Optimal. Leaf size=326 \[ \frac {(a+b) \left (a^2-4 a b+b^2\right ) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {40 a^2 b}{63 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (3 a^2-b^2\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\tan (c+d x)}}-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)} \]
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Rubi [A]
time = 0.30, antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3646, 3709,
3610, 3615, 1182, 1176, 631, 210, 1179, 642} \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}-\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}+\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (3 a^2-b^2\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\tan (c+d x)}}-\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}+\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}-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {40 a^2 b}{63 d \tan ^{\frac {7}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3610
Rule 3615
Rule 3646
Rule 3709
Rubi steps
\begin {align*} \int \frac {(a+b \tan (c+d x))^3}{\tan ^{\frac {11}{2}}(c+d x)} \, dx &=-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {2}{9} \int \frac {10 a^2 b-\frac {9}{2} a \left (a^2-3 b^2\right ) \tan (c+d x)-\frac {1}{2} b \left (7 a^2-9 b^2\right ) \tan ^2(c+d x)}{\tan ^{\frac {9}{2}}(c+d x)} \, dx\\ &=-\frac {40 a^2 b}{63 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {2}{9} \int \frac {-\frac {9}{2} a \left (a^2-3 b^2\right )-\frac {9}{2} b \left (3 a^2-b^2\right ) \tan (c+d x)}{\tan ^{\frac {7}{2}}(c+d x)} \, dx\\ &=-\frac {40 a^2 b}{63 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {2}{9} \int \frac {-\frac {9}{2} b \left (3 a^2-b^2\right )+\frac {9}{2} a \left (a^2-3 b^2\right ) \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x)} \, dx\\ &=-\frac {40 a^2 b}{63 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (3 a^2-b^2\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {2}{9} \int \frac {\frac {9}{2} a \left (a^2-3 b^2\right )+\frac {9}{2} b \left (3 a^2-b^2\right ) \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x)} \, dx\\ &=-\frac {40 a^2 b}{63 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (3 a^2-b^2\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\tan (c+d x)}}-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {2}{9} \int \frac {\frac {9}{2} b \left (3 a^2-b^2\right )-\frac {9}{2} a \left (a^2-3 b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx\\ &=-\frac {40 a^2 b}{63 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (3 a^2-b^2\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\tan (c+d x)}}-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}+\frac {4 \text {Subst}\left (\int \frac {\frac {9}{2} b \left (3 a^2-b^2\right )-\frac {9}{2} a \left (a^2-3 b^2\right ) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{9 d}\\ &=-\frac {40 a^2 b}{63 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (3 a^2-b^2\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\tan (c+d x)}}-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(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 )}{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 )}{d}\\ &=-\frac {40 a^2 b}{63 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (3 a^2-b^2\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\tan (c+d x)}}-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(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 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 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} 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} 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} 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} d}-\frac {40 a^2 b}{63 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (3 a^2-b^2\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\tan (c+d x)}}-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(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} 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} 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} 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} 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} 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} d}-\frac {40 a^2 b}{63 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (3 a^2-b^2\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\tan (c+d x)}}-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.71, size = 104, normalized size = 0.32 \begin {gather*} -\frac {2 \left (7 a \left (a^2-3 b^2\right ) \, _2F_1\left (-\frac {9}{4},1;-\frac {5}{4};-\tan ^2(c+d x)\right )+3 b \left (3 \left (3 a^2-b^2\right ) \, _2F_1\left (-\frac {7}{4},1;-\frac {3}{4};-\tan ^2(c+d x)\right ) \tan (c+d x)+b (7 a+3 b \tan (c+d x))\right )\right )}{63 d \tan ^{\frac {9}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 289, normalized size = 0.89
method | result | size |
derivativedivides | \(\frac {-\frac {2 a^{3}}{9 \tan \left (d x +c \right )^{\frac {9}{2}}}-\frac {2 a \left (a^{2}-3 b^{2}\right )}{\sqrt {\tan \left (d x +c \right )}}+\frac {2 a \left (a^{2}-3 b^{2}\right )}{5 \tan \left (d x +c \right )^{\frac {5}{2}}}-\frac {6 a^{2} b}{7 \tan \left (d x +c \right )^{\frac {7}{2}}}+\frac {2 b \left (3 a^{2}-b^{2}\right )}{3 \tan \left (d x +c \right )^{\frac {3}{2}}}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-a^{3}+3 b^{2} a \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{d}\) | \(289\) |
default | \(\frac {-\frac {2 a^{3}}{9 \tan \left (d x +c \right )^{\frac {9}{2}}}-\frac {2 a \left (a^{2}-3 b^{2}\right )}{\sqrt {\tan \left (d x +c \right )}}+\frac {2 a \left (a^{2}-3 b^{2}\right )}{5 \tan \left (d x +c \right )^{\frac {5}{2}}}-\frac {6 a^{2} b}{7 \tan \left (d x +c \right )^{\frac {7}{2}}}+\frac {2 b \left (3 a^{2}-b^{2}\right )}{3 \tan \left (d x +c \right )^{\frac {3}{2}}}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-a^{3}+3 b^{2} a \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{d}\) | \(289\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 279, normalized size = 0.86 \begin {gather*} -\frac {630 \, \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 ) + 630 \, \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 ) - 315 \, \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 ) + 315 \, \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 ) + \frac {8 \, {\left (315 \, {\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{4} + 135 \, a^{2} b \tan \left (d x + c\right ) - 105 \, {\left (3 \, a^{2} b - b^{3}\right )} \tan \left (d x + c\right )^{3} + 35 \, a^{3} - 63 \, {\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{2}\right )}}{\tan \left (d x + c\right )^{\frac {9}{2}}}}{1260 \, 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. 8178 vs.
\(2 (280) = 560\).
time = 3.48, size = 8178, normalized size = 25.09 \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 {\left (a + b \tan {\left (c + d x \right )}\right )^{3}}{\tan ^{\frac {11}{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 = 10.10, size = 1821, normalized size = 5.59 \begin {gather*} 2\,\mathrm {atanh}\left (\frac {32\,a^6\,d^3\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {3\,a\,b^5}{2\,d^2}+\frac {b^6\,1{}\mathrm {i}}{4\,d^2}-\frac {a^6\,1{}\mathrm {i}}{4\,d^2}+\frac {3\,a^5\,b}{2\,d^2}-\frac {a^2\,b^4\,15{}\mathrm {i}}{4\,d^2}-\frac {5\,a^3\,b^3}{d^2}+\frac {a^4\,b^2\,15{}\mathrm {i}}{4\,d^2}}}{16\,a^9\,d^2+a^8\,b\,d^2\,48{}\mathrm {i}-288\,a^7\,b^2\,d^2-a^6\,b^3\,d^2\,736{}\mathrm {i}+960\,a^5\,b^4\,d^2+a^4\,b^5\,d^2\,960{}\mathrm {i}-736\,a^3\,b^6\,d^2-a^2\,b^7\,d^2\,288{}\mathrm {i}+48\,a\,b^8\,d^2+b^9\,d^2\,16{}\mathrm {i}}-\frac {32\,b^6\,d^3\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {3\,a\,b^5}{2\,d^2}+\frac {b^6\,1{}\mathrm {i}}{4\,d^2}-\frac {a^6\,1{}\mathrm {i}}{4\,d^2}+\frac {3\,a^5\,b}{2\,d^2}-\frac {a^2\,b^4\,15{}\mathrm {i}}{4\,d^2}-\frac {5\,a^3\,b^3}{d^2}+\frac {a^4\,b^2\,15{}\mathrm {i}}{4\,d^2}}}{16\,a^9\,d^2+a^8\,b\,d^2\,48{}\mathrm {i}-288\,a^7\,b^2\,d^2-a^6\,b^3\,d^2\,736{}\mathrm {i}+960\,a^5\,b^4\,d^2+a^4\,b^5\,d^2\,960{}\mathrm {i}-736\,a^3\,b^6\,d^2-a^2\,b^7\,d^2\,288{}\mathrm {i}+48\,a\,b^8\,d^2+b^9\,d^2\,16{}\mathrm {i}}+\frac {480\,a^2\,b^4\,d^3\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {3\,a\,b^5}{2\,d^2}+\frac {b^6\,1{}\mathrm {i}}{4\,d^2}-\frac {a^6\,1{}\mathrm {i}}{4\,d^2}+\frac {3\,a^5\,b}{2\,d^2}-\frac {a^2\,b^4\,15{}\mathrm {i}}{4\,d^2}-\frac {5\,a^3\,b^3}{d^2}+\frac {a^4\,b^2\,15{}\mathrm {i}}{4\,d^2}}}{16\,a^9\,d^2+a^8\,b\,d^2\,48{}\mathrm {i}-288\,a^7\,b^2\,d^2-a^6\,b^3\,d^2\,736{}\mathrm {i}+960\,a^5\,b^4\,d^2+a^4\,b^5\,d^2\,960{}\mathrm {i}-736\,a^3\,b^6\,d^2-a^2\,b^7\,d^2\,288{}\mathrm {i}+48\,a\,b^8\,d^2+b^9\,d^2\,16{}\mathrm {i}}-\frac {480\,a^4\,b^2\,d^3\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {3\,a\,b^5}{2\,d^2}+\frac {b^6\,1{}\mathrm {i}}{4\,d^2}-\frac {a^6\,1{}\mathrm {i}}{4\,d^2}+\frac {3\,a^5\,b}{2\,d^2}-\frac {a^2\,b^4\,15{}\mathrm {i}}{4\,d^2}-\frac {5\,a^3\,b^3}{d^2}+\frac {a^4\,b^2\,15{}\mathrm {i}}{4\,d^2}}}{16\,a^9\,d^2+a^8\,b\,d^2\,48{}\mathrm {i}-288\,a^7\,b^2\,d^2-a^6\,b^3\,d^2\,736{}\mathrm {i}+960\,a^5\,b^4\,d^2+a^4\,b^5\,d^2\,960{}\mathrm {i}-736\,a^3\,b^6\,d^2-a^2\,b^7\,d^2\,288{}\mathrm {i}+48\,a\,b^8\,d^2+b^9\,d^2\,16{}\mathrm {i}}\right )\,\sqrt {\frac {-a^6\,1{}\mathrm {i}+6\,a^5\,b+a^4\,b^2\,15{}\mathrm {i}-20\,a^3\,b^3-a^2\,b^4\,15{}\mathrm {i}+6\,a\,b^5+b^6\,1{}\mathrm {i}}{4\,d^2}}+2\,\mathrm {atanh}\left (\frac {32\,a^6\,d^3\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {a^6\,1{}\mathrm {i}}{4\,d^2}-\frac {b^6\,1{}\mathrm {i}}{4\,d^2}+\frac {3\,a\,b^5}{2\,d^2}+\frac {3\,a^5\,b}{2\,d^2}+\frac {a^2\,b^4\,15{}\mathrm {i}}{4\,d^2}-\frac {5\,a^3\,b^3}{d^2}-\frac {a^4\,b^2\,15{}\mathrm {i}}{4\,d^2}}}{16\,a^9\,d^2-a^8\,b\,d^2\,48{}\mathrm {i}-288\,a^7\,b^2\,d^2+a^6\,b^3\,d^2\,736{}\mathrm {i}+960\,a^5\,b^4\,d^2-a^4\,b^5\,d^2\,960{}\mathrm {i}-736\,a^3\,b^6\,d^2+a^2\,b^7\,d^2\,288{}\mathrm {i}+48\,a\,b^8\,d^2-b^9\,d^2\,16{}\mathrm {i}}-\frac {32\,b^6\,d^3\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {a^6\,1{}\mathrm {i}}{4\,d^2}-\frac {b^6\,1{}\mathrm {i}}{4\,d^2}+\frac {3\,a\,b^5}{2\,d^2}+\frac {3\,a^5\,b}{2\,d^2}+\frac {a^2\,b^4\,15{}\mathrm {i}}{4\,d^2}-\frac {5\,a^3\,b^3}{d^2}-\frac {a^4\,b^2\,15{}\mathrm {i}}{4\,d^2}}}{16\,a^9\,d^2-a^8\,b\,d^2\,48{}\mathrm {i}-288\,a^7\,b^2\,d^2+a^6\,b^3\,d^2\,736{}\mathrm {i}+960\,a^5\,b^4\,d^2-a^4\,b^5\,d^2\,960{}\mathrm {i}-736\,a^3\,b^6\,d^2+a^2\,b^7\,d^2\,288{}\mathrm {i}+48\,a\,b^8\,d^2-b^9\,d^2\,16{}\mathrm {i}}+\frac {480\,a^2\,b^4\,d^3\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {a^6\,1{}\mathrm {i}}{4\,d^2}-\frac {b^6\,1{}\mathrm {i}}{4\,d^2}+\frac {3\,a\,b^5}{2\,d^2}+\frac {3\,a^5\,b}{2\,d^2}+\frac {a^2\,b^4\,15{}\mathrm {i}}{4\,d^2}-\frac {5\,a^3\,b^3}{d^2}-\frac {a^4\,b^2\,15{}\mathrm {i}}{4\,d^2}}}{16\,a^9\,d^2-a^8\,b\,d^2\,48{}\mathrm {i}-288\,a^7\,b^2\,d^2+a^6\,b^3\,d^2\,736{}\mathrm {i}+960\,a^5\,b^4\,d^2-a^4\,b^5\,d^2\,960{}\mathrm {i}-736\,a^3\,b^6\,d^2+a^2\,b^7\,d^2\,288{}\mathrm {i}+48\,a\,b^8\,d^2-b^9\,d^2\,16{}\mathrm {i}}-\frac {480\,a^4\,b^2\,d^3\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {a^6\,1{}\mathrm {i}}{4\,d^2}-\frac {b^6\,1{}\mathrm {i}}{4\,d^2}+\frac {3\,a\,b^5}{2\,d^2}+\frac {3\,a^5\,b}{2\,d^2}+\frac {a^2\,b^4\,15{}\mathrm {i}}{4\,d^2}-\frac {5\,a^3\,b^3}{d^2}-\frac {a^4\,b^2\,15{}\mathrm {i}}{4\,d^2}}}{16\,a^9\,d^2-a^8\,b\,d^2\,48{}\mathrm {i}-288\,a^7\,b^2\,d^2+a^6\,b^3\,d^2\,736{}\mathrm {i}+960\,a^5\,b^4\,d^2-a^4\,b^5\,d^2\,960{}\mathrm {i}-736\,a^3\,b^6\,d^2+a^2\,b^7\,d^2\,288{}\mathrm {i}+48\,a\,b^8\,d^2-b^9\,d^2\,16{}\mathrm {i}}\right )\,\sqrt {\frac {a^6\,1{}\mathrm {i}+6\,a^5\,b-a^4\,b^2\,15{}\mathrm {i}-20\,a^3\,b^3+a^2\,b^4\,15{}\mathrm {i}+6\,a\,b^5-b^6\,1{}\mathrm {i}}{4\,d^2}}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {6\,a\,b^2}{5}-\frac {2\,a^3}{5}\right )-{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (6\,a\,b^2-2\,a^3\right )-{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (2\,a^2\,b-\frac {2\,b^3}{3}\right )+\frac {2\,a^3}{9}+\frac {6\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )}{7}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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