Optimal. Leaf size=249 \[ \frac {\left (a^2+2 a b-b^2\right ) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {\left (a^2-2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {2 \left (a^2-b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {4 a b \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d} \]
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Rubi [A]
time = 0.16, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3624, 3609,
3615, 1182, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {\left (a^2+2 a b-b^2\right ) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \text {ArcTan}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}+\frac {2 \left (a^2-b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {\left (a^2-2 a b-b^2\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {\left (a^2-2 a b-b^2\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {4 a b \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3609
Rule 3615
Rule 3624
Rubi steps
\begin {align*} \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2 \, dx &=\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\int \tan ^{\frac {3}{2}}(c+d x) \left (a^2-b^2+2 a b \tan (c+d x)\right ) \, dx\\ &=\frac {4 a b \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\int \sqrt {\tan (c+d x)} \left (-2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=\frac {2 \left (a^2-b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {4 a b \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\int \frac {-a^2+b^2-2 a b \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx\\ &=\frac {2 \left (a^2-b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {4 a b \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 \text {Subst}\left (\int \frac {-a^2+b^2-2 a b x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}\\ &=\frac {2 \left (a^2-b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {4 a b \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}-\frac {\left (a^2+2 a b-b^2\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}\\ &=\frac {2 \left (a^2-b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {4 a b \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d}-\frac {\left (a^2+2 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d}\\ &=\frac {\left (a^2-2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {\left (a^2-2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {2 \left (a^2-b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {4 a b \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {\left (a^2+2 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^2+2 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}\\ &=\frac {\left (a^2+2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {\left (a^2-2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {2 \left (a^2-b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {4 a b \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b^2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.73, size = 120, normalized size = 0.48 \begin {gather*} \frac {15 \sqrt [4]{-1} (a-i b)^2 \text {ArcTan}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+15 \sqrt [4]{-1} (a+i b)^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+2 \sqrt {\tan (c+d x)} \left (15 a^2-15 b^2+10 a b \tan (c+d x)+3 b^2 \tan ^2(c+d x)\right )}{15 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 238, normalized size = 0.96
method | result | size |
derivativedivides | \(\frac {\frac {2 b^{2} \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}+\frac {4 a b \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+2 a^{2} \left (\sqrt {\tan }\left (d x +c \right )\right )-2 b^{2} \left (\sqrt {\tan }\left (d x +c \right )\right )+\frac {\left (-a^{2}+b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}-\frac {a b \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{2}}{d}\) | \(238\) |
default | \(\frac {\frac {2 b^{2} \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}+\frac {4 a b \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+2 a^{2} \left (\sqrt {\tan }\left (d x +c \right )\right )-2 b^{2} \left (\sqrt {\tan }\left (d x +c \right )\right )+\frac {\left (-a^{2}+b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}-\frac {a b \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{2}}{d}\) | \(238\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 205, normalized size = 0.82 \begin {gather*} \frac {24 \, b^{2} \tan \left (d x + c\right )^{\frac {5}{2}} + 80 \, a b \tan \left (d x + c\right )^{\frac {3}{2}} - 30 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - 30 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - 15 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + 15 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + 120 \, {\left (a^{2} - b^{2}\right )} \sqrt {\tan \left (d x + c\right )}}{60 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5043 vs.
\(2 (211) = 422\).
time = 1.08, size = 5043, normalized size = 20.25 \begin {gather*} \text {Too large to display} \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 )^{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 [B]
time = 5.76, size = 986, normalized size = 3.96 \begin {gather*} \sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\left (\frac {2\,a^2}{d}-\frac {2\,b^2}{d}\right )+\frac {2\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^{5/2}}{5\,d}+\frac {4\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^{3/2}}{3\,d}-\mathrm {atan}\left (\frac {a^4\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {a\,b^3}{d^2}-\frac {b^4\,1{}\mathrm {i}}{4\,d^2}-\frac {a^4\,1{}\mathrm {i}}{4\,d^2}-\frac {a^3\,b}{d^2}+\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2}}\,32{}\mathrm {i}}{\frac {a^6\,16{}\mathrm {i}}{d}-\frac {b^6\,16{}\mathrm {i}}{d}+\frac {32\,a\,b^5}{d}+\frac {32\,a^5\,b}{d}+\frac {a^2\,b^4\,112{}\mathrm {i}}{d}-\frac {192\,a^3\,b^3}{d}-\frac {a^4\,b^2\,112{}\mathrm {i}}{d}}+\frac {b^4\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {a\,b^3}{d^2}-\frac {b^4\,1{}\mathrm {i}}{4\,d^2}-\frac {a^4\,1{}\mathrm {i}}{4\,d^2}-\frac {a^3\,b}{d^2}+\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2}}\,32{}\mathrm {i}}{\frac {a^6\,16{}\mathrm {i}}{d}-\frac {b^6\,16{}\mathrm {i}}{d}+\frac {32\,a\,b^5}{d}+\frac {32\,a^5\,b}{d}+\frac {a^2\,b^4\,112{}\mathrm {i}}{d}-\frac {192\,a^3\,b^3}{d}-\frac {a^4\,b^2\,112{}\mathrm {i}}{d}}-\frac {a^2\,b^2\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {a\,b^3}{d^2}-\frac {b^4\,1{}\mathrm {i}}{4\,d^2}-\frac {a^4\,1{}\mathrm {i}}{4\,d^2}-\frac {a^3\,b}{d^2}+\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2}}\,192{}\mathrm {i}}{\frac {a^6\,16{}\mathrm {i}}{d}-\frac {b^6\,16{}\mathrm {i}}{d}+\frac {32\,a\,b^5}{d}+\frac {32\,a^5\,b}{d}+\frac {a^2\,b^4\,112{}\mathrm {i}}{d}-\frac {192\,a^3\,b^3}{d}-\frac {a^4\,b^2\,112{}\mathrm {i}}{d}}\right )\,\sqrt {-\frac {a^4\,1{}\mathrm {i}+4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}-4\,a\,b^3+b^4\,1{}\mathrm {i}}{4\,d^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {a^4\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {a^4\,1{}\mathrm {i}}{4\,d^2}+\frac {b^4\,1{}\mathrm {i}}{4\,d^2}+\frac {a\,b^3}{d^2}-\frac {a^3\,b}{d^2}-\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2}}\,32{}\mathrm {i}}{\frac {32\,a\,b^5}{d}+\frac {b^6\,16{}\mathrm {i}}{d}-\frac {a^6\,16{}\mathrm {i}}{d}+\frac {32\,a^5\,b}{d}-\frac {a^2\,b^4\,112{}\mathrm {i}}{d}-\frac {192\,a^3\,b^3}{d}+\frac {a^4\,b^2\,112{}\mathrm {i}}{d}}+\frac {b^4\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {a^4\,1{}\mathrm {i}}{4\,d^2}+\frac {b^4\,1{}\mathrm {i}}{4\,d^2}+\frac {a\,b^3}{d^2}-\frac {a^3\,b}{d^2}-\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2}}\,32{}\mathrm {i}}{\frac {32\,a\,b^5}{d}+\frac {b^6\,16{}\mathrm {i}}{d}-\frac {a^6\,16{}\mathrm {i}}{d}+\frac {32\,a^5\,b}{d}-\frac {a^2\,b^4\,112{}\mathrm {i}}{d}-\frac {192\,a^3\,b^3}{d}+\frac {a^4\,b^2\,112{}\mathrm {i}}{d}}-\frac {a^2\,b^2\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {a^4\,1{}\mathrm {i}}{4\,d^2}+\frac {b^4\,1{}\mathrm {i}}{4\,d^2}+\frac {a\,b^3}{d^2}-\frac {a^3\,b}{d^2}-\frac {a^2\,b^2\,3{}\mathrm {i}}{2\,d^2}}\,192{}\mathrm {i}}{\frac {32\,a\,b^5}{d}+\frac {b^6\,16{}\mathrm {i}}{d}-\frac {a^6\,16{}\mathrm {i}}{d}+\frac {32\,a^5\,b}{d}-\frac {a^2\,b^4\,112{}\mathrm {i}}{d}-\frac {192\,a^3\,b^3}{d}+\frac {a^4\,b^2\,112{}\mathrm {i}}{d}}\right )\,\sqrt {\frac {a^4\,1{}\mathrm {i}-4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}+4\,a\,b^3+b^4\,1{}\mathrm {i}}{4\,d^2}}\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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