Integrand size = 18, antiderivative size = 166 \[ \int \sqrt {c+d x} \sinh ^2(a+b x) \, dx=-\frac {(c+d x)^{3/2}}{3 d}+\frac {\sqrt {d} e^{-2 a+\frac {2 b c}{d}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{3/2}}-\frac {\sqrt {d} e^{2 a-\frac {2 b c}{d}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{3/2}}+\frac {\sqrt {c+d x} \sinh (2 a+2 b x)}{4 b} \]
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Time = 0.20 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3393, 3377, 3389, 2211, 2235, 2236} \[ \int \sqrt {c+d x} \sinh ^2(a+b x) \, dx=\frac {\sqrt {\frac {\pi }{2}} \sqrt {d} e^{\frac {2 b c}{d}-2 a} \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {d} e^{2 a-\frac {2 b c}{d}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{3/2}}+\frac {\sqrt {c+d x} \sinh (2 a+2 b x)}{4 b}-\frac {(c+d x)^{3/2}}{3 d} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3377
Rule 3389
Rule 3393
Rubi steps \begin{align*} \text {integral}& = -\int \left (\frac {1}{2} \sqrt {c+d x}-\frac {1}{2} \sqrt {c+d x} \cosh (2 a+2 b x)\right ) \, dx \\ & = -\frac {(c+d x)^{3/2}}{3 d}+\frac {1}{2} \int \sqrt {c+d x} \cosh (2 a+2 b x) \, dx \\ & = -\frac {(c+d x)^{3/2}}{3 d}+\frac {\sqrt {c+d x} \sinh (2 a+2 b x)}{4 b}-\frac {d \int \frac {\sinh (2 a+2 b x)}{\sqrt {c+d x}} \, dx}{8 b} \\ & = -\frac {(c+d x)^{3/2}}{3 d}+\frac {\sqrt {c+d x} \sinh (2 a+2 b x)}{4 b}-\frac {d \int \frac {e^{-i (2 i a+2 i b x)}}{\sqrt {c+d x}} \, dx}{16 b}+\frac {d \int \frac {e^{i (2 i a+2 i b x)}}{\sqrt {c+d x}} \, dx}{16 b} \\ & = -\frac {(c+d x)^{3/2}}{3 d}+\frac {\sqrt {c+d x} \sinh (2 a+2 b x)}{4 b}+\frac {\text {Subst}\left (\int e^{i \left (2 i a-\frac {2 i b c}{d}\right )-\frac {2 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{8 b}-\frac {\text {Subst}\left (\int e^{-i \left (2 i a-\frac {2 i b c}{d}\right )+\frac {2 b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{8 b} \\ & = -\frac {(c+d x)^{3/2}}{3 d}+\frac {\sqrt {d} e^{-2 a+\frac {2 b c}{d}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{3/2}}-\frac {\sqrt {d} e^{2 a-\frac {2 b c}{d}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{3/2}}+\frac {\sqrt {c+d x} \sinh (2 a+2 b x)}{4 b} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.78 \[ \int \sqrt {c+d x} \sinh ^2(a+b x) \, dx=\frac {1}{48} \sqrt {c+d x} \left (-\frac {16 (c+d x)}{d}+\frac {3 \sqrt {2} e^{2 a-\frac {2 b c}{d}} \Gamma \left (\frac {3}{2},-\frac {2 b (c+d x)}{d}\right )}{b \sqrt {-\frac {b (c+d x)}{d}}}-\frac {3 \sqrt {2} e^{-2 a+\frac {2 b c}{d}} \Gamma \left (\frac {3}{2},\frac {2 b (c+d x)}{d}\right )}{b \sqrt {\frac {b (c+d x)}{d}}}\right ) \]
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\[\int \sinh \left (b x +a \right )^{2} \sqrt {d x +c}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (122) = 244\).
Time = 0.26 (sec) , antiderivative size = 590, normalized size of antiderivative = 3.55 \[ \int \sqrt {c+d x} \sinh ^2(a+b x) \, dx=\frac {3 \, \sqrt {2} \sqrt {\pi } {\left (d^{2} \cosh \left (b x + a\right )^{2} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - d^{2} \cosh \left (b x + a\right )^{2} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + {\left (d^{2} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - d^{2} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left (d^{2} \cosh \left (b x + a\right ) \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - d^{2} \cosh \left (b x + a\right ) \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {\frac {b}{d}} \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) + 3 \, \sqrt {2} \sqrt {\pi } {\left (d^{2} \cosh \left (b x + a\right )^{2} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + d^{2} \cosh \left (b x + a\right )^{2} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + {\left (d^{2} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + d^{2} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left (d^{2} \cosh \left (b x + a\right ) \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + d^{2} \cosh \left (b x + a\right ) \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {-\frac {b}{d}} \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) + 4 \, {\left (3 \, b d \cosh \left (b x + a\right )^{4} + 12 \, b d \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 3 \, b d \sinh \left (b x + a\right )^{4} - 8 \, {\left (b^{2} d x + b^{2} c\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (4 \, b^{2} d x - 9 \, b d \cosh \left (b x + a\right )^{2} + 4 \, b^{2} c\right )} \sinh \left (b x + a\right )^{2} - 3 \, b d + 4 \, {\left (3 \, b d \cosh \left (b x + a\right )^{3} - 4 \, {\left (b^{2} d x + b^{2} c\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {d x + c}}{96 \, {\left (b^{2} d \cosh \left (b x + a\right )^{2} + 2 \, b^{2} d \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{2} d \sinh \left (b x + a\right )^{2}\right )}} \]
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\[ \int \sqrt {c+d x} \sinh ^2(a+b x) \, dx=\int \sqrt {c + d x} \sinh ^{2}{\left (a + b x \right )}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.14 \[ \int \sqrt {c+d x} \sinh ^2(a+b x) \, dx=-\frac {\frac {3 \, \sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )}}{b \sqrt {-\frac {b}{d}}} - \frac {3 \, \sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )}}{b \sqrt {\frac {b}{d}}} + 32 \, {\left (d x + c\right )}^{\frac {3}{2}} - \frac {12 \, \sqrt {d x + c} d e^{\left (2 \, a + \frac {2 \, {\left (d x + c\right )} b}{d} - \frac {2 \, b c}{d}\right )}}{b} + \frac {12 \, \sqrt {d x + c} d e^{\left (-2 \, a - \frac {2 \, {\left (d x + c\right )} b}{d} + \frac {2 \, b c}{d}\right )}}{b}}{96 \, d} \]
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\[ \int \sqrt {c+d x} \sinh ^2(a+b x) \, dx=\int { \sqrt {d x + c} \sinh \left (b x + a\right )^{2} \,d x } \]
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Timed out. \[ \int \sqrt {c+d x} \sinh ^2(a+b x) \, dx=\int {\mathrm {sinh}\left (a+b\,x\right )}^2\,\sqrt {c+d\,x} \,d x \]
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