Optimal. Leaf size=493 \[ \frac {(a+b) \left (a^2-4 a b+b^2\right ) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {a^{7/2} \left (35 a^4+102 a^2 b^2+99 b^4\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 b^{9/2} \left (a^2+b^2\right )^3 d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{4 b^4 \left (a^2+b^2\right )^2 d}+\frac {\left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]
[Out]
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Rubi [A]
time = 0.98, antiderivative size = 493, normalized size of antiderivative = 1.00, number
of steps used = 18, number of rules used = 14, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules
used = {3646, 3726, 3728, 3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211}
\begin {gather*} \frac {(a+b) \left (a^2-4 a b+b^2\right ) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \text {ArcTan}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{4 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^3}-\frac {a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{4 b^4 d \left (a^2+b^2\right )^2}+\frac {\left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 b^3 d \left (a^2+b^2\right )^2}+\frac {a^{7/2} \left (35 a^4+102 a^2 b^2+99 b^4\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 b^{9/2} d \left (a^2+b^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 210
Rule 211
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3615
Rule 3646
Rule 3715
Rule 3726
Rule 3728
Rule 3734
Rubi steps
\begin {align*} \int \frac {\tan ^{\frac {11}{2}}(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\int \frac {\tan ^{\frac {5}{2}}(c+d x) \left (\frac {7 a^2}{2}-2 a b \tan (c+d x)+\frac {1}{2} \left (7 a^2+4 b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 b \left (a^2+b^2\right )}\\ &=-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {5}{4} a^2 \left (7 a^2+15 b^2\right )-4 a b^3 \tan (c+d x)+\frac {1}{4} \left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 b^2 \left (a^2+b^2\right )^2}\\ &=\frac {\left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {\sqrt {\tan (c+d x)} \left (-\frac {3}{8} a \left (35 a^4+67 a^2 b^2+8 b^4\right )+3 b^3 \left (a^2-b^2\right ) \tan (c+d x)-\frac {3}{8} a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{3 b^3 \left (a^2+b^2\right )^2}\\ &=-\frac {a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{4 b^4 \left (a^2+b^2\right )^2 d}+\frac {\left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {2 \int \frac {\frac {3}{16} a^2 \left (35 a^4+67 a^2 b^2+24 b^4\right )+3 a b^5 \tan (c+d x)+\frac {3}{16} \left (35 a^6+67 a^4 b^2+32 a^2 b^4-8 b^6\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{3 b^4 \left (a^2+b^2\right )^2}\\ &=-\frac {a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{4 b^4 \left (a^2+b^2\right )^2 d}+\frac {\left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {2 \int \frac {-\frac {3}{2} a b^4 \left (a^2-3 b^2\right )+\frac {3}{2} b^5 \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{3 b^4 \left (a^2+b^2\right )^3}+\frac {\left (a^4 \left (35 a^4+102 a^2 b^2+99 b^4\right )\right ) \int \frac {1+\tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{8 b^4 \left (a^2+b^2\right )^3}\\ &=-\frac {a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{4 b^4 \left (a^2+b^2\right )^2 d}+\frac {\left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {4 \text {Subst}\left (\int \frac {-\frac {3}{2} a b^4 \left (a^2-3 b^2\right )+\frac {3}{2} b^5 \left (3 a^2-b^2\right ) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{3 b^4 \left (a^2+b^2\right )^3 d}+\frac {\left (a^4 \left (35 a^4+102 a^2 b^2+99 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{8 b^4 \left (a^2+b^2\right )^3 d}\\ &=-\frac {a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{4 b^4 \left (a^2+b^2\right )^2 d}+\frac {\left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right )^3 d}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right )^3 d}+\frac {\left (a^4 \left (35 a^4+102 a^2 b^2+99 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{4 b^4 \left (a^2+b^2\right )^3 d}\\ &=\frac {a^{7/2} \left (35 a^4+102 a^2 b^2+99 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 b^{9/2} \left (a^2+b^2\right )^3 d}-\frac {a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{4 b^4 \left (a^2+b^2\right )^2 d}+\frac {\left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right )^3 d}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right )^3 d}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\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} \left (a^2+b^2\right )^3 d}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\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} \left (a^2+b^2\right )^3 d}\\ &=\frac {a^{7/2} \left (35 a^4+102 a^2 b^2+99 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 b^{9/2} \left (a^2+b^2\right )^3 d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{4 b^4 \left (a^2+b^2\right )^2 d}+\frac {\left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}\\ &=\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {a^{7/2} \left (35 a^4+102 a^2 b^2+99 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 b^{9/2} \left (a^2+b^2\right )^3 d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {a \left (35 a^4+67 a^2 b^2+24 b^4\right ) \sqrt {\tan (c+d x)}}{4 b^4 \left (a^2+b^2\right )^2 d}+\frac {\left (35 a^4+67 a^2 b^2+8 b^4\right ) \tan ^{\frac {3}{2}}(c+d x)}{12 b^3 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan ^{\frac {7}{2}}(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (7 a^2+15 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{4 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 6.38, size = 723, normalized size = 1.47 \begin {gather*} \frac {b^2 \tan ^{\frac {13}{2}}(c+d x)}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {-\frac {b \tan ^{\frac {11}{2}}(c+d x)}{d (a+b \tan (c+d x))}+\frac {2 \left (\frac {9 a b \tan ^{\frac {9}{2}}(c+d x)}{2 d (a+b \tan (c+d x))}+\frac {2 \left (-\frac {63 a^2 b \tan ^{\frac {7}{2}}(c+d x)}{4 d (a+b \tan (c+d x))}+\frac {2 \left (\frac {105 a b \left (7 a^2+4 b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{8 d (a+b \tan (c+d x))}+\frac {2 \left (-\frac {315 a^2 b \left (35 a^2+32 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{16 d (a+b \tan (c+d x))}+\frac {2 \left (-\frac {945 a b \left (35 a^4+32 a^2 b^2-4 b^4\right ) \sqrt {\tan (c+d x)}}{32 d (a+b \tan (c+d x))}-\frac {2 \left (\frac {\frac {2 \left (\frac {945 a^4 b^8}{16}-\frac {945}{256} a^4 b^4 \left (35 a^4+67 a^2 b^2+24 b^4\right )-\frac {945}{256} a^4 b^2 \left (35 a^6+67 a^4 b^2+32 a^2 b^4-8 b^6\right )\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \left (a^2+b^2\right ) d}+\frac {-\frac {\sqrt [4]{-1} \left (\frac {945}{32} a^3 b^6 \left (a^2-3 b^2\right )+\frac {945}{32} i a^2 b^7 \left (3 a^2-b^2\right )\right ) \text {ArcTan}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}-\frac {\sqrt [4]{-1} \left (\frac {945}{32} a^3 b^6 \left (a^2-3 b^2\right )-\frac {945}{32} i a^2 b^7 \left (3 a^2-b^2\right )\right ) \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {\left (-\frac {945}{128} a^2 b^4 \left (35 a^4+32 a^2 b^2-4 b^4\right )-a \left (-\frac {945 a b^8}{32}+\frac {945}{128} a^5 b^2 \left (35 a^2+32 b^2\right )\right )\right ) \sqrt {\tan (c+d x)}}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))}\right )}{b}\right )}{b}\right )}{3 b}\right )}{5 b}\right )}{7 b}\right )}{9 b}}{2 a \left (a^2+b^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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Maple [A]
time = 0.19, size = 373, normalized size = 0.76
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (-\frac {b \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+3 a \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{b^{4}}+\frac {2 a^{4} \left (\frac {\left (-\frac {13}{8} a^{4} b -\frac {17}{4} a^{2} b^{3}-\frac {21}{8} b^{5}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )-\frac {a \left (11 a^{4}+30 a^{2} b^{2}+19 b^{4}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{8}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\left (35 a^{4}+102 a^{2} b^{2}+99 b^{4}\right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{4} \left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (-a^{3}+3 b^{2} a \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(373\) |
default | \(\frac {-\frac {2 \left (-\frac {b \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+3 a \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{b^{4}}+\frac {2 a^{4} \left (\frac {\left (-\frac {13}{8} a^{4} b -\frac {17}{4} a^{2} b^{3}-\frac {21}{8} b^{5}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )-\frac {a \left (11 a^{4}+30 a^{2} b^{2}+19 b^{4}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{8}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\left (35 a^{4}+102 a^{2} b^{2}+99 b^{4}\right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{4} \left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (-a^{3}+3 b^{2} a \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(373\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 449, normalized size = 0.91 \begin {gather*} \frac {\frac {3 \, {\left (35 \, a^{8} + 102 \, a^{6} b^{2} + 99 \, a^{4} b^{4}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}\right )} \sqrt {a b}} - \frac {3 \, {\left (2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {3 \, {\left ({\left (13 \, a^{6} b + 21 \, a^{4} b^{3}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + {\left (11 \, a^{7} + 19 \, a^{5} b^{2}\right )} \sqrt {\tan \left (d x + c\right )}\right )}}{a^{6} b^{4} + 2 \, a^{4} b^{6} + a^{2} b^{8} + {\left (a^{4} b^{6} + 2 \, a^{2} b^{8} + b^{10}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{5} + 2 \, a^{3} b^{7} + a b^{9}\right )} \tan \left (d x + c\right )} + \frac {8 \, {\left (b \tan \left (d x + c\right )^{\frac {3}{2}} - 9 \, a \sqrt {\tan \left (d x + c\right )}\right )}}{b^{4}}}{12 \, 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. 11465 vs.
\(2 (437) = 874\).
time = 15.24, size = 23042, normalized size = 46.74 \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(-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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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 = 20.45, size = 2500, normalized size = 5.07 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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