Optimal. Leaf size=106 \[ -\frac {\sqrt {a-i b} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {\sqrt {a+i b} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 \sqrt {a+b \tan (c+d x)}}{d} \]
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Rubi [A]
time = 0.12, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3609, 3620,
3618, 65, 214} \begin {gather*} \frac {2 \sqrt {a+b \tan (c+d x)}}{d}-\frac {\sqrt {a-i b} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {\sqrt {a+i b} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 3609
Rule 3618
Rule 3620
Rubi steps
\begin {align*} \int \tan (c+d x) \sqrt {a+b \tan (c+d x)} \, dx &=\frac {2 \sqrt {a+b \tan (c+d x)}}{d}+\int \frac {-b+a \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx\\ &=\frac {2 \sqrt {a+b \tan (c+d x)}}{d}+\frac {1}{2} (-i a-b) \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx+\frac {1}{2} (i a-b) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx\\ &=\frac {2 \sqrt {a+b \tan (c+d x)}}{d}+\frac {(a-i b) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 d}+\frac {(a+i b) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 d}\\ &=\frac {2 \sqrt {a+b \tan (c+d x)}}{d}-\frac {(i a-b) \text {Subst}\left (\int \frac {1}{-1+\frac {i a}{b}-\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{b d}+\frac {(i a+b) \text {Subst}\left (\int \frac {1}{-1-\frac {i a}{b}+\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{b d}\\ &=-\frac {\sqrt {a-i b} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {\sqrt {a+i b} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 \sqrt {a+b \tan (c+d x)}}{d}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 100, normalized size = 0.94 \begin {gather*} -\frac {\sqrt {a-i b} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )+\sqrt {a+i b} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )-2 \sqrt {a+b \tan (c+d x)}}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(326\) vs.
\(2(88)=176\).
time = 0.12, size = 327, normalized size = 3.08
method | result | size |
derivativedivides | \(\frac {2 \sqrt {a +b \tan \left (d x +c \right )}-\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{4}+\frac {\left (a -\sqrt {a^{2}+b^{2}}\right ) \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \ln \left (-b \tan \left (d x +c \right )-a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-\sqrt {a^{2}+b^{2}}\right )}{4}+\frac {\left (\sqrt {a^{2}+b^{2}}-a \right ) \arctan \left (\frac {-2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{d}\) | \(327\) |
default | \(\frac {2 \sqrt {a +b \tan \left (d x +c \right )}-\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{4}+\frac {\left (a -\sqrt {a^{2}+b^{2}}\right ) \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \ln \left (-b \tan \left (d x +c \right )-a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-\sqrt {a^{2}+b^{2}}\right )}{4}+\frac {\left (\sqrt {a^{2}+b^{2}}-a \right ) \arctan \left (\frac {-2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{d}\) | \(327\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 1740 vs.
\(2 (84) = 168\).
time = 1.21, size = 1740, normalized size = 16.42 \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 \sqrt {a + b \tan {\left (c + d x \right )}} \tan {\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.79, size = 290, normalized size = 2.74 \begin {gather*} \frac {2\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{d}-\mathrm {atan}\left (\frac {b^4\,\sqrt {\frac {a}{4\,d^2}-\frac {b\,1{}\mathrm {i}}{4\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,32{}\mathrm {i}}{\frac {b^5\,16{}\mathrm {i}}{d}+\frac {a^2\,b^3\,16{}\mathrm {i}}{d}}+\frac {32\,a\,b^3\,\sqrt {\frac {a}{4\,d^2}-\frac {b\,1{}\mathrm {i}}{4\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{\frac {b^5\,16{}\mathrm {i}}{d}+\frac {a^2\,b^3\,16{}\mathrm {i}}{d}}\right )\,\sqrt {\frac {a-b\,1{}\mathrm {i}}{4\,d^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {b^4\,\sqrt {\frac {a}{4\,d^2}+\frac {b\,1{}\mathrm {i}}{4\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,32{}\mathrm {i}}{\frac {b^5\,16{}\mathrm {i}}{d}+\frac {a^2\,b^3\,16{}\mathrm {i}}{d}}-\frac {32\,a\,b^3\,\sqrt {\frac {a}{4\,d^2}+\frac {b\,1{}\mathrm {i}}{4\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{\frac {b^5\,16{}\mathrm {i}}{d}+\frac {a^2\,b^3\,16{}\mathrm {i}}{d}}\right )\,\sqrt {\frac {a+b\,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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