Optimal. Leaf size=216 \[ \frac{a \left (a^2-3 b^2\right ) \sin (c+d x)}{d \left (a^2+b^2\right )^3}+\frac{b \left (3 a^2-b^2\right ) \cos (c+d x)}{d \left (a^2+b^2\right )^3}+\frac{b^4 \sin (c+d x)}{2 a d \left (a^2+b^2\right )^2 (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{b^3 \left (8 a^2+b^2\right )}{2 a d \left (a^2+b^2\right )^3 (a \cos (c+d x)+b \sin (c+d x))}-\frac{3 b^2 \left (4 a^2-b^2\right ) \tanh ^{-1}\left (\frac{b-a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2+b^2}}\right )}{d \left (a^2+b^2\right )^{7/2}} \]
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Rubi [B] time = 1.74227, antiderivative size = 492, normalized size of antiderivative = 2.28, number of steps used = 15, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6742, 639, 203, 638, 618, 206, 614} \[ -\frac{3 b^4 \left (a^2+2 b^2\right ) \left (b-a \tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^3 d \left (a^2+b^2\right )^3 \left (-a \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+2 b \tan \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{2 b^4 \left (\left (a^2+2 b^2\right ) \tan \left (\frac{1}{2} (c+d x)\right )+a b\right )}{a^3 d \left (a^2+b^2\right )^2 \left (-a \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+2 b \tan \left (\frac{1}{2} (c+d x)\right )\right )^2}-\frac{4 b^3 \left (a b \left (3 a^2+2 b^2\right ) \tan \left (\frac{1}{2} (c+d x)\right )+2 a^4-b^4\right )}{a^3 d \left (a^2+b^2\right )^3 \left (-a \tan ^2\left (\frac{1}{2} (c+d x)\right )+a+2 b \tan \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{2 \left (a \left (a^2-3 b^2\right ) \tan \left (\frac{1}{2} (c+d x)\right )+b \left (3 a^2-b^2\right )\right )}{d \left (a^2+b^2\right )^3 \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right )}-\frac{3 b^4 \left (a^2+2 b^2\right ) \tanh ^{-1}\left (\frac{b-a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2+b^2}}\right )}{a^2 d \left (a^2+b^2\right )^{7/2}}+\frac{4 b^4 \left (3 a^2+2 b^2\right ) \tanh ^{-1}\left (\frac{b-a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2+b^2}}\right )}{a^2 d \left (a^2+b^2\right )^{7/2}}-\frac{2 b^2 \left (3 a^2 b^2+6 a^4+b^4\right ) \tanh ^{-1}\left (\frac{b-a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2+b^2}}\right )}{a^2 d \left (a^2+b^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 639
Rule 203
Rule 638
Rule 618
Rule 206
Rule 614
Rubi steps
\begin{align*} \int \frac{\cos ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^4}{\left (1+x^2\right )^2 \left (a+2 b x-a x^2\right )^3} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (\frac{2 \left (a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) x\right )}{\left (a^2+b^2\right )^3 \left (1+x^2\right )^2}-\frac{a \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3 \left (1+x^2\right )}+\frac{4 b^3 \left (-b \left (a^2+b^2\right )-a \left (2 a^2+b^2\right ) x\right )}{a^3 \left (a^2+b^2\right )^2 \left (a+2 b x-a x^2\right )^2}-\frac{4 b^4 (a+2 b x)}{a^3 \left (a^2+b^2\right ) \left (-a-2 b x+a x^2\right )^3}-\frac{b^2 \left (6 a^4+3 a^2 b^2+b^4\right )}{a^2 \left (a^2+b^2\right )^3 \left (-a-2 b x+a x^2\right )}\right ) \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{d}\\ &=\frac{4 \operatorname{Subst}\left (\int \frac{a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) x}{\left (1+x^2\right )^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{\left (a^2+b^2\right )^3 d}-\frac{\left (2 a \left (a^2-3 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{\left (a^2+b^2\right )^3 d}+\frac{\left (8 b^3\right ) \operatorname{Subst}\left (\int \frac{-b \left (a^2+b^2\right )-a \left (2 a^2+b^2\right ) x}{\left (a+2 b x-a x^2\right )^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^2 d}-\frac{\left (8 b^4\right ) \operatorname{Subst}\left (\int \frac{a+2 b x}{\left (-a-2 b x+a x^2\right )^3} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right ) d}-\frac{\left (2 b^2 \left (6 a^4+3 a^2 b^2+b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-a-2 b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^2 \left (a^2+b^2\right )^3 d}\\ &=-\frac{a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}+\frac{2 \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan \left (\frac{1}{2} (c+d x)\right )\right )}{\left (a^2+b^2\right )^3 d \left (1+\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )}+\frac{2 b^4 \left (a b+\left (a^2+2 b^2\right ) \tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^2 d \left (a+2 b \tan \left (\frac{1}{2} (c+d x)\right )-a \tan ^2\left (\frac{1}{2} (c+d x)\right )\right )^2}-\frac{4 b^3 \left (2 a^4-b^4+a b \left (3 a^2+2 b^2\right ) \tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^3 d \left (a+2 b \tan \left (\frac{1}{2} (c+d x)\right )-a \tan ^2\left (\frac{1}{2} (c+d x)\right )\right )}+\frac{\left (2 a \left (a^2-3 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{\left (a^2+b^2\right )^3 d}+\frac{\left (6 b^4 \left (a^2+2 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\left (-a-2 b x+a x^2\right )^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^2 d}-\frac{\left (4 b^4 \left (3 a^2+2 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x-a x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^2 \left (a^2+b^2\right )^3 d}+\frac{\left (4 b^2 \left (6 a^4+3 a^2 b^2+b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,-2 b+2 a \tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^2 \left (a^2+b^2\right )^3 d}\\ &=-\frac{2 b^2 \left (6 a^4+3 a^2 b^2+b^4\right ) \tanh ^{-1}\left (\frac{b-a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{7/2} d}+\frac{2 \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan \left (\frac{1}{2} (c+d x)\right )\right )}{\left (a^2+b^2\right )^3 d \left (1+\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )}+\frac{2 b^4 \left (a b+\left (a^2+2 b^2\right ) \tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^2 d \left (a+2 b \tan \left (\frac{1}{2} (c+d x)\right )-a \tan ^2\left (\frac{1}{2} (c+d x)\right )\right )^2}-\frac{3 b^4 \left (a^2+2 b^2\right ) \left (b-a \tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^3 d \left (a+2 b \tan \left (\frac{1}{2} (c+d x)\right )-a \tan ^2\left (\frac{1}{2} (c+d x)\right )\right )}-\frac{4 b^3 \left (2 a^4-b^4+a b \left (3 a^2+2 b^2\right ) \tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^3 d \left (a+2 b \tan \left (\frac{1}{2} (c+d x)\right )-a \tan ^2\left (\frac{1}{2} (c+d x)\right )\right )}-\frac{\left (3 b^4 \left (a^2+2 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-a-2 b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^2 \left (a^2+b^2\right )^3 d}+\frac{\left (8 b^4 \left (3 a^2+2 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^2 \left (a^2+b^2\right )^3 d}\\ &=\frac{4 b^4 \left (3 a^2+2 b^2\right ) \tanh ^{-1}\left (\frac{b-a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{7/2} d}-\frac{2 b^2 \left (6 a^4+3 a^2 b^2+b^4\right ) \tanh ^{-1}\left (\frac{b-a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{7/2} d}+\frac{2 \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan \left (\frac{1}{2} (c+d x)\right )\right )}{\left (a^2+b^2\right )^3 d \left (1+\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )}+\frac{2 b^4 \left (a b+\left (a^2+2 b^2\right ) \tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^2 d \left (a+2 b \tan \left (\frac{1}{2} (c+d x)\right )-a \tan ^2\left (\frac{1}{2} (c+d x)\right )\right )^2}-\frac{3 b^4 \left (a^2+2 b^2\right ) \left (b-a \tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^3 d \left (a+2 b \tan \left (\frac{1}{2} (c+d x)\right )-a \tan ^2\left (\frac{1}{2} (c+d x)\right )\right )}-\frac{4 b^3 \left (2 a^4-b^4+a b \left (3 a^2+2 b^2\right ) \tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^3 d \left (a+2 b \tan \left (\frac{1}{2} (c+d x)\right )-a \tan ^2\left (\frac{1}{2} (c+d x)\right )\right )}+\frac{\left (6 b^4 \left (a^2+2 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,-2 b+2 a \tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^2 \left (a^2+b^2\right )^3 d}\\ &=-\frac{3 b^4 \left (a^2+2 b^2\right ) \tanh ^{-1}\left (\frac{b-a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{7/2} d}+\frac{4 b^4 \left (3 a^2+2 b^2\right ) \tanh ^{-1}\left (\frac{b-a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{7/2} d}-\frac{2 b^2 \left (6 a^4+3 a^2 b^2+b^4\right ) \tanh ^{-1}\left (\frac{b-a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{7/2} d}+\frac{2 \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan \left (\frac{1}{2} (c+d x)\right )\right )}{\left (a^2+b^2\right )^3 d \left (1+\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )}+\frac{2 b^4 \left (a b+\left (a^2+2 b^2\right ) \tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^2 d \left (a+2 b \tan \left (\frac{1}{2} (c+d x)\right )-a \tan ^2\left (\frac{1}{2} (c+d x)\right )\right )^2}-\frac{3 b^4 \left (a^2+2 b^2\right ) \left (b-a \tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^3 d \left (a+2 b \tan \left (\frac{1}{2} (c+d x)\right )-a \tan ^2\left (\frac{1}{2} (c+d x)\right )\right )}-\frac{4 b^3 \left (2 a^4-b^4+a b \left (3 a^2+2 b^2\right ) \tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^3 d \left (a+2 b \tan \left (\frac{1}{2} (c+d x)\right )-a \tan ^2\left (\frac{1}{2} (c+d x)\right )\right )}\\ \end{align*}
Mathematica [C] time = 1.03486, size = 211, normalized size = 0.98 \[ \frac{\frac{2 a \left (a^2-3 b^2\right ) \sin (c+d x)}{\left (a^2+b^2\right )^3}-\frac{2 b \left (b^2-3 a^2\right ) \cos (c+d x)}{\left (a^2+b^2\right )^3}-\frac{b^3 \left (8 a^2+b^2\right )}{a \left (a^2+b^2\right )^3 (a \cos (c+d x)+b \sin (c+d x))}-\frac{6 b^2 \left (b^2-4 a^2\right ) \tanh ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )-b}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}+\frac{b^4 \sin (c+d x)}{a (a-i b)^2 (a+i b)^2 (a \cos (c+d x)+b \sin (c+d x))^2}}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.213, size = 283, normalized size = 1.3 \begin{align*}{\frac{1}{d} \left ( -2\,{\frac{{b}^{2}}{ \left ({a}^{2}+{b}^{2} \right ) ^{3}} \left ({\frac{1}{ \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a-2\,\tan \left ( 1/2\,dx+c/2 \right ) b-a \right ) ^{2}} \left ( -1/2\,{\frac{{b}^{2} \left ( 9\,{a}^{2}+2\,{b}^{2} \right ) \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}}{a}}-1/2\,{\frac{b \left ( 8\,{a}^{4}-15\,{a}^{2}{b}^{2}-2\,{b}^{4} \right ) \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{{a}^{2}}}+1/2\,{\frac{{b}^{2} \left ( 23\,{a}^{2}+2\,{b}^{2} \right ) \tan \left ( 1/2\,dx+c/2 \right ) }{a}}+4\,{a}^{2}b+1/2\,{b}^{3} \right ) }-3/2\,{\frac{4\,{a}^{2}-{b}^{2}}{\sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,a\tan \left ( 1/2\,dx+c/2 \right ) -2\,b}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) } \right ) }-2\,{\frac{ \left ( -{a}^{3}+3\,a{b}^{2} \right ) \tan \left ( 1/2\,dx+c/2 \right ) -3\,{a}^{2}b+{b}^{3}}{ \left ({a}^{6}+3\,{a}^{4}{b}^{2}+3\,{a}^{2}{b}^{4}+{b}^{6} \right ) \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.609099, size = 1073, normalized size = 4.97 \begin{align*} \frac{4 \,{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{3} - 3 \,{\left (4 \, a^{2} b^{4} - b^{6} +{\left (4 \, a^{4} b^{2} - 5 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (4 \, a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \sqrt{a^{2} + b^{2}} \log \left (\frac{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} - 2 \, \sqrt{a^{2} + b^{2}}{\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) + 2 \,{\left (4 \, a^{6} b - 10 \, a^{4} b^{3} - 17 \, a^{2} b^{5} - 3 \, b^{7}\right )} \cos \left (d x + c\right ) + 2 \,{\left (2 \, a^{5} b^{2} - 11 \, a^{3} b^{4} - 13 \, a b^{6} + 2 \,{\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{4 \,{\left ({\left (a^{10} + 3 \, a^{8} b^{2} + 2 \, a^{6} b^{4} - 2 \, a^{4} b^{6} - 3 \, a^{2} b^{8} - b^{10}\right )} d \cos \left (d x + c\right )^{2} + 2 \,{\left (a^{9} b + 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} + 4 \, a^{3} b^{7} + a b^{9}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{8} b^{2} + 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} + 4 \, a^{2} b^{8} + b^{10}\right )} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37751, size = 539, normalized size = 2.5 \begin{align*} -\frac{\frac{3 \,{\left (4 \, a^{2} b^{2} - b^{4}\right )} \log \left (\frac{{\left | 2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt{a^{2} + b^{2}}} - \frac{4 \,{\left (a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, a^{2} b - b^{3}\right )}}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}} - \frac{2 \,{\left (9 \, a^{3} b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, a b^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 8 \, a^{4} b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, a^{2} b^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 2 \, b^{7} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 23 \, a^{3} b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, a b^{6} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 8 \, a^{4} b^{3} - a^{2} b^{5}\right )}}{{\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )}{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 2 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - a\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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