15}{4} a^2 (3 i A-B) \tan (c+d x)\right )}{\sqrt {a+i a \tan (c+d x)}} \, dx}{15 a^4} \\ & = \frac {A+i B}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {3 A+i B}{6 a d (a+i a \tan (c+d x))^{3/2}}+\frac {7 A+i B}{4 a^2 d \sqrt {a+i a \tan (c+d x)}}+\frac {\int \cot (c+d x) \sqrt {a+i a \tan (c+d x)} \left (15 a^3 A-\frac {15}{8} a^3 (7 i A-B) \tan (c+d x)\right ) \, dx}{15 a^6} \\ & = \frac {A+i B}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {3 A+i B}{6 a d (a+i a \tan (c+d x))^{3/2}}+\frac {7 A+i B}{4 a^2 d \sqrt {a+i a \tan (c+d x)}}+\frac {A \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)} \, dx}{a^4}+\frac {(i A+B) \int \sqrt {a+i a \tan (c+d x)} \, dx}{8 a^3} \\ & = \frac {A+i B}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {3 A+i B}{6 a d (a+i a \tan (c+d x))^{3/2}}+\frac {7 A+i B}{4 a^2 d \sqrt {a+i a \tan (c+d x)}}+\frac {A \text {Subst}\left (\int \frac {1}{x \sqrt {a+i a x}} \, dx,x,\tan (c+d x)\right )}{a^2 d}+\frac {(A-i B) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{4 a^2 d} \\ & = \frac {(A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{4 \sqrt {2} a^{5/2} d}+\frac {A+i B}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {3 A+i B}{6 a d (a+i a \tan (c+d x))^{3/2}}+\frac {7 A+i B}{4 a^2 d \sqrt {a+i a \tan (c+d x)}}-\frac {(2 i A) \text {Subst}\left (\int \frac {1}{i-\frac {i x^2}{a}} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{a^3 d} \\ & = -\frac {2 A \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {(A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{4 \sqrt {2} a^{5/2} d}+\frac {A+i B}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac {3 A+i B}{6 a d (a+i a \tan (c+d x))^{3/2}}+\frac {7 A+i B}{4 a^2 d \sqrt {a+i a \tan (c+d x)}} \\ \end{align*}
Time = 3.28 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.13 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {3 a^{13/2} (A+i B)+(a+i a \tan (c+d x)) \left (\frac {5}{2} a^{11/2} (3 A+i B)+\frac {15}{8} a^4 (a+i a \tan (c+d x)) \left (2 \sqrt {a} (7 A+i B)-16 A \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {a+i a \tan (c+d x)}+\sqrt {2} (A-i B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right ) \sqrt {a+i a \tan (c+d x)}\right )\right )}{15 a^{13/2} d (a+i a \tan (c+d x))^{5/2}} \]
[In]
[Out]
Time = 0.31 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\frac {2 a \left (-\frac {-i B -7 A}{8 a^{3} \sqrt {a +i a \tan \left (d x +c \right )}}-\frac {-i B -3 A}{12 a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {-i B -A}{10 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {\left (i B -A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{16 a^{\frac {7}{2}}}-\frac {A \,\operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{a^{\frac {7}{2}}}\right )}{d}\) | \(153\) |
default | \(\frac {2 a \left (-\frac {-i B -7 A}{8 a^{3} \sqrt {a +i a \tan \left (d x +c \right )}}-\frac {-i B -3 A}{12 a^{2} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {-i B -A}{10 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {\left (i B -A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{16 a^{\frac {7}{2}}}-\frac {A \,\operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{a^{\frac {7}{2}}}\right )}{d}\) | \(153\) |
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 644 vs. \(2 (145) = 290\).
Time = 0.27 (sec) , antiderivative size = 644, normalized size of antiderivative = 3.35 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx=-\frac {{\left (15 \, \sqrt {\frac {1}{2}} a^{3} d \sqrt {\frac {A^{2} - 2 i \, A B - B^{2}}{a^{5} d^{2}}} e^{\left (5 i \, d x + 5 i \, c\right )} \log \left (-\frac {4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{3} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {A^{2} - 2 i \, A B - B^{2}}{a^{5} d^{2}}} + {\left (-i \, A - B\right )} a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{i \, A + B}\right ) - 15 \, \sqrt {\frac {1}{2}} a^{3} d \sqrt {\frac {A^{2} - 2 i \, A B - B^{2}}{a^{5} d^{2}}} e^{\left (5 i \, d x + 5 i \, c\right )} \log \left (-\frac {4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (-i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{3} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {A^{2} - 2 i \, A B - B^{2}}{a^{5} d^{2}}} + {\left (-i \, A - B\right )} a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{i \, A + B}\right ) + 60 \, a^{3} d \sqrt {\frac {A^{2}}{a^{5} d^{2}}} e^{\left (5 i \, d x + 5 i \, c\right )} \log \left (\frac {16 \, {\left (3 \, A a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + A a^{2} + 2 \, \sqrt {2} {\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 )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {A^{2}}{a^{5} d^{2}}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{A}\right ) - 60 \, a^{3} d \sqrt {\frac {A^{2}}{a^{5} d^{2}}} e^{\left (5 i \, d x + 5 i \, c\right )} \log \left (\frac {16 \, {\left (3 \, A a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + A a^{2} - 2 \, \sqrt {2} {\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 )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {A^{2}}{a^{5} d^{2}}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{A}\right ) - \sqrt {2} {\left ({\left (123 \, A + 23 i \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + 2 \, {\left (72 \, A + 17 i \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, {\left (12 \, A + 7 i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 3 \, A + 3 i \, B\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-5 i \, d x - 5 i \, c\right )}}{120 \, a^{3} d} \]
[In]
[Out]
\[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx=\int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \cot {\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {5}{2}}}\, dx \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.97 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx=-\frac {\frac {15 \, \sqrt {2} {\left (A - i \, B\right )} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{a^{\frac {5}{2}}} - \frac {240 \, A \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}}} - \frac {4 \, {\left (15 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} {\left (7 \, A + i \, B\right )} + 10 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} {\left (3 \, A + i \, B\right )} a + 12 \, {\left (A + i \, B\right )} a^{2}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2}}}{240 \, d} \]
[In]
[Out]
\[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \cot \left (d x + c\right )}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Time = 7.83 (sec) , antiderivative size = 528, normalized size of antiderivative = 2.75 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^{5/2}} \, dx=\frac {\frac {A+B\,1{}\mathrm {i}}{5\,d}+\frac {\left (3\,A+B\,1{}\mathrm {i}\right )\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}{6\,a\,d}+\frac {\left (7\,A+B\,1{}\mathrm {i}\right )\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2}{4\,a^2\,d}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}}-\frac {2\,A\,\mathrm {atanh}\left (\frac {127\,A^3\,a\,d\,\sqrt {\frac {1}{a^3}}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{127\,d\,A^3+2{}\mathrm {i}\,d\,A^2\,B+d\,A\,B^2}+\frac {A\,B^2\,a\,d\,\sqrt {\frac {1}{a^3}}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{127\,d\,A^3+2{}\mathrm {i}\,d\,A^2\,B+d\,A\,B^2}+\frac {A^2\,B\,a\,d\,\sqrt {\frac {1}{a^3}}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,2{}\mathrm {i}}{127\,d\,A^3+2{}\mathrm {i}\,d\,A^2\,B+d\,A\,B^2}\right )\,\sqrt {\frac {1}{a^3}}}{a\,d}+\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,A^3\,a\,d\,\sqrt {-\frac {1}{a^3}}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,127{}\mathrm {i}}{32\,\left (\frac {127\,d\,A^3}{16}-\frac {125{}\mathrm {i}\,d\,A^2\,B}{16}+\frac {3\,d\,A\,B^2}{16}-\frac {1{}\mathrm {i}\,d\,B^3}{16}\right )}+\frac {\sqrt {2}\,B^3\,a\,d\,\sqrt {-\frac {1}{a^3}}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{32\,\left (\frac {127\,d\,A^3}{16}-\frac {125{}\mathrm {i}\,d\,A^2\,B}{16}+\frac {3\,d\,A\,B^2}{16}-\frac {1{}\mathrm {i}\,d\,B^3}{16}\right )}+\frac {\sqrt {2}\,A\,B^2\,a\,d\,\sqrt {-\frac {1}{a^3}}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,3{}\mathrm {i}}{32\,\left (\frac {127\,d\,A^3}{16}-\frac {125{}\mathrm {i}\,d\,A^2\,B}{16}+\frac {3\,d\,A\,B^2}{16}-\frac {1{}\mathrm {i}\,d\,B^3}{16}\right )}+\frac {125\,\sqrt {2}\,A^2\,B\,a\,d\,\sqrt {-\frac {1}{a^3}}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{32\,\left (\frac {127\,d\,A^3}{16}-\frac {125{}\mathrm {i}\,d\,A^2\,B}{16}+\frac {3\,d\,A\,B^2}{16}-\frac {1{}\mathrm {i}\,d\,B^3}{16}\right )}\right )\,\left (B+A\,1{}\mathrm {i}\right )\,\sqrt {-\frac {1}{a^3}}}{8\,a\,d} \]
[In]
[Out]