Optimal. Leaf size=422 \[ \frac {(i a-b)^{3/2} (i A-B) \text {ArcTan}\left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {(i a+b)^{3/2} (i A+B) \tanh ^{-1}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {2 \left (315 a^4 A-63 a^2 A b^2+8 A b^4-420 a^3 b B-18 a b^3 B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{315 a^3 d}+\frac {2 \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a^2 d}+\frac {2 \left (21 a^2 A-A b^2-24 a b B\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}-\frac {2 (10 A b+9 a B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{63 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{9 d} \]
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Rubi [A]
time = 1.32, antiderivative size = 422, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {4326, 3686,
3730, 3697, 3696, 95, 209, 212} \begin {gather*} \frac {2 \left (21 a^2 A-24 a b B-A b^2\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}+\frac {2 \left (105 a^3 B+126 a^2 A b-9 a b^2 B+4 A b^3\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a^2 d}-\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 {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{315 a^3 d}+\frac {(-b+i a)^{3/2} (-B+i A) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {ArcTan}\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) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{63 d}-\frac {(b+i a)^{3/2} (B+i A) \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) \sqrt {a+b \tan (c+d x)}}{9 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 209
Rule 212
Rule 3686
Rule 3696
Rule 3697
Rule 3730
Rule 4326
Rubi steps
\begin {align*} \int \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (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))^{3/2} (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) \sqrt {a+b \tan (c+d x)}}{9 d}+\frac {1}{9} \left (2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \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 (10 A b+9 a B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{63 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{9 d}-\frac {\left (4 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \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 \left (21 a^2 A-A b^2-24 a b B\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}-\frac {2 (10 A b+9 a B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{63 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{9 d}+\frac {\left (8 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \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 \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a^2 d}+\frac {2 \left (21 a^2 A-A b^2-24 a b B\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}-\frac {2 (10 A b+9 a B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{63 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{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-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 \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 {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{315 a^3 d}+\frac {2 \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a^2 d}+\frac {2 \left (21 a^2 A-A b^2-24 a b B\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}-\frac {2 (10 A b+9 a B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{63 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{9 d}+\frac {\left (32 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \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 \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 {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{315 a^3 d}+\frac {2 \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a^2 d}+\frac {2 \left (21 a^2 A-A b^2-24 a b B\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}-\frac {2 (10 A b+9 a B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{63 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{9 d}-\frac {1}{2} \left ((a+i b)^2 (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 {1}{2} \left ((a-i b)^2 (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 {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 {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{315 a^3 d}+\frac {2 \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a^2 d}+\frac {2 \left (21 a^2 A-A b^2-24 a b B\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}-\frac {2 (10 A b+9 a B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{63 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{9 d}-\frac {\left ((a+i b)^2 (i A-B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\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) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\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 \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 {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{315 a^3 d}+\frac {2 \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a^2 d}+\frac {2 \left (21 a^2 A-A b^2-24 a b B\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}-\frac {2 (10 A b+9 a B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{63 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{9 d}-\frac {\left ((a+i b)^2 (i A-B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\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) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\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) \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)}}{\sqrt {i a-b} d}-\frac {(i a+b)^{3/2} (i A+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-63 a^2 A b^2+8 A b^4-420 a^3 b B-18 a b^3 B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{315 a^3 d}+\frac {2 \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{315 a^2 d}+\frac {2 \left (21 a^2 A-A b^2-24 a b B\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{105 a d}-\frac {2 (10 A b+9 a B) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{63 d}-\frac {2 a A \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{9 d}\\ \end {align*}
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Mathematica [A]
time = 6.00, size = 411, normalized size = 0.97 \begin {gather*} -\frac {\cot ^{\frac {9}{2}}(c+d x) \left (315 a^4 b B \sqrt {a+b \tan (c+d x)}+35 a^4 (8 a A-9 b B) \sqrt {a+b \tan (c+d x)}+40 a^4 (10 A b+9 a B) \tan (c+d x) \sqrt {a+b \tan (c+d x)}-4 \tan ^2(c+d x) \left (6 a^3 \left (21 a^2 A-A b^2-24 a b B\right ) \sqrt {a+b \tan (c+d x)}+a \tan (c+d x) \left (315 \sqrt [4]{-1} a^3 \left ((-a+i b)^{3/2} (A-i B) \text {ArcTan}\left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )-(a+i b)^{3/2} (A+i B) \text {ArcTan}\left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )\right ) \tan ^{\frac {3}{2}}(c+d x)+2 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)}-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 ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}\right )\right )\right )}{1260 a^4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 31.51, size = 74462, normalized size = 176.45
method | result | size |
default | \(\text {Expression too large to display}\) | \(74462\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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