Integrand size = 38, antiderivative size = 194 \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx=\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (A-i B) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{a^{3/2} d}+\frac {A+i B}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}+\frac {11 A+5 i B}{6 a d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {(25 A+7 i B) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d \sqrt {\tan (c+d x)}} \]
[Out]
Time = 0.91 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {3677, 3679, 12, 3625, 211} \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx=\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (A-i B) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{a^{3/2} d}-\frac {(25 A+7 i B) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d \sqrt {\tan (c+d x)}}+\frac {A+i B}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}+\frac {11 A+5 i B}{6 a d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \]
[In]
[Out]
Rule 12
Rule 211
Rule 3625
Rule 3677
Rule 3679
Rubi steps \begin{align*} \text {integral}& = \frac {A+i B}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}+\frac {\int \frac {\frac {1}{2} a (7 A+i B)-2 a (i A-B) \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)}} \, dx}{3 a^2} \\ & = \frac {A+i B}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}+\frac {11 A+5 i B}{6 a d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {\int \frac {\sqrt {a+i a \tan (c+d x)} \left (\frac {1}{4} a^2 (25 A+7 i B)-\frac {1}{2} a^2 (11 i A-5 B) \tan (c+d x)\right )}{\tan ^{\frac {3}{2}}(c+d x)} \, dx}{3 a^4} \\ & = \frac {A+i B}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}+\frac {11 A+5 i B}{6 a d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {(25 A+7 i B) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d \sqrt {\tan (c+d x)}}+\frac {2 \int \frac {3 a^3 (i A+B) \sqrt {a+i a \tan (c+d x)}}{8 \sqrt {\tan (c+d x)}} \, dx}{3 a^5} \\ & = \frac {A+i B}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}+\frac {11 A+5 i B}{6 a d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {(25 A+7 i B) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d \sqrt {\tan (c+d x)}}+\frac {(i A+B) \int \frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {\tan (c+d x)}} \, dx}{4 a^2} \\ & = \frac {A+i B}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}+\frac {11 A+5 i B}{6 a d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {(25 A+7 i B) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d \sqrt {\tan (c+d x)}}+\frac {(A-i B) \text {Subst}\left (\int \frac {1}{-i a-2 a^2 x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{2 d} \\ & = \frac {\left (\frac {1}{4}+\frac {i}{4}\right ) (A-i B) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{a^{3/2} d}+\frac {A+i B}{3 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^{3/2}}+\frac {11 A+5 i B}{6 a d \sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {(25 A+7 i B) \sqrt {a+i a \tan (c+d x)}}{6 a^2 d \sqrt {\tan (c+d x)}} \\ \end{align*}
Time = 2.67 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.86 \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx=\frac {\frac {3 \sqrt {2} (i A+B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \tan (c+d x)}{\sqrt {i a \tan (c+d x)}}-\frac {2 i \left (-12 A+(-39 i A+9 B) \tan (c+d x)+(25 A+7 i B) \tan ^2(c+d x)\right )}{(-i+\tan (c+d x)) \sqrt {a+i a \tan (c+d x)}}}{12 a d \sqrt {\tan (c+d x)}} \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 922 vs. \(2 (156 ) = 312\).
Time = 0.14 (sec) , antiderivative size = 923, normalized size of antiderivative = 4.76
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(923\) |
default | \(\text {Expression too large to display}\) | \(923\) |
parts | \(\text {Expression too large to display}\) | \(967\) |
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 511 vs. \(2 (146) = 292\).
Time = 0.27 (sec) , antiderivative size = 511, normalized size of antiderivative = 2.63 \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx=\frac {3 \, \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{3} d^{2}}} \log \left (\frac {2 \, \sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{3} d^{2}}} e^{\left (i \, d x + i \, c\right )} + \sqrt {2} {\left ({\left (i \, A + B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, A + B\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{4 i \, A + 4 \, B}\right ) - 3 \, \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{3} d^{2}}} \log \left (-\frac {2 \, \sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{3} d^{2}}} e^{\left (i \, d x + i \, c\right )} - \sqrt {2} {\left ({\left (i \, A + B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, A + B\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{4 i \, A + 4 \, B}\right ) - \sqrt {2} {\left (2 \, {\left (19 i \, A - 4 \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} - {\left (-25 i \, A + B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, {\left (-7 i \, A + 4 \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, A + B\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{12 \, {\left (a^{2} d e^{\left (5 i \, d x + 5 i \, c\right )} - a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}\right )}} \]
[In]
[Out]
\[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {A + B \tan {\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}} \tan ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
Timed out. \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {A+B\,\mathrm {tan}\left (c+d\,x\right )}{{\mathrm {tan}\left (c+d\,x\right )}^{3/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]
[In]
[Out]