Integrand size = 35, antiderivative size = 314 \[ \int \frac {\sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x))}{\tan ^{\frac {9}{2}}(c+d x)} \, dx=-\frac {\sqrt {i a-b} (i A-B) \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {\sqrt {i a+b} (i A+B) \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^3 d \sqrt {\tan (c+d x)}} \]
[Out]
Time = 1.87 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3689, 3730, 3697, 3696, 95, 209, 212} \[ \int \frac {\sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x))}{\tan ^{\frac {9}{2}}(c+d x)} \, dx=\frac {2 \left (35 a^2 A-7 a b B+4 A b^2\right ) \sqrt {a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (105 a^3 B+35 a^2 A b+14 a b^2 B-8 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{105 a^3 d \sqrt {\tan (c+d x)}}-\frac {\sqrt {-b+i a} (-B+i A) \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {\sqrt {b+i a} (B+i A) \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 (7 a B+A b) \sqrt {a+b \tan (c+d x)}}{35 a d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)} \]
[In]
[Out]
Rule 95
Rule 209
Rule 212
Rule 3689
Rule 3696
Rule 3697
Rule 3730
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2}{7} \int \frac {\frac {1}{2} (-A b-7 a B)+\frac {7}{2} (a A-b B) \tan (c+d x)+3 A b \tan ^2(c+d x)}{\tan ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx \\ & = -\frac {2 A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {4 \int \frac {\frac {1}{4} \left (-35 a^2 A-4 A b^2+7 a b B\right )-\frac {35}{4} a (A b+a B) \tan (c+d x)-b (A b+7 a B) \tan ^2(c+d x)}{\tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx}{35 a} \\ & = -\frac {2 A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {8 \int \frac {\frac {1}{8} \left (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right )-\frac {105}{8} a^2 (a A-b B) \tan (c+d x)-\frac {1}{4} b \left (35 a^2 A+4 A b^2-7 a b B\right ) \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx}{105 a^2} \\ & = -\frac {2 A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^3 d \sqrt {\tan (c+d x)}}+\frac {16 \int \frac {\frac {105}{16} a^3 (a A-b B)+\frac {105}{16} a^3 (A b+a B) \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{105 a^3} \\ & = -\frac {2 A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^3 d \sqrt {\tan (c+d x)}}+\frac {1}{2} ((a-i b) (A-i B)) \int \frac {1+i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx+\frac {1}{2} ((a+i b) (A+i B)) \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx \\ & = -\frac {2 A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^3 d \sqrt {\tan (c+d x)}}+\frac {((a-i b) (A-i B)) \text {Subst}\left (\int \frac {1}{(1-i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {((a+i b) (A+i B)) \text {Subst}\left (\int \frac {1}{(1+i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d} \\ & = -\frac {2 A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^3 d \sqrt {\tan (c+d x)}}+\frac {((a-i b) (A-i B)) \text {Subst}\left (\int \frac {1}{1-(i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {((a+i b) (A+i B)) \text {Subst}\left (\int \frac {1}{1-(-i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d} \\ & = -\frac {\sqrt {i a-b} (i A-B) \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {\sqrt {i a+b} (i A+B) \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^3 d \sqrt {\tan (c+d x)}} \\ \end{align*}
Time = 4.42 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x))}{\tan ^{\frac {9}{2}}(c+d x)} \, dx=\frac {105 \sqrt [4]{-1} \sqrt {-a+i b} (i A+B) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )-105 (-1)^{3/4} \sqrt {a+i b} (A+i B) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+\frac {2 \sqrt {a+b \tan (c+d x)} \left (-15 a^3 A-3 a^2 (A b+7 a B) \tan (c+d x)+a \left (35 a^2 A+4 A b^2-7 a b B\right ) \tan ^2(c+d x)+\left (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right ) \tan ^3(c+d x)\right )}{a^3 \tan ^{\frac {7}{2}}(c+d x)}}{105 d} \]
[In]
[Out]
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 1.62 (sec) , antiderivative size = 2185304, normalized size of antiderivative = 6959.57
\[\text {output too large to display}\]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 8039 vs. \(2 (259) = 518\).
Time = 1.42 (sec) , antiderivative size = 8039, normalized size of antiderivative = 25.60 \[ \int \frac {\sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x))}{\tan ^{\frac {9}{2}}(c+d x)} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x))}{\tan ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {\sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x))}{\tan ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \sqrt {b \tan \left (d x + c\right ) + a}}{\tan \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x))}{\tan ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x))}{\tan ^{\frac {9}{2}}(c+d x)} \, dx=\int \frac {\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{{\mathrm {tan}\left (c+d\,x\right )}^{9/2}} \,d x \]
[In]
[Out]