Optimal. Leaf size=114 \[ \frac {10 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{21 a d e^2}+\frac {10 \sin (c+d x)}{21 a d e \sqrt {e \sec (c+d x)}}+\frac {2 i}{7 d (a+i a \tan (c+d x)) (e \sec (c+d x))^{3/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3502, 3769, 3771, 2641} \[ \frac {10 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{21 a d e^2}+\frac {10 \sin (c+d x)}{21 a d e \sqrt {e \sec (c+d x)}}+\frac {2 i}{7 d (a+i a \tan (c+d x)) (e \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3502
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))} \, dx &=\frac {2 i}{7 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))}+\frac {5 \int \frac {1}{(e \sec (c+d x))^{3/2}} \, dx}{7 a}\\ &=\frac {10 \sin (c+d x)}{21 a d e \sqrt {e \sec (c+d x)}}+\frac {2 i}{7 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))}+\frac {5 \int \sqrt {e \sec (c+d x)} \, dx}{21 a e^2}\\ &=\frac {10 \sin (c+d x)}{21 a d e \sqrt {e \sec (c+d x)}}+\frac {2 i}{7 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))}+\frac {\left (5 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 a e^2}\\ &=\frac {10 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \sec (c+d x)}}{21 a d e^2}+\frac {10 \sin (c+d x)}{21 a d e \sqrt {e \sec (c+d x)}}+\frac {2 i}{7 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.60, size = 125, normalized size = 1.10 \[ -\frac {\sec ^3(c+d x) \left (5 i \sin (c+d x)+5 i \sin (3 (c+d x))-14 \cos (c+d x)+2 \cos (3 (c+d x))+20 i \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (\cos (c+d x)+i \sin (c+d x))\right )}{42 a d (\tan (c+d x)-i) (e \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ \frac {{\left (84 \, a d e^{2} e^{\left (4 i \, d x + 4 i \, c\right )} {\rm integral}\left (-\frac {5 i \, \sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{21 \, a d e^{2}}, x\right ) + \sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-7 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 9 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 19 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{84 \, a d e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \sec \left (d x + c\right )\right )^{\frac {3}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.28, size = 218, normalized size = 1.91 \[ \frac {2 \cos \left (d x +c \right ) \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \left (-1+\cos \left (d x +c \right )\right )^{2} \left (1+\cos \left (d x +c \right )\right )^{2} \left (3 i \left (\cos ^{4}\left (d x +c \right )\right )+5 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \cos \left (d x +c \right )+5 i \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+3 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+5 \cos \left (d x +c \right ) \sin \left (d x +c \right )\right )}{21 a d \,e^{3} \sin \left (d x +c \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]
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 \[ - \frac {i \int \frac {1}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \tan {\left (c + d x \right )} - i \left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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