Optimal. Leaf size=418 \[ \frac {2 \left (21 a^2 A-45 a b B-25 A b^2\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 d}+\frac {2 \left (105 a^3 B+231 a^2 A b-135 a b^2 B-5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a d}-\frac {2 \left (315 a^4 A-735 a^3 b B-483 a^2 A b^2+45 a b^3 B-10 A b^4\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{315 a^2 d}-\frac {2 a (3 a B+4 A b) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{21 d}+\frac {(-b+i a)^{5/2} (A+i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {(b+i a)^{5/2} (A-i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{9 d} \]
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Rubi [A] time = 2.03, antiderivative size = 418, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {4241, 3605, 3645, 3649, 3616, 3615, 93, 203, 206} \[ \frac {2 \left (21 a^2 A-45 a b B-25 A b^2\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 d}+\frac {2 \left (231 a^2 A b+105 a^3 B-135 a b^2 B-5 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a d}-\frac {2 \left (-483 a^2 A b^2+315 a^4 A-735 a^3 b B+45 a b^3 B-10 A b^4\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{315 a^2 d}-\frac {2 a (3 a B+4 A b) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{21 d}+\frac {(-b+i a)^{5/2} (A+i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {(b+i a)^{5/2} (A-i B) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{9 d} \]
Antiderivative was successfully verified.
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Rule 93
Rule 203
Rule 206
Rule 3605
Rule 3615
Rule 3616
Rule 3645
Rule 3649
Rule 4241
Rubi steps
\begin {align*} \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx &=\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\tan ^{\frac {11}{2}}(c+d x)} \, dx\\ &=-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{9 d}+\frac {1}{9} \left (2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\sqrt {a+b \tan (c+d x)} \left (\frac {3}{2} a (4 A b+3 a B)-\frac {9}{2} \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)-\frac {3}{2} b (2 a A-3 b B) \tan ^2(c+d x)\right )}{\tan ^{\frac {9}{2}}(c+d x)} \, dx\\ &=-\frac {2 a (4 A b+3 a B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{21 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{9 d}+\frac {1}{63} \left (4 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {-\frac {3}{4} a \left (21 a^2 A-25 A b^2-45 a b B\right )-\frac {63}{4} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)-\frac {3}{4} b \left (38 a A b+18 a^2 B-21 b^2 B\right ) \tan ^2(c+d x)}{\tan ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx\\ &=\frac {2 \left (21 a^2 A-25 A b^2-45 a b B\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 d}-\frac {2 a (4 A b+3 a B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{21 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{9 d}-\frac {\left (8 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\frac {3}{8} a \left (231 a^2 A b-5 A b^3+105 a^3 B-135 a b^2 B\right )-\frac {315}{8} a \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \tan (c+d x)-\frac {3}{2} a b \left (21 a^2 A-25 A b^2-45 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}\\ &=\frac {2 \left (231 a^2 A b-5 A b^3+105 a^3 B-135 a b^2 B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a d}+\frac {2 \left (21 a^2 A-25 A b^2-45 a b B\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 d}-\frac {2 a (4 A b+3 a B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{21 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{9 d}+\frac {\left (16 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\frac {3}{16} a \left (315 a^4 A-483 a^2 A b^2-10 A b^4-735 a^3 b B+45 a b^3 B\right )+\frac {945}{16} a^2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)+\frac {3}{8} a b \left (231 a^2 A b-5 A b^3+105 a^3 B-135 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^2}\\ &=-\frac {2 \left (315 a^4 A-483 a^2 A b^2-10 A b^4-735 a^3 b B+45 a b^3 B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{315 a^2 d}+\frac {2 \left (231 a^2 A b-5 A b^3+105 a^3 B-135 a b^2 B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a d}+\frac {2 \left (21 a^2 A-25 A b^2-45 a b B\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 d}-\frac {2 a (4 A b+3 a B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{21 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{9 d}-\frac {\left (32 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {-\frac {945}{32} a^3 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )+\frac {945}{32} a^3 \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{945 a^3}\\ &=-\frac {2 \left (315 a^4 A-483 a^2 A b^2-10 A b^4-735 a^3 b B+45 a b^3 B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{315 a^2 d}+\frac {2 \left (231 a^2 A b-5 A b^3+105 a^3 B-135 a b^2 B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a d}+\frac {2 \left (21 a^2 A-25 A b^2-45 a b B\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 d}-\frac {2 a (4 A b+3 a B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{21 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{9 d}+\frac {1}{2} \left ((a-i b)^3 (i A+B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {1+i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx-\frac {\left (16 \left (-\frac {945}{32} a^3 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )+\frac {945}{32} i a^3 \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{945 a^3}\\ &=-\frac {2 \left (315 a^4 A-483 a^2 A b^2-10 A b^4-735 a^3 b B+45 a b^3 B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{315 a^2 d}+\frac {2 \left (231 a^2 A b-5 A b^3+105 a^3 B-135 a b^2 B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a d}+\frac {2 \left (21 a^2 A-25 A b^2-45 a b B\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 d}-\frac {2 a (4 A b+3 a B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{21 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{9 d}+\frac {\left ((a-i b)^3 (i A+B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {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 (16 \left (-\frac {945}{32} a^3 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )+\frac {945}{32} i a^3 \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{(1+i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{945 a^3 d}\\ &=-\frac {2 \left (315 a^4 A-483 a^2 A b^2-10 A b^4-735 a^3 b B+45 a b^3 B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{315 a^2 d}+\frac {2 \left (231 a^2 A b-5 A b^3+105 a^3 B-135 a b^2 B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a d}+\frac {2 \left (21 a^2 A-25 A b^2-45 a b B\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 d}-\frac {2 a (4 A b+3 a B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{21 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{9 d}+\frac {\left ((a-i b)^3 (i A+B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {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 (32 \left (-\frac {945}{32} a^3 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )+\frac {945}{32} i a^3 \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {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 )}{945 a^3 d}\\ &=\frac {(i a-b)^{5/2} (A+i B) \tan ^{-1}\left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {(i a+b)^{5/2} (A-i B) \tanh ^{-1}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {2 \left (315 a^4 A-483 a^2 A b^2-10 A b^4-735 a^3 b B+45 a b^3 B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{315 a^2 d}+\frac {2 \left (231 a^2 A b-5 A b^3+105 a^3 B-135 a b^2 B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a d}+\frac {2 \left (21 a^2 A-25 A b^2-45 a b B\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 d}-\frac {2 a (4 A b+3 a B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{21 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{9 d}\\ \end {align*}
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Mathematica [A] time = 6.82, size = 564, normalized size = 1.35 \[ \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {b B (a+b \tan (c+d x))^{3/2}}{3 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {1}{3} \left (-\frac {3 b (a B+2 A b) \sqrt {a+b \tan (c+d x)}}{8 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {1}{4} \left (-\frac {\left (16 a^2 A-33 a b B-18 A b^2\right ) \sqrt {a+b \tan (c+d x)}}{6 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2 \left (\frac {6 a \left (18 a^2 B+38 a A b-21 b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 \left (\frac {18 a^2 \left (21 a^2 A-45 a b B-25 A b^2\right ) \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 \left (-\frac {3 a^2 \left (105 a^3 B+231 a^2 A b-135 a b^2 B-5 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (-\frac {9 a^2 \left (315 a^4 A-735 a^3 b B-483 a^2 A b^2+45 a b^3 B-10 A b^4\right ) \sqrt {a+b \tan (c+d x)}}{2 d \sqrt {\tan (c+d x)}}-\frac {2835 a^4 \left (\sqrt [4]{-1} (-a+i b)^{5/2} (A-i B) \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+\sqrt [4]{-1} (a+i b)^{5/2} (A+i B) \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )\right )}{4 d}\right )}{3 a}\right )}{5 a}\right )}{7 a}\right )}{9 a}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 6.24, size = 101072, normalized size = 241.80 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{\frac {11}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {cot}\left (c+d\,x\right )}^{11/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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