Optimal. Leaf size=250 \[ \frac {i \text {ArcTan}\left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(i a-b)^{3/2} d}-\frac {3 a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{b^{5/2} d}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(i a+b)^{3/2} d}-\frac {2 a^2 \tan ^{\frac {3}{2}}(c+d x)}{b \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {\left (3 a^2+b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{b^2 \left (a^2+b^2\right ) d} \]
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Rubi [A]
time = 1.16, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3646, 3728,
3736, 6857, 65, 223, 212, 95, 211, 214} \begin {gather*} -\frac {2 a^2 \tan ^{\frac {3}{2}}(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (3 a^2+b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{b^2 d \left (a^2+b^2\right )}+\frac {i \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)^{3/2}}-\frac {3 a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{b^{5/2} d}+\frac {i \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)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 95
Rule 211
Rule 212
Rule 214
Rule 223
Rule 3646
Rule 3728
Rule 3736
Rule 6857
Rubi steps
\begin {align*} \int \frac {\tan ^{\frac {7}{2}}(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx &=-\frac {2 a^2 \tan ^{\frac {3}{2}}(c+d x)}{b \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {2 \int \frac {\sqrt {\tan (c+d x)} \left (\frac {3 a^2}{2}-\frac {1}{2} a b \tan (c+d x)+\frac {1}{2} \left (3 a^2+b^2\right ) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{b \left (a^2+b^2\right )}\\ &=-\frac {2 a^2 \tan ^{\frac {3}{2}}(c+d x)}{b \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {\left (3 a^2+b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{b^2 \left (a^2+b^2\right ) d}+\frac {2 \int \frac {-\frac {1}{4} a \left (3 a^2+b^2\right )-\frac {1}{2} b^3 \tan (c+d x)-\frac {3}{4} a \left (a^2+b^2\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{b^2 \left (a^2+b^2\right )}\\ &=-\frac {2 a^2 \tan ^{\frac {3}{2}}(c+d x)}{b \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {\left (3 a^2+b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{b^2 \left (a^2+b^2\right ) d}+\frac {2 \text {Subst}\left (\int \frac {-\frac {1}{4} a \left (3 a^2+b^2\right )-\frac {b^3 x}{2}-\frac {3}{4} a \left (a^2+b^2\right ) x^2}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{b^2 \left (a^2+b^2\right ) d}\\ &=-\frac {2 a^2 \tan ^{\frac {3}{2}}(c+d x)}{b \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {\left (3 a^2+b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{b^2 \left (a^2+b^2\right ) d}+\frac {2 \text {Subst}\left (\int \left (-\frac {3 a \left (a^2+b^2\right )}{4 \sqrt {x} \sqrt {a+b x}}+\frac {a b^2-b^3 x}{2 \sqrt {x} \sqrt {a+b x} \left (1+x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{b^2 \left (a^2+b^2\right ) d}\\ &=-\frac {2 a^2 \tan ^{\frac {3}{2}}(c+d x)}{b \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {\left (3 a^2+b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{b^2 \left (a^2+b^2\right ) d}-\frac {(3 a) \text {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 b^2 d}+\frac {\text {Subst}\left (\int \frac {a b^2-b^3 x}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{b^2 \left (a^2+b^2\right ) d}\\ &=-\frac {2 a^2 \tan ^{\frac {3}{2}}(c+d x)}{b \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {\left (3 a^2+b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{b^2 \left (a^2+b^2\right ) d}-\frac {(3 a) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {\tan (c+d x)}\right )}{b^2 d}+\frac {\text {Subst}\left (\int \left (\frac {i a b^2+b^3}{2 (i-x) \sqrt {x} \sqrt {a+b x}}+\frac {i a b^2-b^3}{2 \sqrt {x} (i+x) \sqrt {a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{b^2 \left (a^2+b^2\right ) d}\\ &=-\frac {2 a^2 \tan ^{\frac {3}{2}}(c+d x)}{b \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {\left (3 a^2+b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{b^2 \left (a^2+b^2\right ) d}-\frac {\text {Subst}\left (\int \frac {1}{(i-x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 (i a-b) d}-\frac {(3 a) \text {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 )}{b^2 d}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {x} (i+x) \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 (i a+b) d}\\ &=-\frac {3 a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{b^{5/2} d}-\frac {2 a^2 \tan ^{\frac {3}{2}}(c+d x)}{b \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {\left (3 a^2+b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{b^2 \left (a^2+b^2\right ) d}-\frac {\text {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 )}{(i a-b) d}-\frac {\text {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 )}{(i a+b) d}\\ &=\frac {i \tan ^{-1}\left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(i a-b)^{3/2} d}-\frac {3 a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{b^{5/2} d}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(i a+b)^{3/2} d}-\frac {2 a^2 \tan ^{\frac {3}{2}}(c+d x)}{b \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {\left (3 a^2+b^2\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{b^2 \left (a^2+b^2\right ) d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 3.18, size = 270, normalized size = 1.08 \begin {gather*} \frac {\frac {5 (-1)^{3/4} \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}}-\frac {5 (-1)^{3/4} \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}}-\frac {5 \sqrt {\tan (c+d x)}}{(a-i b) \sqrt {a+b \tan (c+d x)}}-\frac {5 \sqrt {\tan (c+d x)}}{(a+i b) \sqrt {a+b \tan (c+d x)}}+\frac {2 \, _2F_1\left (\frac {3}{2},\frac {5}{2};\frac {7}{2};-\frac {b \tan (c+d x)}{a}\right ) \tan ^{\frac {5}{2}}(c+d x) \sqrt {1+\frac {b \tan (c+d x)}{a}}}{a \sqrt {a+b \tan (c+d x)}}}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] result has leaf size over 500,000. Avoiding possible recursion issues.
time = 2.18, size = 764550, normalized size = 3058.20 \[\text {output too large to display}\]
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 \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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {tan}\left (c+d\,x\right )}^{7/2}}{{\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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