Optimal. Leaf size=107 \[ -\frac{15 \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{10 i \sec ^3(c+d x)}{d \left (a^4+i a^4 \tan (c+d x)\right )}-\frac{15 \tan (c+d x) \sec (c+d x)}{2 a^4 d}+\frac{2 i \sec ^5(c+d x)}{a d (a+i a \tan (c+d x))^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.100898, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3500, 3768, 3770} \[ -\frac{15 \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{10 i \sec ^3(c+d x)}{d \left (a^4+i a^4 \tan (c+d x)\right )}-\frac{15 \tan (c+d x) \sec (c+d x)}{2 a^4 d}+\frac{2 i \sec ^5(c+d x)}{a d (a+i a \tan (c+d x))^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3500
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \frac{\sec ^7(c+d x)}{(a+i a \tan (c+d x))^4} \, dx &=\frac{2 i \sec ^5(c+d x)}{a d (a+i a \tan (c+d x))^3}-\frac{5 \int \frac{\sec ^5(c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{a^2}\\ &=\frac{2 i \sec ^5(c+d x)}{a d (a+i a \tan (c+d x))^3}+\frac{10 i \sec ^3(c+d x)}{d \left (a^4+i a^4 \tan (c+d x)\right )}-\frac{15 \int \sec ^3(c+d x) \, dx}{a^4}\\ &=-\frac{15 \sec (c+d x) \tan (c+d x)}{2 a^4 d}+\frac{2 i \sec ^5(c+d x)}{a d (a+i a \tan (c+d x))^3}+\frac{10 i \sec ^3(c+d x)}{d \left (a^4+i a^4 \tan (c+d x)\right )}-\frac{15 \int \sec (c+d x) \, dx}{2 a^4}\\ &=-\frac{15 \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{15 \sec (c+d x) \tan (c+d x)}{2 a^4 d}+\frac{2 i \sec ^5(c+d x)}{a d (a+i a \tan (c+d x))^3}+\frac{10 i \sec ^3(c+d x)}{d \left (a^4+i a^4 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [B] time = 6.12355, size = 988, normalized size = 9.23 \[ \frac{15 \cos (4 c) \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right ) \sec ^4(c+d x) (\cos (d x)+i \sin (d x))^4}{2 d (i \tan (c+d x) a+a)^4}-\frac{15 \cos (4 c) \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right ) \sec ^4(c+d x) (\cos (d x)+i \sin (d x))^4}{2 d (i \tan (c+d x) a+a)^4}+\frac{\cos (d x) \sec ^4(c+d x) (8 i \cos (3 c)-8 \sin (3 c)) (\cos (d x)+i \sin (d x))^4}{d (i \tan (c+d x) a+a)^4}+\frac{\sec (c) \sec ^4(c+d x) (4 i \cos (4 c)-4 \sin (4 c)) (\cos (d x)+i \sin (d x))^4}{d (i \tan (c+d x) a+a)^4}+\frac{15 i \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right ) \sec ^4(c+d x) \sin (4 c) (\cos (d x)+i \sin (d x))^4}{2 d (i \tan (c+d x) a+a)^4}-\frac{15 i \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right ) \sec ^4(c+d x) \sin (4 c) (\cos (d x)+i \sin (d x))^4}{2 d (i \tan (c+d x) a+a)^4}+\frac{\sec ^4(c+d x) (8 \cos (3 c)+8 i \sin (3 c)) \sin (d x) (\cos (d x)+i \sin (d x))^4}{d (i \tan (c+d x) a+a)^4}+\frac{4 \sec ^4(c+d x) \left (\frac{1}{2} \cos \left (4 c-\frac{d x}{2}\right )-\frac{1}{2} \cos \left (4 c+\frac{d x}{2}\right )+\frac{1}{2} i \sin \left (4 c-\frac{d x}{2}\right )-\frac{1}{2} i \sin \left (4 c+\frac{d x}{2}\right )\right ) (\cos (d x)+i \sin (d x))^4}{d \left (\cos \left (\frac{c}{2}\right )+\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right ) (i \tan (c+d x) a+a)^4}+\frac{4 \sec ^4(c+d x) \left (-\frac{1}{2} \cos \left (4 c-\frac{d x}{2}\right )+\frac{1}{2} \cos \left (4 c+\frac{d x}{2}\right )-\frac{1}{2} i \sin \left (4 c-\frac{d x}{2}\right )+\frac{1}{2} i \sin \left (4 c+\frac{d x}{2}\right )\right ) (\cos (d x)+i \sin (d x))^4}{d \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right ) (i \tan (c+d x) a+a)^4}+\frac{\sec ^4(c+d x) \left (\frac{1}{4} \cos (4 c)+\frac{1}{4} i \sin (4 c)\right ) (\cos (d x)+i \sin (d x))^4}{d \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^2 (i \tan (c+d x) a+a)^4}+\frac{\sec ^4(c+d x) \left (-\frac{1}{4} \cos (4 c)-\frac{1}{4} i \sin (4 c)\right ) (\cos (d x)+i \sin (d x))^4}{d \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^2 (i \tan (c+d x) a+a)^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.095, size = 192, normalized size = 1.8 \begin{align*}{\frac{1}{2\,{a}^{4}d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{4\,i}{{a}^{4}d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{1}{2\,{a}^{4}d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}-{\frac{15}{2\,{a}^{4}d}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+16\,{\frac{1}{{a}^{4}d \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) }}+{\frac{1}{2\,{a}^{4}d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{4\,i}{{a}^{4}d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+{\frac{1}{2\,{a}^{4}d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}+{\frac{15}{2\,{a}^{4}d}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.61453, size = 630, normalized size = 5.89 \begin{align*} \frac{{\left (30 \, \cos \left (5 \, d x + 5 \, c\right ) + 60 \, \cos \left (3 \, d x + 3 \, c\right ) + 30 \, \cos \left (d x + c\right ) + 30 i \, \sin \left (5 \, d x + 5 \, c\right ) + 60 i \, \sin \left (3 \, d x + 3 \, c\right ) + 30 i \, \sin \left (d x + c\right )\right )} \arctan \left (\cos \left (d x + c\right ), \sin \left (d x + c\right ) + 1\right ) +{\left (30 \, \cos \left (5 \, d x + 5 \, c\right ) + 60 \, \cos \left (3 \, d x + 3 \, c\right ) + 30 \, \cos \left (d x + c\right ) + 30 i \, \sin \left (5 \, d x + 5 \, c\right ) + 60 i \, \sin \left (3 \, d x + 3 \, c\right ) + 30 i \, \sin \left (d x + c\right )\right )} \arctan \left (\cos \left (d x + c\right ), -\sin \left (d x + c\right ) + 1\right ) -{\left (-15 i \, \cos \left (5 \, d x + 5 \, c\right ) - 30 i \, \cos \left (3 \, d x + 3 \, c\right ) - 15 i \, \cos \left (d x + c\right ) + 15 \, \sin \left (5 \, d x + 5 \, c\right ) + 30 \, \sin \left (3 \, d x + 3 \, c\right ) + 15 \, \sin \left (d x + c\right )\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right ) -{\left (15 i \, \cos \left (5 \, d x + 5 \, c\right ) + 30 i \, \cos \left (3 \, d x + 3 \, c\right ) + 15 i \, \cos \left (d x + c\right ) - 15 \, \sin \left (5 \, d x + 5 \, c\right ) - 30 \, \sin \left (3 \, d x + 3 \, c\right ) - 15 \, \sin \left (d x + c\right )\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right ) + 60 \, \cos \left (4 \, d x + 4 \, c\right ) + 100 \, \cos \left (2 \, d x + 2 \, c\right ) + 60 i \, \sin \left (4 \, d x + 4 \, c\right ) + 100 i \, \sin \left (2 \, d x + 2 \, c\right ) + 32}{{\left (-4 i \, a^{4} \cos \left (5 \, d x + 5 \, c\right ) - 8 i \, a^{4} \cos \left (3 \, d x + 3 \, c\right ) - 4 i \, a^{4} \cos \left (d x + c\right ) + 4 \, a^{4} \sin \left (5 \, d x + 5 \, c\right ) + 8 \, a^{4} \sin \left (3 \, d x + 3 \, c\right ) + 4 \, a^{4} \sin \left (d x + c\right )\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.41002, size = 455, normalized size = 4.25 \begin{align*} -\frac{15 \,{\left (e^{\left (5 i \, d x + 5 i \, c\right )} + 2 \, e^{\left (3 i \, d x + 3 i \, c\right )} + e^{\left (i \, d x + i \, c\right )}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 15 \,{\left (e^{\left (5 i \, d x + 5 i \, c\right )} + 2 \, e^{\left (3 i \, d x + 3 i \, c\right )} + e^{\left (i \, d x + i \, c\right )}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right ) - 30 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 50 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 16 i}{2 \,{\left (a^{4} d e^{\left (5 i \, d x + 5 i \, c\right )} + 2 \, a^{4} d e^{\left (3 i \, d x + 3 i \, c\right )} + a^{4} d e^{\left (i \, d x + i \, c\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.20373, size = 155, normalized size = 1.45 \begin{align*} -\frac{\frac{15 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac{15 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} - \frac{2 \,{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 8 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 8 i\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{2} a^{4}} - \frac{32}{a^{4}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]