Optimal. Leaf size=128 \[ -\text{Unintegrable}\left (\frac{\tan (a+b x)}{(c+d x)^3},x\right )-\frac{4 b^2 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b c}{d}+2 b x\right )}{d^3}-\frac{4 b^2 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{d^3}-\frac{2 b \cos (2 a+2 b x)}{d^2 (c+d x)}-\frac{\sin (2 a+2 b x)}{d (c+d x)^2} \]
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Rubi [A] time = 0.397779, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sec (a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\sec (a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx &=\int \left (\frac{3 \cos (a+b x) \sin (a+b x)}{(c+d x)^3}-\frac{\sin ^2(a+b x) \tan (a+b x)}{(c+d x)^3}\right ) \, dx\\ &=3 \int \frac{\cos (a+b x) \sin (a+b x)}{(c+d x)^3} \, dx-\int \frac{\sin ^2(a+b x) \tan (a+b x)}{(c+d x)^3} \, dx\\ &=3 \int \frac{\sin (2 a+2 b x)}{2 (c+d x)^3} \, dx+\int \frac{\cos (a+b x) \sin (a+b x)}{(c+d x)^3} \, dx-\int \frac{\tan (a+b x)}{(c+d x)^3} \, dx\\ &=\frac{3}{2} \int \frac{\sin (2 a+2 b x)}{(c+d x)^3} \, dx+\int \frac{\sin (2 a+2 b x)}{2 (c+d x)^3} \, dx-\int \frac{\tan (a+b x)}{(c+d x)^3} \, dx\\ &=-\frac{3 \sin (2 a+2 b x)}{4 d (c+d x)^2}+\frac{1}{2} \int \frac{\sin (2 a+2 b x)}{(c+d x)^3} \, dx+\frac{(3 b) \int \frac{\cos (2 a+2 b x)}{(c+d x)^2} \, dx}{2 d}-\int \frac{\tan (a+b x)}{(c+d x)^3} \, dx\\ &=-\frac{3 b \cos (2 a+2 b x)}{2 d^2 (c+d x)}-\frac{\sin (2 a+2 b x)}{d (c+d x)^2}-\frac{\left (3 b^2\right ) \int \frac{\sin (2 a+2 b x)}{c+d x} \, dx}{d^2}+\frac{b \int \frac{\cos (2 a+2 b x)}{(c+d x)^2} \, dx}{2 d}-\int \frac{\tan (a+b x)}{(c+d x)^3} \, dx\\ &=-\frac{2 b \cos (2 a+2 b x)}{d^2 (c+d x)}-\frac{\sin (2 a+2 b x)}{d (c+d x)^2}-\frac{b^2 \int \frac{\sin (2 a+2 b x)}{c+d x} \, dx}{d^2}-\frac{\left (3 b^2 \cos \left (2 a-\frac{2 b c}{d}\right )\right ) \int \frac{\sin \left (\frac{2 b c}{d}+2 b x\right )}{c+d x} \, dx}{d^2}-\frac{\left (3 b^2 \sin \left (2 a-\frac{2 b c}{d}\right )\right ) \int \frac{\cos \left (\frac{2 b c}{d}+2 b x\right )}{c+d x} \, dx}{d^2}-\int \frac{\tan (a+b x)}{(c+d x)^3} \, dx\\ &=-\frac{2 b \cos (2 a+2 b x)}{d^2 (c+d x)}-\frac{3 b^2 \text{Ci}\left (\frac{2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac{2 b c}{d}\right )}{d^3}-\frac{\sin (2 a+2 b x)}{d (c+d x)^2}-\frac{3 b^2 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{d^3}-\frac{\left (b^2 \cos \left (2 a-\frac{2 b c}{d}\right )\right ) \int \frac{\sin \left (\frac{2 b c}{d}+2 b x\right )}{c+d x} \, dx}{d^2}-\frac{\left (b^2 \sin \left (2 a-\frac{2 b c}{d}\right )\right ) \int \frac{\cos \left (\frac{2 b c}{d}+2 b x\right )}{c+d x} \, dx}{d^2}-\int \frac{\tan (a+b x)}{(c+d x)^3} \, dx\\ &=-\frac{2 b \cos (2 a+2 b x)}{d^2 (c+d x)}-\frac{4 b^2 \text{Ci}\left (\frac{2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac{2 b c}{d}\right )}{d^3}-\frac{\sin (2 a+2 b x)}{d (c+d x)^2}-\frac{4 b^2 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{d^3}-\int \frac{\tan (a+b x)}{(c+d x)^3} \, dx\\ \end{align*}
Mathematica [A] time = 5.89447, size = 0, normalized size = 0. \[ \int \frac{\sec (a+b x) \sin (3 a+3 b x)}{(c+d x)^3} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.504, size = 0, normalized size = 0. \begin{align*} \int{\frac{\sec \left ( bx+a \right ) \sin \left ( 3\,bx+3\,a \right ) }{ \left ( dx+c \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sec \left (b x + a\right ) \sin \left (3 \, b x + 3 \, a\right )}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}, x\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 = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (b x + a\right ) \sin \left (3 \, b x + 3 \, a\right )}{{\left (d x + c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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