Optimal. Leaf size=226 \[ \frac {(i a-b)^{3/2} \text {ArcTan}\left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {\left (3 a^2-8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{4 \sqrt {b} d}+\frac {(i a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {3 a \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{4 d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 d} \]
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Rubi [A]
time = 1.09, antiderivative size = 226, 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 = {3651, 3728,
3736, 6857, 65, 223, 212, 95, 211, 214} \begin {gather*} \frac {\left (3 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 )}{4 \sqrt {b} d}+\frac {(-b+i a)^{3/2} \text {ArcTan}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 d}+\frac {3 a \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{4 d}+\frac {(b+i a)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 95
Rule 211
Rule 212
Rule 214
Rule 223
Rule 3651
Rule 3728
Rule 3736
Rule 6857
Rubi steps
\begin {align*} \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} \, dx &=\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 d}+\frac {1}{2} \int \frac {\sqrt {a+b \tan (c+d x)} \left (-\frac {a}{2}-2 b \tan (c+d x)+\frac {3}{2} a \tan ^2(c+d x)\right )}{\sqrt {\tan (c+d x)}} \, dx\\ &=\frac {3 a \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{4 d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 d}+\frac {1}{2} \int \frac {-\frac {5 a^2}{4}-4 a b \tan (c+d x)+\frac {1}{4} \left (3 a^2-8 b^2\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx\\ &=\frac {3 a \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{4 d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 d}+\frac {\text {Subst}\left (\int \frac {-\frac {5 a^2}{4}-4 a b x+\frac {1}{4} \left (3 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 )}{2 d}\\ &=\frac {3 a \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{4 d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 d}+\frac {\text {Subst}\left (\int \left (\frac {3 a^2-8 b^2}{4 \sqrt {x} \sqrt {a+b x}}-\frac {2 \left (a^2-b^2+2 a b x\right )}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac {3 a \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{4 d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 d}-\frac {\text {Subst}\left (\int \frac {a^2-b^2+2 a b x}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}+\frac {\left (3 a^2-8 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{8 d}\\ &=\frac {3 a \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{4 d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 d}-\frac {\text {Subst}\left (\int \left (\frac {-2 a b+i \left (a^2-b^2\right )}{2 (i-x) \sqrt {x} \sqrt {a+b x}}+\frac {2 a b+i \left (a^2-b^2\right )}{2 \sqrt {x} (i+x) \sqrt {a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{d}+\frac {\left (3 a^2-8 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {\tan (c+d x)}\right )}{4 d}\\ &=\frac {3 a \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{4 d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 d}-\frac {\left (i (a-i b)^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} (i+x) \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}-\frac {\left (i (a+i b)^2\right ) \text {Subst}\left (\int \frac {1}{(i-x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {\left (3 a^2-8 b^2\right ) \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 )}{4 d}\\ &=\frac {\left (3 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 )}{4 \sqrt {b} d}+\frac {3 a \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{4 d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 d}-\frac {\left (i (a-i b)^2\right ) \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 )}{d}-\frac {\left (i (a+i b)^2\right ) \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 )}{d}\\ &=\frac {(i a-b)^{3/2} \tan ^{-1}\left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {\left (3 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 )}{4 \sqrt {b} d}+\frac {(i a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {3 a \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{4 d}+\frac {\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 d}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(850\) vs. \(2(226)=452\).
time = 6.12, size = 850, normalized size = 3.76 \begin {gather*} \frac {2 a \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)} \left (1+\frac {b \tan (c+d x)}{a}\right )^2 \left (\frac {3 \sqrt {a} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{8 \sqrt {b} \sqrt {\tan (c+d x)} \left (1+\frac {b \tan (c+d x)}{a}\right )^{5/2}}+\frac {1}{4} \left (\frac {3}{2 \left (1+\frac {b \tan (c+d x)}{a}\right )^2}+\frac {1}{1+\frac {b \tan (c+d x)}{a}}\right )\right )-i \left (-2 b \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)} \left (1+\frac {b \tan (c+d x)}{a}\right ) \left (\frac {\sqrt {a} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{2 \sqrt {b} \sqrt {\tan (c+d x)} \left (1+\frac {b \tan (c+d x)}{a}\right )^{3/2}}+\frac {1}{2 \left (1+\frac {b \tan (c+d x)}{a}\right )}\right )-(-a-i b) \left (\frac {2 \sqrt [4]{-1} (-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 {2 \sqrt {a} \sqrt {b} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {1+\frac {b \tan (c+d x)}{a}}}{\sqrt {a+b \tan (c+d x)}}\right )\right )}{2 d}+\frac {2 a \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)} \left (1+\frac {b \tan (c+d x)}{a}\right )^2 \left (\frac {3 \sqrt {a} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{8 \sqrt {b} \sqrt {\tan (c+d x)} \left (1+\frac {b \tan (c+d x)}{a}\right )^{5/2}}+\frac {1}{4} \left (\frac {3}{2 \left (1+\frac {b \tan (c+d x)}{a}\right )^2}+\frac {1}{1+\frac {b \tan (c+d x)}{a}}\right )\right )-i \left (2 b \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)} \left (1+\frac {b \tan (c+d x)}{a}\right ) \left (\frac {\sqrt {a} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{2 \sqrt {b} \sqrt {\tan (c+d x)} \left (1+\frac {b \tan (c+d x)}{a}\right )^{3/2}}+\frac {1}{2 \left (1+\frac {b \tan (c+d x)}{a}\right )}\right )-(-a+i b) \left (2 \sqrt [4]{-1} \sqrt {-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 )+\frac {2 \sqrt {a} \sqrt {b} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {1+\frac {b \tan (c+d x)}{a}}}{\sqrt {a+b \tan (c+d x)}}\right )\right )}{2 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 = 1.18, size = 1346578, normalized size = 5958.31 \[\text {output too large to display}\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \tan {\left (c + d x \right )}\right )^{\frac {3}{2}} \tan ^{\frac {3}{2}}{\left (c + d x \right )}\, dx \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 {\mathrm {tan}\left (c+d\,x\right )}^{3/2}\,{\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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