Optimal. Leaf size=337 \[ \frac {\left (11 a^2-8 b^2\right ) \sqrt {a+b \tan (c+d x)}}{8 d \sqrt {\cot (c+d x)}}+\frac {5 a \left (a^2-8 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{8 \sqrt {b} d}+\frac {b^2 \sqrt {a+b \tan (c+d x)}}{3 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {13 a b \sqrt {a+b \tan (c+d x)}}{12 d \cot ^{\frac {3}{2}}(c+d x)}-\frac {i (-b+i a)^{5/2} \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {i (b+i a)^{5/2} \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d} \]
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Rubi [A] time = 2.13, antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {4241, 3566, 3647, 3655, 6725, 63, 217, 206, 93, 205, 208} \[ \frac {\left (11 a^2-8 b^2\right ) \sqrt {a+b \tan (c+d x)}}{8 d \sqrt {\cot (c+d x)}}+\frac {5 a \left (a^2-8 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{8 \sqrt {b} d}+\frac {b^2 \sqrt {a+b \tan (c+d x)}}{3 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {13 a b \sqrt {a+b \tan (c+d x)}}{12 d \cot ^{\frac {3}{2}}(c+d x)}-\frac {i (-b+i a)^{5/2} \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {i (b+i a)^{5/2} \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 205
Rule 206
Rule 208
Rule 217
Rule 3566
Rule 3647
Rule 3655
Rule 4241
Rule 6725
Rubi steps
\begin {align*} \int \frac {(a+b \tan (c+d x))^{5/2}}{\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} \, dx\\ &=\frac {b^2 \sqrt {a+b \tan (c+d x)}}{3 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {1}{3} \left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {1}{2} a \left (6 a^2-5 b^2\right )+3 b \left (3 a^2-b^2\right ) \tan (c+d x)+\frac {13}{2} a b^2 \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx\\ &=\frac {b^2 \sqrt {a+b \tan (c+d x)}}{3 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {13 a b \sqrt {a+b \tan (c+d x)}}{12 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\sqrt {\tan (c+d x)} \left (-\frac {39}{4} a^2 b^2+6 a b \left (a^2-3 b^2\right ) \tan (c+d x)+\frac {3}{4} b^2 \left (11 a^2-8 b^2\right ) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{6 b}\\ &=\frac {b^2 \sqrt {a+b \tan (c+d x)}}{3 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {13 a b \sqrt {a+b \tan (c+d x)}}{12 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {\left (11 a^2-8 b^2\right ) \sqrt {a+b \tan (c+d x)}}{8 d \sqrt {\cot (c+d x)}}+\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {-\frac {3}{8} a b^2 \left (11 a^2-8 b^2\right )-6 b^3 \left (3 a^2-b^2\right ) \tan (c+d x)+\frac {15}{8} a b^2 \left (a^2-8 b^2\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{6 b^2}\\ &=\frac {b^2 \sqrt {a+b \tan (c+d x)}}{3 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {13 a b \sqrt {a+b \tan (c+d x)}}{12 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {\left (11 a^2-8 b^2\right ) \sqrt {a+b \tan (c+d x)}}{8 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}{8} a b^2 \left (11 a^2-8 b^2\right )-6 b^3 \left (3 a^2-b^2\right ) x+\frac {15}{8} a b^2 \left (a^2-8 b^2\right ) x^2}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{6 b^2 d}\\ &=\frac {b^2 \sqrt {a+b \tan (c+d x)}}{3 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {13 a b \sqrt {a+b \tan (c+d x)}}{12 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {\left (11 a^2-8 b^2\right ) \sqrt {a+b \tan (c+d x)}}{8 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 {15 a b^2 \left (a^2-8 b^2\right )}{8 \sqrt {x} \sqrt {a+b x}}-\frac {6 \left (a b^2 \left (a^2-3 b^2\right )+b^3 \left (3 a^2-b^2\right ) x\right )}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{6 b^2 d}\\ &=\frac {b^2 \sqrt {a+b \tan (c+d x)}}{3 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {13 a b \sqrt {a+b \tan (c+d x)}}{12 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {\left (11 a^2-8 b^2\right ) \sqrt {a+b \tan (c+d x)}}{8 d \sqrt {\cot (c+d x)}}-\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {a b^2 \left (a^2-3 b^2\right )+b^3 \left (3 a^2-b^2\right ) x}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{b^2 d}+\frac {\left (5 a \left (a^2-8 b^2\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 )}{16 d}\\ &=\frac {b^2 \sqrt {a+b \tan (c+d x)}}{3 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {13 a b \sqrt {a+b \tan (c+d x)}}{12 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {\left (11 a^2-8 b^2\right ) \sqrt {a+b \tan (c+d x)}}{8 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 {i a b^2 \left (a^2-3 b^2\right )-b^3 \left (3 a^2-b^2\right )}{2 (i-x) \sqrt {x} \sqrt {a+b x}}+\frac {i a b^2 \left (a^2-3 b^2\right )+b^3 \left (3 a^2-b^2\right )}{2 \sqrt {x} (i+x) \sqrt {a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{b^2 d}+\frac {\left (5 a \left (a^2-8 b^2\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 )}{8 d}\\ &=\frac {b^2 \sqrt {a+b \tan (c+d x)}}{3 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {13 a b \sqrt {a+b \tan (c+d x)}}{12 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {\left (11 a^2-8 b^2\right ) \sqrt {a+b \tan (c+d x)}}{8 d \sqrt {\cot (c+d x)}}+\frac {\left ((i a-b)^3 \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 d}+\frac {\left ((i a+b)^3 \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 (5 a \left (a^2-8 b^2\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 )}{8 d}\\ &=\frac {5 a \left (a^2-8 b^2\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)}}{8 \sqrt {b} d}+\frac {b^2 \sqrt {a+b \tan (c+d x)}}{3 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {13 a b \sqrt {a+b \tan (c+d x)}}{12 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {\left (11 a^2-8 b^2\right ) \sqrt {a+b \tan (c+d x)}}{8 d \sqrt {\cot (c+d x)}}+\frac {\left ((i a-b)^3 \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 ((i a+b)^3 \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 {i (i a-b)^{5/2} \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 {5 a \left (a^2-8 b^2\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)}}{8 \sqrt {b} d}-\frac {i (i a+b)^{5/2} \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 {b^2 \sqrt {a+b \tan (c+d x)}}{3 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {13 a b \sqrt {a+b \tan (c+d x)}}{12 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {\left (11 a^2-8 b^2\right ) \sqrt {a+b \tan (c+d x)}}{8 d \sqrt {\cot (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 4.31, size = 320, normalized size = 0.95 \[ \frac {\sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (3 \left (11 a^2-8 b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}+\frac {15 a^{3/2} \left (a^2-8 b^2\right ) \sqrt {\frac {b \tan (c+d x)}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {b} \sqrt {a+b \tan (c+d x)}}+8 b^2 \tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}-24 (-1)^{3/4} (-a+i b)^{5/2} \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+24 (-1)^{3/4} (a+i b)^{5/2} \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+26 a b \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}\right )}{24 d} \]
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 = 2.30, size = 18203, normalized size = 54.01 \[ \text {output too large to display} \]
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 \[ \int \frac {{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cot \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2}}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}} \,d x \]
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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