Optimal. Leaf size=500 \[ \frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}} d}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}} d}-\frac {b \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {b \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 b^3 d}-\frac {8 a \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{15 b^2 d}+\frac {2 \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b d} \]
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Rubi [A]
time = 0.57, antiderivative size = 500, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3647, 3728,
3712, 3566, 722, 1108, 648, 632, 212, 642} \begin {gather*} -\frac {b \log \left (-\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \tan (c+d x)}+\sqrt {a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt {2} d \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}+\frac {b \log \left (\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \tan (c+d x)}+\sqrt {a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt {2} d \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+a}}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a^2+b^2}+a}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} d \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a^2+b^2}+a}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} d \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}}}+\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 b^3 d}-\frac {8 a \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{15 b^2 d}+\frac {2 \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 722
Rule 1108
Rule 3566
Rule 3647
Rule 3712
Rule 3728
Rubi steps
\begin {align*} \int \frac {\tan ^4(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx &=\frac {2 \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b d}+\frac {2 \int \frac {\tan (c+d x) \left (-2 a-\frac {5}{2} b \tan (c+d x)-2 a \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{5 b}\\ &=-\frac {8 a \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{15 b^2 d}+\frac {2 \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b d}+\frac {4 \int \frac {2 a^2+\frac {1}{4} \left (8 a^2-15 b^2\right ) \tan ^2(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{15 b^2}\\ &=\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 b^3 d}-\frac {8 a \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{15 b^2 d}+\frac {2 \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b d}+\int \frac {1}{\sqrt {a+b \tan (c+d x)}} \, dx\\ &=\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 b^3 d}-\frac {8 a \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{15 b^2 d}+\frac {2 \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b d}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a+x} \left (b^2+x^2\right )} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 b^3 d}-\frac {8 a \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{15 b^2 d}+\frac {2 \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b d}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{a^2+b^2-2 a x^2+x^4} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{d}\\ &=\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 b^3 d}-\frac {8 a \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{15 b^2 d}+\frac {2 \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b d}+\frac {b \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}-x}{\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {b \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+x}{\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}\\ &=\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 b^3 d}-\frac {8 a \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{15 b^2 d}+\frac {2 \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b d}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {a^2+b^2} d}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {a^2+b^2} d}-\frac {b \text {Subst}\left (\int \frac {-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 x}{\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {b \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 x}{\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}\\ &=-\frac {b \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {b \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 b^3 d}-\frac {8 a \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{15 b^2 d}+\frac {2 \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b d}-\frac {b \text {Subst}\left (\int \frac {1}{2 \left (a-\sqrt {a^2+b^2}\right )-x^2} \, dx,x,-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 \sqrt {a+b \tan (c+d x)}\right )}{\sqrt {a^2+b^2} d}-\frac {b \text {Subst}\left (\int \frac {1}{2 \left (a-\sqrt {a^2+b^2}\right )-x^2} \, dx,x,\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 \sqrt {a+b \tan (c+d x)}\right )}{\sqrt {a^2+b^2} d}\\ &=\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}} d}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a^2+b^2} \sqrt {a-\sqrt {a^2+b^2}} d}-\frac {b \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {b \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a^2+b^2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 b^3 d}-\frac {8 a \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{15 b^2 d}+\frac {2 \tan ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{5 b d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 17.48, size = 144, normalized size = 0.29 \begin {gather*} \frac {-\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b}}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b}}+\frac {2 \sqrt {a+b \tan (c+d x)} \left (8 a^2-18 b^2+3 b^2 \sec ^2(c+d x)-4 a b \tan (c+d x)\right )}{15 b^3}}{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. \(845\) vs.
\(2(409)=818\).
time = 0.14, size = 846, normalized size = 1.69
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-\frac {4 a \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 a^{2} \sqrt {a +b \tan \left (d x +c \right )}-2 b^{2} \sqrt {a +b \tan \left (d x +c \right )}+2 b^{4} \left (\frac {\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\right ) \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 )}{2}+\frac {2 \left (2 a^{2} b^{2}+2 b^{4}-\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{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}}}{4 b^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {-\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\right ) \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 )}{2}+\frac {2 \left (-2 a^{2} b^{2}-2 b^{4}+\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{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}}}{4 b^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{d \,b^{3}}\) | \(846\) |
default | \(\frac {\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-\frac {4 a \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 a^{2} \sqrt {a +b \tan \left (d x +c \right )}-2 b^{2} \sqrt {a +b \tan \left (d x +c \right )}+2 b^{4} \left (\frac {\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\right ) \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 )}{2}+\frac {2 \left (2 a^{2} b^{2}+2 b^{4}-\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{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}}}{4 b^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {-\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\right ) \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 )}{2}+\frac {2 \left (-2 a^{2} b^{2}-2 b^{4}+\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{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}}}{4 b^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{d \,b^{3}}\) | \(846\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1886 vs.
\(2 (411) = 822\).
time = 0.96, size = 1886, normalized size = 3.77 \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 \frac {\tan ^{4}{\left (c + d x \right )}}{\sqrt {a + b \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(-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 = 7.94, size = 791, normalized size = 1.58 \begin {gather*} \left (\frac {4\,a^2}{b^3\,d}-\frac {2\,\left (a^2+b^2\right )}{b^3\,d}\right )\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}+\frac {\ln \left (16\,b^2\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}+16\,b^3\,d\,\sqrt {-\frac {1}{d^2\,\left (a-b\,1{}\mathrm {i}\right )}}-\frac {16\,a\,b^2\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{a-b\,1{}\mathrm {i}}\right )\,\sqrt {-\frac {1}{a\,d^2-b\,d^2\,1{}\mathrm {i}}}}{2}-\ln \left (-16\,b^2\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}+16\,b^3\,d\,\sqrt {-\frac {1}{d^2\,\left (a-b\,1{}\mathrm {i}\right )}}+\frac {16\,a\,b^2\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{a-b\,1{}\mathrm {i}}\right )\,\sqrt {-\frac {1}{4\,\left (a\,d^2-b\,d^2\,1{}\mathrm {i}\right )}}+\frac {2\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2}}{5\,b^3\,d}-\frac {4\,a\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}}{3\,b^3\,d}+\mathrm {atan}\left (-\frac {b^2\,\sqrt {-\frac {a}{4\,a^2\,d^2+4\,b^2\,d^2}+\frac {b\,1{}\mathrm {i}}{4\,a^2\,d^2+4\,b^2\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,32{}\mathrm {i}}{-\frac {64\,a\,b^3\,d^2}{4\,a^2\,d^3+4\,b^2\,d^3}+\frac {b^4\,d^2\,64{}\mathrm {i}}{4\,a^2\,d^3+4\,b^2\,d^3}}+\frac {128\,a\,b^3\,\sqrt {-\frac {a}{4\,a^2\,d^2+4\,b^2\,d^2}+\frac {b\,1{}\mathrm {i}}{4\,a^2\,d^2+4\,b^2\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{-\frac {256\,a^3\,b^3\,d^2}{4\,a^2\,d^3+4\,b^2\,d^3}-\frac {256\,a\,b^5\,d^2}{4\,a^2\,d^3+4\,b^2\,d^3}+\frac {b^6\,d^2\,256{}\mathrm {i}}{4\,a^2\,d^3+4\,b^2\,d^3}+\frac {a^2\,b^4\,d^2\,256{}\mathrm {i}}{4\,a^2\,d^3+4\,b^2\,d^3}}+\frac {a^2\,b^2\,\sqrt {-\frac {a}{4\,a^2\,d^2+4\,b^2\,d^2}+\frac {b\,1{}\mathrm {i}}{4\,a^2\,d^2+4\,b^2\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,128{}\mathrm {i}}{-\frac {256\,a^3\,b^3\,d^2}{4\,a^2\,d^3+4\,b^2\,d^3}-\frac {256\,a\,b^5\,d^2}{4\,a^2\,d^3+4\,b^2\,d^3}+\frac {b^6\,d^2\,256{}\mathrm {i}}{4\,a^2\,d^3+4\,b^2\,d^3}+\frac {a^2\,b^4\,d^2\,256{}\mathrm {i}}{4\,a^2\,d^3+4\,b^2\,d^3}}\right )\,\sqrt {-\frac {a-b\,1{}\mathrm {i}}{4\,a^2\,d^2+4\,b^2\,d^2}}\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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