Optimal. Leaf size=457 \[ \frac {\left (-5 a^2 B+40 a A b-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt {\cot (c+d x)}}+\frac {\left (-5 a^3 B+40 a^2 A b-112 a b^2 B-64 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{64 b d \sqrt {\cot (c+d x)}}+\frac {\left (-5 a^4 B+40 a^3 A b-240 a^2 b^2 B-320 a A b^3+128 b^4 B\right ) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{64 b^{3/2} d}-\frac {(-b+i a)^{5/2} (-B+i A) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}-\frac {(b+i a)^{5/2} (B+i A) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}} \]
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Rubi [A] time = 3.01, antiderivative size = 457, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {4241, 3607, 3647, 3655, 6725, 63, 217, 206, 93, 205, 208} \[ \frac {\left (-5 a^2 B+40 a A b-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt {\cot (c+d x)}}+\frac {\left (40 a^2 A b-5 a^3 B-112 a b^2 B-64 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{64 b d \sqrt {\cot (c+d x)}}+\frac {\left (40 a^3 A b-240 a^2 b^2 B-5 a^4 B-320 a A b^3+128 b^4 B\right ) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{64 b^{3/2} d}-\frac {(-b+i a)^{5/2} (-B+i A) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}-\frac {(b+i a)^{5/2} (B+i A) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 205
Rule 206
Rule 208
Rule 217
Rule 3607
Rule 3647
Rule 3655
Rule 4241
Rule 6725
Rubi steps
\begin {align*} \int \frac {(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx &=\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\\ &=\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {(a+b \tan (c+d x))^{5/2} \left (-\frac {a B}{2}-4 b B \tan (c+d x)+\frac {1}{2} (8 A b-a B) \tan ^2(c+d x)\right )}{\sqrt {\tan (c+d x)}} \, dx}{4 b}\\ &=\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {(a+b \tan (c+d x))^{3/2} \left (-\frac {1}{4} a (8 A b+5 a B)-12 b (A b+a B) \tan (c+d x)+\frac {1}{4} \left (40 a A b-5 a^2 B-48 b^2 B\right ) \tan ^2(c+d x)\right )}{\sqrt {\tan (c+d x)}} \, dx}{12 b}\\ &=\frac {\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt {\cot (c+d x)}}+\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\sqrt {a+b \tan (c+d x)} \left (-\frac {3}{8} a \left (24 a A b+5 a^2 B-16 b^2 B\right )-24 b \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)+\frac {3}{8} \left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \tan ^2(c+d x)\right )}{\sqrt {\tan (c+d x)}} \, dx}{24 b}\\ &=\frac {\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{64 b d \sqrt {\cot (c+d x)}}+\frac {\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt {\cot (c+d x)}}+\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {-\frac {3}{16} a \left (88 a^2 A b-64 A b^3+5 a^3 B-144 a b^2 B\right )-24 b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)+\frac {3}{16} \left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{24 b}\\ &=\frac {\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{64 b d \sqrt {\cot (c+d x)}}+\frac {\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt {\cot (c+d x)}}+\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {-\frac {3}{16} a \left (88 a^2 A b-64 A b^3+5 a^3 B-144 a b^2 B\right )-24 b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) x+\frac {3}{16} \left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right ) x^2}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{24 b d}\\ &=\frac {\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{64 b d \sqrt {\cot (c+d x)}}+\frac {\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt {\cot (c+d x)}}+\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \left (\frac {3 \left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right )}{16 \sqrt {x} \sqrt {a+b x}}-\frac {24 \left (b \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )+b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) x\right )}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{24 b d}\\ &=\frac {\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{64 b d \sqrt {\cot (c+d x)}}+\frac {\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt {\cot (c+d x)}}+\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}}-\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {b \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )+b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) x}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{b d}+\frac {\left (\left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{128 b d}\\ &=\frac {\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{64 b d \sqrt {\cot (c+d x)}}+\frac {\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt {\cot (c+d x)}}+\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}}-\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \left (\frac {-b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )+i b \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )}{2 (i-x) \sqrt {x} \sqrt {a+b x}}+\frac {b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )+i b \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )}{2 \sqrt {x} (i+x) \sqrt {a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{b d}+\frac {\left (\left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {\tan (c+d x)}\right )}{64 b d}\\ &=\frac {\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{64 b d \sqrt {\cot (c+d x)}}+\frac {\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt {\cot (c+d x)}}+\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}}+\frac {\left ((i a+b)^3 (A-i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} (i+x) \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {\left (\left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{64 b d}-\frac {\left (\left (-b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )+i b \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}{(i-x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 b d}\\ &=\frac {\left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{64 b^{3/2} d}+\frac {\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{64 b d \sqrt {\cot (c+d x)}}+\frac {\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt {\cot (c+d x)}}+\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}}+\frac {\left ((i a+b)^3 (A-i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{i-(-a+i b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {\left (\left (-b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )+i b \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}{i-(a+i b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{b d}\\ &=-\frac {(i a-b)^{5/2} (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)}}{d}+\frac {\left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{64 b^{3/2} d}-\frac {(i a+b)^{5/2} (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 {\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{64 b d \sqrt {\cot (c+d x)}}+\frac {\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt {\cot (c+d x)}}+\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 5.58, size = 431, normalized size = 0.94 \[ \frac {\sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-2 \left (5 a^2 B-40 a A b+48 b^2 B\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}-3 \left (5 a^3 B-40 a^2 A b+112 a b^2 B+64 A b^3\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}-\frac {3 \sqrt {a} \left (5 a^4 B-40 a^3 A b+240 a^2 b^2 B+320 a A b^3-128 b^4 B\right ) \sqrt {\frac {b \tan (c+d x)}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {b} \sqrt {a+b \tan (c+d x)}}-192 \sqrt [4]{-1} b (-a+i b)^{5/2} (B+i A) \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+192 (-1)^{3/4} b (a+i b)^{5/2} (A+i B) \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+8 (8 A b-a B) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}+48 B \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}\right )}{192 b d} \]
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 = 6.78, size = 39803, normalized size = 87.10 \[ \text {output too large to display} \]
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 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cot \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2}}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}} \,d x \]
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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