Integrand size = 35, antiderivative size = 382 \[ \int \frac {(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac {11}{2}}(c+d x)} \, dx=\frac {(i a-b)^{3/2} (i A-B) \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {(i a+b)^{3/2} (i A+B) \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2 (10 A b+9 a B) \sqrt {a+b \tan (c+d x)}}{63 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (21 a^2 A-A b^2-24 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{315 a^2 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (315 a^4 A-63 a^2 A b^2+8 A b^4-420 a^3 b B-18 a b^3 B\right ) \sqrt {a+b \tan (c+d x)}}{315 a^3 d \sqrt {\tan (c+d x)}} \]
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Time = 2.60 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3686, 3730, 3697, 3696, 95, 209, 212} \[ \int \frac {(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac {11}{2}}(c+d x)} \, dx=\frac {2 \left (21 a^2 A-24 a b B-A b^2\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (105 a^3 B+126 a^2 A b-9 a b^2 B+4 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{315 a^2 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (315 a^4 A-420 a^3 b B-63 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \tan (c+d x)}}{315 a^3 d \sqrt {\tan (c+d x)}}+\frac {(-b+i a)^{3/2} (-B+i A) \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {(b+i a)^{3/2} (B+i A) \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 (9 a B+10 A b) \sqrt {a+b \tan (c+d x)}}{63 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)} \]
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Rule 95
Rule 209
Rule 212
Rule 3686
Rule 3696
Rule 3697
Rule 3730
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {2}{9} \int \frac {\frac {1}{2} a (10 A b+9 a B)-\frac {9}{2} \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)-\frac {1}{2} b (8 a A-9 b B) \tan ^2(c+d x)}{\tan ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx \\ & = -\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2 (10 A b+9 a B) \sqrt {a+b \tan (c+d x)}}{63 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {4 \int \frac {\frac {3}{4} a \left (21 a^2 A-A b^2-24 a b B\right )+\frac {63}{4} a \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)+\frac {3}{2} a b (10 A b+9 a B) \tan ^2(c+d x)}{\tan ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx}{63 a} \\ & = -\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2 (10 A b+9 a B) \sqrt {a+b \tan (c+d x)}}{63 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (21 a^2 A-A b^2-24 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {8 \int \frac {-\frac {3}{8} a \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right )+\frac {315}{8} a^2 \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)+\frac {3}{2} a b \left (21 a^2 A-A b^2-24 a b B\right ) \tan ^2(c+d x)}{\tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx}{315 a^2} \\ & = -\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2 (10 A b+9 a B) \sqrt {a+b \tan (c+d x)}}{63 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (21 a^2 A-A b^2-24 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{315 a^2 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {16 \int \frac {-\frac {3}{16} a \left (315 a^4 A-63 a^2 A b^2+8 A b^4-420 a^3 b B-18 a b^3 B\right )-\frac {945}{16} a^3 \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)-\frac {3}{8} a b \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx}{945 a^3} \\ & = -\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2 (10 A b+9 a B) \sqrt {a+b \tan (c+d x)}}{63 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (21 a^2 A-A b^2-24 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{315 a^2 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (315 a^4 A-63 a^2 A b^2+8 A b^4-420 a^3 b B-18 a b^3 B\right ) \sqrt {a+b \tan (c+d x)}}{315 a^3 d \sqrt {\tan (c+d x)}}+\frac {32 \int \frac {\frac {945}{32} a^4 \left (2 a A b+a^2 B-b^2 B\right )-\frac {945}{32} a^4 \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{945 a^4} \\ & = -\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2 (10 A b+9 a B) \sqrt {a+b \tan (c+d x)}}{63 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (21 a^2 A-A b^2-24 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{315 a^2 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (315 a^4 A-63 a^2 A b^2+8 A b^4-420 a^3 b B-18 a b^3 B\right ) \sqrt {a+b \tan (c+d x)}}{315 a^3 d \sqrt {\tan (c+d x)}}-\frac {1}{2} \left ((a+i b)^2 (i A-B)\right ) \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx+\frac {1}{2} \left ((a-i b)^2 (i A+B)\right ) \int \frac {1+i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx \\ & = -\frac {2 a A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2 (10 A b+9 a B) \sqrt {a+b \tan (c+d x)}}{63 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (21 a^2 A-A b^2-24 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{315 a^2 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (315 a^4 A-63 a^2 A b^2+8 A b^4-420 a^3 b B-18 a b^3 B\right ) \sqrt {a+b \tan (c+d x)}}{315 a^3 d \sqrt {\tan (c+d x)}}-\frac {\left ((a+i b)^2 (i A-B)\right ) \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 {\left ((a-i b)^2 (i A+B)\right ) \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 A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2 (10 A b+9 a B) \sqrt {a+b \tan (c+d x)}}{63 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (21 a^2 A-A b^2-24 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{315 a^2 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (315 a^4 A-63 a^2 A b^2+8 A b^4-420 a^3 b B-18 a b^3 B\right ) \sqrt {a+b \tan (c+d x)}}{315 a^3 d \sqrt {\tan (c+d x)}}-\frac {\left ((a+i b)^2 (i A-B)\right ) \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 {\left ((a-i b)^2 (i A+B)\right ) \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)^2 (i A-B) \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {i a-b} d}-\frac {(i a+b)^{3/2} (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 A \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2 (10 A b+9 a B) \sqrt {a+b \tan (c+d x)}}{63 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (21 a^2 A-A b^2-24 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{315 a^2 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (315 a^4 A-63 a^2 A b^2+8 A b^4-420 a^3 b B-18 a b^3 B\right ) \sqrt {a+b \tan (c+d x)}}{315 a^3 d \sqrt {\tan (c+d x)}} \\ \end{align*}
Time = 6.78 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.24 \[ \int \frac {(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac {11}{2}}(c+d x)} \, dx=-\frac {b B \sqrt {a+b \tan (c+d x)}}{4 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {1}{4} \left (-\frac {(8 a A-9 b B) \sqrt {a+b \tan (c+d x)}}{9 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (-\frac {4 a (10 A b+9 a B) \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 \left (-\frac {6 a \left (21 a^2 A-A b^2-24 a b B\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 \left (\frac {a \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (-\frac {945 a^4 \left (\sqrt [4]{-1} (-a+i b)^{3/2} (A-i B) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )-\sqrt [4]{-1} (a+i b)^{3/2} (A+i B) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )\right )}{4 d}+\frac {3 a \left (315 a^4 A-63 a^2 A b^2+8 A b^4-420 a^3 b B-18 a b^3 B\right ) \sqrt {a+b \tan (c+d x)}}{2 d \sqrt {\tan (c+d x)}}\right )}{3 a}\right )}{5 a}\right )}{7 a}\right )}{9 a}\right ) \]
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result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 0.90 (sec) , antiderivative size = 2403057, normalized size of antiderivative = 6290.73
\[\text {output too large to display}\]
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Leaf count of result is larger than twice the leaf count of optimal. 12202 vs. \(2 (324) = 648\).
Time = 2.43 (sec) , antiderivative size = 12202, normalized size of antiderivative = 31.94 \[ \int \frac {(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac {11}{2}}(c+d x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac {11}{2}}(c+d x)} \, dx=\int \frac {\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}}{{\mathrm {tan}\left (c+d\,x\right )}^{11/2}} \,d x \]
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