Optimal. Leaf size=341 \[ \frac {(i A-B) \text {ArcTan}\left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{(i a-b)^{5/2} d}-\frac {(i A+B) \tanh ^{-1}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{(i a+b)^{5/2} d}-\frac {2 b \left (3 a^2 A+4 A b^2-a b B\right )}{3 a^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac {2 A \sqrt {\cot (c+d x)}}{a d (a+b \tan (c+d x))^{3/2}}-\frac {2 b \left (3 a^4 A+17 a^2 A b^2+8 A b^4-8 a^3 b B-2 a b^3 B\right )}{3 a^3 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}} \]
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Rubi [A]
time = 0.87, antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {4326, 3690,
3730, 3697, 3696, 95, 209, 212} \begin {gather*} -\frac {2 b \left (3 a^2 A-a b B+4 A b^2\right )}{3 a^2 d \left (a^2+b^2\right ) \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac {2 b \left (3 a^4 A-8 a^3 b B+17 a^2 A b^2-2 a b^3 B+8 A b^4\right )}{3 a^3 d \left (a^2+b^2\right )^2 \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}+\frac {(-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 (-b+i a)^{5/2}}-\frac {(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 (b+i a)^{5/2}}-\frac {2 A \sqrt {\cot (c+d x)}}{a d (a+b \tan (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 209
Rule 212
Rule 3690
Rule 3696
Rule 3697
Rule 3730
Rule 4326
Rubi steps
\begin {align*} \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx &=\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2}} \, dx\\ &=-\frac {2 A \sqrt {\cot (c+d x)}}{a d (a+b \tan (c+d x))^{3/2}}-\frac {\left (2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\frac {1}{2} (4 A b-a B)+\frac {1}{2} a A \tan (c+d x)+2 A b \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}} \, dx}{a}\\ &=-\frac {2 b \left (3 a^2 A+4 A b^2-a b B\right )}{3 a^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac {2 A \sqrt {\cot (c+d x)}}{a d (a+b \tan (c+d x))^{3/2}}-\frac {\left (4 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\frac {1}{4} \left (9 a^2 A b+8 A b^3-3 a^3 B-2 a b^2 B\right )+\frac {3}{4} a^2 (a A+b B) \tan (c+d x)+\frac {1}{2} b \left (3 a^2 A+4 A b^2-a b B\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx}{3 a^2 \left (a^2+b^2\right )}\\ &=-\frac {2 b \left (3 a^2 A+4 A b^2-a b B\right )}{3 a^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac {2 A \sqrt {\cot (c+d x)}}{a d (a+b \tan (c+d x))^{3/2}}-\frac {2 b \left (3 a^4 A+17 a^2 A b^2+8 A b^4-8 a^3 b B-2 a b^3 B\right )}{3 a^3 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}-\frac {\left (8 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\frac {3}{8} a^3 \left (2 a A b-a^2 B+b^2 B\right )+\frac {3}{8} a^3 \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}{3 a^3 \left (a^2+b^2\right )^2}\\ &=-\frac {2 b \left (3 a^2 A+4 A b^2-a b B\right )}{3 a^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac {2 A \sqrt {\cot (c+d x)}}{a d (a+b \tan (c+d x))^{3/2}}-\frac {2 b \left (3 a^4 A+17 a^2 A b^2+8 A b^4-8 a^3 b B-2 a b^3 B\right )}{3 a^3 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}-\frac {\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}{2 \left (a^2+b^2\right )^2}+\frac {\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}{2 \left (a^2+b^2\right )^2}\\ &=-\frac {2 b \left (3 a^2 A+4 A b^2-a b B\right )}{3 a^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac {2 A \sqrt {\cot (c+d x)}}{a d (a+b \tan (c+d x))^{3/2}}-\frac {2 b \left (3 a^4 A+17 a^2 A b^2+8 A b^4-8 a^3 b B-2 a b^3 B\right )}{3 a^3 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}-\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 \left (a^2+b^2\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 \left (a^2+b^2\right )^2 d}\\ &=-\frac {2 b \left (3 a^2 A+4 A b^2-a b B\right )}{3 a^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac {2 A \sqrt {\cot (c+d x)}}{a d (a+b \tan (c+d x))^{3/2}}-\frac {2 b \left (3 a^4 A+17 a^2 A b^2+8 A b^4-8 a^3 b B-2 a b^3 B\right )}{3 a^3 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}-\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 )}{\left (a^2+b^2\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 a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\left (a^2+b^2\right )^2 d}\\ &=-\frac {(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} (a+i b)^2 d}-\frac {(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)}}{(i a+b)^{5/2} d}-\frac {2 b \left (3 a^2 A+4 A b^2-a b B\right )}{3 a^2 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac {2 A \sqrt {\cot (c+d x)}}{a d (a+b \tan (c+d x))^{3/2}}-\frac {2 b \left (3 a^4 A+17 a^2 A b^2+8 A b^4-8 a^3 b B-2 a b^3 B\right )}{3 a^3 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 2.69, size = 334, normalized size = 0.98 \begin {gather*} \frac {\sqrt {\cot (c+d x)} \left (-\frac {6 A}{(a+b \tan (c+d x))^{3/2}}-\frac {2 b \left (3 a^2 A+4 A b^2-a b B\right ) \tan (c+d x)}{a \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {\sqrt {\tan (c+d x)} \left (3 \sqrt [4]{-1} a^3 \left (\frac {(a+i b)^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 )}{\sqrt {-a+i b}}-\frac {(a-i b)^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 )}{\sqrt {a+i b}}\right )-\frac {2 b \left (3 a^4 A+17 a^2 A b^2+8 A b^4-8 a^3 b B-2 a b^3 B\right ) \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{a^2 \left (a^2+b^2\right )^2}\right )}{3 a 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 = 31.54, size = 54581, normalized size = 160.06
method | result | size |
default | \(\text {Expression too large to display}\) | \(54581\) |
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 \frac {{\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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